Modelling Water Droplet Movement on a Leaf Surfaceath_Oqielat_Thesis.pdf · Surface Fitting with Application to Leaf Data. Applied Mathematical Modelling, 33:2582-2595, 2009. M. Oqielat,

School of Mathematical Sciences

Queensland University of Technology

Modelling Water Droplet

Movement on a Leaf Surface

Moa’ath Nasser Oqielat

Bachelor of Applied Mathematics,

Jordan University of Science and Technology (JUST)

Master of Pure Mathematics,

National University of Malaysia (UKM)

A thesis submitted for the degree of Doctor of Philosophy in the Faculty of

Science, Queensland University of Technology according to QUT requirements.

Principal supervisor: Professor Ian W. Turner

Associate supervisors: Professor John A. Belward

2009

Keywords

Surface fitting, finite elements methods, radial basis functions, Clough-Tocher

method, Interpolation method, extrapolation method, virtual leaf, virtual plants,

truncated Taylor series, weighted least squares, physical based modelling, thin-film

approximation.

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Abstract

The central aim for the research undertaken in this PhD thesis is the development

of a model for simulating water droplet movement on a leaf surface and to compare

the model behavior with experimental observations. A series of five papers has been

presented to explain systematically the way in which this droplet modelling work

has been realised. Knowing the path of the droplet on the leaf surface is important

for understanding how a droplet of water, pesticide, or nutrient will be absorbed

through the leaf surface.

An important aspect of the research is the generation of a leaf surface represen-

tation that acts as the foundation of the droplet model. Initially a laser scanner is

used to capture the surface characteristics for two types of leaves in the form of a

large scattered data set. After the identification of the leaf surface boundary, a set

of internal points is chosen over which a triangulation of the surface is constructed.

We present a novel hybrid approach for leaf surface fitting on this triangulation

that combines Clough-Tocher (CT) and radial basis function (RBF) methods to

achieve a surface with a continuously turning normal. The accuracy of the hy-

brid technique is assessed using numerical experimentation. The hybrid CT-RBF

method is shown to give good representations of Frangipani and Anthurium leaves.

Such leaf models facilitate an understanding of plant development and permit the

modelling of the interaction of plants with their environment.

The motion of a droplet traversing this virtual leaf surface is affected by various

forces including gravity, friction and resistance between the surface and the droplet.

The innovation of our model is the use of thin-film theory in the context of droplet

movement to determine the thickness of the droplet as it moves on the surface.

Experimental verification shows that the droplet model captures reality quite well

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and produces realistic droplet motion on the leaf surface. Most importantly, we

observed that the simulated droplet motion follows the contours of the surface and

spreads as a thin film.

In the future, the model may be applied to determine the path of a droplet

of pesticide along a leaf surface before it falls from or comes to a standstill on

the surface. It will also be used to study the paths of many droplets of water or

pesticide moving and colliding on the surface.

List of Publications and Manuscripts

M. Oqielat, I. Turner, and J. Belward. A Hybrid Clough-Tocher Method for

Surface Fitting with Application to Leaf Data. Applied Mathematical Modelling,

33:2582-2595, 2009.

M. Oqielat, J. Belward, I. Turner, and B. Loch. A hybrid Clough-Tocher radial

basis function method for modelling leaf surfaces. In Oxley, L. and Kulasiri, D.

(eds) MODSIM 2007 International Congress on Modelling and Simulation. Mod-

elling and Simulation Society of Australia and New Zealand, December 2007, pages

400-406, 2007.

M. Oqielat, I. Turner, J. Belward, and S. McCue. Water Droplet Movement on

a Leaf Surface. Mathematics and Computer in Simulation. Paper has now been

revised and resubmitted to the journal as requested by the editor on 19/04/09

taking into consideration the comments and suggestions by the reviewers, 2009.

J. Belward, I. Turner, and M. Oqielat. Numerical Investigations of Linear Least

Squares Methods for Derivatives Estimation. CTAC 08 Computational Techniques

and applications conference, Australia, July 2008.

I. Turner, J. Belward, and M. Oqielat. Error Bounds for Least Squares Gradient

Estimates. SIAM Journal on Scientic Computing, Under review, 2008.

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Contents

1 Introduction and Literature Review 1

1.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Surface Fitting Techniques . . . . . . . . . . . . . . . . . . . 7

1.2.2 Virtual Leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.3 The path of a droplet . . . . . . . . . . . . . . . . . . . . . . 26

1.3 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.3.1 Construction of Virtual Surface . . . . . . . . . . . . . . . . 38

1.3.2 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.3.3 Droplet Model . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.4.1 Outline of Chapter 2 for the Paper Published in the Applied

Mathematical Modelling Journal, 2009 . . . . . . . . . . . . 51

1.4.2 Outline of Chapter 3 for Paper Published in the proceedings

of the MODSIM07 Conference, 2007 . . . . . . . . . . . . . 52

1.4.3 Outline of Chapter 4 for the Paper Published in the proceed-

ings of the CTAC08 Conference, 2008 . . . . . . . . . . . . . 54

1.4.4 Outline of Chapter 5 for the Paper Submitted to the SIAM

Journal on Scientific Computing, 2008 . . . . . . . . . . . . 55

1.4.5 Outline of Chapter 6 for the Paper Submitted to the Journal

of Mathematics and Computer in Simulation, 2009 . . . . . 56

1.4.6 Outline of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . 57

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2 A Hybrid Clough-Tocher Method for Surface Fitting with Appli-

cation to Leaf Data 58

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2 Surface Fitting Methods . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.1 The Clough-Tocher finite element method . . . . . . . . . . 60

2.2.2 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . 63

2.2.3 Hybrid Method . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3 Numerical Experimentation for the Franke Data Set. . . . . . . . . 69

2.3.1 Clough-Tocher method . . . . . . . . . . . . . . . . . . . . . 70

2.3.2 Hybrid Clough-Tocher Radial basis function method . . . . 71

2.4 Application of the Hybrid method to a Leaf Data Set . . . . . . . . 74

2.4.1 Leaf reference plane . . . . . . . . . . . . . . . . . . . . . . . 74

2.4.2 Triangulation of the leaf surface . . . . . . . . . . . . . . . . 76

2.4.3 Numerical Experiments for the Leaf Surface . . . . . . . . . 79

2.5 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . 84

2.6 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3 A Hybrid Clough-Tocher Radial Basis Function Method for Mod-

elling Leaf Surfaces 86

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.1.1 Clough-Tocher finite element method . . . . . . . . . . . . . 87

3.1.2 Radial basis functions . . . . . . . . . . . . . . . . . . . . . 88

3.2 Hybrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.3 Application of the Hybrid Method for the Frangipani and Anthurium

Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3.1 Data from laser scanner . . . . . . . . . . . . . . . . . . . . 91


3.3.3 Triangulation method . . . . . . . . . . . . . . . . . . . . . . 92

3.3.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . 96

3.4 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . 98

3.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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4 Numerical investigations of linear least squares methods for deriva-

tive estimation 99

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Construction of a leaf surface . . . . . . . . . . . . . . . . . . . . . 100

4.3 Theoretical error bounds . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5 Error Bounds for Least Squares Gradient Estimates 109

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Least Squares Gradient Estimation . . . . . . . . . . . . . . . . . . 111

5.3 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3.1 Classical Least Squares Gradient Estimates . . . . . . . . . . 113

5.3.2 Weighted Least Squares Gradient Estimates . . . . . . . . . 116

5.4 Tighter Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.5.1 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . 121

5.5.2 Scattered Data Results . . . . . . . . . . . . . . . . . . . . . 122

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Water Droplet Movement on a Leaf Surface 135

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2 Relevant Literature and Experiments . . . . . . . . . . . . . . . . . 137

6.3 Leaf surface model . . . . . . . . . . . . . . . . . . . . . . . . . . . 140


6.3.2 Triangulation of the leaf surface . . . . . . . . . . . . . . . . 140

6.4 Droplet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4.1 External and internal forces . . . . . . . . . . . . . . . . . . 141

6.4.2 Thin-film flow down a slope . . . . . . . . . . . . . . . . . . 143

6.4.3 Motion of a droplet over the leaf surface . . . . . . . . . . . 145

6.5 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . 150

6.6 Conclusions and future research . . . . . . . . . . . . . . . . . . . . 162

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6.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7 Summary and Discussion 164

7.1 Directions for Future Research . . . . . . . . . . . . . . . . . . . . . 173

7.1.1 Droplet Modelling . . . . . . . . . . . . . . . . . . . . . . . 173

7.1.2 Surface Fitting . . . . . . . . . . . . . . . . . . . . . . . . . 174

List of Figures

1.1 The Clough-Tocher triangle showing subdivision into three subtri-

angles. The directional derivatives at triangle vertices and normal

derivatives at edge midpoints are pictured as arrows. . . . . . . . . 8

1.2 Mapping a triangle on a standard triangle. . . . . . . . . . . . . . . 10

1.3 The Maple leaf, taken from [84]. The shape of a maple leaf and the

texture image projected onto the shape. . . . . . . . . . . . . . . . 20

1.4 (a) A single compound leaf model taken from [84]. (b) The branch-

ing skeleton and contour, reproduced from [62]. . . . . . . . . . . . 20

1.5 The scanned Lobed leaves, taken from [84]. . . . . . . . . . . . . . 21

1.6 Frangipani leaf: (a) point sets and (b) triangulation . . . . . . . . 22

1.7 Photographs of the scanned (a) Frangipani leaf and (b) Anthurium

leaf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.8 Photographs of the scanned (a) Elephant’s ear tree leaf and (b)

Flame tree leaf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.9 (a) The laser scanner. (b) The sonic digitiser. . . . . . . . . . . . . 26

1.10 Discrete surface model. . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.11 Photos of the scanned (a) Frangipani and (b) Anthurium leaves. . . 38

1.12 This figure exhibits a droplet movement across the leaf surface. . . 50

2.1 The Clough-Tocher triangle showing subdivision into three subtri-

angles. The directional derivatives at triangle vertices and normal

derivatives at side midpoints are pictured as arrows . . . . . . . . . 61

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2.2 Anthurium Leaf data points. There are 4,688 surface points (rep-

resented by the smaller dots) and 79 boundary points (represented

by the larger dots). . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.3 The Anthurium leaf surface model constructed from the data points

(shown in figure 2.2) using the hybrid CT-RBF method. . . . . . . 76

2.4 (a) The 79 Anthurium leaf boundary points. (b) The 49 points

generated from the convex hull algorithm. (c) The square points

represent the final 38 boundary points. (d) The vertices of the mesh

structure generated using Easymesh. The square points represent

the 38 boundary points that are given to Easymesh; the dot points

represent the 28 extra points added by Easymesh, while the × points

represent the 146 internal points. . . . . . . . . . . . . . . . . . . . 78

2.5 Triangulation of the 212 points of the Anthurium leaf surface gen-

erated using EasyMesh. . . . . . . . . . . . . . . . . . . . . . . . . 79

2.6 The triangulation of (a) coarser grid of 103 points and (b) a refined

grid using 762 points of the Anthurium leaf surface generated using

EasyMesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.1 (a) The 17 Frangipani leaf boundary points. (b) The 27 points

generated from the convex hull algorithm. (c) The square points

represent the final 11 boundary points. (d) The vertices of the mesh

structure generated using Easymesh. The square points represent

the 11 boundary points that are given to Easymesh; the dot points

represent the 58 extra points added by Easymesh, while the x points

represent the 93 internal points. . . . . . . . . . . . . . . . . . . . 94

3.2 (a) Triangulation of 151 points of Frangipani leaf surface generated

using EasyMesh. (b) Triangulation of 141 points of Frangipani leaf

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 Second order least squares errors and error bounds. Varying radius

on the left, varying weightings on the right. . . . . . . . . . . . . . . 125

5.2 Third order least squares errors and error bounds. Varying radius

on the left, varying weightings on the right. . . . . . . . . . . . . . . 126

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5.3 Relative error (line style ) and error bounds (line style and ) for

function F1 using (a) first order, (c) second order and (e) third order

least squares estimates. The corresponding weighted least squares

estimates and their bounds are depicted in (b), (d) and (f). . . . . . 131









6.1 The direction of movement dp with normal N and gravity fext. . . 142

6.2 Thin-film flow down a slope. . . . . . . . . . . . . . . . . . . . . . . 143

6.3 The droplet movement within the kth triangle. . . . . . . . . . . . . 146

6.4 (a) exhibits the first orientation of the leaf, (b) shows the second

orientation of the leaf, (c) shows the six dots captured using the

sonic digitizer and (d) depicts the sonic digitizer device. . . . . . . 151

6.5 (a) shows the boundary points of the leaf, the string points and the

six dots for the first orientation; (b) shows the the second orientation

of the data; (c) shows the leaf surface points that were captured

using the scanner; (d) depicts the leaf surface points after rotation

to the reference plane and its normal; (e) exhibits the sonic digitizer

leaf boundary points after rotation to the reference plane and its

normal; (f) shows both data sets in the same reference plane. . . . 153

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6.6 (a) and (b) show the transformation of both data sets into the

xy−plane; (c) is the projection of both data sets into the xy−plane;

(d) is the rotation of the data to become coincident; (e) depicts the

inverse rotation of both data sets into the original position that we

have in the experiment, where data set 1 is represented by circles

while data set 2 is represented by dots; (f) exhibits the final rotation

of the first orientation data set and the string. . . . . . . . . . . . 154

6.7 (a,c,e) show the droplet movement across the leaf surface from three

different starting positions for the fist orientation. (b,d,f) exhibit

the corresponding droplet movement generated by the model for

the three different starting locations shown in (a,c,e). . . . . . . . . 156

6.8 (a,c,e) show the droplet movement across the leaf surface from three

different starting positions for the second orientation. (b,d,f) exhibit

the corresponding droplet movement generated by the model for the

three different starting locations shown in (a,c,e). . . . . . . . . . . 157

6.9 The figures show a comparison of the thin-film model results against

the experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.10 (a) and (b) exhibit the six dots on the final transformed first and

second orientation data sets; (c) and (e) represent the triangulation

and the refined triangulation respectively of the first orientation

data set; (d) and (f) represent the triangulation and the refined

triangulation respectively of the second orientation data set. . . . . 160

6.11 Each of these figures show two paths of the same droplet on the

refined triangulation. One represents the path on the unrefined

triangulation, given in figures 6.10 (c,d), while the other represents

the path on the refined triangulation, given in figures 6.10 (e,f). . . 161

List of Tables

1.1 Choices of R for which the interpolation matrix is invertible. . . . 18

2.1 Choices of R given by Rippa [121] for which the interpolation matrix

Λ is invertible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.2 A comparison of the RMS error for the six test functions using the

CT method. The results in column 2 uses exact gradients and the

results given in columns 3-5 use respectively, 1st, 2nd and 3rd order

Taylor series expansions to estimate the gradient at the vertices and

edge midpoints of the triangle. . . . . . . . . . . . . . . . . . . . . 70

2.3 A comparison of the RMS error for the six test functions using

the 1st, 2nd and 3rd order Taylor series to estimate the gradient at

the vertices of the triangle. The gradients at edge midpoints are

estimated by taking the mean of the gradients at the two vertices

on the same edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


hybrid global (n = 100 points) and hybrid local multiquadric RBF

interpolants (m = 20 or m = 40 points). The parameter c was

computed globally using the n = 100 points. . . . . . . . . . . . . . 72


hybrid local multiquadric RBF interpolant (m = 20 or m = 40

points). The parameter c was computed locally using the same

(m = 20 or m = 40) points. . . . . . . . . . . . . . . . . . . . . . . 72

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2.6 RMS error computed using the local and global hybrid CT-RBF

method for the Anthurium leaf data points together with the max-

imum error associated with the surface fit. . . . . . . . . . . . . . . 81

2.7 The relative error of the estimated gradients at the common mid-

point of the six triangles using local hybrid RBF method. . . . . . 83

2.8 RMS and maximum error computed using 1st and 3rd order Taylor

series. In the 1st and 3rd columns, the gradient was estimated at the

vertices and edge midpoints, while in the 2nd and 4th columns, the

gradient at the edge midpoints was estimated by taking the mean

of the gradients at the two vertices at the same edge. . . . . . . . . 83

3.1 RMS computed using hybrid local and global RBF for the Frangi-

pani leaf data points as well as the maximum error associated with

the surface fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.2 RMS computed using hybrid local and global RBF for the An-

thurium leaf data points as well as the maximum error associated

with the surface fit. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1 Norms of the errors in the gradient and Hessian with differing radii 105

4.2 Error bound and modified error bound values for varying radii . . . 105

4.3 Singular values (sv’s) of the least squares and elimination matrices

for various radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4 Cumulative sums of the singular expansion solution for the deriva-

tives for the point (3,4) at radii between .25 and .5 . . . . . . . . . 107

5.1 A comparison of the relative error and the error bounds using second

and third order least squares gradient estimates for the function F1. 123





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5.4 A comparison of the relative error and the error bounds using weighted

second and third order least squares gradient estimates for the func-

tion F1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124



tion F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124



tion F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.7 A comparison of the relative error and the error bounds using first,


tion F1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


first, second and third order least squares gradient estimates for the

function F1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128



tion F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129



function F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129



tion F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130



function F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher educational institution. To

the best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made.

Signed:

Date:

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Acknowledgements

My first and foremost thanks to ALLAH for the opportunities that He has given

to me throughout my life, especially those that have brought me to the position of

finishing this thesis.

I would like to express my great thanks to my principal supervisor Professor Ian

Turner for his contribution to the research, for his encouragement and consistent

support throughout, for reading an endless number of drafts and providing valuable

feedback for each of them, and for assisting me to secure the necessary financial

support for my studies by obtaining a QUT fee waiver scholarship. Without his

help this thesis would not have been finished. I believe he has been a great and

a wonderful supervisor. No matter how many times I knock at his door daily or

how many questions or what the questions, he always had time to meet me, to

hear the questions and the knowledge to answer them. He never failed to reply to

my thousands of e-mails. His supervision has been exciting and a perfect learning

curve I have gained from him during my PhD.

I would like also to acknowledge my associate supervisor Professor John Bel-

ward for his knowledge, expertise, invaluable advice and guidance in scientific

writing throughout my study. His support and comments are greatly appreciated.

I would like to thanks the High Performance Computing center especially Mark

Barry and Mark Dwyer for their leaf and droplet visualisations. I also Acknowledge

Dr Jim Hanan from Queensland University for allowing me to use the equipment

to perform the droplet experiments. Many thanks also goes to Peter Nelson from

ISS for his assistance and advice in English language.

Many thanks to the Queensland University of Technology (QUT) for awarding

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me a fee waiver and the School of Mathematical Sciences at QUT for providing me

a living allowance for my final semester that enabled me to pursue my research.

Thanks to QUT also for providing me with financial support to present a research

paper at the conference MODSIM07 (University of Canterbury, New Zealand), to

attend the QANZIAM meeting 2006 (Stanthorpe, QLD, Australia) and also to

attend the Australian Partnership for Advance Computing 2006 workshop (QUT).

Thanks also due to postgraduate students and staff in the School of Mathe-

matical Sciences for providing the perfect environment for mathematical research.

This line is to memories my parents, brothers and sisters who always wished

my success.

To the most important person in my life, my mother ”Gazeah Ahmad” receive

my grateful appreciation for believing in me, for her incredible support and love

through my many years in education and for being there whenever I needed her;

”without your love and encouragement you gave me each day and your endless

patience, I would not have found an opportunity to finish my thesis”. Thanks to

my oldest brother Khaled who took over as my Dad after he passed away when I

was 10 years old. He always supported me financially, encouraged me to do my

PhD and told me that I must spend most of my time on my studies. Thanks

are also due to my brother Amjad, and to my sisters, Manal, Tharwah, Khawlah,

Hend, Wisam and Abtesam, who loved, encouraged me to do higher education and

supported me during my PhD studies.

Special thanks goes to my two wonderful brothers who supported me and lived

with me during my studies in Australia. To Ahmad, who guided and helped me

to come to Australia and to receive my PhD offer. To Osama who supported

me through my studies and provided me with regular delicious foods. My three

sisters-in-law, Manal, Maha and Lara, and my four brothers-in-law, Ali, Ahmad,

Waleed and Gazee, are remembered appreciatively. I should not forget my relatives

(Uncles and Aunts) and friends.

This thesis is dedicated to my mother; and to my father (ALLAH bless him).

I wish he was alive to see what i have achieved and to share my happiness for

completing this thesis. He encouraged and directed my education and provided for

me financially before and after he passed away.

Chapter 1

Introduction and Literature Review

The aim of the research presented in this thesis is to develop a new model for

generating a realistic movement of a water droplet on a virtual leaf surface and

to compare the model behavior with experimental observation. Knowledge of the

path of the droplet on the leaf surface is significant for understanding how a droplet

of water or nutrient will interact with and subsequently be absorbed through the

surface. This knowledge is important for many applications, particularly the sim-

ulation of a pesticide application to plant surfaces. This model can be used to

determine the effectiveness of a treatment, and then to develop certain pesticides

that have the ability to protect leaves for longer periods of time.

The reconstruction of the leaf surface forms the foundation for our theoreti-

cal study of a water droplet path on the leaf. It will be essential to produce a

smooth surface for this purpose, therefore, an adequate representation of the leaf

is required. Surface fitting techniques can be used to reconstruct the leaf surface.

From the literature review carried out in this thesis on mathematical methods for

surface fitting, it seems that the Clough-Tocher (CT) method is an economical and

accurate method for the construction of continuously differentiable surfaces from

three-dimensional data points. We used a laser scanner to sample data points from

a real leaf surface, which generates a large number (x, y and z coordinates) of data

points in R3.

1

Introduction and Literature Review 2

The Clough-Tocher approach is an interpolating finite element method based

on a triangular domain. We faced two main issues in the construction of the leaf

surface using the CT method. The first issue was to determine a new reference

plane for the leaf data since the coordinate system used by the scanner may not be

suitable for interpolation due to the possibility of multivalued and vertical surfaces.

This problem was solved by constructing a linear least squares plane to the leaf

data. The second issue was the construction of the triangulation of the leaf surface

mesh that is required for the CT method. We constructed a triangulation of the

leaf using the EasyMesh Generator, software written in the C language. EasyMesh

generates two-dimensional Delaunay and constrained Delaunay triangulations of

general domains.

Since the number of data points that represent the leaf surface is large, the

simulation of droplet movement could be computationally demanding if thousands

of triangles have to be considered, so a coarser mesh based on a smaller subset

of data points is used that is representative of the major surface features. This

choice of a subset of the data also helped in avoiding undesirably shaped triangles,

which is important when using the CT method to produce an accurate (smooth)

surface representation. As mentioned before, the virtual surface is comprised of

a triangular mesh structure over which the CT seamed element interpolant is

constructed. The CT method requires derivative estimates at the vertices and

midpoints of these triangles. This motivated us to introduce a new novel hybrid

method for gradient estimation that combines the CT method and the radial basis

functions (RBF) method to achieve a surface with a continuously turning normal.

As we will see, this hybrid method produces a leaf surface having characteristics

that are very suitable for simulating surface droplet movement. The hybrid CT-

RBF method uses a locally constructed surface based on a multiquadratic RBF,

which is then used to estimate the gradients at the vertices and midpoints of the

CT triangle.

The hybrid method has been successfully applied to real world data samples of

Frangipani and Anthurium leaves for the purpose of developing the desired leaf sur-

face representation. The method has been shown to provide good representations

of these leaves.


After the leaf surface model was developed, we started to derive a new model

for a water droplet traversing a virtual Frangipani leaf. To model the water droplet

motion there are many dominant factors and forces that need to be considered.

However, in our model the forces that affect the droplet movement on the leaf sur-

face are divided into two groups. An internal force, which consists of a friction and

a resistance component between the surface and the droplet, and an external force

due to gravity. Newton’s second law is used to determine the features of the droplet

motion and then by including these two forces into the Newton law the final model

was produced. As mentioned before the leaf surface comprises a mesh of triangles

over which the hybrid CT-RBF interpolant method is constructed. This represen-

tation of the surface offers many advantages, for example, it is easy to control the

droplet movement on the surface as well as to determine the droplet position on

the surface at any given time. We have derived an equation for droplet velocity

and droplet position at any time and the motion of the droplet was computed over

each triangle.

The innovation of our model is the use of thin-film theory in the context of

droplet movement to estimate the thickness of the droplet as it moves on the

surface, which facilitates the development of a stopping criterion for the droplet.

This idea enables the droplet to meander down an inclined leaf surface and under

certain conditions, stop. In order to calculate the film height, we computed the

location of the droplet front as it evolved in time. As a result, the model shows

that the droplet stays on the surface, or it leaves the surface depending on the

parameters.

To highlight the power of our theory, an experimental verification of the water

droplet model was presented. A freshly cut Frangipani leaf was used to perform

experiments whereby a number of water droplet paths were observed. We used a

clamp to hold the leaf in place and then we used a mass to hold a string attached

to this clamp to determine the direction of gravity with respect to the reference

plane for use in the model. A video camera was used to record the path that the

droplet traversed on the leaf surface.

At the beginning of the experiment we marked five artificial dots on the leaf

surface to locate some reference points for the droplet motion on the surface. Then,


we used a sonic digitizer device to measure the location of these five dots as well as

four additional data points on the string. As mentioned before, the laser scanner

was also used here to capture the data points of the leaf surface for the purpose of

surface reconstruction.

One issue raised here was that we could not scan the leaf while it was held by

the metal clamp. Thus, we had to scan the leaf in a different position from the

clamped leaf and then it was necessary to apply a rotation to the laser scanner

data point set to bring it in line with the points recorded by the sonic digitizer.

Then, we located the corresponding five points from the scanned set to the five

dots that were captured using the sonic device as starting points for the droplet

movement. After that, we started rolling the droplet on the leaf surface and at the

same time recorded the droplet path using the video camera.

In our experiment we have used two different masses of the droplet measured

using a syringe, along with two different leaf orientations to simulate the droplet

motion. The reason for this choice is to test if the path of the same droplet would

change by changing its mass and orientation. It was noted that the droplet moves

along the leaf vein if the leaf surface is horizontal, or close to horizontal; while on

other occasions the droplet moves and then falls from the surface, or it stays on

the surface depending on the forces that affect the droplet movement as well as

the leaf orientation.

By comparing the droplet paths produced by our model with the paths that we

have seen in the experiment, we observed that the model is able to capture reality

quite well and produced realistic droplet motion. After surveying the literature

for water droplet models used to simulate movement on a leaf surface, or models

that take into account the thickness of the water droplet as it moves along the leaf

surface, we found that our model is the most inclusive of any that have appeared

to date.

This thesis consists of 7 chapters. Chapters 2 to 6 are identical to five journal

papers, 3 already published, 1 under revision and the other submitted to a journal

for review. In chapter 2, a survey of surface interpolation methods and the CT

and RBF methods is presented. Then, a new hybrid method for modelling leaf

surfaces is proposed. Gradient estimates are essential for the hybrid method, and


an investigation of Taylor series expansions and the multiquadratic RBF method

was carried out. A numerical study of these methods for a data set taken from

Franke [55] is presented to assess the accuracy of these methods. The result of this

study is a continuous, smooth leaf surface that is suitable for simulating droplet

motion. In chapter 3, the hybrid CT-RBF method is applied to laser scanned

Frangipani and Anthurium leaves to construct the surface. A detailed description

is given of the construction of a triangulation of the leaf surface and a reference

plane for the laser scanner data points.

In chapter 4, a numerical investigation of the linear least squares method for

derivative estimation is given. The least squares problem is extracted using a

truncated Taylor series expansion from which the desired gradient approximation

can be estimated. The error bound (for the case n = 2) associated with this

method is derived to assess its accuracy. This bound contains in its denominator

the smallest singular value of the constructed least squares coefficient matrix. It

is found that the smallest singular value affects the accuracy of the bound, so it is

conjectured that rather than using the smallest singular value in the error bound

it appears more appropriate to use the smallest singular value of a reduced form

of the least squares matrix given in [13]. In chapter 5, an error bound theory

for the least squares and the weighted least squares gradient estimates is derived

for order n Taylor series expansions. This error bound is supported by numerical

experiments performed on a data taken from Franke [55]. The derived error bound

is found to be pessimistic by several orders, thus a tighter error bound is derived

that appeared much better than the initial bound.

In chapter 6, a new model for simulating water droplet movement on a virtual

leaf surface is presented. This model is verified against experimental measurement

performed on a fresh Frangipani leaf. Finally, the conclusions of this research and

some directions for future research are given in chapter 7.

1.1 Research Objectives

In this section the main research objectives of this thesis are stated and briefly

reviewed.


• Develop a new model to determine the path of a droplet on the

surface of a virtual leaf

The overall objective of this research is to develop a new model to produce realistic

motion of a water droplet moving on a virtual leaf surface and to contrast the

model behavior with experimental observations. This droplet model is based on

using Newton’s second law to identify the features of the droplet motion coupled

with a new idea of thin-film theory to estimate the thickness of the droplet as

it traverses the surface so that a stopping criterion for the droplet motion can

be proposed. The droplet model is verified and calibrated using experimental

measurements. The effectiveness of the model is considered in terms of capturing

a realistic motion. This model helps with the understanding of water droplet,

pesticide and nutrient interaction or absorption through the leaf surface.

It is anticipated that certain surface representations will be found to be more

appropriate in the context of surface droplet movement. The application of sur-

face fitting techniques to reconstruct the leaf images forms the foundation for a

theoretical study of water droplet paths on leaves. It will be essential to produce

a smooth surface for this purpose and this is our next objective.

• Survey existing methods for surface fitting and propose new tech-

niques for modelling leaf surfaces

A primary objective of the thesis is to survey existing methods for surface fit-

ting and propose new techniques for modelling the leaf surface based on hybrid

strategies that combine the CT method with an RBF technique. The new hybrid

CT-RBF method is an interpolating finite element method, which has the advan-

tage of resulting in a smooth surface over the whole domain. The virtual surface

is comprised of a mesh of triangles over which the interpolant is constructed from

scattered data captured by a laser scanner.

1.2 Literature Review

This section consists of three subsections. In §1.2.1, techniques for reconstructing

surfaces from scattered data are surveyed. The concept of a virtual leaf surface and


the use of three-dimensional digitising devices, for example, the laser scanner and

the sonic digitiser are described in §1.2.2. In §1.2.3 an overview of some models

for simulating the water droplet movement on surfaces is given.

1.2.1 Surface Fitting Techniques

The techniques investigated in this thesis are mathematical methods of surface

fitting applied to scattered leaf surface data sets that are sampled from real leaves.

These methods are interpolation methods based on the CT method and RBF

methods and a review of these methods is given in the following two sections.

Clough-Tocher method

The Clough-Tocher method (CTM), introduced originally by Clough and Tocher

for structural mechanics applications [29] is used to minimize the degree of the

polynomial interpolant without losing the continuity of the gradient over the whole

domain. In this way, the requirement of second derivatives for the construction

of the interpolant is avoided and the number of nodes is reduced. The CTM is

a seamed element approach, whereby each triangle is treated as a macro-element

split into subtriangles, which are called micro-elements, as shown in Figure 1.1.

The Clough-Tocher approach has the advantage of producing a smooth surface over

the whole domain. A more detailed description of CTM can be found elsewhere

[80, 84, 110, 122].

The CTM sees an interpolating cubic polynomial constructed on each subtri-

angle to enable a bivariate piecewise cubic interpolant to be devised over the entire

mesh that is continuously differentiable. The key result is that only twelve degrees

of freedom are required for the CTM, the function values and the gradient at each

vertex, as well as the normal directional derivative information at the midpoint of

the edges.

For many applications, the derivatives at the midpoints of each side and the

derivative information at the vertices are usually unavailable. The vertex gradient

estimates are often generated from neighbouring data information, and thereafter

the edge normal derivatives are determined as the mean of the normal derivatives


Figure 1.1: The Clough-Tocher triangle showing subdivision into three subtriangles. The

directional derivatives at triangle vertices and normal derivatives at edge midpoints are pictured

as arrows.

estimated at the two vertices associated with the edge. This approximation, which

produces a quadratic fit, is based on the assumption that the normal slope along

the sides of the triangle changes linearly, see Lancaster [80]. Such an approximation

is only exact for quadratic functions. The CTM interpolant has the form:

ϕ(x, y) =3∑

i=1

(fibi + (ci, di)

T · ∇fi

)+

3∑

j=1

∂f

∂nj

ej . (1.1)

In this representation the twelve functions bi(x, y), ci(x, y), di(x, y) and ej(x, y), i =

1, 2, 3 are cardinal basis functions (see Appendix A), having the property that just

one of them is unity and the reminder zero at each of the node points. Thus, twelve

independent pieces of information are needed to determine ϕ, which comprise of

the function values and the gradient at each vertex, together with the normal

directional derivative information along the edges (refer to figure 1.1). One notes

from equation (1.1) that the basis is a dual mixture of both point evaluations and

directional derivative functionals. A more detailed description of this approach,

together with the precise set of cardinal basis functions can be found in [80].


For the purposes of implementation, the Clough-Tocher triangle is at first trans-

ferred to a standard triangle T0 having the vertices located at P1 = (0, 2√

33

), P2 =

(−1,−√

33

) and P3 = (1,−√

33

), and mid-points P4 = (12,√

36

), P5 = (−12

,√

36

) and

P6 = (0,√

36

). T0 is then split into three subtriangles (see figure 1.1) T1, T2 and T3.

Interpolation on the Clough-Tocher triangle can always be carried out on a stan-

dard triangle and then transformed to any given triangle T . This transformation

is known as an affine transformation.

The mapping of an arbitrary triangle T having vertices (x0, y0), (x1, y1) and

(x2, y2) onto a standard triangle, see Figure 1.2, can be achieved using the mapping:

ξ(x, y) = a11x + a12y + b1,

η(x, y) = a21x + a22y + b2, (1.2)

where

a11 = 2y2−y1−y0

∆ , a12 = −2x2+x0+x1

∆ ,

b1 = x2(y1+y0)−y2(x1+x0)∆ ,

a21 =−√

3(y1−y(0))

∆, a22 =

√3(x1−x(0))

∆,

b2 = 2√3

+√

3(x2(y1−y0)−y2(x1−x0))∆ ,

and

∆ = det

1 x0 y0

1 x1 y1

1 x2 y2

(1.3)

= x1y2 − x2y1 − x0(y2 − y1) + y0(x2 − x1).

The mapping in the reverse direction, from the standard triangle onto T , is


given by:

x(ξ, η) = b11(ξ − b1) + b12(η − b2),

y(ξ, η) = b21(ξ − b1) + b22(η − b2), (1.4)

where

b11 =1

2(x1 − x0), b12 =

1

2√

3(2x2 − x0 − x1),

b21 =1

2(y1 − y0), b22 =

1

2√

3(2y2 − y0 − y1).

The nodal values at the vertices and midpoints of the standard triangle T0 are

(x2, y2)

(x0, y0)

(x1, y1)

x

y

Triangle T

(0, 2√

3/3)

(−1,−√

3/3) (1,−√

3/3)

ξ

η

Standard Triangle

Figure 1.2: Mapping a triangle on a standard triangle.

obtained from the transformed node values of the arbitrary triangle T in two steps.

First, at any point of T0, we determine

uξ = b11fx + b21fy,

uη = b12fx + b22fy, (1.5)


where fx and fy are given at each vertex of T , and b11, b21, b12 and b22 are given

in equation (1.4). Second, the nodal values at the vertices of T0 are determined in

terms of the values of uξ and uη according to the following relations:

∂u∂z1

(P1) = uξ(P1),∂u∂z2

(P2) = −12

uξ(P2) +√

32

uη(P2),

∂u∂z3

(P2) = −12

uξ(P3) −√

32

uη(P3),

∂u∂w1

(P1) = uη(P1),∂u∂w2

(P2) = −√

32

uξ(P2) − 12uη(P2),

∂u∂w3

(P3) =√

32

uξ(P3) − 12uη(P3),

∂u∂n4

(P4) =√

32

uξ(P4) + 12uη(P4),

∂u∂n5

(P5) = −√

32

uξ(P5) + 12uη(P5),

∂u∂n6

(P6) = −uη(P6).

In these equations the partial derivatives with respect to z and w give the gradient

vector at each vertex of T0 from which the gradients ∇fi, i = 1, 2 and 3, (which

appear in figure 1.1) can be calculated.

Breslin [18] estimated the directional derivatives necessary for the CT method

using the difference quotients, see equation (1.7), that is extracted from the Taylor

series expansion given in equation (1.6). Breslin computed the difference quotients

from nearby scattered data points, and then applied a least squares fit to estimate

the CT gradients. Loch [84] estimated the gradients at the vertices for use in the

CT method following Breslin [18] and then estimated directional derivatives at

each edge midpoint by taking the mean of the gradients at the two vertices on the

same edge.

In this thesis we adopted the approach given in [18] for our gradient estimation

and used the set of nearest neighbours closest to each vertex and edge midpoint to

generate approximate directional derivatives using a truncated multivariable Tay-

lor series expansion. We investigated first, second and third derivative information

in the Taylor series expansion for the gradients estimates. This procedure enables

an overdetermined linear system to be constructed that can be solved in the least

squares sense to extract the required gradient approximation.

We now outline the process of gradient estimation in the following paragraphs.


Suppose we require an estimate of the gradient ∇f(a) at some point a ∈ D

and a is surrounded by m scattered data points vi = a + hiνi, i = 1, . . . , m with

hi = ‖vi − a‖ and νi is a unit vector. Then, Taylor’s Theorem for several variables

states that

f(a + hν) = f(a) + h(ν · ∇)f(a)

1!+ · · ·+ hn (ν · ∇)nf(a)

n!+ Rn, (1.6)

where Rn is the Taylor remainder. If the Taylor series is truncated (for example

at n = 1) and the first two terms of the right hand side of (1.6) are evaluated at a

scattered set of points vi, the difference quotient in the direction of the unit vector

νi is obtained by transfering the term f(a) to the left hand side and then dividing

the equation by hi to obtain

f(a + hiνi) − f(a)

hi= (νi · ∇)f(a) + O(hi), (1.7)

where O(hi) represents the error of the gradient estimate. Equation (1.7) is ap-

plied at each of a neighbouring set of points vi, i = 1, . . . , m near a to obtain the

overdetermined linear system

Aγ ≈ q, (1.8)

where A, γ and q are given (for the case n = 1 as an example) by:

A =

νx1 νy1

νx2 νy2

......

νxmνym

, γ =

∂f(a)

∂x

∂f(a)∂y

, q =

(f(a + h1ν1) − f(a))/h1

(f(a + h2ν2) − f(a))/h2

...

(f(a + hmνm) − f(a))/hm

.

The least squares approximation of equation (1.8) for γ = argminx∈R2‖Ax − q‖2

yields estimates of the gradient of f at a that are O(h) accurate.

In general, the Taylor series approach offers a gradient estimate with accuracy

of O(hnmax), where n is the number of terms taken in the Taylor expansion and

hmax = max1≤i≤m ‖hi‖2 is the maximum distance from the point of interest say

a and any of the cloud of neighbouring points used for estimating the gradient;

numerical experiments reported by Belward et al. [13] are consistent with this

generalisation for the case n = 2.


In this work it was shown that the smallest singular value of the least squares

gradient coefficient matrix can impact the accuracy of the gradient estimate. An

error bound for a gradient approximation of O(hnmax) of the least square gradient

estimate was derived as

‖∇f(a) − γ‖‖∇f(a)‖ ≤ ϑmaxh

nmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 , (1.9)

where σ1 is the smallest singular value of A and ϑmax is the Lipschitz constant,

which can be estimated (see [135]) by an application of the mean value value

theorem to the mixed partial derivatives in the Taylor series remainder as

ϑmax =√

2 maxξ∈D

(∣∣∣∣∂n+1f(ξ)

∂xn−i+1∂yi

∣∣∣∣ , i = 0, . . . , n

).

The difficulty in the use of the CT method concerns the absence of the gradients.

We developed a new, accurate, and efficient technique to estimate the required

gradients based on using Radial basis functions, which is discussed in the next

section.

Radial Basis Functions

The simple structure of the radial basis function (RBF) surface makes the RBF

approximation method straightforward to apply, and successful in many areas.

In order to obtain a smooth surface representation to estimate the function

values at points other than data points, radial basis function schemes have found

applications in areas such as geodesy [69], hydrology [16], and medical imaging [24].

Hardy [63] presents applications of RBFs in geodesy, geophysics, photogrammetry,

remote signal processing, geography, surveying and mapping, hydrology and the

solution of parabolic, elliptic and hyperbolic partial differential equations. A major

contribution of the theory of RBF approximation can be found in Beatson [10,11,

12], Powell [113] and Buhmann [19].


A Radial Basis Function (RBF) approximation to f is a function S of the form:

S(x) =N∑

i=1

aiR (‖x − xi‖) , x ∈ R2 (1.10)

where R(r) is a fixed real-valued function of ri = ‖x − xi‖ with ‖.‖ denoting the

Euclidean norm. The points xi, i = 1, 2, . . . , N are called the centres of the RBF

approximation. The function S(x) interpolates f at x1, . . . , xN if ai, i = 1, . . . , N

satisfies the system

Λa = F with Λij = R (‖xj − xi‖) , i, j = 1, . . . , N (1.11)

and F = (f1, . . . , fN)T .

A main problem of the RBF method concerns its application to large sets of

data points where the costs of the computation included in fitting and evaluating

the RBF can become time-consuming. This cost manifests because, in order to

calculate the RBF coefficients ai, i = 1, 2, .., N in equation (1.10) a large, dense,

matrix system of size N × N has to be solved. Typically, this system can become

severely ill-conditioned with several, very small in magnitude singular values evi-

dent. Franke [55], for example, suggested that the application of global methods

be restricted to sets of up to 100-200 data points and compared around 30 inter-

polation schemes in two-dimensions, the purpose being to evaluate these schemes

for scattered data interpolation. Franke found that two of the most accurate

schemes were based on fitting RBFs. Beatson et al. reduced the cost of evaluat-

ing the radial basis functions considerably by applying fast evaluation techniques

for example, hierarchical and fast multipole-like methods in [11] and the GMRES

iterative method and fast matrix-vector method in [10]. Beatson [12] developed a

multivariate momentary evaluation scheme by generalising the fast multipole-like

methods for fast evaluation of the RBF’s. The developed algorithm was valid for

different choices of the RBF’s (see also Cherrie [27]). Radial basis functions are

nowadays applied in software to drive laser scanners (Carr [22, 23]).

Well known examples of radial basis function methods include the Hardy’s


multiquadric and thin plate splines. The Hardy’s multiquadratic RBF is given by:

R(r) =√

r2 + c2. (1.12)

The parameter c is specified by the user. Thin plate splines were developed by

Duchon [35]. These are also called surface splines [61] because they minimise the

bending energy of a thin, infinite elastic plate. They reproduce linear polynomials

and can be expressed in the form of equation (1.10) with

R(r) = r2 log r.

Many radial basis functions have an associated width parameter c specified by the

user, which is related to the spread of the function around its center. The default

width is the average over the centers of the distance of each center to its nearest

neighbour, which is a heuristic given in Hassoun [64] for Gaussians (refer table

1.1).

The accuracy for interpolating scattered data with radial basis functions de-

pends on this parameter c. Theoretical results show that multiquadric interpola-

tion becomes more accurate as the multiquadric parameter c increases [89]. For

some values of c the problem may become ill-conditioned [39,73,93]. Many meth-

ods for selecting c for the multiquadric and inverse multiquadric interpolants in

two-dimensions have been introduced in the literature [6, 17, 21, 41, 43, 45, 96, 111].

Franke [55] used c = 1.25 D√n

where D is the diameter of the minimal circle enclos-

ing all data points. A similar suggestion was also made by Foley [44]. Hardy [62]

suggested a value of c = 0.815d where d =Pn

j=1 dj

nand dj is the distance between

the jth data point and its closest neighbour. For an example of different choices of

the parameter c, see [65].

The accuracy of the multiquadric and inverse multiquadric interpolant was

studied by Carlson et al. [21], as well as Franke [55], and they concluded that

the accuracy depends on the choice of the parameter c. Carlson et al. proposed

two methods based on observations from numerical experiments for the use of

multiquadric and inverse multiquadric interpolants in two dimensions. They use

six different test functions and six different sets of data points. A data vector F =


(f1, f2, ..., fN)T for each set of data points and each test function F was calculated

by computing F over the set of data points such that F (xi) = fi, i = 1, 2, ..., N.

The coefficients ai for the interpolating radial basis function S were determined by

solving the equations:

S(xi) = fi, i = 1, 2, ..., N. (1.13)

A root mean square (RMS) error was computed between the interpolating radial

basis function and the test function, where the RMS error depends on the choice of

the parameter c. Carlson et al. specified the optimal value of c that minimises the

RMS by repeating the computation of the RMS error with different choices of the

c value. They introduced a scheme based on the residual error of a quadratic least

squares fit to the data points for selecting an effective value for c, and concluded

that the value of c should be taken proportional to the inverse of the residual error.

Compared to the previous methods that depend on the number and distribution of

data points, Carlson et al. observed that the selection of c improves the accuracy

of the approximation. However, in many cases, the choice of the value of c is

still far from the optimal value. Foley [43] introduced a scheme for evaluating

a better value for the parameter c using some observations from [21], where the

multiquadric and inverse multiquadric interpolants have the same optimal value

of c. Furthermore, the approximation error of these two interpolants is about

the same for the optimal value of c. Foley concluded that the proposed scheme

produced better values for c than the value of c selected in [21].

Rippa [121] repeated some of the experiments preformed by Carlson et al. for

the influence of the parameter c on the quality of the approximation by multi-

quadric, inverse multiquadric, and Gaussian interpolants, and confirmed that the

accuracy of these three RBF interpolants depends on the choice of c.

Rippa proposes an algorithm for selecting a good value for the parameter c in

the sense that the quality of the approximation of the interpolation defined with

the value of c is comparable to the quality of the approximation of the interpolant

defined with the optimal value (the value of c that minimises the RMS error be-

tween the interpolant RBF and the unknown function from which the data vector


F was sampled). He concluded that the optimal value of c depends on the number

and distribution of the data points, on the data vector f , and on the computation

precision. The Rippa algorithm was based on minimising a cost function that rep-

resents the (RMS) error between the interpolating radial basis function and the

unknown function. The cost function is defined by taking the first norm of the

error vector

E = (E1, . . . , EN)T with Es = fs − Ss(xs), s = 1, . . . , N,

where

Ss(x) =N∑

i=1,i6=s

asiR (‖x − xi‖) . (1.14)

is the interpolation to a reduced data set obtained by removing the point xs and

the corresponding data value fs from the original data set, and Es is a function of

c since it requires translation of a basis function that depends on c.

Rippa showed that

Es =as

ass

, (1.15)

with as as defined in equation (1.11) and as is the solution of

Λas = es, (1.16)

where es is the sth column of the N ×N identity matrix. Finally, the cost function

C(c) is given by

C(c) = ‖E(c)‖1 , (1.17)

and

copt = arg minc∈R

‖E(c)‖1 . (1.18)

Rippa shows numerically that the error obtained by using the good value of

c is similar to the error obtained by using the optimal value of c. He also shows

that the graph of the cost function is similar to the graph of the RMS error [121].

Hybrid RBF’s combine a radial basis function model with a more standard linear


Table 1.1: Choices of R for which the interpolation matrix is invertible.

Name R(r)

Multiquadric (r2 + c2)1/2, c ≥ 0Inverse multiquadric (r2 + c2)−1/2, c > 0

Gaussian e−r2/c2 , c > 0

model such as polynomials. For example:

S(x) = P (x) +N∑

i=1

aiR(ri), x ∈ R2 (1.19)

where P (x) =∑p

k=0 ckxk is a low degree polynomial, for instance, linear p = 1 or

cubic p = 3.

The two terms are added together to form the overall model. To avoid solving

large linear systems and large scale optimisation problems arising in the interpo-

lation, a smoothing or minimal energy spline is directly computed from the data.

Davydov et al. [30] used a local hybrid method based on a linear combination of

polynomials and radial basis functions to a modified scattered data set introduced

in [31].

The accuracy and shape recovery capability of the hybrid method was deter-

mined by a numerical experiment based on two test functions. Davydov et al. [31]

extended the idea to achieve a good approximation of the radial basis function

(RBF) method and presented a standard local (RBF) approximation based on

interpolation or least squares, with the local knots selected using a thinning algo-

rithm. The performance of this method was compared with the method of Davydov

et al. [30] for some real world data sets. The least-squares method was used in

the local approximation since it consistently produced better results than inter-

polation. The thin plate spline and the multiquadric were chosen for the exper-

iments. In both methods, they considered local approximation schemes defining

non-polynomial approximations that were later converted into polynomials and


then extended to splines.

1.2.2 Virtual Leaf

The modelling of plant architecture has been researched extensively over the last

decades [7, 31, 114, 123] and models of leaf surfaces have not been generated with

great accuracy or level of detail, until recently when Loch [84] used two methods

to model accurate leaf surfaces. Leaves play an important role in the development

of a plant, and therefore some adequate representation of the leaf is required. A

representation may be used to study biological processes such as photosynthesis

[126] and a canopy light environment [7, 38].

Prior to the work of Loch [84], visual leaf surfaces were designed by trial and er-

ror until a realistic representation that captured the leaf surface and leaf boundary

was reached. Few of the past leaf models were based on extensive measurements

until 3D digitisers and faster computers with improved graphics capabilities be-

came available, see for example Room et al. [123]. Virtual leaf models may be

displayed in an abstract way, by a disk [127], or polygons [42], or more realistically

by a surface model that captures the surface shape and boundary (Prusinkiewicz

et al. [115]). Smith [127] generated visualisations of plants by representing the leaf

models in an abstract way, as disks.

Bloomenthal in [15] represented a maple tree by generating a maple leaf using

a video camera; see figure 1.3. The leaf was represented by three polygons and the

photograph was projected onto the surface model to increase the visual realism.

Bicubic patch methods, in particular Bezier patches, give more realistic visualisa-

tions of individual leaves to represent the surface. In [8, 115], L-systems terminol-

ogy (a formal mathematical approach to describe branching systems) was included

as a predefined surface object to which maps could be applied. Prusinkiewicz et

al. [116] generated a leaf model consisting of leaflets attached to a single stem

(single compound leaves), see figure 1.4(a), while individual leaflets were described

by predefined Bezier patches that allow the shape of the leaflet to be changed by

parameter selection with a graphical function editor. Hammel et al. [58] presented

a method for modelling a compound leaf. A skeleton branching given in figure


Figure 1.3: The Maple leaf, taken from [84]. The shape of a maple leaf and the texture

image projected onto the shape.

(a) (b)

Figure 1.4: (a) A single compound leaf model taken from [84]. (b) The branching skeleton

and contour, reproduced from [62].


1.4(b) was used to capture the layout of the lobes, and the margin of the leaf

was represented by an implicit contour that was traced around the skeleton. The

constructed surface can be bent or complemented for increased realism. Bound-

ary algorithms were applied by Mundermann et al. [97], for modelling lobed leaves.

The model consists of a two-dimensional leaf silhouette, which can either be defined

interactively by a curve editor, or derived from a scan of the leaf. More information

was made available from the leaf scanned image. The surface was constructed by

sweeping a planar curve between the silhouette and the skeleton [128]. The model

was two-dimensional, so the third dimension was determined either by turning the

surface in three dimensions, or by introducing noise, see figure 1.5. Lintermann

Figure 1.5: The scanned Lobed leaves, taken from [84].

and Deussen [82, 83] presented a model that considers leaves as components, the

leaf was defined by its outline, axis, the curves that defined the curvature, and its

material. They used splines to generate the outline of the leaf as well as the curves.

To produce a realistic leaf model of the generation of the dandelion, a mid-rib is

added at first and then curvature is introduced. Finally, the jagged boundary is

captured by a spline.

Maddonni et al. [88] used piecewise linear triangles to represent the leaf surface,

where vertices along the boundary are estimated by allometric relationships and


Espana et al. [38] modelled the undulations of the boundary.

(a) (b)

Figure 1.6: Frangipani leaf: (a) point sets and (b) triangulation

None of the previous models presented are based on detailed three-dimensional

real world leaf surface data. If the data points were collected, they were used to

determine the position, orientation, and size of a leaf; not to define its surface

shape. Loch in her thesis [84], used two finite element based methods (piecewise

linear triangular and piecewise cubic Clough-Tocher triangular) to model detailed

and accurate leaf surfaces in three dimensions for a large number of data points

sampled by a laser scanner from real leaf surfaces. An incremental algorithm was

presented to reduce the size of the set of data points, the algorithm is stopped

when the surface fit has reached 5% or 1% accuracy compared to the total set

of available data points. From these results, guidelines were deduced to describe

where data points should be positioned when a single point-device is used. A

triangulation corresponding to a reduced data set was also presented (see figure

1.6), where the colour map represents the average surface height of three vertices

of each triangle. The model development for Frangipani, Anthurium, Elephants

ear and Flame Leaves is shown in figures 1.7 and 1.8.

Fitting leaf surfaces in the manner described to this point produces a boundary

represented by a piecewise linear curve. To smooth the boundary, Loch [85] applied


(a) (b)

Figure 1.7: Photographs of the scanned (a) Frangipani leaf and (b) Anthurium leaf.

(a) (b)

Figure 1.8: Photographs of the scanned (a) Elephant’s ear tree leaf and (b) Flame tree leaf.


a method based on the interpolation of a three-dimensional parametric piecewise

cubic curve through the boundary points with a projection onto the reference plane.

Three methods were presented to extend the surface into curvilinear triangles along

the boundary. Two methods were based on extrapolation, and the third was based

on an interpolation method. These methods were found to be sufficient to model

the smooth boundary of the leaf surface.

In an approach to simulate and visualise the spray distribution within canopies,

Hanan [60] illustrated an architecture model of maize plants for a chaotic system at

the spray-canopy onto the silk of corn cobs. This method was based on L-system

models of a plant development and a particle system model for capturing spray

droplet flight, impact, and splash. The way in which different droplet arrangements

affect spray deposition is also discussed.

Digitising leaf surfaces

To reconstruct the shape of the leaf using the surface fitting techniques described

in section 1.2.1, we require a set of data points. The process of sampling data

points from the leaf surface using a measuring device is called digitising. This

process ensures that the visible exterior data points of the leaf are sufficient to

capture the surface of the leaf. The most suitable method for digitising depends

on the leaf type, as well as the application for which the data will be used. The

aim of this section is to describe the fitting of leaf surfaces to the laser-scanned

data points.

Three different methods were described in [84] for sampling leaf surface data

for generating a three-dimensional model of the surface. The first method uses a

sonic digitiser (a point-by-point method), see figure 1.9(b), which is an inexpensive

and popular sampling device, and more suitable than the other two methods if the

structure of the whole leaf needs to be captured. The sonic digitiser allows a

selection of point positions and controls the number of points collected. However,

collecting a large set of data points using a sonic digitiser takes a long time, because

this device only samples one data point at a time, and can be inaccurate due to

external factors including hand movement. The second sampling method involves


a laser scanner (a multiple-point method), see figure 1.9(a), which is adopted in

this research. It is the most expensive device of the three methods and consists of a

processing unit, a transmitter and a wand. The wand contains two video cameras

mounted at an angle to the centrally positioned laser line generator [3, 84].

The laser scanner enables a large set of data points to be collected in a short

time by sweeping a line of red laser light over the leaf surface. When the laser

line is projected onto the surface, the cameras capture the intersection of this

line with the surface. In this way, each sweep contains several lines and each line

returns a number of data points, see [3,84,91]. By joining these lines we obtain the

set of data points for the entire surface. The third sampling method is based on

a photogrammetric approach (either the point-by-point method, or the multiple-

point method). This approach is the cheapest method for sampling leaf data,

however it does not produce useful sets of data points because it is more suitable

for capturing volumes than surfaces. For more details on this technique see [1,84].

Loch [84] collected data points for different types of leaves using the second

sampling method described above. Leaves were chosen according to their bound-

ary shape to include a variety of different boundaries. The leaf boundaries were

classified either into a simple boundary type, for example, the Frangipani and An-

thurium leaves that are adopted in this research (as shown in figure 1.7); or for a

re-entrant boundary type, for example, Flame tree and Elephant’s Ear as shown

in figure 1.8. The boundary of the leaf is important for a realistic representation

of the leaf because it is used for the purpose of constructing a triangulation of

the leaf surface. The laser scanner returns a large set of data points on the leaf

surface without identification of data points situated on the boundary. The po-

sition of data points along the boundary can, however, be selected using a sonic

digitiser. To determine particular boundary information using a laser scanner, two

ways of modifying the laser scanner technique were explored, see [2, 3]. The first

is a pen-like device attached to the laser scanner processing unit that replaces the

wand. The second way was suggested by ARANZ (Applied Research Associates

New Zealand) based on blocking part of the laser scanner and modifying the soft-

ware to record only one data point at a time. One notes, however, that there are

some drawbacks associated with these processes (see [84] for more information),


(a) (b)

Figure 1.9: (a) The laser scanner. (b) The sonic digitiser.

and to avoid these drawbacks McAleer [59] has written the software PointPicker

to handle large sets of data points. Loch [84] selected the boundary points by hand

from the complete set of points chosen by the PointPicker software.

1.2.3 The path of a droplet

Several researchers have studied the simulation and modelling of fluids, in partic-

ular water. Most of these models concerned water motion in the form of waves

[53,90,109,134] and realistic liquid animation has also been modelled [25,51,52,74].

However, only a limited number of methods, during the 1990’s, address the natural

phenomenon of water droplets flowing on surfaces [54,68,70,71,72,137]. There are

different factors that affect how a water droplet flows on the surface of a leaf, includ-

ing gravity, interfacial tension, surface tension, and air resistance. Consequently,

a physically correct simulation of water droplet flow on structured surfaces is not

found. Meta-balls in a gravitational field were used [139] to model static droplet

shapes on flat surfaces. Tong et al. [132] modelled and animated water flows using

meta-balls by proposing a volume-preserving approach. Lanfen [81] presented a

physically plausible method for two, or more, large water droplets morphing on

a plane. The droplet on the plane is characterised by contact area, between the

droplet and the plane in two-dimensions and a profile curve [139], where this profile


is used to express the droplet height (along the y-axis) in the contact area. The

morphing process is driven from the contact area boundary (distance field) and

combined with a rigid transformation to produce a natural effect. For providing

a realistic scene, the ray-tracing method is implemented. Kaneda et al. [71] pro-

i, j

i − 1, j + 1

i, j + 1

i + 1, j + 1

Figure 1.10: Discrete surface model.

posed a method for generating a realistic animation of water droplets and streams

on a glass plate, such as a windowpane or windshield, taking into account the

dominant parameters of the dynamical system, which include gravity, interfacial

tensions and the collision of droplets. A sphere was used to model the droplet. A

high-speed rendering was also developed, which takes into account reflection and

refraction of light. Their method reduced the computation cost of animations that

contains scenes involving a rain covered windshield, or windowpane. The speed of

the droplet was assumed to depend on the wetness of the direction (i + k, j + 1),

k = −1, 0, 1 (see Figure 1.10) and the angle of inclination of the glass plate (θ)

instead of depending on the mass of the droplet. The form of the velocity is

v = v0 + ai+k,j+1(θ)t, k = −1, 0, 1,


where v0 and t are the initial speed of the droplet and the time respectively when

it is rolled on the glass plate, or after the collision happens between droplets. The

acceleration of the droplet is denoted ai+k,j+1, where the inclination angle is θ and

i, j, k are the indices of the droplet position on the surface.

The stream of the droplets does not run straight down the glass plate but

meanders down the plate, along a path determined by impurities on the surface

and inside the droplet itself. To simulate the water droplets and their stream,

Kaneda et al. [71] developed a discrete surface model where the glass plate was

divided into a small mesh (see Figure 1.10). To every lattice point on the glass

plate, an affinity for water, 0 ≤ ci,j ≤ 1, is assigned in advance. A water droplet at

the lattice point (i, j) begins to meander down a surface when its mass mi,j exceeds

a static critical weight mSc (θ), where the inclination angle of the surface is θ. The

droplet, at lattice point (i, j) is assumed to move to one of three neighbouring

points (i − 1, j + 1), (i, j + 1) and (i + 1, j + 1), as shown in Figure 1.10. If some

water exists at any of these three different locations then the droplet will move

in the direction in which the water exists, with the direction (i, j + 1) given the

highest priority. The droplet moves in the direction in which the water mass is

largest for the case where there is no water at point (i, j + 1). If there is no water

present at any of these three points, a decision parameter is used to identify the

next position.

The speed of the new droplet, v′

0, colliding and merging with another droplet

is calculated using the law for conservation of momentum; namely

v′

0 =m1v1 + m2v2

m1 + m2,

where m1, m2 and v1, v2 are, respectively, the mass and the speed of two droplets

before they merge, while the mass of the new droplet is assumed to be m1 + m2.

A meandering droplet that has no water ahead of it will decelerate and eventually

stop when its static critical weight is larger than the mass of the droplet.

Kaneda et al. [72] proposed an extended method based on previous work given

in [71] for generating a realistic animation of water droplets as well as their streams

on curved surfaces, taking into account effects such as gravity, interfacial tensions,


and water merging. A discrete surface model is used to simulate the flow of droplets

running down the curved surface along with their stream. The surface is divided

into small quadrilateral elements with a normal vector at the centre. Contribution

is made by the affinity to the meander of the streams and to the wetting phe-

nomenon, with the degree of affinity, ci,j, randomly assigned to each element in

advance based on a normal distribution, which is assumed to depend on the in-

terfacial tension. The motion of the water droplets on the surface depends on the

external forces (f ext), gravity and wind. When these forces exceed a static critical

force (internal force f int), the water droplet starts to meander down the surface.

The critical force f int originates from the interfacial tension between the water

and the surface, and it is the resistance that prevents the droplet from moving.

The force of resistance is calculated using the degree of affinity by the following

equation:

f inti,j = −βdci,jd

∗p, (1.20)

where βd is a coefficient for converting the degree of affinity into the resistance

(and is set experimentally) and d∗p is the unit vector that indicates the direction of

movement. When the droplet travels from one element to another, it has an initial

speed of v0. The initial speed is projected in the direction of movement, which is

given by the equation:

v0p = (v0 · d∗p)d

∗p. (1.21)

This initial speed is updated when the droplet arrives at the next element as

v′

0 = ap∆t + v0p. (1.22)

In equation (1.22), ∆t is the time for the droplet to travel to the next mesh

element and ap is the projected acceleration of the droplet in the direction of

movement, which is given by:

ap =(fext

i,j + f inti,j ) · d∗

p

mi,jd∗

p. (1.23)

Eight different directions classify the direction of movement. Kaneda et al. [72]


calculated the probabilities for movement in each direction based on three different

factors. The first factor is the direction of movement. The second factor is the

degree of affinity for water to move to the neighbouring elements, and the third

factor is the wet or dry condition of the eight neighbouring elements. When the

direction of movement is determined, the water droplet is moved to the next ele-

ment. The time taken for the droplet to move to the next element is calculated and

the accumulative time is stored. If this accumulated time exceeds a user specified

time frame, the droplet motion is stopped. The stream meanders down the surface

because of impurities and small scratches on the surface, which are expressed by

the interfacial tension of the surface. The route of the stream is determined by

external forces (gravity and wind). Two rendering methods are proposed: a fast

rendering method using spheres, and a more sophisticated method that pursues

the photo-reality using meta-balls.

When the external forces that act on the droplet exceed a static critical force,

the droplet flows on the surface and some amount of water remains behind because

of the wetting, and later the water flow merges with the remaining water. There-

fore, a solution to the wetting phenomenon and the problem with two droplets

merging, is also addressed. The mass of the remaining water, m′

i,j, when the

droplet moves from one element to another again depends on the affinity, so that

the larger the affinity the larger the remaining water. The mass is calculated using

the function h(ci,j), which gives the coefficient for the remaining water as:

m′

i,j = h(ci,j)mi,j .

When two droplets merge, the speed and the mass of the new droplet is computed

using the law of conservation of momentum. Kaneda et al. [72] calculated the mass,

m′

i+k,j+l, and speed, v′

i+k,j+l, of the new droplet after merging on the element

Mi+k,j+l (k, l = −1, 0, 1; with k and l not both zero at the same time) using the

following equations:

m′

i+k,j+l = mi+k,j+l + mi,j − m′

i,j, (1.24)

v′

i+k,j+l =mi+k,j+lvi+k,j+l + (mi,j − m

′

i,j)v′

0

m′

i,j

, (1.25)


where v′

0 is a new initial speed for the droplet given in equation (1.22) and mi+k,j+l

and vi+k,j+l are the mass and the speed of the water droplet respectively on

the mesh element Mi+k,j+l when the droplet moved from mesh Mi,j to the mesh

Mi+k,j+l before merging.

Kaneda et al. [70] proposed a method for generating realistic animations of wa-

ter droplets that meander down a transparent surface based on the work presented

in [71,72]. This work is useful for applications such as drive simulators and anima-

tion of water droplets on a windshield. The main difference between this work and

the previous work is the modelling of obstacles that move against water droplets

on a surface; for example, a windshield wiper. Further, Kaneda et al. considered

the contact angle of the water on the surface. The droplet is represented by a

single particle system in a discrete environment where the motion of the droplet

is generated by a particle system. The droplet is modelled as a sphere and the

contact angle between the droplet and the surface is also taken into account.

The curved surface is divided into small quadrilateral elements. The external

forces and obstacles affect the droplet movements when the droplets travel from

one element to another. The external forces are assumed to be gravity and wind,

and the affinity is assigned for each element in advance (randomly in most cases)

based on a normal distribution in order to capture the meandering and wetting

phenomenon. This affinity describes the lack of uniformity of an object surface

because of impurities and small scratches. The direction of movement is classified

into eight different directions as mentioned in [72]. The probability of movement

of a water droplet in each direction is calculated based on the probability of the

three dominant factors mentioned before, in addition to the existence of obstacles

on the neighbouring elements.

To produce a physically plausible animation of the droplet flow, Jonsson [68]

proposed a new model based on using normals of the bump map surface in the

computation of water droplet flow based on the model presented in [71]. As a

result, the water droplets meander down the micro-structured surface described

by the bump map. Jonsson assumed that the external force that affects the water

droplet flow is due to gravity, while the internal force of resistance originates from

the interfacial tension that exists between the water droplet and the surface. The


direction of the internal force is opposite the direction of movement. The direc-

tion of the water droplet movement is computed by applying the Gram Schmidt

orthogonalisation algorithm [102] to orthogonalise fext against N as follows:

dp = fext − (N · fext)N, (1.26)

where N is the unit length normal vector, which is retrieved at every point from

the bump map, and fext is the gravity; N, dp

‖dp‖ then forms an orthonormal set of

vectors. Solid spheres are used to model the droplets and each droplet is a particle

system. Only one normal is retrieved from the bump map to compute the motion

for each droplet. The velocity and acceleration of the droplet are computed (see

Kaneda et al. [72]) along with the new position of the droplet by adding velocity

to the position for each step. The new position is given by:

pi+1 = pi + vp, (1.27)

where p is the droplet position and vp is the projected velocity into the bump

map surface. A maximum speed is used for modelling adhesion, in which case the

speed of the droplet on the bumpy area is reduced.

Fournier et al. [54] presented a model oriented towards an efficient, visually-

satisfying simulation of a water droplet moving down a surface represented by a

mesh of triangles. The efficiency results from the separation between the shape and

the motion of the droplet. The aim was to simulate the shape and motion of large

liquid droplets travelling down a surface when it is affected by surface roughness,

adhesion, gravity and friction forces. The adhesion is a function of the interfacial

area between the surface and the droplet, which is a force that works along the

normal surface. A droplet will fall from the surface if the component of the droplet

acceleration force that is normal to the surface is larger than the adhesion force

of the droplet. Fournier et al. assumed that the roughness of the surface only

contributes in the tangential force. The motion of the droplet is generated by a

particle system, with the droplet represented by a single particle [117].

The surface is defined by a mesh of triangles. A “neighbourhood” graph is


built at the beginning of the simulation so that each triangle is linked to adjacent

triangles. Throughout the entire simulation, one knows exactly in which triangle

a droplet is located. A droplet might cross several triangles between two time

steps. The motion is computed over each individual triangle to ensure the droplet

is properly affected by the deformations on the surface it has traversed. The

neighbourhood graph is used to quickly identify to which triangle the droplet

moves, and the time and position of the droplet are computed when it exits this

triangle.

Fournier considered in his model the two forces that affect the droplet move-

ment. These are gravity Fg and friction Ff . The friction force is modelled as a

linear viscous force with a constant negative factor due to surface roughness. Both

of these forces are assumed to be constant over a triangle and can be used to derive

an equation for the velocity and position of a droplet at any time t as follows:

v(t) = −Fr

kf+

(v(ti) +

Fr

kf

)e

kf t

m , (1.28)

p(t) = p(ti) −Fr(t)

kf

+ m

(v(ti) +

Fr

kf

)(e

kf t

m − 1

kf

), (1.29)

where Fr = Fg +Ff . Fournier et al. [54] simulated the effect of a streak that is left

behind a droplet when it passes over a triangle by reducing the roughness along

this path. The shape of a droplet is characterised by a small set of properties; for

example, volume conservation and surface tension.

Computational fluid dynamics has been successfully applied to simulate realis-

tic animation of fluid. Foster [52] presented a model based on the Navier-Stokes

equations to explain the fluid motion. A few years after that, Stam [129] presented

the stable fluid method in which the liquid velocity is controlled by using the

semi-lagrangian method. Enright [36,37] and Foster [51] used the level set method

to develop liquid surfaces so they can simulate more complex liquid motions. To

reduce volume loss and increase the surface accuracy, they combined the level set

evolution with particles (particle level set method). This combination also allows

the equation of motion for a liquid to be solved. A similar method, called the level

set method, was developed by Osher [107, 108] but does not use particles. Losas-


son [87] simulated water on a refined grid, such as an octree structure instead of

a regular grid, to capture more surface details using the Navier-Stokes equations.

In this thesis, the Navier-Stokes equations were not used to calculate the motion

equation of the droplet because of the computation expense needed to solve this

system on each element in the virtual leaf mesh.

A virtual surface model is proposed by Wang [137] to simulate various small-

scale fluid phenomena such as, flattened drops, stretched and separating drops and

capillary action. This method is based on selecting the contact angle between the

solid objects and the fluid surfaces. The selected contact angle uses a dynamical

model based on some properties, including the fluid front motion. The interfacial

tension between the solid and liquid surfaces is captured by calculating the surface

tension along the contact angle. Simulation of flow of free surface and interfaces is

presented by Ruben [124]. Here, the interface simulation is based on a predefined

grid that does not move with the interface because it is a fixed-grid method.

Various experiments have been performed in science and physics to understand

the interfacial tension between a liquid and a solid and many methods have been

developed to simulate liquid-solid interaction. Korlie [79] simulated a liquid drop

on a horizontal solid surface in three-dimensional space using quasi-molecular par-

ticles. Korlie used water as the liquid and graphite as the solid surface. Feng et

al. [40] simulated the drop impact and flattening process onto a solid surface using

Lagrangian meshing by the finite element method. The drop falling and depositing

effect was also studied by Zhao et al. [142] using a variational level set evolution

equation constructed by minimizing the energy of the surface tension. Bussman et

al. [20] developed a three-dimensional model of droplet impact and splashing onto

an asymmetric surface and then on curved shapes. A volume tracking algorithm

has also been developed for the volume-of-fluid method to track the droplet free

surface. Sussman et al. [130] presented an efficient 2D method based on a level

set method for computing the spreading of a body of oil underneath a sheet of

ice. The proposed solid surface was implemented for the contact angles. Later,

Renardy et al. [119] used a volume-of-fluid method for simulating a moving con-

tact line and their algorithm was limited to flat solid surfaces in two-dimensions.

O’Brien [100] presented a model for simulating the dynamic behaviour of splashing


fluid when objects impact on its surface. This model, which does not deal with

the flow of droplets on the surface, treats the liquid either as a volume, a particle

or a surface. The forces (objects) produce waves and splashes when objects collide

with the liquid surface. The model shows that there is no interaction between

liquid drops, however they merged with the liquid volume when they fall back into

it. Dorsey et al. [34] simulated weathering effects such as water sedimentation of

deposits while flowing on surfaces. The flow is modeled as a particle system with

each water drop treated as a particle. The model does not provide any motion

equation of the droplets because the focus is on the staining effects with time.

A diffuse-interface model was used to investigate liquid droplet spreading in

a partially wet regime on a smooth solid surface by Khatavkar et al. [76]. The

ambient fluid (air or fluid) considered in this model operates by displacing the

liquid droplet spread. The velocity of the ambient fluid affects the droplet spread

as well as the process time scale. The contact angle is taken into account in the

model and expresses the wettability, which also affects the spreading kinetics from

the initial stage. Later, Khatavkar et al. [75] used the diffuse-interface model to

study the impact of micron size drops on a smooth solid surface. The spreading

of a droplet on a wall under an inertial effect was investigated by Hang Ding [32]

using two numerical methods, the diffuse interface and a slip-length-based level-set

method.

Forster et al. [46] studied droplet impact behaviour and spreading on different

plant surfaces with small droplets between 100-1000 µm. They presented a model

for pesticide spray droplet adhesion [46]. The adhesion model is derived from

empirical data and is not process driven. However, the main point is that with

a small number of measurable parameters the adhesion was calculated. They

presented a set of behaviours of a 600 µm droplet comprised of water, water plus

Citowett, water plus bond adheres and water plus L-77 onto the upper or lower

surface of an avocado leaf. This leaf has the properties that the upper surface is

easily wetted, due to the low contact angle and the lower surface is harder to wet

as a result of the high contact angle. Forster et al. used a single droplet generator,

and water formulations with a fluor and UV light to visualise the droplets. The

avocado leaf is seated at a 45o angle as the larger the angle of incidence, the more


likely there will be difficulty in droplets adhering.

Dorr et al. [33] modeled pesticide spray droplets on plant surfaces. The model

combined a plant architectural model with a particle trajectory model. The particle

model was implemented to model spray droplet movement. Dorr et al. verified his

model by comparing it with an experiment of deposition of spray droplets on three

types of plants, cotton, sow thistle and wild oats at two growth stages (5-leaf and

2-leaf). They found that the architecture of the plant affects the amount of spray

deposition, for example deposition on the wild oats (5-leaf stage) was more than

the deposition on wild oats (2-leaf stage). In conclusion, the experimental data

was consistent with the model.

1.3 Research Methodology

Recall from §1.1 that the main objective of the research presented in this thesis is

to develop a new model for simulating a water droplet moving on a leaf surface.

In this section we describe how we derived this model. The main components of

this model are the construction of a virtual leaf surface from a scattered data set

and the interaction of a single droplet moving down this surface.

We have investigated two types of real leaves namely, a Frangipani leaf and an

Anthurium leaf, for the purpose of surface reconstruction. Note however that only

the Frangipani leaf was used for studying the behaviour of the droplet. A laser

scanner was used to capture the leaf data points, which returned a large number of

scattered points in three-dimensions. The coordinates returned by the scanner may

not necessarily coincide with the xy−plane in the data point coordinate system.

A solution of this problem is explained in section §1.3.1.

After collecting the leaf surface data, we have investigated surface fitting strate-

gies for the surface reconstruction and then proposed a new hybrid technique for

modelling the leaf surface. The hybrid method combines the Clough-Tocher (CT)

method with the radial basis function (RBF) method to achieve a smooth surface

over the whole domain, which is essential for simulating the path of the droplet.

The CT approach is an interpolating finite element method based on a triangular

domain. This triangulation was constructed using the EasyMesh generator, more


information of this process is given in §1.3.1.

The CT method requires derivative estimates at the vertices and midpoints of

the triangular mesh. This motivated us to investigate a number of techniques to

estimate the gradients of the CT triangle and finally to introduce the hybrid CT-

RBF method, which is based on using a multiquadric RBF to estimate the gradients

at the vertices and midpoints of the CT triangle. More details of this strategy are

explained in section §1.3.1. Another technique to estimate the gradients was based

on using a truncated Taylor series expansion. In this technique the gradients are

either estimated at the vertices and midpoints of the triangles, or estimated at

the vertices of the triangle and then averaged to obtain the gradient at the edge

midpoints.

Numerical experiments were carried out to assess the accuracy of these tech-

niques. These experiments were based on two data sets and six test functions taken

from Franke [55]. The numerical results show that the RBF method produced a

better gradient estimate for use in the CT method than the Taylor series method.

Thereafter, the new hybrid CT-RBF method was derived and then applied to the

leaf data to reconstruct the surface of the leaf, this process can be found in §1.3.1.

Using the truncated Taylor series method to estimate the gradients for the CT

triangles motivated us to derive an error bound for the least square gradient esti-

mates, as well as for the weighted least square gradients estimates. This theory was

supported by numerical experiments that are performed on the Franke data. More

details of this error bound analysis, along with the numerical investigations are

given in §1.3.2. The initially derived error bounds turned out to be several orders

pessimistic. This finding motivated us to improve the bound and to derive tighter

bounds. The tighter bounds were also supported by a numerical investigation of a

least squares problem, see §1.3.2 for more information.

After we constructed the leaf surface, we developed a model for a droplet of

water moving across the leaf surface. The droplet motion in our model was de-

termined using Newton’s second law, together with thin-film theory to estimate

the thickness of the droplet on the surface. Two forces are considered to affect the

droplet movement on the leaf surface, namely an internal force due to a friction and

a resistance force and a gravitational force. As mentioned before, the virtual leaf


surface consists of a mesh of triangles and the motion of the droplet is computed

over each individual triangle.

In order to verify our theoretical study of the water droplet path on a leaf, we

decided to perform experimental measurements of a droplet traversing a Frangipani

leaf surface. We marked five dots on the leaf surface as reference points of the

droplet and then a sonic digitizer was used to identify the location of these dots on

the leaf. A video camera was used to record the droplet movements. More details

of this model and the experiments are given in §1.3.3.

1.3.1 Construction of Virtual Surface

The work presented in this section discusses three surface fitting techniques, namely,

the CT method, the RBF method and then a new hybrid CT-RBF technique that

combines the CT method with the RBF method for modelling leaf surfaces. This

work is reported in the two papers [104, 105]. The reconstruction of the shape of

a leaf using surface fitting techniques requires a set of representative data points

sampled from the leaf surface using a measuring device. We used a laser scanner

to capture the leaf data, which returned a large number of three-dimensional data

points, see figure 1.11.

(a) (b)

Figure 1.11: Photos of the scanned (a) Frangipani and (b) Anthurium leaves.

Recall that the CT method is an interpolating finite element method based on

a triangular domain. In this method the function value is known at the triangle

vertices, however the derivative information at the vertices and at the midpoints of


each side of the triangular elements are unavailable. As mentioned in the literature

review, the vertex gradients can be approximated from the neighbouring data

information and then the edge normal derivatives can be estimated as the mean of

the normal derivatives from the two vertices associated with the same edge [80].

Breslin [18] used this approach when applying the CTM for rainfall data and

Loch [84] used it for leaf surface construction.

The research presented in our publication [105] examines different techniques

of estimating the gradients for use in the CT triangle interpolant. In the first

technique, we adopted the same approach given in [80] for the gradient estimation

and used the set of nearest neighbours closest to each vertex and edge midpoint

to generate approximate directional derivatives using a truncated multivariable

Taylor series expansion. The required gradient approximation is then extracted

by solving a constructed overdetermined linear system using the method of least

squares. First, second and third order derivative information in the Taylor series

expansion was investigated for the gradients estimates.

Two methods were used to estimate the gradients at the edge midpoints; the

first method estimates the gradient at the midpoints using the least square method.

The second method, given in Lancaster [80], used the average of the gradients of

the two vertices associated with the same edge. This approach allows three less

gradient estimates to be needed for the CT element and therefore is less compu-

tationally expensive than the first method, but only offers quadratic accuracy.

Intuitively, first, second and third order accuracy of the gradients estimate can

be obtained respectively by truncating the Taylor series after the first, second and

third terms. The estimated gradients using third order Taylor series was found to

be not much better than the gradient obtained using second order and first order

Taylor series. This made us question what is the error associated with this gradient

estimate. Then, we derived the error bound for the least square square gradients

estimates, which contained in the denominator the smallest singular value of the

least squares matrix. We found that this singular value affects the accuracy of

the gradient estimate. It is known that this singular value can become smaller

when the number of columns is increased in the least square matrix. A numerical

investigation of linear least squares methods for derivative estimation is given and


we found that the derived error bound was pessimistic by several orders. Thus, we

improved the derived bound by introducing a tighter error bound.

We developed a new hybrid technique that combines the CT method with the

RBF method to reduce the computational cost involved in the gradient estimates

using the truncated Taylor series, which result due to the construction and solution

of the overdetermined linear system by the method of least squares at each of the

triangle vertices and midpoints. This strategy is based on using the multiquadratic

RBF to estimate the gradients at the vertices and midpoints of the CT triangle.

The multiquadric RBF interpolant is given by:

S(x) =N∑

i=1

aiR (‖x − xi‖) , x ∈ R2 (1.30)

where R(r), given in equation (1.12), is a fixed real-valued function of ri = ‖x − xi‖with ‖.‖ denoting the Euclidean norm. The function S(x) approximates a function

f at r1, . . . , rN if ai, i = 1, . . . , N satisfies the system

Λa = F with Λij = R(rij), i, j = 1, . . . , N (1.31)

where rij = ‖xj − xi‖ and F = (f1, . . . , fN)T .

The gradient of S based on using n which is typically less than N , data points

is then given by:

∇Sn(x) =

n∑

i=1

ai∇R(ri), (1.32)

where

∇R(ri) =x − xi

ri

R′

(ri) (1.33)

and R′

represents the derivative of the radial basis function.

Our method of surface reconstruction is based on selecting a subset of n points

from the complete leaf data set N to construct a triangulation of the surface. Two

particular methods of the hybrid method were considered in our research. We

refered to these methods as the global hybrid method and the local hybrid method.

In the global hybrid method we use the same subset of points to construct a global

multiquadratic RBF interpolant from which equation (1.32) is used to compute


the gradients (∇Sn(x)) for all CT triangles in the mesh. For the local hybrid

method, only a ‘local’ subset of m points from the N is used to construct a local

RBF interpolant for each triangle and then equation (1.32) is implemented to

compute the required gradients (∇Sm(x)). These m points represent the closest

points to each of the vertices and midpoints for the CT triangle of interest. The

RBF linear system (1.31), constructed for either the global RBF interpolant or the

local RBF interpolant, is solved approximately using the truncated singular value

decomposition (TSVD) method to avoid problems with the ”near” singularity of

Λ. The TSVD method is based on disregarding the small singular values of the

system Λ [95] according to the criterion whereby the singular values that are less

than, or equal to, the product of the largest singular value with a chosen target ε

are ignored.

As mentioned in the literature review, the multiquadratic RBF has associated

with it a width parameter c, see equation (1.12), which is specified by the user.

The literature shows that the accuracy of the RBF interpolant depends strongly on

this parameter. Carlson et al. [21] proposed an optimal value of c that minimizes

the root mean square error (RMS) that represents the error between the RBF

interpolant and the test function. Rippa [121] improved the optimal value by

suggesting method for estimating the optimal value of this parameter based on

minimizing a cost function that approximates the RMS error with considerable

economy. For more details, see §1.2.1.

In this thesis we proposed two different strategies for the choice of the c param-

eter based on either a local or global implementation of the Rippa algorithm [121].

The local and global values of c were constructed from the same neighboring data

points that we used to construct the local and global RBF in the hybrid method.

The accuracy of the different surface fitting methods discussed above were

demonstrated by performing numerical experiments based on two data sets and

six test functions taken from Franke [55]. The numerical results are analysed using

the root mean square error (RMS), which is given by

RMS =

√∑qi=1[S(ai, bi) − f(ai, bi)]2

q, (1.34)


where f(ai, bi) represents the exact value of the function for the set of data points

and S(ai, bi) represents the algorithmic estimate at the same data points. An

extensive comparison of the different surface fitting approaches indicates for the

Franke data set that the hybrid CT-RBF method produces a more accurate surface

representation for the CT method than the Taylor series approach. In particular

using the multiquadratic RBF produced more accurate gradient estimates for the

CT triangle than using first, second and third order Taylor series expansions. Next,

we applied the hybrid method and the CT method using Taylor series to a data

set sampled from real Anthurium and Frangipani leaves for the purpose of surface

reconstruction.

To apply the hybrid method (using the multiquadric RBF to estimate the

gradients at the vertices and edge midpoints of the triangles) or the CT method

(using Taylor series to estimate the gradients at the vertices and edge midpoints

of the triangles) to the leaf data sets, preprocessing steps are required, which

include the determination of a new reference plane for the leaf data and then the

triangulation of the leaf surface mesh. The coordinate system used by the scanner

may not be suitable for interpolation due to the possibility of multivalued and

vertical surfaces. A new reference plane that is a linear least squares plane to

the leaf data was constructed and then the coordinate system was rotated so that

the reference plane becomes the xy−plane. These rotations can be achieved by at

first rotating the normal vector of the reference plane about the y-axis into the

yz-plane, and then rotating about the x-axis into the xz-plane

Given the data points Pi = (xi, yi, zi)T , i = 1, . . . , N , the least squares plane is

the function p(x, y) = a1x + a2y + a3, for which

E(p) =∑N

i=1(zi − p(xi, yi))2,

is minimized as a function of a1, a2 and a3 in the least square sense to obtain the

best fit.

The second issue was the triangulation of the leaf surface mesh. Since the

number of data points that represents the leaf surface is large (3,388 points for the

Frangipani and 4,688 for the Anthurium), the computational expense is reduced

by selecting only a subset of the data, which avoids undesirably shaped triangles


and it is economical for computing the droplet motion on the mesh, to generate a

triangulation of the leaf surface. We used EasyMesh, a software package written in

the C language by Niceno [99], to generate the triangulations. EashMesh returns a

good quality triangulation, because the triangular elements are close to equilateral

if the domain is convex. However, the boundary points of the Frangipani and

Anthurium leaves do not form a convex hull. To overcome this problem, we applied

an algorithm given in Sedgewick [125] to generate the convex hull of the whole leaf

data set. After that, the closest points to the leaf boundary points from these

convex hull points were found using the Matlab command dsearch. This process

produces the boundary points that define the convex domain.

In the interior of the convex hull (leaf surface) we can define either a hori-

zontal or vertical line in the domain to enable EasyMesh to coarsen the mesh

(produce fewer and better shaped triangles) in these locations. For the Frangipani

and Anthurium leaves it appears that the vertical line produces a more suitable

triangulation than the horizontal line.

EasyMesh was then provided with an input file that contains the boundary

points, together with the vertical line and the desired triangle edge length for the

mesh elements. Then, a node file is returned by EasyMesh that contains the same

boundary points along with a set of points (interior points) distributed inside the

leaf that represent the vertices of the mesh structure. We imported this node file

into Matlab and then the closest points in the leaf data set were located from

the interior points using dsearch. These resulting points were used as the triangle

vertices of the leaf surface mesh structure. Eventually, we use the Matlab command

delaunay to triangulate the leaf points.

Finally, after the least squares plane and the triangulation of the leaf surface

were constructed, the hybrid CT-RBF method (local and global) and the CT

method using Taylor series to estimate the gradients were applied to construct

the surfaces of the Frangipani and the Anthurium leaves. The quality of the

approximation of these two methods was measured using two error metrics. The

first error metric is the root mean square error RMS, see equation (1.34), while the

second error metric measured the quality in terms of the maximum error associated

with the surface fit in relation to the maximum variation in z as


maximum error = max(|S(ai,bi)−zi|)max(zi)−min(zi)

,

where S(ai, bi), i = 1, 2, . . . , N are the CT estimated values at the data points (N)

and f(ai, bi) = zi, i = 1, 2, . . . , N are the given function values at the same data

points.

As a result, more accurate RMS values and maximum errors were obtained

using the local hybrid method than using the global hybrid method in all three

cases. Also, the CT method using Taylor series to estimate the gradients at the

vertices and edge midpoints of the triangles produces similar RMS values and

maximum errors to that offered by the local and global hybrid methods.

The research outlined in this section provides an accurate model of the leaf

surface that forms the basis for a theoretical study of water droplet paths on

leaves, which is the topic of section §1.3.3.

1.3.2 Error Bounds

As mentioned in the previous section, the Taylor series method was employed to

estimate the gradient of the CT triangle. A natural question that arises is what

is the spatial error associated with this type of estimation strategy? Intuitively

it seems quite plausible that this error will be O(hnmax), where n is the number

of terms taken in the Taylor expansion and hmax is the maximum distance from

the point of interest say a and any of the cloud of neighbouring points used for

estimating the gradient.

This motivated us to derive the error bound theory of not only least square

gradients estimates but also for the weighted least square gradient estimates, which

led to our paper [135]. As mentioned in section §1.2.1, the multivariable Taylor

series expansion given in equation (1.6) is used to estimate the gradients locally,

say at point a, from the scattered data values vi = a + hiνi, i = 1, . . . , m with

hi = ‖vi − a‖ to obtain the overdetermined system of equations Aγ ≈ q, see

equation (1.8). Here, the vector q has as its ith elements qi = f(a+hiνi)−f(a)hi

the

difference quotients for f in the direction of the unit vectors νi = (νxi, νyi

)T . The

elements of γ are the partial derivatives of f . The solution of the least squares

problem (1.8) γ = arg minγ∈R‖Aγ − q‖, then enables us to obtain the gradient


estimate from the first two components of γ as

∇f(a) ≈ E1A†q, (1.35)

where A† is the pseudoinverse of A and

E1 =

1 0 0 · · · 0

0 1 0 · · · 0

∈ R

2×p. (1.36)

The gradient estimate can also be obtained via a weighted least squares approach.

The weighted least squares method is based on a row scaled system in the sense that

more importance is given to points closer to the point of interest by introducing

a diagonal matrix W = diag(w1, w2, . . . , ws), where wi = ‖a − vi‖−d, d = 1, 2

for inverse distance or inverse distance squared weights respectively. In this case,

the overdetermined system becomes WAγ ≈ Wq, and the least squares solution

becomes γ = arg minγ∈R‖WAγ − Wq‖, which then enables the gradient estimate

to be extracted as the first two components of γ as ∇f(a) ≈ E1(WA)†Wq, where

(WA)† is the pseudoinverse of WA.

Belward et al. in previous work [13] estimated the gradient ∇f(a) (for the case

n = 2 in equation (1.6)) by applying an orthogonal reduction of the columns 3− 5

in A = (A1|A2) using a QR-factorization of the matrix A2 as QT A2 =

(A12

0

), to

obtain QT A =

(A11 A12

A21 0

). Then ∇f(a) is estimated by g = argminy∈R2‖A21y−

q‖2, where q2 represents the last m − 3 entries in QT q. Belward et al. compared

this method with the direct approach given in (1.35) and showed that although

both strategies appear different, they produce the same least squares error and

gradient estimates.

In the work presented in [14], we derived an error bound for a quadratic n = 2

least squares gradient approximation. An important component of this bound is

the ratio of hnmax (the maximum distance from the point of interest to any neigh-

bouring point in the least squares stencil raised to the order of the method) to the

smallest singular value σ1 of the least squares matrix A. Numerical experimen-

tation is performed for the purpose of examining the reliability and utility of the


derived error bound. It was found that the derived bound is quite pessimistic and

the cause was due to the smallest singular value of the least squares matrix A,

which depends on hi. It is conjectured that rather than using the smallest singular

value of the matrix A in the error bound it appears more appropriate to use the

smallest singular value of the reduced matrix A21.

The error bound was then modified by applying the elimination method de-

scribed in [13]. We observed from the singular values of the least squares matrix A

that the singular values could be divided into two groups identified by their mag-

nitudes. The first group is associated with the gradient A1 and the second group

is associated with the higher order terms in A2. Therefore, a closer bound would

be obtained if the smallest singular value of the first group (A21) were used instead

of the smallest singular value of the whole system. This modification provides a

much tighter bound on the gradient estimates.

In [135], we generalised the results presented in [14] to an order n least squares

and weighted least squares gradient approximation and a proof of the above con-

jecture is given. At first, we showed that the smallest singular value of the least

squares gradient coefficient matrix plays an important role – if this matrix is ill-

conditioned the more the impact on the overall error. By invoking the Cauchy-

Schwarz inequality and Lipschitz continuity, we derived the error bound of the

classical least squares gradient estimate in the form

‖∇f(a) − E1γ‖‖∇f(a)‖ ≤ ϑmaxh

nmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 , (1.37)

where σ1 is the smallest singular value of A, which is assumed to have rank(A) = p

and p = (n+1)(n+2)2

−1, ϑmax is Lipschitz constant, hmax = max1≤k≤m hk, E1 is given

in (1.36) and νi = (νxi, νyi

)T is defined as hiνxi= xi − ax, hiνyi

= yi − ay. The

bound on the error of the weighted least squares gradient estimate was derived as

‖∇f(a) − E1γ‖‖∇f(a)‖ ≤ ϑmaxh

nmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 , (1.38)

where σ1 is the smallest nonzero singular value of A = WA, which is assumed to


have rank(A) = p and wmax = max1≤k≤m wk.

The Lipschitz constant ϑmax (which are the largest magnitude partial deriva-

tives of order n for an nth order approximation) in the error bounds can be esti-

mated by an application of the mean value theorem to the mixed partial derivatives

in the Taylor series remainder as

ϑmax =√

2 maxξ∈D

(∣∣∣∣∂n+1f(ξ)

∂xn−i+1∂yi

∣∣∣∣ , i = 0, . . . , n

),

where these maxima were determined with the help of Maple.

To improve and tighten the error bounds of the classical and weighted least

squares gradient estimate given in equations (1.38) and (1.37), we proved the

conjectured results given in [14] and derived the tighter error bound of the least

squares problem

‖∇f(a) − E1A†q‖

‖∇f(a)‖ ≤ ϑmaxhnmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 , (1.39)

where σ1 is the smallest singular value of A21, which is assumed to have full column

rank. A similar result also holds for the weighted least square problem given

in (1.38) where the orthogonal decomposition is now performed on WA2 so that

QT WA2 =

A12

0

where A12 is upper triangular. In this case QT (WA1|WA2) =

A11 A12

A21 0

. Thus, the derived tighter error bound of the weighted least squares

problem is

‖∇f(a) − E1A†Wq‖

‖∇f(a)‖ ≤ ϑmaxhnmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 ,

where σ1 ≤ σ1 and σ1 is the smallest singular value of A21, which has full column

rank.

We also proved that equations (1.39) and (1.40) are, respectively, lower bounds

of the classical and weighted least squares gradient estimate given in equations

(1.37) and (1.38). The tighter error bounds were assessed by performing numerical


experimentation concerning a practical scattered data set taken from Franke [55].

The experiment shows that the tighter error bound (1.39) and (1.40) is superior to

the error bound (1.37) and (1.38), which is consistent with the theory presented.

1.3.3 Droplet Model

As stated earlier, the main objective in this research is to develop a new model for

generating a realistic movement of a water droplet traversing a virtual leaf surface

and to compare the model behavior with experimental observation, the result of

this aim is led to the paper [106]. The leaf surface model described in §1.3.1 forms

the basis for the droplet model. Although there is a large literature on modelling

the spreading of droplets on surfaces, a literature search of papers that describe

the simulation of droplet motion on leaf surfaces has found an absence of thin-film

theoretic models to determine the spread of the droplet.

As mentioned before the leaf surface model forms the basis for the droplet

model. The virtual surface is comprised of a mesh of triangles built using Easymesh

over which the hybrid CT-RBF interpolant is constructed from scattered data cap-

tured by a laser scanner. The simulation of droplet movement could be compu-

tationally demanding if thousands of triangles have to be considered, and conse-

quently a coarser mesh based on a smaller subset of data points is used that is

representative of the major surface features.

To model the droplet motion there are many important factors and forces

that need to be taken into account. Here, two forces are assumed to affect the

droplet movement on the leaf surface namely an internal force, which consists of a

friction and a resistance component between the surface and the droplet, and an

external force due to gravity. The resistance force originates from the interfacial

tension that exists between the water droplets and the leaf surface [68,72], and its

direction is opposite the direction of the droplet movement (dp). The friction force

Ff is modelled as a linear retarding force with a constant negative factor kf due

to surface roughness [54]. For a droplet of constant mass, the droplet motion is

determined using Newton’s second law, coupled with the idea of thin-film theory,


which results in the model

mdv

dt= mdp − kfv(t) − αdp, (1.40)

where αdp is the resistance force, kfv(t) is the frictional force and m is the mass

of the droplet.

The motion of the droplet is computed over each triangle and the equations for

velocity and position of the water droplet at any time are derived from equation

(1.40). We determine the direction of the droplet movement on entering a triangle

and project the gravitational force and the droplet velocity in this direction in

order to ensure that the droplet remains on the leaf surface. When the droplet

enters a triangle at time t, we compute the exit position and the exit time as well

as the velocity at this time, with the position and time found by intersecting the

droplet path with each triangle edges using a Newton algorithm.

The originality of our model is the use of thin-film theory to estimate the

thickness of the droplet as it moves on the surface for the development of a stopping

criterion. If this thickness is less than a set tolerance the droplet movement is

stopped, otherwise it continues to move to the triangle edge.

In order to verify our theoretical study of the water droplet movement, we

calibrated the droplet model using experimental measurement. A series of water

droplet experiments were performed on a freshly cut Frangipani leaf. A video

camera was used to record the droplet traversing the leaf surface. We used the

sonic digitizer device shown in figure 1.9(b) to measure the locations of the droplets

on the fresh leaf and then we found the corresponding starting locations from the

scanned leaf to use them in our model as the starting points of the different droplets.

A syringe was used also to measure the droplet mass.

To test if the droplet path would change if we changed the orientation, we

have chosen two different masses and two different leaf orientations to simulate

the droplet movement. One of them was at a steeper angle than the other. As

a result, we observed from our experiment that the droplet moves and then falls

from the surface if the leaf orientation was steep; on other occasions the droplet

moves along the leaf vein, if the leaf surface is horizontal or close to horizontal, or


it comes to rest on the surface. By comparing these observations with the droplet

movement in our model, we found that our model captures these movements quite

well and produces realistic droplet motion.

In this paragraph, a brief description of the visualisation techniques that were

applied to produce the leaf image and the droplet path on this image is given,

see for example figure 1.12. The droplet paths that were produced by our model

Figure 1.12: This figure exhibits a droplet movement across the leaf surface.

were imported into the reverse engineering software package Rapidform2006 (Inus

Technology Inc., Seoul, Korea) as a set of points. Then, a smoothed path curve

was created by connecting these points. The path curve was converted to a tube,

to enhance the path, by extruding a circular curve along the path. After that, the

leaf and the tube surfaces were exported from Rapidform2006 and imported into

Autodesk Maya for rendering. The stand shown in figure 1.12 was also added to

provide a reference for the leaf orientation. The leaf was textured from a photo-

graph taken of the actual experiment to highlight the surface features and to pro-

vide more visual queues on the actual leaf orientation. Finally, the background,

shadowing and conversion of the colour images into greyscale was completed in

Adobe Photoshop CS3.


1.4 Thesis Outline

This thesis is presented by publications. Our contribution to the literature is listed

in five papers that represent the content of this thesis. The outlines of these papers

are given in the following subsections.

1.4.1 Outline of Chapter 2 for the Paper Published in the

Applied Mathematical Modelling Journal, 2009

The work on surface fitting techniques for leaf data presented in this chapter ap-

peared in the paper:

M. Oqielat, I. Turner, and J. Belward. A Hybrid Clough-Tocher Method for

Surface Fitting with Application to Leaf Data. Applied Mathematical Modelling,

33:2582-2595, 2009.

Statement of Join Authorship

Moa’ath N. Oqielat (Candidate) Introduced a new surface fitting technique

for modelling leaf surfaces, developed all of the Matlab codes and interpreted the

numerical results, wrote the manuscript and acted as the corresponding author.

Ian W. Turner Suggested the surface fitting techniques and gradient approxima-

tion method, directed and guided the research, assisted with the interpretation of

results and preparation of the paper and proof read the manuscript.

John A. Belward Suggested the surface fitting techniques and gradient approxi-

mation method, directed and guided the research, assisted with the interpretation

of results and preparation of the paper and proof read the manuscript.

Paper Abstract

The leaves play an important role in the development of a plant and are an inte-

gral component of any plant model. Mathematical models of leaves are therefore

essential for their accurate representation and may be used only for visualization

purposes, or for the purposes of studying biological processes such as photosynthe-

sis [126], or a canopy light environment [7, 38]. This paper presents a brief survey

of surface fitting strategies and then a new hybrid technique for modelling a leaf


surface is proposed, which is based on combining the Clough-Tocher and radial

basis function methods. We demonstrate the accuracy of this hybrid approach by

applying it to two scattered data sets. The first set is taken from Franke [55], while

the second set is sampled from an Anthurium leaf using a laser scanner [84]. It is

found that the new hybrid surface fitting methodology produces an accurate and

realistic leaf surface representation.

1.4.2 Outline of Chapter 3 for Paper Published in the pro-

ceedings of the MODSIM07 Conference, 2007

The work on virtual leaf models presented in this chapter appeared in the paper:

M. Oqielat, J. Belward, I. Turner, and B. Loch. A hybrid Clough-Tocher radial

basis function method for modelling leaf surfaces. In Oxley, L. and Kulasiri, D.

(eds) MODSIM 2007 International Congress on Modelling and Simulation. Mod-

elling and Simulation Society of Australia and New Zealand, December 2007, pages

400406, 2007.


Moa’ath N. Oqielat (Candidate) Developed a model for leaf surface recon-

struction, developed the Matlab codes, interpreted all numerical results, wrote

the manuscript, acted as the corresponding author and presented the work at the

MODSIM07 conference.

Ian W. Turner Directed and guided the work, assisted with the interpretation

of results and proof read the manuscript.

John A. Belward Directed and guided the work, assisted with the interpretation

of results and proof read the manuscript.

Birgit I. Loch Provided us with the leaf data and proof read the manuscript.

Paper Abstract

We present a novel hybrid approach for leaf surface fitting that combines Clough-

Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a

continuously turning normal. The hybrid CT-RBF method is shown to give good

representations of a Frangipani leaf and an Anthurium leaf.


The development of the algorithm has been made to facilitate the understand-

ing of leaf surface properties. By identifying and quantifying the response of plants

to the inputs via their leaves information will be obtained for application to prac-

tical and theoretical issues of scientific and sociological importance. The use of

pesticides to assist agricultural production has ecological effects; avoidance of the

overuse of water is of critical importance and a measured use of resources is of

economic importance.

An understanding of the mechanisms of the development of a plant will, gener-

ally, include the an understanding of the role played by its leaves. This subject has

attracted considerable interest over the last decade as summarised in the introduc-

tion (Room et al. 1996, Prusinkiewicz 1998) . Their shape, size, and position are

important in several ways. For example energy uptake is assumed to be a function

of light interception. This influences plants both individually and collectively, the

latter through competition for resources. Similarly, the amount of precipitation,

nutrients or pesticide can be better quantified if a detailed model of a leaf is ac-

cessible. Thus important aspects of leaf modelling can be facilitated with accurate

knowledge of the leaf surface. This can be obtained from a surface fit to a set of

measurements made by a data collection device such as a laser scanner or a sonic

digitiser (Loch 2004).

This work will form the basis for a theoretical study of pathways of water

droplets on leaves. The initial investigation will assume that the leaf is smooth

and the droplet experiences, at most, gravitational, surface tension and viscous

forces. It will be necessary to produce a surface fit with a continuously varying

gradient. This is assured by interpolation of data values and gradient values on

a triangulation of the data points using piecewise bivariate cubics (Clough 1965).

Derivative values are obtained by computing the gradient of an RBF which inter-

polates the data values (Powell 1991).

The issues reported here include:

-The selection of points from the data set. The choice of a subset of the data which

avoids undesirably shaped triangles was aided by the use of EasyMesh a software

package which generates Delaunay triangulations.

-Choice of RBF and suitable width parameter c. Hardy’s multiquadrics were se-


lected in conjunction with the use of Rippa’s algorithm to determine the width

parameter.

-The use of local and global RBF interpolates. Numerical experiments investigated

the use of local, less costly RBF interpolates compared with global, more expensive

and more robust RBF counterparts. The results favoured the former approach.

The method reported is generally applicable to scattered data and has the potential

for application to the numerical solution of partial differential equations.

1.4.3 Outline of Chapter 4 for the Paper Published in the

proceedings of the CTAC08 Conference, 2008

The work on numerical investigations of linear least squares methods for derivative

estimation presented in this chapter appeared in the paper:

J. Belward, I. Turner, and M. Oqielat. Numerical Investigations of Linear Least

Squares Methods for Derivatives Estimation. CTAC 08 Computational Techniques

and applications conference, Australia, July 2008.


Moa’ath N. Oqielat (Candidate) Developed the numerical results and the Matlab

codes, assisted with the interpretation of results and proof read the manuscript.

Ian W. Turner Suggested the error bound theory, assisted with interpretation of

results, preparation of the paper and proof read the manuscript.

John A. Belward Conjectured a tighter error bound, developed the numerical

results and the Matlab codes, assisted with the interpretation of results and prepa-

ration of the paper and presented the work at the CTAC08 conference.

Paper Abstract

In this work the results of a numerical investigation into the errors for least squares

estimates of function gradients are presented. The underlying algorithm amounts

to setting up a least squares problem using a truncated Taylor expansion from

which the desired gradient approximation can be extracted. The error bound

associated with this method contains in its numerator terms related to the Taylor

series remainder, while its denominator contains the smallest singular value of the


least squares matrix. Perhaps for this reason the error bounds are often found to

be pessimistic by several orders of magnitude.

In this paper the circumstances in which these poor estimates arise is elucidated

and an empirical correction of the theoretical error bounds is conjectured. This is

followed by an indication of how this idea can be supported by a rigorous argument.

The particular methods analysed were chosen to provide accurate gradient esti-

mates combined with the opportunity for computational efficiency with large data

sets. The current target of the work is in the construction of models of leaf sur-

faces, which will assist the study of local effects such as the movement of droplets

on leaf surfaces. The work also has the potential for input into growth mechanisms

of whole plants.

1.4.4 Outline of Chapter 5 for the Paper Submitted to the

SIAM Journal on Scientific Computing, 2008

The work on error bounds for least squares gradient estimates presented in this

chapter has been submitted for review:

I. Turner, J. Belward, and M. Oqielat. Error Bounds for Least Squares Gradient

Estimates. SIAM Journal on Scientic Computing, under review, 2008.


Moa’ath N. Oqielat (Candidate) Posed the question of what is the error bound

associated with the least square gradient estimate, developed all of the Mat-

lab codes, assisted with the interpretation of results, wrote and submitted the

manuscript.

Ian W. Turner Introduced the new error bound theory, proved the theory, as-

sisted with interpretation of results and the preparation of the paper, proof read

the manuscript and acted as the corresponding author.

John A. Belward Assisted with the interpretation of results and the preparation

of the paper, proof read the manuscript.

Paper Abstract


Least squares gradient estimates find application in many fields of computational

science, particularly for the purposes of surface fitting and gradient reconstruction

in computational fluid dynamics and data visualization. In this paper we derive

error bounds for classical and weighted least squares gradient estimates. The

bounds reflect how the number of points used in the least squares stencil and

the smallest singular value of the least squares matrix impact the accuracy of

these estimates. We show how an extrapolation method based on Householder

transformations provides substantially tighter bounds. Numerical case studies are

presented to elucidate our theory for a data set taken from Franke [55].

1.4.5 Outline of Chapter 6 for the Paper Submitted to the

Journal of Mathematics and Computer in Simula-

tion, 2009

The work on water droplet movement on a leaf surface presented in this chapter

has been submitted for review:

M. Oqielat, I. Turner, J. Belward, and S. McCue. Water Droplet Movement on

a Leaf Surface. Mathematics and Computer in Simulation. Paper has now been

revised and resubmitted to the journal as requested by the editor on 19/04/09

taking into consideration the comments and suggestions by the reviewers.


Moa’ath N. Oqielat (Candidate) Introduced a new model for the simulation

of a water droplet movement on a leaf surface, performed the model experiment,

developed all of the Matlab codes, wrote the manuscript and acted as corresponding

author.

Ian W. Turner Directed and guided the research, assisted with the interpretation

of the model, assisted with performing the model experiments and proof read

manuscript.

John A. Belward Directed and guided the research, assisted with the interpreta-

tion of the model, assisted with performing the model experiments and proof read

manuscript.


Scott McCue Helped with introducing the thin-film theory into the droplet

model, proof the theory and proof read manuscript.

Paper Abstract

Modelling droplet movement on leaf surfaces is an important component in under-

standing how water, pesticide or nutrient is absorbed through the leaf surface. A

new model is proposed in this paper for generating a realistic, or natural looking

trajectory of a water droplet traversing a virtual leaf surface. The virtual surface

is comprised of a triangular mesh structure over which a hybrid Clough-Tocher

seamed element interpolant is constructed from real-life scattered data captured

by a laser scanner. The motion of the droplet is assumed to be affected by gravita-

tional, frictional and surface resistance forces and the innovation of our approach

is the use of thin-film theory to develop a stopping criterion for the droplet as it

moves on the surface. The droplet model is verified and calibrated using experi-

mental measurement; the results are promising and appear to capture reality quite

well.

1.4.6 Outline of Chapter 7

In this chapter the new contributions of the PhD research are given and the main

conclusions drawn from the work are summarised. The chapter is concluded with

recommendations for future research.

Chapter 2

A Hybrid Clough-Tocher Method for Surface Fitting

with Application to Leaf Data

2.1 Introduction

The application of surface fitting techniques to the construction and reconstruction

of leaf images is one of the primary aims of this paper. This is an important

research topic because the accurate representation of leaves is required for the

development of a virtual plant model. In this research we investigate scattered

data interpolation methods based on radial basis functions and Clough-Tocher

methods for the purposes of developing the desired leaf surface representation.

These methods have the advantage of providing good accuracy near the boundary

of their domain.

Although the modelling of plant architecture has been researched extensively

over the last few decades [7, 31, 114, 123], one notes that models of leaf surfaces

have not been generated with great accuracy or level of detail until recently when

Loch [84] presented two finite element based methods (piecewise linear triangular

and piecewise cubic Clough-Tocher triangular). Loch used these methods to model

accurate leaf surfaces for Frangipani, Anthurium, Flame, and Elephant’s ear leaves

in three dimensions. These methods used a large number of data points sampled

from the real leaf surface using a laser scanner.

58

A Hybrid Clough-Tocher Method for Surface Fitting with Application to Leaf Data 59

The research presented in this paper at first surveys existing interpolation tech-

niques based on the Clough-Tocher methodology for surface fitting. Then, a new

hybrid surface fitting technique that combines Clough-Tocher with radial basis

function techniques is proposed. Finally, this hybrid strategy is applied to a large

number of three-dimensional data points captured from an Anthurium leaf surface.

This work forms the foundation for which future research can be built, for exam-

ple, accurate leaf surface representation may be used in the context of modelling

surface droplet movement.

The research is presented over four main sections of the paper. In §2 a brief

overview of surface fitting methods is presented. These methods are interpola-

tion methods based on the Clough-Tocher (CT) method, the radial basis function

(RBF) method and a hybrid (CT-RBF) method that combines the Clough-Tocher

method with radial basis functions. The choice of the RBF, together with a suit-

able width parameter c that is associated with the RBF, are also described in this

section. In §3, the accuracy of the different surface fitting methods is assessed

using the set of data points taken from Franke [55] for six test functions. The

numerical results are analysed using the root mean square error (RMS) and the

maximum error as metrics to measure the quality of the approximation for each

of the methods considered. In §4 these surface fitting methods are applied to con-

struct a surface representation for an Anthurium leaf surface. Finally, the work

is concluded in §5, where future work and further applications of our research are

also discussed.

2.2 Surface Fitting Methods

Three interpolation methods, CT, RBF and the hybrid CT-RBF will be described

in this section together with the application of these methods to a set of data points.

The bivariate interpolation problem of scattered data points is stated formally as

follows:

Given N scattered data points (xi, yi)T , and corresponding function values

zi, i = 1, 2, . . . , N, find a function f : D ⊂ R2 → R that interpolates these data


satisfying

f(xi, yi) = zi, i = 1, . . . , N. (2.1)

The data points (xi, yi)T are assumed to be distinct and not all collinear, D is the

domain of the function.

Finite element methods are based on dividing the domain on which the data

points are given into subdomains and then applying either a triangulation or rect-

angulation to the data points to form elements on which interpolants can be con-

structed in a piecewise manner. Triangulation is perhaps the most commonly used

approach and will also be adopted in this paper. In this method the function value

is assigned at the triangle vertices and a surface function (polynomial) is fitted for

the interpolation in each triangle. Derivatives may need to be estimated if they

are not provided with the data. The complete surface is then generated by joining

the polynomials on each subdomain. The interested reader is referred to [80] for

more information on this topic.

2.2.1 The Clough-Tocher finite element method

The Clough-Tocher (CT) method, introduced originally by Clough and Tocher [29],

is used to minimize the degree of the polynomial interpolant without losing the

continuity of the gradient over the whole domain. The method has the advantage

that it results in a smooth surface. The CT method is a seamed element approach,

whereby each triangle is treated as a macro-element that is split into subtrian-

gles, which are called micro-elements. An interpolating cubic polynomial is then

constructed on each subtriangle to enable a bivariate piecewise cubic interpolant

to be devised that is continuously differentiable over the entire domain. The CT

interpolant has the form:

ϕ(x, y) =3∑

i=1

(fibi + (ci, di)

T · ∇fi

)+

3∑

j=1

∂f

∂nj

ej . (2.2)

In this representation the twelve functions bi(x, y), ci(x, y), di(x, y) and ej(x, y), i =

1, 2, 3 are cardinal basis functions (see Lancaster et al. [80]), having the property

that just one of them is unity and the reminder zero at each of the node points.


Figure 2.1: The Clough-Tocher triangle showing subdivision into three subtriangles. The

directional derivatives at triangle vertices and normal derivatives at side midpoints are pictured

as arrows

Thus twelve independent pieces of information are needed to determine ϕ, which

comprise of the function values and the gradient at each vertex together with the

normal directional derivative information along the edges (refer to Figure 2.1). A

more detailed description of this approach, together with the precise set of cardinal

basis functions can be found in [80].

It is often the case that the derivatives at the midpoints of each side and the

derivative information at the vertices of the triangular elements are unavailable.

One possibility for overcoming this missing data is to approximate the vertex gra-

dients from neighbouring data information and thereafter estimate the edge normal

derivatives as the mean of the normal derivatives from the two vertices associated

with the edge [80]. This approximation is based on the assumption that the normal

slope along the sides of the triangle changes linearly. Breslin [18] and Loch [84]

used this approximation for rainfall data and leaf surface construction. Recently

the authors [13] presented an analysis of least squares gradient approximation

methods and assessed the accuracy of this approach.


In the present work we adopted the same approach for our gradient estimation

and used the set of nearest neighbours closest to each vertex and edge midpoint

to generate approximate directional derivatives using a truncated multivariable

Taylor series expansion. This procedure enables an overdetermined linear system to

be constructed that can be solved in the least squares sense to extract the required

gradient approximation. The closest points were used to estimate respectively the

first, second and third derivative information in the Taylor series approximation.

Two strategies were used; the first strategy estimates the gradient at the midpoints

using the least squares method and the second strategy used the average of the

gradients of the two vertices associated with the same edge. The latter method

requires less computation than the first.

We now give a brief overview of this process. Let Z = f(x), x ∈ R2 be the

surface of interest. The aim is to approximate the gradient of f at some point

xj by computing the difference quotients from nearby scattered data points. Let

xi, i = 1, . . . , m be the set of spatial locations of the m neighbours of the point xj

where the gradient estimate is required. Consider the truncated Taylor expansion:

f(xj + hi) = f(xj) +

k∑

ℓ=1

1

ℓ!(hi · ∇)ℓf(xj), i = 1, . . . , m, (2.3)

where

hi = (∆xi, ∆yi)T , ∆xi = xi − xj , ∆yi = yi − yj.

By truncating at k = 1 (linear), k = 2 (quadratic) or k = 3 (cubic), an overde-

termined linear system is obtained that can be solved in the least square sense to

estimate the required derivatives. This system can be expressed as

Au ≈ q, (2.4)

where A, u and q are given (for the case k = 1 as an example) by:


A =

∆x1 ∆y1

∆x2 ∆y2

......

∆xm ∆ym

, u =

∂f∂x

∂f∂y

, q =

f(xj + h1) − f(xj)

f(xj + h2) − f(xj)...

f(xj + hm) − f(xj)

.

Intuitively, one might expect that this approach offers a gradient estimate with

accuracy of O(hkmax), where hmax = max1≤i≤m ‖hi‖2; numerical experiments re-

ported by Belward et al. [13] are consistent with this conjecture for k = 2.

2.2.2 Radial Basis Functions

Suppose that Z = f(x), x ∈ R2 is a real-valued function that must be approximated

in some region Ω. A Radial Basis Function (RBF) approximation to f is a function

S of the form:

S(x) =N∑

i=1

aiR (‖x − xi‖) , x ∈ R2 (2.5)

where R(r) is a fixed real-valued function of ri = ‖x − xi‖ with ‖.‖ denoting the

Euclidean norm. The points xi, i = 1, 2, . . . , N are called the centres of the RBF

approximation. The function S(x) interpolates f at x1, . . . , xN if ai, i = 1, . . . , N

satisfies the system

Λa = F with Λij = R (‖xj − xi‖) i, j = 1, . . . , N (2.6)

and

F = (f1, . . . , fN)T .

Radial basis function schemes are often used to obtain a smooth surface repre-

sentation that allows the function values to be estimated at points other than

data points. The method has found application in areas such as geodesy [69],

hydrology [16] and medical imaging [24]. Hardy [63] also presents applications of

RBFs in geodesy, geophysics, photogrammetry, remote signal processing, geogra-

phy, surveying and mapping, hydrology and the solution of parabolic, elliptic and

hyperbolic partial differential equations. A review of the theory of RBF approxi-

mation is given by Powell [113]. Radial basis functions are nowadays also applied


in software to drive laser scanners [22, 23].

A major problem of the radial basis function method concerns its application

to large sets of data points where the computational costs involved in fitting and

evaluating the RBF can become time consuming. This cost manifests because

in order to compute the RBF coefficients ai, i = 1, 2, .., N in equation (2.6), a

large dense matrix system of size N × N has to be solved. Typically, this system

can become severely ill-conditioned with several very small in magnitude singular

values evident [55]. Franke [55], for example, suggested that the application of

global methods be restricted to sets of up to 100-200 data points, and he compared

around 30 interpolation schemes in two-dimensions to reach this conclusion. Franke

found that two of the most accurate schemes were based on fitting RBFs. Cherrie

[27] showed that a way to considerably reduce the cost of evaluating the radial

basis function was by applying fast evaluation techniques (see also [10, 11]). A

second drawback with the use of an RBF approximation is a degradation of its

accuracy near the boundary of its domain.

Two well known examples of radial basis function methods include Hardy’s

multiquadric and thin plate splines. The Hardy’s multiquadric RBF [63] is given

by:

R(r) =√

r2 + c2. (2.7)

The parameter c is specified by the user, however, it is well known that the accuracy

for interpolating scattered data with radial basis functions depends strongly on this

parameter, see for example [21,55,121]. Theoretical results show that multiquadric

interpolation becomes more accurate as the multiquadric parameter c increases

[89]. For some values of c the problem may become ill-conditioned [73, 39, 93].

Franke [55] used c = 1.25 D√N

where D is the diameter of the minimal circle

enclosing all data points. A similar suggestion was made also by Foley [44]. Hardy

[62] suggested a value of c = 0.815d where d =Pn

j=1 dj

Nand dj is the distance

between the jth data point and its closest neighbour.

The accuracy of the multiquadric and inverse multiquadric interpolant was

studied by Carlson and Foley [21] as well as by Franke [55]. They concluded that

the accuracy greatly depends on the choice of the parameter c. They used six


different test functions and six different sets of data points for their analyses. A

root mean square (RMS) error was computed between the interpolating radial basis

function and the test function, where the RMS error depends on the choice of the

parameter c. Carlson and Foley specified the optimal value of c that minimizes

the RMS by repeating the computation of the RMS error with different choices of

c. Rippa [121] repeated some of the experiments performed by Carlson and Foley

on the influence that the parameter c has on the quality of the approximation

obtained using multiquadric, inverse multiquadric and Gaussian interpolants (see

Table 2.1). They confirmed that the accuracy of these three RBF interpolants

depends significantly on the choice of c. Rippa considered two sets of data points

and nine different test functions defined on the unit square. A data vector F =

(f1, f2, . . . , fN )T was constructed by evaluating each test function over the set of

data points so that

S(xi) = fi, i = 1, 2, . . . , N. (2.8)

Rippa suggested an algorithm for selecting a good value for the parameter c in the

sense that the quality of the approximation of the interpolant defined with this

value is comparable to the quality of the approximation of the interpolant defined

with the optimal value. The latter is defined as the value of c that minimizes the

RMS error between the RBF interpolant and the unknown function from which

the data vector F was sampled. The Rippa algorithm was based on minimizing a

cost function that approximates the RMS error with considerable economy.

This cost function is defined as follows:

Let E be the vector

E = (E1, . . . , EN)T (2.9)

with

Es = fs − Ss(xs), s = 1, . . . , N, (2.10)

where Ss is the interpolant to the data set with the point (xs, fs) removed, so that:

Ss(x) =N∑

i=1,i6=s

asiR (‖x − xi‖) . (2.11)


Rippa showed that

Es =as

ass

, (2.12)

where as is as defined in equation (2.6) and as is the solution of

Λas = es, (2.13)

where es is the sth column of the N × N identity matrix.

Finally, the cost function C(c) is given by:

C(c) = ‖E(c)‖1 , (2.14)

and

copt = arg minc∈R

‖E(c)‖1 . (2.15)

For more details on this process the interested reader is referred to Rippa [121].

The solution of the linear system Λa = F using truncated singular value

decomposition

The interpolation problem of equation (2.6) has a unique solution if and only if

the matrix Λ in (2.6) is invertible. Micchelli [94] gave conditions on Λ that can

be checked for many problems. In particular, the RBFs listed in Table 2.1 ensure

invertibility. However, due to the poor conditioning of Λ for a wide variety of c

Table 2.1: Choices of R given by Rippa [121] for which the interpolation matrix Λ is invertible.

Name R(r)

Multiquadric (r2 + c2)1/2, c ≥ 0Inverse multiquadric (r2 + c2)−1/2, c > 0

Gaussian e−r2c2 , c > 0


values, the truncated singular value decomposition (TSVD) [133] was applied to

compute an approximate solution of the linear system. This method is based on

the singular value decomposition of Λ:

Λ = UΣV T =

N∑

i=1

uiσivTi , (2.16)

where the left and right singular vectors ui and vi are the columns of the matrices

U and V , respectively, and σi are the singular values of Λ.

Small singular values are discarded by applying TSVD [95] according to the

criterion whereby the singular values that are less than, or equal to, the product

of the largest singular value with a chosen target ε are ignored. Thus, if σi ≤ σ1ε

we ignore σi, i = 2, . . . , N . The target ε is a tolerance used to determine near

singularity and rank, which is taken here as the machine epsilon. A new matrix

Λt is then formed with rank t defined by:

Λt =t∑

i=1

uiσivTi , t ≤ rank(Λ) (2.17)

and the solution to (2.6) is then approximated by:

a = Λ†tF =

t∑

i=1

uTi F

σi

vi, (2.18)

where the matrix Λ†t is the pseudoinverse of the matrix Λt.

2.2.3 Hybrid Method

As was mentioned above, the CT method requires derivative estimates at the

vertices and midpoints of the elements for its evaluation. We propose here a new

hybrid approach for surface fitting that is based on using a multiquadric RBF

(either local or global) to estimate the gradient at the vertices and midpoints of

the Clough-Tocher triangle. The multiquadric RBF interpolant is given by:

S(x) =N∑

i=1

aiR(ri), (2.19)


where R(r) is given in equation (2.7). The gradient of S based on using N data

points is then given by:

∇S(x) =

N∑

i=1

ai∇R(ri), (2.20)

where

∇R(ri) =x − xi

riR

′

(ri), (2.21)

and R′

represents the derivative of the radial basis function.

Local and Global Hybrid approximations

Our method of surface reconstruction proceeds by selecting a subset of n points

from the complete data set to generate a triangulation of the surface. These n

points form the vertices of the triangular mesh elements that are used for the CT

method. We then consider two particular variants of the hybrid method outlined

above, which we refer to as the global and local variants. In the global hybrid

method we use these n points to construct a global multiquadric RBF interpolant

Sn(x), from which ∇Sn(x) is subsequently used to evaluate the gradients for all

CT triangles in the mesh. However, for the local hybrid method only a local subset

of size m (typically m = 20 or m = 40 in our numerical experiments) of the N

points is used to construct a local RBF interpolant Sm(x) for each triangle. Then,

∇Sm(x) is used to evaluate the required gradients for the CT element. Note that

these m points typically represent the closest points to each of the vertices and

edge midpoints for the CT element of interest.

The procedure that uses this hybrid approach for the purpose of surface fitting is

summarised in the following algorithm:

Algorithm 1: Surface Fitting using the Hybrid RBF-CT Method

INPUT: N data points (xi, fi), i = 1, . . . , N

Step 1: Choose a subset of n data points from the given N points to triangulate the surface.

Step 2: Using either a global multiquadric RBF interpolant constructed from the n triangulation

points OR, a local multiquadric RBF interpolant constructed on each triangle using a local

subset of m points, generate the RBF linear system (2.6).

Step 3: Approximately solve this linear system using the TSVD method.

Step 4: Use the RBF coefficients to construct either the global or local gradient.


Step 5: Apply the hybrid CT-RBF method to construct the surface using either ∇Sn(x) (global)

or ∇Sm(x) (local) to provide the necessary derivative information for the construction of

the CT interpolant.

We investigate two different strategies for the choice of the width parameter c

for use in the multiquadric RBF. Our strategies are based on either local or global

implementations of the Rippa algorithm outlined in §2.2. For the global strategy,

all points (n = 100 points in total) are used to apply the algorithm to produce one

global value of copt that is used for all CT elements; whilst in the local strategy

the algorithm was applied many times (using either m = 20 or m = 40 points) to

obtain a local estimate of copt that could then be used with each CT element.

2.3 Numerical Experimentation for the Franke

Data Set.

In this section we present the results of our numerical experiments for each of the

mathematical methods of surface fitting discussed in §2. We chose the data set

from Franke [55] to assess the accuracy of these methods. This data set consists of

two subsets and six test functions (see appendix) defined on the unit square [0, 1]2.

The first subset contains 100 data points distributed more or less uniformly over

the unit square, while the second subset contains 33 points with larger variations

in the density of the data points. The 100 points are used for triangulating the

surface for the CT method, while the q = 33 points are used to measure the

quality of the algorithmic estimate by finding the root mean square error (RMS)

(see Rippa [121]), which is given by

RMS =

√∑qi=1[S(ai, bi) − f(ai, bi)]2

q, (2.22)

where f(ai, bi) represents the exact value of the function for the set of data points

and S(ai, bi) represents the algorithmic estimate at the same data points.


Table 2.2: A comparison of the RMS error for the six test functions using the CT method.

The results in column 2 uses exact gradients and the results given in columns 3-5 use respectively,

1st, 2nd and 3rd order Taylor series expansions to estimate the gradient at the vertices and edge

midpoints of the triangle.

Function Exact 1st order 2ndorder 3rdorder

F1 2.6e-3 1.1e-2 1.5e-2 0.9e-2F2 2.1e-3 7.7e-3 6.8e-3 5.7e-3F3 1.4e-4 1.8e-3 1.7e-3 1.1e-3F4 4.1e-5 8.3e-4 4.4e-4 4.4e-4F5 2.6e-4 1.9e-3 1.8e-3 0.1e-2F6 8.7e-5 2.5e-3 8.7e-4 4.7e-4

Table 2.3: A comparison of the RMS error for the six test functions using the 1st, 2nd and

3rd order Taylor series to estimate the gradient at the vertices of the triangle. The gradients

at edge midpoints are estimated by taking the mean of the gradients at the two vertices on the

same edge.

Function 1st order 2ndorder 3rdorder

F1 1.1e-2 1.5e-2 1.0e-2F2 7.9e-3 6.8e-3 5.6e-3F3 1.9e-3 1.8e-3 1.2e-3F4 8.9e-4 4.3e-4 4.5e-4F5 1.9e-3 1.8e-3 1.2e-3F6 2.6e-3 8.8e-4 4.7e-4

2.3.1 Clough-Tocher method

Table 2.2 shows the RMS errors obtained for the CT method for each of the six test

functions when the exact function gradient and the first (k = 1), second (k = 2)

and third (k = 3) order Taylor series methods given in equation (2.3) are used to

estimate the gradients at the vertices and edge midpoints of the triangle. Here,

the gradients were approximated in the least squares sense by choosing the closest


7, 12 and 20 points to each of the vertices and edge midpoints respectively to

generate the overdetermined system (2.4). In Table 2.3 the gradients at each of

the edge midpoints were estimated by taking the average of the gradient at the

two vertices along the same edge, implying that three less gradient estimates are

required for each CT element. Note that the exact gradient entries in Table 2.2

produce the best RMS errors one could expect to achieve using the CT method and

as a consequence, these should therefore be used as the benchmark for comparison

of all methods discussed in this section.

We observe from Tables 2.2 and 2.3 that the RMS errors obtained for the third

order Taylor series method was only slightly more accurate than that produced

using first and second order Taylor series. Furthermore, using second order Taylor

series offered only minor improvements in RMS error over the first order Taylor

series, except in the case of the first function F1, which is probably related to the

behaviour of the function as well as to the selection of the data points. Note,

however, that when increasing the order of the Taylor series, the computational

effort for the solution of the least squares also increases. One must therefore

question whether this additional computational effort is justified given the minimal

improvements observed.

It is also apparent from these tests that the additional work required to estimate

the gradient at the edge midpoints (Table 2.2) rather than simply to use the average

(Table 2.3) did not produce a sufficient improvement to justify its usage.

2.3.2 Hybrid Clough-Tocher Radial basis function method

The global multiquadric RBF interpolant that uses all N = 100 data points (here

N = n) and the local multiquadric RBF interpolants that use m = 20 or m = 40

data points are now applied in the hybrid framework to construct the gradient of

the CT triangle for the Franke data set. The parameter c in the three cases was

estimated either globally using the n = 100 data points (Table 2.4), or locally using

a selection of m = 20 or m = 40 neighbouring data points for each CT element

(Table 2.5).

Tables 2.4 and 2.5 show the RMS errors for the six test functions using the


Table 2.4: A comparison of the RMS error for the six test functions using the hybrid global

(n = 100 points) and hybrid local multiquadric RBF interpolants (m = 20 or m = 40 points).

The parameter c was computed globally using the n = 100 points.

Function c Hybrid Global Hybrid Local RBFRBF m=40 m=20

F1 0.2506 3.3e-3 3.5e-3 3.7e-3F2 0.1560 3.7e-3 3.5e-3 3.4e-3F3 0.5907 1.7e-4 1.8e-4 2.3e-4F4 1.1974 4.1e-5 3.9e-5 4.3e-5F5 0.4909 2.6e-4 2.8e-4 4.0e-4F6 8.9018 6.9e-4 3.4e-4 2.3e-4

Table 2.5: A comparison of the RMS error for the six test functions using the hybrid local

multiquadric RBF interpolant (m = 20 or m = 40 points). The parameter c was computed

locally using the same (m = 20 or m = 40) points.

Function Hybrid Local RBF (m=40) Hybrid Local RBF (m=20)[cmin cmax] RMS [cmin cmax] RMS

F1 [0.1706 3.3519] 3.4e-3 [0.1282 11.1452] 5.6e-3F2 [0.0544 3.8690] 3.5e-3 [0.0258 26.1493] 4.2e-3F3 [0.4655 3.9275] 3.3e-4 [0.3533 10.3444] 4.3e-4F4 [0.8555 2.5426] 3.9e-5 [0.8288 10.2650] 1.9e-4F5 [0.2873 0.7660] 3.1e-4 [0.2360 9.7754] 1.2e-3F6 [1.5453 16.180] 3.8e-5 [2.2192 10.0326] 2.2e-4

global and local hybrid approaches. We remark that, in both cases, the RMS

errors are almost as good as the (exact) benchmark values given in Table 2.2. It

is also possible to observe that the RMS errors produced using the global hybrid

method appear similar to those produced using the local hybrid method. Note

that using c locally is more computationally costly than using c globally because

each time the local RBF is constructed a new value of c must be calculated. Table

2.5 shows the range of c values obtained using the local approach. As one would


hope, the global values of c were always contained in the local ranges of c given

for each of the functions.

Another observation from Tables 2.4 and 2.5 was that the RMS error produced

using the local hybrid method constructed with m = 40 points was always found

to be more accurate than the RMS produced for the surface representation con-

structed from m = 20 points (for both cases whether c is approximated globally

or locally). Furthermore, it appears from our numerical experimentation for the

Franke data that the local gradient estimates obtained when m = 40 using a glob-

ally determined value of c would be the most computationally competitive of all

of our methods when m is large.

In summary, a comparison of the different surface fitting approaches highlights

for the Franke data set that the hybrid CT-RBF method (see Tables 2.4 and

2.5) produces a more accurate surface representation for the CT method than the

Taylor series approach (see Tables 2.2 and 2.3). In fact the hybrid method gives

RMS errors quite close to the case where the exact gradient is used (see Table 2.2).

We now carry this finding to the next section and explore the suitability of the

hybrid surface fitting strategy for a real leaf data set.

The computational expense of the different methods mentioned above was as-

sessed by profiling the codes in Matlab. In the Clough-Tocher method when the

Taylor series was used to estimate the gradients for the CT triangle, an overdeter-

mined system was constructed for each of the triangle vertices and midpoints (six

systems in total). Then the pseudoinverse was used to obtain the desired gradients

in the least squares sense. Note however, that using the average of the gradient at

the edge midpoints implies that only three systems are required for the gradient

estimate, which represents a computational saving. In the local hybrid CT-RBF

method a single local RBF was constructed for each triangle and then the TSVD

was used to estimate the gradients at the vertices and midpoints of that particular

triangle. For the global hybrid CT-RBF method only one large system was con-

structed to obtain the global RBF and then this global RBF was used to estimate

the gradients at the vertices and midpoints for all triangles in the mesh.

By profiling the codes in Matlab, we observed for all methods that most of the

computational time was spent in solving the gradient least squares/RBF problems


either via the pseudoinverse or TSVD. In conclusion, if the cost of computing the

parameter c is neglected, we found that the global hybrid CT-RBF method was

the most efficient of all methods tested, followed by the local CT-RBF method and

then the Taylor series method.

2.4 Application of the Hybrid method to a Leaf

Data Set

The reconstruction of the shape of a leaf using surface fitting techniques requires a

set of representative data points sampled from the leaf surface. Loch [84] sampled

data points using a laser scanner for Frangipani, Anthurium, Flame and Elephant’s

Ear leaves. We now assess the accuracy of the hybrid CT-RBF method and the CT

method based on using Taylor series to estimate the gradients for the CT triangle

for a laser scanned Anthurium leaf data set [84]. This leaf data set consists of two

sets of data. The first set contains 4,688 points, which represent the entire set

of leaf surface points, while the second set contains 79 points that represent the

boundary points of the Anthurium leaf surface, see figure 2.2. In order to apply the

two CT based surface fitting methods to the Anthurium leaf data a preprocessing

phase is necessary, which includes the determination of a new reference plane for

the data and the subsequent triangulation for the leaf surface mesh.

2.4.1 Leaf reference plane

The reference plane of the set of measured leaf data may not necessarily coincide

with the xy-plane in the data point coordinate system. A solution is to use a

reference plane that is a least squares fit to these data points and then to rotate

the coordinate system so that the reference plane becomes the xy-plane. These

rotations can be achieved by at first rotating the normal vector of the reference

plane about the y-axis into the yz-plane and then rotating about the x-axis into

the xz-plane (see equation 2.23).

Given the data points Pi = (xi, yi, zi)T , i = 1, . . . , N , the least square plane is

the function p(x, y) = a1x + a2y + a3, for which


Figure 2.2: Anthurium Leaf data points. There are 4,688 surface points (represented by the

smaller dots) and 79 boundary points (represented by the larger dots).

E(p) =∑N

i=1(zi − p(xi, yi))2

is minimized as a function of a1, a2 and a3 in the least square sense to obtain the

best fit. The data points (P′

), after they are projected to the new reference plane,

are given by:

P′

= R(xi, yi, zi)T , i = 1, 2, . . . , N. (2.23)

In equation (2.23) R = Rx ·Ry represents the rotation matrix that rotates the unit

normal vector of the least square plane about the x−axis and y−axis, where Rx

and Ry are defined respectively as:

Rx =

1 0 0

0 cos α sin α

0 − sin α cos α

, Ry =

cos β 0 sin β

0 1 0

− sin β 0 cos β

(2.24)

and cos α = 1√a22+1

, sin α = a2√a22+1

, cos β =√

a22+1

a21+a2

2+1, sin β = a1√

a21+a2

2+1.


Figure 2.3: The Anthurium leaf surface model constructed from the data points (shown in

figure 2.2) using the hybrid CT-RBF method.

2.4.2 Triangulation of the leaf surface

Given that the Anthurium leaf data set is large (here N = 4, 688 points), the

computational expense for surface fitting can be reduced by selecting only a subset

of these data to generate a triangulation of the leaf surface. Here, this triangulation

is generated using the EasyMesh mesh generator, which is software written in the

C language by Bojan Niceno [99]. EasyMesh generates two-dimensional Delaunay

and constrained Delaunay triangulations in general domains. The software returns

a good quality triangulation if the domain is convex. However, because the 79

chosen boundary points of the Anthurium leaf shown in Figure 2.4(a) do not enclose

a convex set, EasyMesh was unable to produce the desired triangulation in this

case. To overcome this problem an algorithm (Sedgewick [125]) was employed to

generate the convex hull from the entire set of leaf data points, which provided the

49 points shown in Figure 2.4(b). Next, the closest from the original 79 boundary

points to these 49 points were then found using the Matlab command dsearch,


resulting in a set of 38 boundary points that defined the convex domain shown in

Figure 2.4(c).

In the interior of the convex hull either a horizontal or vertical line can be

defined in the domain to enable EasyMesh to produce fewer and better shaped

triangles. For the Anthurium leaf the vertical line exhibited in 2.4(c) produced a

more suitable triangulation than the horizontal line.

Thus, in summary, the following steps are applied to construct the triangulation

of the Anthurium leaf using EasyMesh:

Step 1: EasyMesh is provided with an input file that contains the 38 boundary

points, together with the vertical line description and the desired triangle edge

length for the mesh elements. EasyMesh then returns a node file that contained

the same boundary points along with some additional boundary points (28 points

in this case) introduced during the meshing procedure. Easymesh also provides

the set of points distributed inside the leaf (146 internal points in this case), which

represent the vertices of the mesh structure shown in Figure 2.4(d).

Step 2: This node file was then imported to Matlab and the closest points in

the leaf data set were located from the internal points generated in step 1 using

dsearch. These resulting points were used as the triangle vertices of the leaf surface

mesh structure.

Step 3: To obtain the boundary points of the leaf for which we do not have

surface values, we find the closest points from the leaf data set to the EasyMesh

boundary points and use their surface values.

Step 4: Finally, we use the Matlab command delaunay to triangulate the leaf

points that were obtained from steps 2 and 3.

These four steps produce the final triangulation for the leaf surface shown in

Figure 2.5. After the triangulation of the Anthurium leaf surface was constructed,

the hybrid CT-RBF method (shown in figure 2.3 using the multiquadric RBF to

estimate the gradients at the vertices and edge midpoints of the triangles) and the

CT method (using Taylor series to estimate the gradients at the vertices and edge

midpoints of the triangles) were applied to construct the surface of the leaf. The

local hybrid approach for the leaf surface reconstruction is based on choosing the

set of 30 nearest neighbors closest to each of the vertices and to the center of the


−20 0 20 40 60 80 100−150

−100

−50

0

50

100

79 boundary points

−20 0 20 40 60 80 100−150

−100

−50

0

50

100

49 convex hull points

(a) (b)

−20 0 20 40 60 80 100−150

−100

−50

0

50

100

49 cv points38 boundary pointsvertical line

−20 0 20 40 60 80 100−150

−100

−50

0

50

100

28 points146 internal points38 points

(c) (d)

Figure 2.4: (a) The 79 Anthurium leaf boundary points. (b) The 49 points generated from

the convex hull algorithm. (c) The square points represent the final 38 boundary points. (d) The

vertices of the mesh structure generated using Easymesh. The square points represent the 38

boundary points that are given to Easymesh; the dot points represent the 28 extra points added

by Easymesh, while the × points represent the 146 internal points.


triangle. Then, a local radial basis function is built from these 120 points on each

triangle and used to estimate the gradient at the triangle vertices and midpoints

for the CT method. The global hybrid approach was also applied by constructing

one global RBF from the triangulation points and then using it to evaluate the

gradients at the vertices and midpoints of all triangles in the mesh. The parameter

c in both cases was estimated globally using the triangulation points following the

Rippa [121] framework. The results obtained for these surface fitting methods are

shown in Table 3.2.

−20 0 20 40 60 80 100−150

−100

−50

0

50

100

Figure 2.5: Triangulation of the 212 points of the Anthurium leaf surface generated using

EasyMesh.

For the CT method, the gradients are estimated in a least squares sense by

choosing either 6, or 20, neighbours to produce linear or cubic gradient estimates

at the vertices and midpoints of the CT elements.

2.4.3 Numerical Experiments for the Leaf Surface

In this section we present the results of applying the hybrid method and the CT

method to the Anthurium leaf data. After the triangulation points were selected,

the remaining data points (say r) were used to measure the quality of the approx-


−20 0 20 40 60 80 100−150

−100

−50

0

50

100

−20 0 20 40 60 80 100−150

−100

−50

0

50

100

(a) (b)

Figure 2.6: The triangulation of (a) coarser grid of 103 points and (b) a refined grid using

762 points of the Anthurium leaf surface generated using EasyMesh.

imation of the methods using two error metrics. The first error metric is the root

mean square error RMS (see equation 3.6), while the second error metric mea-

sured the quality in terms of the maximum error associated with the surface fit in

relation to the maximum variation in z as

maximum error = max(|S(ai,bi)−zi|)max(zi)−min(zi)

,

where

S(ai, bi), i = 1, 2, . . . , r

are the CT estimated values at the data points (r) and

f(ai, bi) = zi, i = 1, 2, . . . , r

are the given function values at the same data points.

Three different sets of the surface triangulation points were constructed using

EasyMesh for the purpose of obtaining a more accurate surface representation

and to check that our findings were consistent as the mesh was refined. These


triangular meshes depicted in Figures 2.5 and 2.6(a)-(b) respectively, consisted of

178, 391 and 1,486 triangles. Note that some of the leaf data points occur outside

of the virtual leaf mesh; these points were ignored in the quality analysis.

Estimation of the gradients of the CT Method (Hybrid method) using

the multiquadric RBF.

Table 2.6: RMS error computed using the local and global hybrid CT-RBF method for the

Anthurium leaf data points together with the maximum error associated with the surface fit.

Hybrid Local Hybrid globalRBF RBF

Relative RMS 0.0119 0.0156maximum error 0.0761 0.0910No. of Triangles 178 178Relative RMS 0.0038 0.0065maximum error 0.0293 0.0382No. of Triangles 391 391Relative RMS 0.0017 0.0022maximum error 0.0244 0.0229No. of Triangles 1486 1486

Table 3.2 shows the relative RMS and the maximum errors using the local and

global hybrid methods for the three different triangulations of the Anthurium leaf

data set shown in Figures 2.5 and 2.6. The relative RMS given in the table was

computed using:

Relative RMS = RMSmax(zi)−min(zi)

, i = 1, 2, . . . , r.

Note that the exhibited triangulations of the leaf consisted respectively of 178,

391 and 1,486 triangles, giving a total of 4,427, 4,460 and 3,793 data points to

assess the accuracy of the surface representation in each case. Note also that

these EasyMesh triangulations comprised respectively 103 vertices including 52

boundary points for the first mesh; 212 vertices including 66 boundary points for


the second mesh; and 762 vertices including 106 boundary points for the third

mesh. There were respectively 166, 59 and 144 points ignored in the analysis

because these points were deemed to lie outside of the leaf mesh structure.

One observes from the table that using the local hybrid method produced

more accurate RMS values and maximum errors than using the global hybrid

method in all three cases. Furthermore, when the number of triangular elements

increase, the RMS errors and the maximum errors decrease, resulting in a more

accurate surface representation. This observation is one that is expected and one

that provides a good validation for the hybrid methodology for obtaining the leaf

surface representation.

It is important to note that the selection of the local set of points used for

the construction of the local RBF is crucial for the accuracy of the estimated

gradients. For the results reported in Table 3.2 we have used the closest 30 points

to each vertex and to the centroid of the triangle to construct this local RBF.

Using this point set produced the best results of all numerical experimentation for

a reasonable computational expense. Using less than 30 points, say for example

10 or 20, reduced the accuracy of the fit because of insufficient points being used

to provide a good local representation of the surface to ensure reasonable gradient

estimates. On the other hand, using too many points increases the computational

overheads considerably for only a moderate improvement in accuracy.

Finally, we would like to point out to the reader that care must be taken with

the implementation of the local hybrid method to ensure continuity of the CT

surface. For example, the gradient ∇Sm(x) obtained from one local point set for

a given triangle used to evaluate the gradients along the common edge need not

necessarily match the values estimated from a neighbouring triangle due to the

different point sets being used.

To investigate this discrepancy we carried out a numerical experiment on func-

tion F6 from the Franke data set [55]. We selected a subset of six triangles and

then estimated (using the local hybrid method) the gradient at the midpoint of

the common edges. We also evaluated the exact gradient at the same points and

used these to measured the relative error of the estimated gradients. The relative

error at these common midpoints is shown in Table 2.7. Clearly these differences


are insignificant, highlighting that there is no major concern for this strategy.

Table 2.7: The relative error of the estimated gradients at the common midpoint of the six

triangles using local hybrid RBF method.

fx fy

1st triangle 4.4e-3 02nd triangle 0 1.0e-33rd triangle 2.2e-3 0.6e-34th triangle 0 1.2e-35th triangle 3.2e-3 2.5e-3

Estimation of the gradients for the CT Method using Taylor series

Table 2.8: RMS and maximum error computed using 1st and 3rd order Taylor series. In the

1st and 3rd columns, the gradient was estimated at the vertices and edge midpoints, while in the

2nd and 4th columns, the gradient at the edge midpoints was estimated by taking the mean of

the gradients at the two vertices at the same edge.

1st order 1st(Average) 3rd order 3rd(Av.)

Relative RMS 0.0102 0.0110 0.0127 0.0139maximum error 0.0554 0.0573 0.0875 0.0897No. of Triangles 178 178 178 178Relative RMS 0.0036 0.0039 0.0036 0.0039maximum error 0.0288 0.0288 0.0290 0.0286No. of Triangles 391 391 391 391Relative RMS 0.0015 0.0015 0.0010 0.0010maximum error 0.0162 0.0160 0.0115 0.0094No. of Triangles 1486 1486 1486 1486

Table 2.8 shows the relative RMS and the maximum errors for the three dif-

ferent triangulations of the Anthurium leaf data sets using the 1st and 3rd order


Taylor series expansions. The trends depicted in Table 3.2 appear consistent with

the observations from Table 2.8.

Note in this case however that estimating the gradients at the vertices and mid-

points of the triangles produces a slightly more accurate RMS error than taking the

average of the gradients at the edge midpoints. Moreover, a more accurate surface

representation is obtained when the number of triangular elements increases.

In conclusion, it appears from the results given in the table that this approach

produces similar accuracy for the CT method to that offered by the local and

global hybrid method results given in Table 3.2.

2.5 Conclusions and Future Research

The work presented in this paper discusses different mathematical techniques for

surface fitting that allow the user to construct accurate leaf surface representations

from three-dimensional data sets. A new mathematical surface fitting technique

based on a hybrid CT-RBF methodology has been successfully applied and com-

pared with other interpolation methods and shown to produce a good accuracy for

the leaf surface representation compared with the other methods.

The research described here provides a basis on which future research can be

built. For example, the surface representation can be extended to generate not

only realistic images of leaves but it can also be applied in models that determine

water droplet, or pesticide paths along a leaf surface before it falls from or comes

to a standstill on the surface. The latter will help with the evaluation of differing

pesticide formulations and the effectiveness of a treatment. This model develop-

ment will form the basis of future research.


2.6 Appendix.

In the numerical experiments two sets of data points and six test functions defined

on the unit square [0, 1]2 were taken from Franke [55]. The six test functions are

given by:

F1(x, y) = 0.75 exp

(− (9x − 2)2 + (9y − 2)2

4

)+ 0.75 exp

(− (9x + 1)2

49− 9y + 1

10

)

+ 0.5 exp

(− (9x − 7)2 + (9y − 3)2

4

)− 0.2 exp

(−(9x − 4)2 − (9y − 7)2

);

F2(x, y) =tanh(9y − 9x) + 1

9;

F3(x, y) =1.25 + cos(5.4y)

6(1 + (3x − 1)2);

F4(x, y) =exp (− 81

16 ((x − 0.5)2 + (y − 0.5)2))

3;

F5(x, y) =exp (− 81

4 ((x − 0.5)2 + (y − 0.5)2))

3;

F6(x, y) =

√64 − 81((x − 0.5)2 + (y − 0.5)2)

9− 0.5.

Acknowledgment The authors wish to thank the reviewer for the insightful

comments on the initial version of the manuscript that improved the overall final

presentation of the work.

Chapter 3

A Hybrid Clough-Tocher Radial Basis Function

Method for Modelling Leaf Surfaces

3.1 Introduction

There are many situations in science for which surface observations of a biological

system are made. Surface data can often be collected at a discrete set of points

and a key problem is to reconstruct the surface, or perhaps capture important

features of the surface from a discrete set of measurements. The modelling of

plant architecture has been researched extensively over the last decades (Room et

al. [123], Prusinkiewicz [114]) and models of leaf surfaces have generally not been

generated with great accuracy or level of detail, until recently when (Loch [84])

presented two methods to accurately model leaf surfaces. Leaves play an important

role in the development of a plant, and therefore some adequate representation of

the leaf is required. This representation may be used for visualization purposes

only (Loch [84]) or may be used to study biological processes such as photosynthesis

(Sinoquet et al. [126]) and canopy light environments (Espana et al. [38]).

Virtual plants are developmental plant models that combine geometrical and

topological information that can be used to produce a visualization (Room et

al. [123]). Few of the past leaf models were based on accurate measurements

until 3D digitizers and faster computers with improved graphic capabilities became

86

A Hybrid Clough-Tocher Radial Basis Function Method for Modelling Leaf Surfaces 87

available. Virtual leaf models may be displayed in an abstract way, where the leaf is

represented by a disk (Smith [127]), polygons and texture maps (Foley et al. [42])

or, more realistically, by a surface model that captures the surface shape and

boundary (Prusinkiewicz et al. [115]). Hammel et al. [58] used branching skeletons

for compound leaves and boundary algorithms were applied by Mundermann et al.

[97] for modelling lobed leaves. Maddonni et al. [88] used piecewise linear triangles

to represent the leaf surface, where vertices along the boundary are estimated

by allometric relationships. Espana et al. [38] modeled the undulations of the

boundary. Finally, Frey [56] based his approach on splines and texture maps.

Two methods have been presented (Loch et al. [84,86]) based on finite elements

methods (piecewise linear triangular and piecewise cubic Clough-Tocher triangu-

lar) to model accurate leaf surfaces in three dimensions. Here a large number of

data points sampled by a laser scanner extracted from the real leaf surface were

used in an incremental algorithm to reduce the size of the set of data points.

The research presented in this paper introduces a new surface fitting method

based on hybrid strategies that combine Clough-Tocher with radial basis tech-

niques for modelling the leaf surface, which is based on a large number of three-

dimensional data points captured from the real leaf surface.

This paper consists of four sections. In this section we briefly review surface

fitting methods, including the Clough-Tocher and the radial basis function method.

In section 2 a new surface fitting method is presented that combines the CT and

RBF methods for modelling leaf surfaces. The application of the new method to

a Frangipani leaf and Anthurium leaf is presented in section 3, where a processing

methodology is detailed. Future work and further applications of the model are

discussed in section 4.

3.1.1 Clough-Tocher finite element method

The Clough-Tocher method (CTM) is an interpolating finite element method that

was introduced originally by Clough and Tocher [29]. This method is used to mini-

mize the degree of the polynomial interpolant fitted across the triangular elements

without losing the continuity of the gradient over the whole domain.


The CTM is a seamed element approach, whereby each triangle is treated as a

macro-element that is split into subtriangles, which are called micro-elements. The

CTM, has the advantage that it results in a smooth surface over the whole domain.

It approximates the surface as an interpolating cubic polynomial constructed on

each subtriangle which enables a bivariate piecewise cubic interpolant to be devised

over the entire triangle that is continuously differentiable. The key result is that

only twelve degrees of freedom are required for the CTM, namely the function

values and the gradient at each vertex, as well as the normal derivative along the

edges.

In the context of leaf surface fitting, the function value is assigned at the triangle

vertices. However, the derivative information at the vertices and at the midpoints

of each side is unavailable and needs to be estimated. The vertex gradient estimates

are generated from neighbouring data information and thereafter the edge normal

derivatives are determined as the mean of the normal derivatives estimated at

the two vertices associated with the edge. This approximation is based on the

assumption that the normal slope along the sides of the triangle changes linearly

(Lancaster et al. [80]). A more detailed description of CTM including the list

of cardinal basis functions for the standard triangular element can be found in

(Lancaster et al. [80], Loch et al. [86], Ritchie [122]).

3.1.2 Radial basis functions

A Radial Basis Function (RBF) approximation to f is a function S of the form:

S(x) =n∑

i=1

aiΦi(x) x ∈ R2 (3.1)

where Φi(x) = R (‖x − xi‖) , R(r) is a non-negative real-valued function with

non-negative argument r and ‖.‖ denotes the Euclidean norm. The points xibelonging to R

2 are called the centres of the RBF approximation. The expansion

coefficients ai are determined by satisfying some approximation criterion; in this

application by interpolation (see equation 3.3).

In order to obtain a smooth surface representation to estimate the function


values at points other than data points, radial basis function schemes have found

applications in areas such as geodesy (Junkins et al. [69]) and medical imaging

(Carr et al. [24]). A major problem of the radial basis function method concerns

large sets of data points where the computational costs involved in fitting and

evaluating the RBF can become time-consuming. A review of the theory of RBF

approximation is given by Powell [113].

Well known examples of radial basis function methods include Hardy’s multi-

quadric RBF which is adopted in this paper:

R (‖x − xi‖) =

√c2 + ‖x − xi‖2. (3.2)

The parameter c must be specified by the user; it is related to the spread of the

function around its centers. The accuracy of the multiquadric interpolant depends

heavily on the choice of c (Franke [55]).

Thus, we face the problem of how to select a good value for the parameter c.

Many methods for selecting c for the multiquadric interpolants in two-dimensions

have been introduced in the literature. Franke [55] used c = 1.25 D√n

where D is

the diameter of the minimal circle enclosing all data points. Hardy [62] suggested

a value of c = 0.815d where d =Pn

j=1 dj

nand dj is the distance between the jth data

point and its closest neighbour.

Rippa [121] studied the influence of the parameter c on the quality of the

approximation of the multiquadric interpolant and concluded that the accuracy

depends on the choice of the parameter c. Rippa considered two sets of data

points and nine different test functions defined on the unit square. A data vector

f = (f1, f2, ..., fn)T was constructed by evaluating each test function over the set

of data points so that

S(xj) = fj , j = 1, 2, ..., n. (3.3)

Rippa [121] suggests an algorithm for selecting a good value for the parameter c

based on minimizing a cost function that represents the error between the inter-

polating radial basis function and the unknown function (RMS), see equation 3.6.

This cost function is defined as follows:


Let the error vector E = (E1, ..., En)T where Ek = fk − Sk(xk) = ak

x[k]k

, k =

1, ..., n and Sk(x) =∑n

i=1,i6=k aki R (‖x − xi‖) , and then

cgood = argminc∈R

‖E(c)‖1 . (3.4)

Here, Sk is the interpolant to a reduced data set obtained by removing the point

xk and the corresponding data value fk from the original data set and Ek is a

function of c since it requires translates of a basis function that depends on c. For

more details see Rippa [121].

3.2 Hybrid Method

We propose a new hybrid approach for surface fitting based on the CTM that uses

a multiquadric RBF to estimate the gradient at the vertices and mid-points of

the Clough-Tocher triangle. The multiquadric RBF interpolant S(x) is given by

equation 3.1. The gradient of S is then given by

∇S(x) =n∑

i=1

ai∇Φi(x), (3.5)

where ∇Φi(x) = ∇R (‖x − xi‖) = x−xi

‖x−xi‖R′

(‖x−xi‖) (R′

denotes the derivative of

R(r)).

The hybrid method is essentially an interpolating finite element method. We

outline this procedure in the following steps.

Step 1: Given n data points xi, i = 1, ..., n and a data vector fi, i = 1, ..., n,choose a subset of m data points from the n data points for the purpose of a

triangulation of the leaf surface.

Step 2: Find c using Rippa’s method (section 1.2).

Step 3: A global multiquadric RBF interpolant that uses the triangulation

points is then constructed and used to estimate the gradients for all triangles.

OR

A local multiquadric RBF interpolant that uses a local set of points constructed

on each triangle is used to estimate the gradients for a particular Clough-Tocher


triangle.

Step 4: In both methods, global and local RBF, the truncated singular value

decomposition TSVD (Tony et al. [133]) is applied to solve the linear system (3.3)

for the coefficients ai.Step 5: The CTM is applied to construct the leaf surface.

3.3 Application of the Hybrid Method for the

Frangipani and Anthurium Leaves

Reconstruction of the shape of a leaf using surface fitting techniques requires a set

of representative data points sampled from the surface. The process of sampling

data points from the leaf surface using a measuring device is called digitizing such

that the visible exterior data points of the leaf are enough to capture the surface

of the leaf. Loch [84] collected data points for different types of leaves (such as,

Frangipani, Anthurium, Flame and Elephant’s Ear) using a laser scanner. The

boundary points were selected by hand from the complete set of points using the

PointPicker, software written by McAleer (Hanan et al. [59]).

3.3.1 Data from laser scanner

In this research the hybrid Clough-Tocher Radial basis function interpolation

method was applied to the laser scanned Frangipani and Anthurium leaf data

taken from Loch [84] to construct the surface of those two leaves. The Frangi-

pani leaf data set contains two subsets of data. The first set consists of 3,388

points, which represents the entire leaf surface scanned points; while the second

set consists of 17 points representing the boundary points of the Frangipani leaf

surface. The Anthurium leaf data set consists of a set containing 4,688 points,

which represent the entire leaf surface points and a second set containing 79 points

representing the boundary points of the Anthurium leaf surface. These point sets

are displayed in Figures 1.11 (a) and (b).



The coordinate system used by the scanner, which returns the coordinates of points

on the leaf, may not be suitable for interpolation due to the possibility of multi-

valued and vertical surfaces. A solution is to use a reference plane that is a least

squares fit to the data. We construct a reference plane by making a linear least

squares fit to the data and rotating the coordinate system so that the reference

plane becomes the xy−plane. This rotation can be achieved by rotating the normal

vector of the reference plane about the x−axis into the xz−plane and then about

the y−axis into the yz−plane (Oqielat et al. [105]). This procedure is successful if

the vertical height of the data points is single valued in the transformed coordinate

system.

3.3.3 Triangulation method

In order to apply the hybrid method to the leaf data sets a triangulation of the

leaf surface needs to be constructed. Since the number of data points that repre-

sent the surface is large, the computational expense is reduced by selecting only

a subset of this set to generate a triangulation of the leaf. In this work the trian-

gulation of the leaf is constructed using the EasyMesh generator, software written

in the C language by Bojan [99]. EasyMesh generates two-dimensional Delaunay

and constrained Delaunay triangulations in general domains. We will explain the

triangulation process for only the Frangipani leaf because the process is the same

for the Anthurium leaf.

An input file that must be provided to EasyMesh is one that contains the 17

boundary points (nodes) and the desired length of the triangle sides. EasyMesh

returns a good triangulation if the domain is convex. However, because the piece-

wise linear boundary defined by the 17 chosen points do not enclose a convex set,

e.g see Figure 3.1 (a), EasyMesh was unable to produce a triangulation with the

required properties. To overcome this problem, an algorithm was used to generate

a convex hull from the entire set of leaf data points. This process provided a total

of 27 points, and the next closest points to the given 17 boundary points from

these points were found using the Matlab command dsearch. This process resulted


in 11 boundary points being identified as defining the convex domain exhibited in

Figure 3.1 (c).

In the interior of the convex hull (leaf surface) we can define either a horizontal,

or vertical, line in the domain to enable EasyMesh to produce fewer and better

shaped triangles. For the Frangipani and Anthurium leaves (Oqielat et al. [105])

it appears that the vertical line produces a more suitable triangulation than the

horizontal line, see for example Figure 3.1(c).

In summary, we applied the following steps to construct the triangulation of

the Frangipani leaf using EasyMesh:

Step 1: EasyMesh was provided with an input file that contains the 11 bound-

ary points, the vertical line and the desired triangle edge length. EasyMesh re-

turned the node file that contained the same boundary points, together with ad-

ditional boundary points (58 point) and a set of points distributed inside the leaf

(93 internal points). These represented the triangle vertices of the mesh structure,

see Figure 3.1 (d).

Step 2: Import the node file to Matlab and then locate the closest points in

the leaf data set from the internal points generated in Step 1 using dsearch. These

resulting points represent the triangle vertices of the leaf surface mesh structure.

Step 3: To obtain the boundary points of the leaf for which we do not have

surface values, we find the closest points from the leaf data set to the EasyMesh

boundary points and use their surface values.

Step 4: Use the Matlab command Delaunay to triangulate the leaf points

obtained from step 2 and 3.

This process gives the final triangulation for the leaf surface illustrated in Fig-

ure 3.2 (a). After the triangulation of the leaf surface is constructed the hybrid

Clough-Tocher Radial Basis Function method is applied to construct the leaf sur-

face. The local hybrid approach applied here is based on choosing the set of 5

nearest neighbours to each vertex and to the center of the triangle. Next, a local

radial basis function is built from the 20 points for each triangle, which is then

used to estimate the directional derivative at the triangle vertices and midpoints.

A global hybrid approach is also applied, which is based on building one single

global RBF from the triangulation points and then using it to evaluate the gradi-


−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

17 boundary points

−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

27 convex hull points

(a) (b)

−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

27 cv points11 pointsvertical line

−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

58 boundary points11 points93 interior points

(c) (d)

Figure 3.1: (a) The 17 Frangipani leaf boundary points. (b) The 27 points generated from

the convex hull algorithm. (c) The square points represent the final 11 boundary points. (d) The

vertices of the mesh structure generated using Easymesh. The square points represent the 11

boundary points that are given to Easymesh; the dot points represent the 58 extra points added

by Easymesh, while the x points represent the 93 internal points.


−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

−60 −40 −20 0 20 40 60−200

−100

0

100

200

300

400

(a) (b)

Figure 3.2: (a) Triangulation of 151 points of Frangipani leaf surface generated using

EasyMesh. (b) Triangulation of 141 points of Frangipani leaf surface.

ents at the vertices and midpoints of all triangles. The parameter c in both cases

was estimated globally using the triangulation points following the Rippa [121]

framework.

One problem that arose when applying the local RBF method to the Frangipani

leaf concerned the poor interpolant values arising at the “tail” of the leaf located

near the stem. The reason for the poor interpolant values occurred because there

was insufficient data to construct these interpolants. To overcome this problem

we needed to delete some of the smaller triangles from the mesh at the leaf tail

(by deleting 10 points from the boundary points added from EasyMesh at the tail)

to form larger triangles that contained enough data to proceed with the hybrid

method. Triangulations determined from this construction process are illustrated

in Figure 3.2 (b). This problem did not arise for the Anthurium leaf.


3.3.4 Numerical experiments

In this section we present the results of applying the hybrid method to the Frangi-

pani and Anthurium leaf data. After the triangulation points were selected, the

rest of the m data points (denoted by fk = f(xk), k = 1, ..., m) from the leaf data

set were used to measure the quality of the approximation of the hybrid method.

We noted that some of the m data points occurred outside of the virtual leaf mesh

and these points were ignored in the quality analysis. We then applied the hybrid

method to estimate the surface values for the data points occurring inside the tri-

angulation to construct the leaf surface, see Figure 1.11 (c) and (d).

The error metric we used was the root mean square error RMS, given by:

RMS =

√∑k=mk=1 [S(xk) − fk]2

m. (3.6)

S(xk) represents the CT estimated value at the m data points and fk represents the

given function values at the same data points. The second error metric measured

the quality in terms of the maximum error associated with the surface fit in relation

to the maximum variation in f .

scaled max error =max (|S(xk) − fk|)max(fk) − min(fk)

,

with k = 1, 2, ..., m.

Tables 3.1 and 3.2 show the scaled maximum errors and the scaled RMS =

RMSmax(fk)−min(fk)

for the Frangipani and the Anthurium leaf data sets respectively

using the local and global hybrid method. For the Frangipani leaf there were a

total of 3,155 data points used to assess the accuracy of the surface. Note the

EasyMesh triangulation comprised 141 vertices. There were more than 100 points

ignored in the analysis because these points were deemed to lie outside the leaf

mesh structure.

One observes for the Frangipani leaf that using the local hybrid RBF method

produced slightly more accurate RMS value than using the global hybrid RBF

method while it is the converse for the maximum error. The trends depicted in

Table 3.1 for the Frangipani leaf appear consistent with observations from Table

3.2 for the Anthurium leaf.


Table 3.1: RMS computed using hybrid local and global RBF for the Frangipani leaf data

points as well as the maximum error associated with the surface fit.


Scaled RMS 0.0086 0.0139Scaled maximum error 0.0700 0.0655boundary points 48 48points tested 3155 3155Triangulation points 141 141outside points 104 104No. of Triangles 257 257

Table 3.2: RMS computed using hybrid local and global RBF for the Anthurium leaf data

points as well as the maximum error associated with the surface fit.


Scaled RMS 0.0043 0.0068Scaled maximum error 0.0537 0.0435boundary points 66 66points tested 4460 4460Triangulation points 212 212outside points 59 59No. of Triangles 387 387


3.4 Conclusions and Future Research

The work presented in this paper describes a new mathematical surface fitting tech-

nique for modelling the leaf surface. It allows the user to construct an accurate leaf

surface based on three-dimensional data points.It provides a basis on which future

research can be built. Surface representations can be extended to generate not

only realistic images of leaves but also be applied to models determining a droplet

path on the leaf surface. Knowing this path is important for many application; for

example, in the simulation of a pesticide application to plant surfaces (Hanan et

al. [60], Reichard et al. [118]) Knowledge of this behaviour may be used to deter-

mine the effectiveness of a treatment, and then to develop certain pesticides that

have the ability to protect leaves for longer periods of time. Similar models may

treat moisture precipitation and energy uptake through photosynthesis enabled by

ray tracing techniques.

At present projections of the image boundaries in the reference plane are piece-

wise linear. Work on genuinely curved boundaries is in progress.

An advantage of the leaf models described in this paper is that they may be

used in different plant modelling environments such as AMAP (Godin et al. [57]),

xfrog (Lintermann et al. [83]) or LStudio (Prusinkiewicz et al. [116]).

3.5 Acknowledgments

This paper was carried out thanks to funding from the School of Mathematical

Sciences. Many thanks to Dr. Joseph Young and Mr. Mark Dwyer from the

Queensland University of Technology HPC centre for the leaf visualizations.

Chapter 4

Numerical investigations of linear least squares

methods for derivative estimation

4.1 Introduction

This paper presents the results of numerical investigations of an algorithm to deter-

mine the gradient of a function from planar scattered data values. The particular

methods analysed were chosen to provide accurate gradient estimates combined

with the opportunity for computational efficiency with large data sets. The cur-

rent target of the work is in the construction of models of leaf surfaces, which will

assist the study of local effects such as the movement of droplets on leaf surfaces.

The work also has the potential for input into growth mechanisms of whole plants.

Theoretical error bounds set out in §4.3 are evaluated and found to be several

orders of magnitude pessimistic. An examination of the terms which make up the

estimates suggests a modification to the bounds. Although the evidence at this

stage is experimental, a heuristic argument developed in §4.4 suggests the means

by which a more accurate bound may be derived.

The current work has been undertaken to contribute to simulation models for

the growth of plants. This has been facilitated by the development of L-systems

that may be used to capture the architecture of a plant, or tree, as it grows. Since

leaves are an essential part of a plant, the understanding of how they respond to

99

Numerical investigations of linear least squares methods for derivative estimation 100

physical inputs is crucial. Knowing where the surface is located is essential for

light interception and deposition of water, liquids and perhaps powders. While a

piecewise linear representation may be adequate in certain situations, the appli-

cations envisaged in this work call for a smoothly varying surface normal, thus

continuity of the gradient will be built into the algorithms described in §4.2.

An important observation is the occurrence of the smallest non-zero singular

value of the least squares matrix in the denominator of the error bound. The

progressive accumulation of the approximants by computation of the singular value

expansion is tabulated. The results quantify the contributions of the elements of

the singular value decomposition and demonstrate the contribution of each term

to the gradient approximations. As a result of this numerical investigation it is

conjectured that rather than using the smallest singular value in the error bound

it appears more appropriate to use the smallest singular value of reduced form of

the least squares matrix. The results of this conjecture are quite promising and

point to how this new result may be justified analytically.

4.2 Construction of a leaf surface

Both sonic and laser scanner devices are used to capture leaf surfaces, in both cases

the raw data comprises three-dimensional coordinates of points. A recent presen-

tation by Oqielat, Belward, Turner and Loch, [104], describes the use of piecewise

polynomial basis functions on a triangulation of the data points to generate smooth

representations of the scanned surface. The image is constructed by a surface fit to

the scattered data and realism may be conveyed by adding texture to the surface

plots. Clough Tocher elements are triangular elements that are comprised of three

triangular micro-elements, which each have their common interior vertex at the

incentre of the element. With this configuration a piecewise cubic approximant

may be constructed with a continuous gradient provided that gradient values are

provided at the nodes, together with the normal derivative at the midpoints of the

edges. The work in reference [105] contains some recent results for leaf surface

fitting that utilise the Clough-Tocher representation to interpolate the scattered

data to ensure this continuous gradient. The approximant is expressed as a lin-


ear combination of twelve basis functions; a cardinal basis for a standard triangle

is given by Lancaster and Salkauskas [80] . Since in this case only data values

are available some estimates of gradients are needed and a brief description of an

accurate method to provide these estimates is given below.

Taylor’s Theorem for several variables states that for f : R2 → R, in open

convex set D ⊂ R2 then

f(a + hν) = f(a) + h(ν · ∇)f(a)

1!+ · · ·+ hn (ν · ∇)nf(a)

n!+ Rn , (4.1)

where the remainder Rn has the integral form

hn+1

n!

∫ 1

0

(1 − t)n(ν · ∇)n+1f(a + thν)dt .

In this work the Taylor series is truncated and the first three terms of the right

hand side of (5.1) evaluated at a scattered set of points vi = a + hiνi, i = 1, . . . , m.

The term f(a) is transferred to the left hand side and the equation divided by

hi = ‖vi − a‖ to obtain


hi

= (νi · ∇)f(a) +hi

2νT

i Hf (a)νi ,

where Hf (a) is the Hessian of f evaluated at a. This equation is applied at each of

a neighbouring set of m points vi, i = 1, . . . , m near a to obtain the overdetermined

linear system

Aγ ≈ q , (4.2)

where vector q has as its elements the difference quotients for f in the direction of

the unit vectors νi = (νxi, νyi

)T . The elements of γ are approximations to the five

partial derivatives of f , namely

γ ≈ (∂f(a)

∂x,∂f(a)

∂y,∂2f(a)

∂x2,∂2f(a)

∂x∂y,∂2f(a)

∂y2)T ,


and the matrix A ∈ Rm×5 takes the following form:

A =

νx1νy1

12h1ν

2x1

h1νx1νy1

12h1ν

2y1

......

......

...

......

......

...

νxmνym

12hmν2

xm

hmνxmνym

12hmν2

ym

.

The least squares solution of equation (4.2) for γ = argminx∈R5‖Ax − q‖2 yields

estimates of the gradient of f at a that are O(h2) accurate.

In earlier work by Belward, Turner and Ilic [13] it was observed that one

could either take the direct approach and estimate ∇f(a) by g = E1A†q where

E1 =

(1 0 0 0 0

0 1 0 0 0

), or perform an orthogonal reduction of the columns 3-5 in

A = (A1|A2) as QT A2 =

(A12

0

), to obtain QT A =

(A11 A12

A21 0

). The symbol

A† denotes the pseudoinverse or generalised inverse of A as described by [28] .

Then ∇f(a) is estimated by g = argminy∈R2‖A21y − q‖2, where q2 represents the

last m−3 entries in QT q. It was shown by Belward et al., [13], that these methods

provide the same solution g.

4.3 Theoretical error bounds

The following lemma and proposition establish a bound for the gradient approxi-

mation computed using the algorithms described in the previous section.

Lemma 1. Let f : D ⊂ R2 → R in an open convex set D and f ∈ C2(D). Suppose

that Hf ∈ Lipλ(D). Then for any a + hν ∈ D with ‖ν‖2 = 1 ,

∣∣∣∣h

2νT Hf(a)ν −

f(a + hν) − f(a)

h

+ νT∇f(a)

∣∣∣∣ ≤λh2

6. (4.3)

Proof. Rearranging the multivariable Taylor series for f about the point a gives:

f(a + νh) − f(a)

h− νT∇f(a) = h

∫ 1

0

(1 − t)νT Hf(a + thν)ν dt ,

and therefore


h

2νT Hf(a)ν−

f(a + hν) − f(a)

h

+νT∇f(a) = h

∫ 1

0

(1−t)νT

Hf(a)−Hf (a+thν)

ν dt .

Hence,

∣∣∣∣h

2νT Hf(a)ν−

f(a + hν) − f(a)

h

+νT∇f(a)

∣∣∣∣ ≤ h

∫ 1

0

∣∣∣∣(1−t)νT

Hf(a)−Hf (a+thν)

ν

∣∣∣∣ dt .

The result follows by invoking Cauchy-Schwarz inequality and Lipschitz continuity

and noting that ‖ν‖ = 1.

Methods for estimating the Lipschitz constant λ in (4.3) are elaborated in

reference [135]. In section §4.4 the value of λ was computed as

λ = maxx∈D(| ∂3f

∂x3−i∂yi|, i = 0, 1, 2, 3) , (4.4)

where these maxima were determined with the help of Maple.

Proposition 1. Suppose around point a we have m neighbouring points vk, k =

1, . . . , m with a, v1, . . . , vm ∈ D; D an open convex set ⊂ R2. Suppose further that

f ∈ C2(D) with Hf ∈ Lipλ(D) and we approximate the gradient locally at a by

E1γ via the least squares solution of the overdetermined system Aγ = q

A ≈

νT11 νT

12

νT21 νT

22

......

νTm1 νT

m2

, q =

f(a+h1ν1)−f(a)h1

f(a+h2ν2)−f(a)h2

...

f(a+hmνm)−f(a)hm

, and hi = ‖vi−a‖2 with hiνi = vi−a;

and νTi1 = (νxi

, νyi), νT

i2 = (hi

2ν2

xi, hiνxi

νyi, hi

2ν2

yi). Then,

‖∇f(a) − E1γ‖2

‖∇f(a)‖2

≤ λ√

m h2max

6 σ1‖∇f(a)‖2

, (4.5)

where σ1 is the smallest singular value of matrix A, which is assumed to have

rank(A) = 5.

Proof. Let E2 ∈ R3×5 be the last 3 rows of the identity matrix I5 and U =(

∂f∂x

(a), ∂f∂y

(a), ∂2f∂x2 (a), ∂2f

∂x∂y(a), ∂2f

∂y2 (a)

)T

be the exact values of the derivatives at

a . Now U − γ can be partitioned as

E1(U − γ)

E2(U − γ)

with E1 defined above and


hence it follows that,

‖U − γ‖22 = ‖E1(U − γ)‖2

2 + ‖E2(U − γ)‖22 ≥ ‖E1(U − γ)‖2

2 = ‖∇f(a) − E1γ‖22 .

Next, with γ = A†q and A†A = I5, the following relations are obtained :

‖U − γ‖22 = ‖U − A†q‖2

2 = ‖A†(AU − q)‖22 ≤ ‖A†‖2

2‖AU − q‖22 =

1

σ21

‖AU − q‖22 .

Now using the result in lemma 2, the following upper bound can be derived

‖AU − q‖22 =

m∑

i=1

∣∣∣∣h

2νT

i Hf(a)νi −


hi

+ νT

i ∇f(a)

∣∣∣∣2

≤m∑

i=1

(λh2

i

6

)2

(Using lemma 2)

≤(

λ

6

)2 m∑

i=1

(h2

max

)2

=

(λ

6

)2

h6max m .

The result follows by taking square roots and noting that

‖∇f(a) − E1γ‖2 ≤ ‖U − γ‖2 ≤λ

6

h2max

σ1

√m

and then dividing both sides by ‖∇f(a)‖2.

As can be seen from the bound (5.4) the approximation scheme has a truncation

error of order 2 , however the conclusion of the proposition does not guarantee

2nd order accuracy because of the presence of the smallest singular value σ1 of

the least squares matrix in the error bound. The numerical results presented

here and in earlier work provide strong evidence that such a conclusion would

be valid. Confining the discussion to the gradient approximation, the results of

the next section show that the bound (5.4) is pessimistic and additional numerical

investigation suggests that a better bound is possible by utilising the singular value

decomposition of A21.


4.4 Numerical results

The tests described here were run on the function

sin r

r, r = (x2 + y2)

12 .

One hundred points (ri cos θi, ri sin θi)T were generated with ri and θi dis-

tributed uniformly with 1 ≤ ri ≤ 2 and 0 ≤ θi ≤ 2π. From these points the 30

points nearest the origin were chosen and used as displacements from the point

(3, 4) to generate a scattered data set. The radial distances were scaled to pro-

duce the annuli with inner radii shown in the top row of Table 4.1. The results

given in the table provide the numerical evidence for second order and first order

convergence of, respectively, the gradient and the Hessian approximations.

Table 4.1: Norms of the errors in the gradient and Hessian with differing radii

radius 2.5000e-01 2.5000e-02 2.5000e-03 2.5000e-04 2.5000e-05gradient error 1.3315e-02 1.3363e-04 1.3363e-06 1.3363e-08 1.3537e-10hessian error 2.2085e-02 2.5141e-03 2.5457e-04 2.5486e-05 2.7235e-06

Table 4.2: Error bound and modified error bound values for varying radii

radius error bound modified error bound2.5e-01 .888e+01 .983e+002.5e-02 .888e+00 .983e-022.5e-03 .888e-01 .983e-042.5e-04 .888e-02 .983e-062.5e-05 .888e-03 .983e-08

Table 4.2 contains two sets of results. In column two the error bounds com-


puted from the expression (5.4) are listed while in column three the values are

computed from a proposed amendment to expression (5.4). Column one contains

the corresponding radii. Examination of the components of expression (5.4) reveals

Table 4.3: Singular values (sv’s) of the least squares and elimination matrices for various

radii

radius .25 .25e-1 .25e-2 .25e-3 .25e-4sv’s of 1.3486e-01 1.3476e-01 1.3476e-01 1.3476e-01 1.3476e-01

A 1.2321e-01 1.2318e-01 1.2318e-01 1.2318e-01 1.2318e-011.9399e-02 1.9413e-03 1.9413e-04 1.9413e-05 1.9413e-061.6844e-02 1.6848e-03 1.6848e-04 1.6848e-05 1.6848e-061.3650e-02 1.3651e-03 1.3651e-04 1.3651e-05 1.3651e-06

sv’s of 1.3476e-01 1.3476e-01 1.3476e-01 1.3476e-01 1.3476e-01A21 1.2318e-01 1.2318e-01 1.2318e-01 1.2318e-01 1.2318e-01

that it is the smallest singular value of the least squares matrix that produces the

poor error estimates exhibited in column 1 of Table 4.2. From the singular value

analysis of the least squares matrix A given in Table 4.3 it can be observed that

the singular values fall into a group of two and a group of three identified by their

magnitudes. In terms of the ordering of the columns of the least squares matrix

these may be associated with, respectively, the gradient and the Hessian. It there-

fore seems quite plausible that a closer bound would be obtained if the smallest

singular value of the group of two were used in place of the smallest singular value

of the whole system. This conjecture is apparent in the second column of Table

4.2, which provides a much tighter bound on the gradient estimates given in Table

4.1.

The argument for this amendment can be strengthened by examining the elim-

ination method described in Section 4.2. After the orthogonal reduction of A2

and the omission of the first three equations a new least squares matrix A21 is

obtained that has just two non-zero columns and two non-zero singular values.

Results displayed in Table 4.3 show that the singular values of the new system


Table 4.4: Cumulative sums of the singular expansion solution for the derivatives for the

point (3,4) at radii between .25 and .5

number radius .25of terms fx fy fxx fxy fyy

1 2.5279e-02 -9.1398e-03 -6.7030e-04 8.0706e-04 -2.5922e-042 5.6724e-02 7.7756e-02 6.4653e-05 1.1226e-03 1.4595e-033 5.8838e-02 7.6049e-02 5.9274e-02 -2.6472e-02 2.8863e-024 5.6445e-02 7.5021e-02 6.9913e-02 6.9343e-02 1.0248e-015 5.6462e-02 7.4987e-02 6.7879e-02 6.7568e-02 1.0508e-01





become increasingly close to those of the full least squares matrix as the radius

of the test point set is reduced. Note that the first two columns of the full least

squares matrix A are independent of the radii of the annuli. The columns of the

elimination matrix A21 have a dependence on h through the orthogonal reduction,

although this dependence diminishes as h is reduced.

A further calculation that also demonstrates the importance of the first two

singular values is to use the singular value decomposition of the least squares

matrix to progressively construct the least squares solution. As the terms are

added the progress towards the solution may be observed. Three sets of results are

shown in Table 4.4. The dominance of the first two terms is increasingly marked

with the diminuition of the radius; by the time the radius reaches .000025 only the


first two terms are needed. Progress towards the solution for the Hessian is more

steady; the results of Table 4.4 conform to O(h) convergence as do the bounds in

column 2 of Table 4.2.

4.5 Conclusion

Numerical investigations have been made of the theoretical bounds of the errors

in gradient estimation from scattered data values using a least squares algorithm.

These bounds have been compared with the errors that are observed when the

algorithm is implemented.

It was noted that the smallest singular value of the least squares matrix A has a

large detrimental influence on the realism of the bounds. On the basis of numerical

experimentation, a modification is suggested in which certain singular values are

deemed associated with the gradient. This modification is more apparent when a

version of the algorithm is used wherein the Hessian terms are eliminated.

These results, which were speculative at the commencement of this work, will

be shown to be rigorous in a further paper on this topic [135].

Chapter 5

Error Bounds for Least Squares Gradient Estimates

5.1 Introduction

The accurate estimation of the gradient of a function f : D ⊂ R2 → R, for some

domain of interest D, from a set of scattered function values arises in many im-

portant applications in applied and computational mathematics. One application

of particular interest to the authors is the measurement of leaves of plants to cap-

ture their image for the modelling of droplet movement and absorption on the leaf

surface. In this case the leaf surface representation requires a smooth fit to a set of

scattered data and our preferred method to obtain this surface is to represent the

function by a set of cubic elements defined on a union of triangular domains using

a Delaunay triangulation of the data points [105]. The surface is then represented

using Clough Tocher basis functions (see for example [80]), which enables a piece-

wise cubic surface with continuous gradient to be obtained if the function values

and the gradients are known at the original data points and the gradient is also

known at the midpoints of the edges of the triangulation. Given that only the data

values are known at the scattered points, the implementation of our surface fitting

strategy requires estimation of the gradient at the desired points, namely given

the values zi at the points (xi, yi)T estimate the values of the gradient, ∇f(xi, yi).

Two other important application areas where least squares gradient estimation

are used is in computational fluid dynamics [9,67,103] and data visualization [112].

109

Error Bounds for Least Squares Gradient Estimates 110

In the former gradient reconstruction plays an important role in building accurate

flux approximations at discrete cell faces, while in the latter the gradients are used

for real-time volume rendering by providing a surface normal approximation that

can be used for lighting, shading and assigning opacity.

It is well known that multivariable Taylor expansions relate function values to

derivatives and these generate linear relations amongst the derivatives and function

values. These expansions provide an excellent mechanism for derivative estima-

tion and when the data points are subject to error it seems only natural to form

overdetermined systems of equations and then obtain gradient estimates by min-

imising residuals via a least squares approach. We exploited this approach in [13],

where we considered two different least squares strategies for approximating the

local gradient estimates and analysed the least squares errors associated with each

method. An important question that arose from that work is what is the spatial

error associated with this type of estimation strategy? Intuitively it seems quite

plausible that this error will be O(hnmax), where n is the number of terms taken in

the Taylor expansion and hmax is the maximum distance from the point of interest

say a and any of the cloud of neighbouring points used for estimating the gradient.

This assertion certainly seems to be well accepted in the literature, although the

leading constant does not appear to have been explicitly identified. Many authors

assert the order of piecewise polynomial approximations in R2; reference [5] is a

typical example. Wei, Hon and Wang [136] give estimates for the construction of

numerical derivatives from noisy data and Zuppa [143] has derived error bounds

for derivative approximation based on algorithms using Gaussian elimination for

a modified local Shepard’s approximation. In the current work orthogonal trans-

formations are used for the error analysis, thereby enabling the error bounds to

be given in terms of the singular values of the least squares matrix used in our

algorithm.

In recent work [14], we derived an error bound where this leading constant was

given for a quadratic least squares gradient approximation and it was conjectured,

as a result of numerical experimentation and observation, that this bound could be

further improved. This current work generalises the results presented in [14] to an

order n least squares gradient approximation and a proof of the conjecture is given.


The main contribution is to not only investigate the order of the method, but to

show that the smallest singular value of the least squares gradient coefficient matrix

plays an important role – if this matrix is ill-conditioned the more the impact on

the overall error.

Another well documented approach for local gradient estimation is to use a

weighted least squares method, whereby the system is row scaled in the sense

that more importance is given to points closer to the point of interest a [9, 67].

Interestingly, our theory shows that this approach may not improve the situation

at all, because the smallest singular value of the scaled matrix is usually smaller

than that of the original matrix.

The paper is structured as follows. In §5.2 a brief overview of least squares

gradient estimation is given, and error bounds are derived for both the classical and

weighted least squares estimates in §5.3. We present two analyses of the order of

convergence of these methods. The second in §5.4 exploits a Householder reduction

of certain columns of the least squares matrix to reveal that error estimates of

subsets of the derivatives can be made using the singular values associated with

those columns. An important corollary is that the gradient estimates are genuinely

O(hn) since these singular values are independent of h. In section §5.5.1 numerical

results are given exhibiting the predicted asymptotic behaviour of the derivative

estimates, while in §5.5.2 we turn to a more practical situation and present results

on point sets taken from Franke’s celebrated paper [55]. Finally in §5.6 the main

conclusions of the research are summarised.

5.2 Least Squares Gradient Estimation

Although many representations of surfaces are possible, here we assume the rep-

resentation as a function f : D ⊂ R2 → R, z = f(x, y). Hence, a reference plane

is assumed to exist with a unique ordinate at each data point in the xy−plane.

Unless otherwise stated in the paper ‖ · ‖ is assumed to be the Euclidean norm.

The gradient estimation strategy is now outlined. Suppose that point a =

(ax, ay)T ∈ D is surrounded by m scattered data points vi = a+hiνi = (xi, yi)

T , i =

1, . . . , m with hi = ‖vi − a‖ and we require an estimate of the gradient ∇f(a).


Assuming that I is an open interval in R containing [0, 1], f ∈ Cn(D) and a+thν ∈D, ∀t ∈ I, then Taylor’s Theorem for several variables states that

f(a + hν) = f(a) + h(ν · ∇)f(a)

1!+ · · ·+ hn (ν · ∇)nf(a)

n!+ Rn, (5.1)

where the remainder Rn has the integral form

Rn =hn+1

n!

∫ 1

0

(1 − t)n(ν · ∇)n+1f(a + thν)dt.

We now consider, as an example (see [14]), the use of relation (5.1) to write

an overdetermined system of equations for the case n = 2, where we have the

overdetermined system Aγ ≈ q with γ =(

∂f∂x

(a), ∂f∂y

(a), ∂2f∂x2 (a), ∂2f

∂x∂y(a), ∂2f

∂y2 (a))T

and

A =

νx1νy1

12h1ν

2x1

h1νx1νy1

12h1ν

2y1

......

......

......

......

......

νxmνym

12hmν2

xm

hmνxmνym

12hmν2

ym

∈ Rm×5.

The entries in the matrix are defined as hiνxi= xi − ax, hiνyi

= yi − ay, and the

right hand side vector q ∈ Rm×1 has as its ith component qi = f(a+hiνi)−f(a)hi

. The

solution of the least squares problem is given by γ = arg minγ∈R5‖Aγ − q‖, which

then enables the gradient estimate to be extracted from the first two components

of γ as ∇f(a) ≈ E1γ = E1A†q, where E1 ∈ R

2×5 is defined by the first two rows

of the identity matrix I5 and A† is the pseudoinverse, or generalised inverse of A

(see for example [92]).

Note also that each of the estimates of the directional derivative may be

weighted without loss of accuracy. This follows since the effect of a weight factor wi

is to introduce a diagonal matrix W = (w1, w2, . . . , ws), where typically one would

use inverse distance, or inverse distance squared weights wi = ‖a−vi‖−d, d = 1, 2 to

give more significance to points closer to a. In this case the overdetermined system

becomes WAγ ≈ Wq, the least squares problem gives γ = arg minγ∈R5‖WAγ −Wq‖, which then enables the gradient estimate to be extracted as the first two com-

ponents of γ as ∇f(a) ≈ E1γ = E1(WA)†Wq, where (WA)† is the pseudoinverse

of WA.


Error bounds for both the classical ‖∇f(a)−E1γ‖ and weighted ‖∇f(a)−E1γ‖least squares gradient estimates for a general nth order approximation are derived

in the next section. In both cases it is shown that the bounds take the form

C hnmaxκ2(W )

σ1, where C is an appropriately defined constant, κ2(W ) (which is 1 for

the classical case) is the condition number of the weight matrix W , σ1 is the

smallest singular value of the least squares matrix A, hmax = max1≤i≤m ‖vi − a‖and n is the degree of the truncated Taylor series given in (5.1).

5.3 Error Bounds

In this section we consider a more general setting than that discussed in §5.2

and derive error bounds for both the classical and weighted least squares gradient

estimation methods for an n term Taylor expansion.

5.3.1 Classical Least Squares Gradient Estimates

The error bound derived in proposition 2 below requires the consideration of the

following lemma, which effectively bounds the error in the Taylor series truncated

at term n.

Lemma 2. Let f : D ⊂ R2 → R in an open convex set D and f ∈ Cn(D). Suppose

for i = 0, 1, . . . , n we have ∂nf∂xn−i∂yi ∈ Lipϑi

(D) with ϑmax = max0≤i≤n ϑi. Then for

any a + hν ∈ D with ‖ν‖ = 1

∣∣∣∣n∑

k=1

hk−1

k!(ν · ∇)kf(a) −

f(a + hν) − f(a)

h

∣∣∣∣ ≤hn

(n + 1)!ϑmax‖ν‖n

1 . (5.2)

Proof. Rearranging the multivariable Taylor series (5.1) for f about the point a

we obtain:

f(a + νh) − f(a)

h−

n−1∑

k=1

hk−1

k!(ν ·∇)kf(a) =

hn−1

(n − 1)!

∫ 1

0

(1−t)n−1(ν ·∇)nf(a+thν)dt

and therefore


hn−1

n!(ν · ∇)nf(a) −

f(a + hν) − f(a)

h

+

n−1∑

k=1

hk−1

k!(ν · ∇)kf(a)

=hn−1

(n − 1)!

∫ 1

0

(1 − t)n−1

(ν · ∇)nf(a) − (ν · ∇)nf(a + thν)

dt

=hn−1

(n − 1)!

n∑

i=0

n

i

νn−i

x νiy

∫ 1

0

(1 − t)n−1

∂nf(a)

∂xn−i∂yi− ∂nf(a + thν)

∂xn−i∂yi

dt.

Hence, ∣∣∣∣n∑

k=1

hk−1

k!(ν · ∇)kf(a) −

f(a + hν) − f(a)

h

∣∣∣∣

=

∣∣∣∣hn−1

(n − 1)!

n∑

i=0

n

i

νn−ix νi

y

∫ 1

0

(1 − t)n−1

∂nf(a)


∂xn−i∂yi

dt

∣∣∣∣

≤ hn−1

(n − 1)!

n∑

i=0

n

i

|νx|n−i|νy|i

∫ 1

0

|1 − t|n−1

∣∣∣∣∂nf(a)


∂xn−i∂yi

∣∣∣∣dt.

(5.3)

We now invoke the Lipschitz continuity of the mixed partial derivatives to obtain

a further inequality on (5.3) as

≤ hn−1

(n − 1)!

n∑

i=0

n

i

|νx|n−i|νy|iϑi

∫ 1

0

|1 − t|n−1‖a − a − thν‖dt

≤ hn

(n + 1)!ϑmax

n∑

i=0

n

i

|νx|n−i|νy|i

=hn

(n + 1)!ϑmax

|νx| + |νy|

n

=hn

(n + 1)!ϑmax‖ν‖n

1 .

We are now in a position to prove the main result for this section, which provides

a bound on the error in the classical least squares gradient estimate.


1, . . . , m with a, v1, . . . , vm ∈ D; D ⊂ R2 an open convex set. Suppose further

that f ∈ Cn(D) with ∂nf∂xn−i∂yi ∈ Lipϑi

(D); i = 0, 1, . . . , n and we approximate the

gradient locally at a by E1γ via the least squares solution γ = arg minγ∈Rp‖Aγ−q‖,


where

A =

νT11 νT

12

νT21 νT

22

......

νTm1 νT

m2

∈ Rm×p, q =

f(a+h1ν1)−f(a)h1

f(a+h2ν2)−f(a)h2

...

f(a+hmνm)−f(a)hm

∈ Rm×1,

E1 =

(1 0 0 · · · 0

0 1 0 · · · 0

)∈ R2×p, hk = ‖vk − a‖ with hkνk = vk − a; νT

k1 =

(νxk, νyk

), νTk2 = (hk

2ν2

xk, hkνxk

νyk, hk

2ν2

yk,

h2k

6ν3

xk,

h2k

2ν2

xkνyk

, . . . ,hn−1

k

n!νn

yk) and p = (n+1)(n+2)

2−

1. Then a bound on the relative error in the least squares gradient estimate is given

by

‖∇f(a) − E1γ‖‖∇f(a)‖ ≤ ϑmaxh

nmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 , (5.4)

where σ1 is the smallest singular value of A, which is assumed to have rank(A) = p,

ϑmax is as defined in lemma 2 and hmax = max1≤k≤m hk.

Proof. Let E2 ∈ R(p−2)×p be the last p − 2 rows of the identity matrix Ip and

U =

(∂f∂x

(a), ∂f∂y

(a), ∂2f∂x2 (a), ∂2f

∂x∂y(a), ∂2f

∂y2 (a), ∂3f∂x3 (a), ∂3f

∂x2∂y(a), . . . ∂nf

∂yn (a)

)T

∈ Rp×1.

Now U − γ can be partitioned as

E1(U − γ)

E2(U − γ)

and hence,

‖U − γ‖2 = ‖E1(U − γ)‖2 + ‖E2(U − γ)‖2 ≥ ‖E1(U − γ)‖2 = ‖∇f(a) − E1γ‖2.

Next, with γ = A†q, we have

‖U − γ‖2 = ‖U − A†q‖2 = ‖A†(AU − q)‖2 ≤ ‖A†‖2‖AU − q‖2 =1

σ21

‖AU − q‖2.


Now using the result in lemma 2, the following upper bound can be derived

‖AU − q‖2 =

m∑

i=1

∣∣∣∣n∑

k=1

hk−1i

k!(νi · ∇)kf(a) −


hi

∣∣∣∣2

≤m∑

i=1

(hn

i

(n + 1)!ϑmax‖νi‖n

1

)2

(using lemma 2)

≤(

ϑmax

(n + 1)!

)2 m∑

i=1

(hn

i ‖νi‖n1

)2

(5.5)

≤(

ϑmax

(n + 1)!

)2 m∑

i=1

(hn

max‖νi‖n1

)2

=

(ϑmax

(n + 1)!

)2(hn

max

)2 m∑

i=1

‖νi‖2n1 .

The result follows from

‖∇f(a) − E1γ‖ ≤ ‖U − γ‖ ≤ ϑmax

(n + 1)!

hnmax

σ1

√√√√m∑

i=1

‖νi‖2n1

and then dividing both sides by ‖∇f(a)‖.

5.3.2 Weighted Least Squares Gradient Estimates

The bound presented in the previous section can be naturally extended to the

weighted least squares case via the following proposition.


1, 2, ..., m with a, v1, ..., vm ∈ D; D ⊂ R2 an open convex set. Suppose fur-

ther that f ∈ Cn(D) with ∂nf∂xn−i∂yi ∈ Lipϑi

(D); i = 0, 1, 2, ..., n and we approxi-

mate the gradient locally at a by E1γ via the weighted least squares solution γ =

arg minγ∈Rp‖Aγ − Wq‖, where A = WA with W = diag(w1, w2, ..., wm) ∈ Rm×m

being the weight matrix with positive weights wi. Then, with the same notation

as given in proposition 2 and wmax = max1≤k≤m wk, we have that a bound on the

relative error in the weighted least squares gradient estimate is given by

‖∇f(a) − E1γ‖‖∇f(a)‖ ≤ ϑmaxh

nmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖ν‖2n1 , (5.6)


where σ1 is the smallest nonzero singular value of A, which is assumed to have

rank(A) = p.

Proof. The proof proceeds along the same lines as proposition 2 with now

‖U − γ‖2 ≤ ‖∇f(a) − E1A†Wq‖2

and

‖U − γ‖2 = ‖U − A†Wq‖2

= ‖A†(AU − Wq)‖2

≤ ‖A†‖2‖AU − Wq‖2

≤ ‖A†‖2‖W‖2‖AU − q‖2

=1

σ21

‖W‖2‖AU − q‖2.

Using inequality (5.5) and noting that ‖W‖ = max1≤k≤m wk = wmax gives

‖∇f(a) − E1γ‖ ≤ ‖U − γ‖ ≤ ϑmax

(n + 1)!

hnmaxwmax

σ1

√√√√m∑

i=1

‖νi‖2n1 .

The result then follows by dividing both sides by ‖∇f(a)‖.

Remark 1. Noting that σ1 = inf‖x‖=1 ‖Ax‖ ≤ ‖W−1‖ inf‖x‖=1 ‖Ax‖ = ‖W−1‖σ1,

then the bound (5.6) can be also written in terms of the condition number of the

weight matrix as follows

‖∇f(a) − E1γ‖‖∇f(a)‖ ≤ ϑmaxh

nmaxκ2(W )

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖ν‖2n1 . (5.7)

5.4 Tighter Error Bounds

It will be apparent from the numerical experimentation performed in §5.5 that the

error bounds given in §5.3 are somewhat pessimistic. To derive tighter bounds we

return to the earlier work by Belward, Turner and Ilic [13] where it was observed

that the gradient ∇f(a) could be estimated using an orthogonal reduction of the


matrix A2 in the partitioned matrix A = (A1|A2), A1 ∈ Rm×2 and A2 ∈ Rm×(p−2),

as follows

1. QT A2 =

A12

0

with A12 ∈ R(p−2)×(p−2) upper triangular.

2. Partitioning QT =

QT1

QT2

yields QT A =

A11 A12

A21 0

, with A11 =

QT1 A1, A12 = QT

1 A2 and A21 = QT2 A1.

3. Then ∇f(a) ≈ arg ming∈R2

∥∥A21g − QT2 q∥∥.

It was proven in [13] that this strategy produces the same least squares error and

gradient estimate as the direct approach described in §5.3. Interestingly though, it

is the orthogonal reduction approach that enables tighter bounds than those given

in (5.4) and (5.6) to be derived. We begin with the following proposition.

Proposition 4. Let A ∈ Rm×p as defined above with rank(A) = p be column par-

titioned as A = (A1|A2) with QT A2 =

A12

0

being the orthogonal reduction of

A2, so that QT A =

QT1

QT2

(A1|A2) =

A11 A12

A21 0

, then

(i) A21 has full column rank with rank(A21) = rank(A1).

(ii) σ1 ≤ σ1 where σ1 is the smallest singular value of A and σ1 is the smallest

singular value of A21.

Proof. (i) rank(A21) = rank(QT2 A1) = rank(A1) − dim N (QT

2 ) ∩ R(A1) (see for

example [92]). Now QT2 A2 = 0 ⇒ R(A2) ⊆ N (QT

2 ). Let z ∈ N (QT2 ) and

observe that QT (A2|z) =

A12 QT

1 z

0 0

so that rank(A2|z) = rank(A2) ⇒

z ∈ R(A2), i.e., N (QT2 ) ⊆ R(A2) and hence it follows that N (QT

2 ) = R(A2).

Thus, N (QT2 ) ∩ R(A1) = ∅, since rank(A) = p, and therefore rank(A21) =

rank(A1) ⇒ A21 has full column rank because any subset of a linearly inde-

pendent set of column vectors must be also linearly independent.


(ii) σ21 = inf‖x‖6=0

‖Ax‖2

‖x‖2 = inf‖x‖6=0‖QT Ax‖2

‖x‖2 = inf‖x‖6=0

‚‚‚‚‚‚‚

0B@

A11 A12

A21 0

1CA

0B@

x1

x2

1CA

‚‚‚‚‚‚‚

2

‖x1‖2+‖x2‖2 . In

particular, choose x1 = v1, x2 = −A−112 A11v1, where (σ1, u1, v1) is the singular

triplet of A21 corresponding to the smallest singular value σ1. Hence, σ1 ≤σ1√

1+‖x2‖2≤ σ1.

Proposition 5. Under the hypothesis of Propositions 2 and 4 we have that

‖∇f(a) − E1A†q‖

‖∇f(a)‖ ≤ ϑmaxhnmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1

≤ ϑmaxhnmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 . (5.8)

Proof.

‖∇f(a) − E1A†q‖ = ‖∇f(a) − A†

21QT2 q‖ see [13] for proof

= ‖A†21(A21∇f(a) − QT

2 q)‖ using Proposition 4(i)

≤ 1

σ1

‖QT2 (A1∇f(a) − q)‖

=1

σ1‖QT

2 (A1|A2)U − q ‖ using QT2 A2 = 0

≤ 1

σ1‖AU − q‖ using the result ‖QT

2 u‖ ≤ ‖u‖

≤ ϑmaxhnmax

σ1(n + 1)!

√√√√m∑

i=1

‖νi‖2n1 .

The first inequality holds by dividing both sides by ‖∇f(a)‖. The second inequality

holds from Proposition 4(ii) since σ1 ≤ σ1 ⇒ 1σ1

≤ 1σ1

.

All the above results also hold for the weighted least squares problem given

in §5.3.2 where the orthogonal decomposition is now performed on WA2 so that

QT WA2 =

A12

0

where A12 is upper triangular. In this case QT (WA1|WA2) =

A11 A12

A21 0

and we have the following proposition.


Proposition 6. Under the hypotheses of Proposition 3 and the extension of Propo-

sition 4 to the weighted least squares case for (i) A21 has full column rank and (ii)

σ1 ≤ σ1, where σ1 is the smallest singular value of A21, we have that

‖∇f(a) − E1A†Wq‖

‖∇f(a)‖ ≤ ϑmaxhnmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1

≤ ϑmaxhnmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 . (5.9)

Proof.

‖∇f(a) − E1A†Wq‖ = ‖∇f(a) − A†

21QT2 Wq‖

= ‖A†21(A21∇f(a) − QT

2 Wq)‖

≤ 1

σ1

‖QT2 (WA1∇f(a) − Wq)‖

=1

σ1

‖QT2 (WA1|WA2)U − Wq ‖

≤ 1

σ1‖W (A∇f(a) − q)‖

≤ ‖W‖σ1

‖A∇f(a) − q‖

≤ ϑmaxhnmaxwmax

σ1(n + 1)!‖∇f(a)‖

√√√√m∑

i=1

‖νi‖2n1 .

The first inequality holds by dividing both sides by ‖∇f(a)‖. The second inequality

holds since 1eσ1

≤ 1σ1

.

Remark 2. Proposition 2 appears at first to imply that the method has nth power

accuracy in h, however since the singular values of the least squares matrix A

depend on h this conclusion is false. The numerical results presented in reference

[13] do exhibit a quadratic rate of convergence and this is implied by Proposition 5.

This follows because the singular values of the reduced matrix A21 are independent

of h, being an orthogonal transformation of the first two columns of the least squares

matrix A, which themselves are independent of h.


5.5 Numerical Experiments

Two sets of numerical experiments are described in this section with the purpose

of examining the veracity and utility of the error bounds derived in §5.3-5.4. First

a set of results are described that confirm the error estimates and asymptotic

behaviour of the error as the test points move towards the data point in question.

Then a contrasting situation is described with scattered data points taken from

tests conducted by Franke [55]. The following three functions were chosen from

Franke’s set.

F1(x, y) =1.25 + cos(5.4y)

6(1 + (3x − 1)2);

F2(x, y) =e−

8116

((x−0.5)2+(y−0.5)2)

3;

F3(x, y) =

√64 − 81((x − 0.5)2 + (y − 0.5)2)

9− 0.5.

5.5.1 Asymptotic Results

A set of test points was generated by a pseudo-random selection of radial distances

and polar angles. The polar angles were fixed and the radial distances were varied

by scaling down in factors of 10.

Thirty points (ri cos θi, ri sin θi)T were generated so that ri and θi were

distributed uniformly with 1 ≤ ri ≤ 2 and 0 ≤ θi ≤ 2π. From these points the

14 points nearest the origin were chosen and used as displacements from the point

a = (.2, .1)T . The inner radius of the annulus was scaled from 0.25 down to 0.25e-

5. In addition to varying the distance between the data points and the evaluation

point, the weighting of the data points by inverse powers of their distances was also

investigated with exponents from −1 to −4. The Lipschitz constant ϑmax in the

error bounds can be estimated by an application of the mean value value theorem

to the mixed partial derivatives in the Taylor series remainder as

ϑmax =√

2 maxξ∈D

(∣∣∣∣∂n+1f(ξ)

∂xn−i+1∂yi

∣∣∣∣ , i = 0, . . . , n

),

In this work the maximum for each of the test functions was determined using the


software package Maple.

The results of our numerical experimentation are shown in tables 5.1 to 5.3

for varying radii and in tables 5.4 to 5.6 for the various weightings at the radius

2.5 × 10−3 for each of the test functions F1, F2 and F3. Tables 5.1 to 5.3 exhibit

the relative error in the gradient estimates at the test point a compared with

Bound1 from proposition 2 and Bound2 from proposition 5 for the classical least

squares estimates, while tables 5.4 to 5.6 exhibit the relative error compared to

Bound1 from proposition 3 and Bound2 from proposition 6 for the weighted case.

The results are also plotted on logarithmic scales in figure 5.5.1 for the quadratic

estimates and figure 5.2 for the cubic gradient estimates. The bounds given in

propositions 5 and 6 are clearly better than those in propositions 2 and 3 for all

cases considered.

Tables 5.1 to 5.3 and the corresponding plots show that as the data points move

closer to the evaluation point a the superiority of the gradient estimates becomes

more marked. We conjecture that this is due to the diminution of the columns of

the full matrix from column 3 and beyond as h is reduced. Note also that the slight

inconsistency of the bounds in tables 5.1 to 5.3 for the cubic gradient estimates at

radius 2.5 × 10−5 is thought to be due to roundoff error.

Two further observations that can be gleaned from the results for the three test

functions are that cubic gradient estimates are more accurate than the quadratic

estimates and that weighting appears to offer very little improvement in accuracy.

Moreover, as a result of the scaling by the condition number of the weight matrix,

the bounds are less spectacular for the weighted least squares gradient estimates. In

fact, one might speculate that the insight gained from the derived error bounds may

well indicate that weighted least squares, in this context of gradient estimation, is

not beneficial as a direct consequence of this scaling.

5.5.2 Scattered Data Results

To assess the accuracy of the error bounds for a scattered data set that is more

likely to arise in practice, we chose the data from Franke [55], which consists of

two subsets defined on the unit square [0, 1] × [0, 1]. The first subset contains


Table 5.1: A comparison of the relative error and the error bounds using second and third

order least squares gradient estimates for the function F1.

Second order Third orderRadius Rel. Error Bound1 Bound2 Rel. Error Bound1 Bound2

2.5e-1 4.8981e-01 4.9819e+01 2.6869e+00 1.4095e-01 1.6826e+03 1.8366e+002.5e-2 7.4944e-03 4.9494e+00 2.6869e-02 2.1657e-04 1.6728e+02 1.8366e-032.5e-3 7.5058e-05 4.9491e-01 2.6869e-04 2.2999e-07 1.6727e+02 1.8366e-062.5e-4 7.5038e-07 4.9491e-02 2.6869e-06 2.3235e-10 1.6727e+00 1.8366e-092.5e-5 7.5037e-09 4.9491e-03 2.6869e-08 6.8197e-12 1.6727e-01 1.8366e-12




2.5e-1 2.3760e-01 2.4069e+01 1.2981e+00 4.1537e-02 5.3435e+02 5.8325e-012.5e-2 2.6968e-03 2.3912e+00 1.2981e-02 4.5445e-05 5.3125e+01 5.8325e-042.5e-3 2.6960e-05 2.3910e-01 1.2981e-04 4.5946e-08 5.3120e+00 5.8325e-072.5e-4 2.6956e-07 2.3910e-02 1.2981e-06 4.5712e-11 5.3120e-01 5.8325e-102.5e-5 2.6977e-09 2.3910e-03 1.2981e-08 2.5271e-12 5.3120e-02 5.8325e-13




2.5e-1 7.1965e-02 2.0562e+01 1.1090e+00 8.5304e-02 1.1333e+03 1.2370e+002.5e-2 7.2914e-04 2.0428e+00 1.1090e+02 2.5560e-05 1.1266e+02 1.2370e-032.5e-3 7.3165e-06 2.0426e-01 1.1090e-04 2.3839e-08 1.1266e+01 1.2370e-062.5e-4 7.3197e-08 2.0426e-02 1.1090e-06 2.5055e-11 1.1266e+01 1.2370e-092.5e-5 7.3197e-10 2.0426e-03 1.1090e-08 9.8809e-12 1.1266e-01 1.2370e-12


Table 5.4: A comparison of the relative error and the error bounds using weighted second

and third order least squares gradient estimates for the function F1.

weight Second order Third orderexponent Rel. Error Bound1 Bound2 Rel. Error Bound1 Bound2

0 7.5058e-05 4.9491e-01 2.6869e-04 2.2999e-07 1.6727e+01 1.8366e-06-1 7.3037e-05 5.5877e-01 2.9682e-04 2.2310e-07 1.9309e+01 2.0289e-06-2 7.1068e-05 6.2744e-01 3.2504e-04 2.1729e-07 2.2420e+01 2.2218e-06-3 6.9239e-05 7.0060e-01 3.5272e-04 2.1260e-07 2.6169e+01 2.4110e-06-4 6.7616e-05 7.7780e-01 3.7894e-04 2.0907e-07 3.0684e+01 2.5903e-06










−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−10

−8

−6

−4

−2

0

2

log10(radius) test neighbourhood

log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd

exact errorboundtighter bound

0 0.5 1 1.5 2 2.5 3 3.5 4−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

negative exponent of weighted distance

log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


(a) (b)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−10

−8

−6

−4

−2

0

2


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd

errorboundtighter bound

0 0.5 1 1.5 2 2.5 3 3.5 4−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


(c) (d)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−10

−8

−6

−4

−2

0

2


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd

exact errorbooundtighter boound

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−5

−4

−3

−2

−1

0


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


(e) (f)

Figure 5.1: Second order least squares errors and error bounds. Varying radius on the left,

varying weightings on the right.


−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−12

−10

−8

−6

−4

−2

0

2

4


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


0 0.5 1 1.5 2 2.5 3 3.5 4−7

−6

−5

−4

−3

−2

−1

0

1

2


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd

exacr errorboundtighter bound

(a) (b)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−14

−12

−10

−8

−6

−4

−2

0

2

4


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd

exacr errorboundtighter bound

0 0.5 1 1.5 2 2.5 3 3.5 4−8

−7

−6

−5

−4

−3

−2

−1

0

1


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


(c) (d)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−12

−10

−8

−6

−4

−2

0

2

4


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


0 0.5 1 1.5 2 2.5 3 3.5 4−8

−7

−6

−5

−4

−3

−2

−1

0

1

2


log1

0 er

ror,

bou

nd a

nd th

e tig

hter

bou

nd


(e) (f)

Figure 5.2: Third order least squares errors and error bounds. Varying radius on the left,

varying weightings on the right.


100 data points distributed more or less uniformly over the unit square, while the

second subset contains 33 points with larger variations in the density of the data

points. We decided to use all of the 133 data points for our analysis and the point

of interest was chosen as a = (.2, .1)T . The Lipschitz constants required for the

error bounds were determined using the Maple software across the unit square. We

remark that the constant can become quite large for the third order case which,

as we will see, impacts the performance of the error bounds.

The results of the numerical experimentation are summarised in tables 5.7 to

5.12 and the data is plotted on logarithm scales in figures 5.3-5.5 as hmax, or

equivalently, the number of points in the least squares stencil increases. We have

compared, as this number of points is increased from 10 to 30 in steps of 5, the

relative error for the first, second and third order classical least squares gradient

estimates in tables 5.7,5.9, 5.11 with Bound1 from proposition 2 and Bound2 from

proposition 5. The relative error is compared to Bound1 from proposition 3 and

Bound2 from proposition 6 for the corresponding weighted cases in tables 5.8, 5.10,

5.12. Note that Bound2 is not recorded for the first order estimate because the

matrix A in this case has only two columns and the extrapolation method can not

be employed. Initial conclusions drawn from these results are that for function

F1 the quadratic estimate offers around the same accuracy as the linear estimate,

but there is improvement for the cubic estimates; however for functions F2 and F3

there is a steady improvement in accuracy as the order of the truncated Taylor

series moves from n = 1 through to n = 3.

It is again evident from the results that Bound1 increases as the order of the

estimate increases, which is primarily due to the increase in the size of the Lip-

schitz constant across the domain for the mixed partial derivatives of the func-

tions considered here. Furthermore, for the quadratic and cubic cases, Bound2

is again superior to Bound1, which is consistent with the findings presented in

§5.5.1. There is only slight improvement in the gradient estimations when the

weighted least squares method is employed, and the error bounds are higher than

the bounds observed for the classical least squares estimates as a result of the scal-

ing by the condition number of the weight matrix. Overall, it would appear that

using the classical least squares method with n = 3 (cubic accuracy) with around


m = 15 points in the least squares stencil provides the best gradient estimates for

the functions under consideration. Interestingly, it can be observed from the plots

that using more points than this for the cubic case can lead to an increase in the

relative error for functions F1 and F3. A plausible explanation for this might be

due to the local behaviour of these functions away from the test point.

Table 5.7: A comparison of the relative error and the error bounds using first, second and

third order least squares gradient estimates for the function F1.

No. of First order Second order Third orderpoints Rel. Error Bound1 Rel. Error Bound1 Bound2 Rel. Error Bound1 Bound2

10 0.16 0.31e1 0.18 0.045e3 0.029e2 0.008 0.285e4 0.037e215 0.35 0.35e1 0.21 0.056e3 0.033e2 0.018 0.149e4 0.038e220 0.43 0.44e1 0.22 0.085e3 0.068e2 0.025 0.199e4 0.092e225 0.60 0.51e1 0.22 0.102e3 0.103e2 0.155 0.227e4 0.117e225 0.69 0.55e1 0.23 0.110e3 0.131e2 0.186 0.238e4 0.150e230 0.75 0.57e1 0.23 0.105e3 0.149e2 0.201 0.237e4 0.182e2

Table 5.8: A comparison of the relative error and the error bounds using weighted first,

second and third order least squares gradient estimates for the function F1.






















0.2 0.25 0.3 0.35 0.4 0.45 0.5

100

hmax

Err

or b

ound

s

relative errorbound 1

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−1

100

101

102

hmax

Err

or b

ound

s


(a) (b)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−1

100

101

102

103

hmax

Err

or b

ound

s

relative errorbound 1bound 2

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−1

100

101

102

103

hmax

Err

or b

ound

s


(c) (d)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

104

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

104

105

hmax

Err

or b

ound

s


(e) (f)

Figure 5.3: Relative error (line style ) and error bounds (line style and ) for func-

tion F1 using (a) first order, (c) second order and (e) third order least squares estimates. The

corresponding weighted least squares estimates and their bounds are depicted in (b), (d) and (f).


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

102

hmax

Err

or b

ound

s


(a) (b)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

102

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

102

103

hmax

Err

or b

ound

s


(c) (d)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

102

103

104

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

104

hmax

Err

or b

ound

s


(e) (f)





0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

hmax

Err

or b

ound

s


(a) (b)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−2

10−1

100

101

102

103

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

hmax

Err

or b

ound

s


(c) (d)

0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

104

hmax

Err

or b

ound

s


0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

101

102

103

104

105

hmax

Err

or b

ound

s


(e) (f)





5.6 Conclusion

In this paper we have derived error bounds for the commonly used least squares

gradient estimation strategies that are based on truncated Taylor series. We have

used results from our previous research to improve and tighten these bounds. An

important component of these bounds is the ratio of hnmax (the maximum distance

from the point of interest to any neighbouring point in the least squares stencil

raised to the order of the method) to the smallest singular value σ1 of the least

squares matrix A. The bounds have been tested to assess the error estimates

and asymptotic behaviour of the error as the test points move towards a chosen

data point. Then the theory was analysed for a practical scattered data set taken

from the literature. The numerical experimentation highlights that the tighter

bounds given in propositions 5 and 6 are useful in gauging the accuracy of the least

squares gradient estimates in that they capture the main trends in the relative error

behaviour. They also indicate that using a weighted least squares approach offers

little improvement in accuracy over the classical least squares strategy. Finally, it

appears for the functions studied here that the cubic classical least squares gradient

approximation performs the best of all methods tested when the number of points

in the stencil is around 15.

Chapter 6

Water Droplet Movement on a Leaf Surface

6.1 Introduction

An important research component of agrichemical spray retention by plants is

to model and simulate droplet movement on the surface of a leaf. To this end,

we present a simple mathematical model for this process, report on experimental

results generated with a particular type of leaf (Frangipani leaf), and compare the

results from each of the two studies. A crucial aspect of our approach is to construct

the surface of the leaf using a recently developed surface fitting method [104,105]

based on a combination of the Clough-Tocher method with radial basis functions.

When a single water droplet impacts on a solid surface, it may bounce off or

perhaps spread out along that surface, depending on the nature and inclination of

the surface, the speed and size of the drop, and the properties of the liquid, includ-

ing the viscosity and surface tension. However, in reality there are more options for

the fate of the droplet, and indeed Rioboo et al. [120] report that their experiments

suggest the outcomes include deposition, prompt splash, corona splash, receding

break-up, partial rebound, and complete rebound. These are also described qual-

itatively in the review article by Yarin [138]. Further, spreading drops may be

characterised by instabilities leading to viscous fingering, as studied by Kim et

al. [77] and Thoroddsen and Sakakibara [131], for example. An important point

is that the detailed fluid mechanics of each of these outcomes is quite sophisti-

135

Water Droplet Movement on a Leaf Surface 136

cated, and requires high level mathematical modelling, including asymptotic and

stability analysis and careful computational simulations, as well as an expensive

experimental setup.

At present none of these ideas has been included in mathematical models for

droplet impaction and/or spreading on leaf surfaces. While these issues may be

addressed in further research, the purpose of the present study is to develop a

simplified model based in part on previous studies on droplet movement, in order

to provide, for the first time, a realistic simulation of droplet movement on leaf

surfaces. The gravity-driven model is effectively one-dimensional, with droplet

movement described as a polygonal path of curved arcs. A novel feature of this

approach is that a thin-film model is used to develop a stopping criterion for

droplet. Experimental verification of the droplet model shows that it captures

reality quite well and produces realistic droplet motion on the leaf surface. Most

importantly, it is observed that the simulated droplet motion follows the contours

of the surface and stops moving at times consistent with experimental observation

(see figure 6.9, for example).

While this research makes its contribution through simulation and visualisation

of the realistic movement of a water droplet flowing on a leaf surface, we do not

address certain, possibly important, phenomena such as the effect that the micro-

scopic detail of each different variety of leaf surface has on the droplet motion.

Further, we do not attempt to describe the time-dependent shape of each droplet

via the Navier-Stokes equations, and as such we do not model the actual droplet

motion in any realistic way from a fluid mechanics perspective. We remark, how-

ever, that our simplified model is able to produce quite realistic droplet motion

and is the most inclusive of any that have appeared to date.

In order to simulate water droplet movement on the leaf surface, the “virtual”

surface itself needs to be constructed using surface fitting methods. Loch [84]

uses two such approaches based on the finite element method to model the leaf

surface. In earlier work [104,105] we introduced a new surface fitting method based

on hybrid strategies that combine the Clough-Tocher method [29, 80] with radial

basis functions [63,121] for this purpose. This method is based on a large number

of three-dimensional data points captured from an actual leaf surface using a laser


scanner. To apply the hybrid method to the leaf data sets, preprocessing steps are

required, which include the determination of a reference plane for the data and

the subsequent triangulation for the leaf surface mesh [104,105]. In this paper, the

hybrid method is used to construct the surface of a Frangipani leaf for the purpose

of simulating water droplet movement on that surface.

The outline of the paper is as follows. In §6.2 a relevant literature review

of droplet simulation is presented. A brief description of the leaf surface model is

explained in §6.3. In §6.4 an overview of the droplet model is presented. Two forces

are assumed to affect the droplet movement on the leaf surface namely an internal

force, which consists of a friction and resistance component between the surface

and the droplet, and an external force due to gravity. The surface is divided into

a mesh of triangles [104,105] and the motion of the droplet is computed over each

triangle. The inclusion of a thin-film concept enables the motion of the droplet to

be stopped at a point where the height of the thin-film along the polygonal path

is less than some specified tolerance. As a result, we observe using our model that

if the leaf surface is horizontal, or close to horizontal, the droplet moves along the

leaf vein; on other occasions the droplet moves and then falls from the surface.

The model also shows that the droplet stops moving on the surface or it leaves the

surface depending on the model parameters. In §6.5, an experimental verification

of the water droplet model for a Frangipani leaf is presented. Finally, the work

is concluded in §6.6, where future work and other applications of our research are

discussed.

6.2 Relevant Literature and Experiments

Several researchers have studied the animation of water droplets since the 1980’s

[51, 52, 53, 109]. However, only a limited number of methods, during the 1990’s,

address the natural phenomenon of water droplets flowing on surfaces where, typ-

ically, meta-balls in a gravitational field were used [139] to model static droplet

shapes on flat surfaces. Tong et al. [132] modelled water flows using meta-balls by

proposing a volume-preserving approach. Lanfen [81] presented a physical model

for two, or more, large water droplets morphing on a plane. Kaneda et al. [71]


proposed a method for generating an animation of water droplets and streams on

a glass plate (divided into a small mesh composed of quadrilateral elements), such

as a windowpane or windshield. This model takes into account the dominant pa-

rameters of the dynamical system, which include gravity, interfacial tensions and

the collision of droplets. To every lattice point on the glass plate, an affinity for

water (0 ≤ ci,j ≤ 1) is assigned in advance. A sphere was used to model the droplet

on a plate.

The method in [71] is not able to simulate flow of a droplet on a curved surface.

Kaneda et al. [72] proposed an extended method for generating a realistic anima-

tion of water droplets as well as their streams on curved surfaces. The motion of

water droplets on the surface depends on the external forces due to gravity and

wind and an internal force due to resistance. The droplet flows on the surface and

some amount of water remains behind because of the wetting, and later the water

flow merges with the remaining water. Therefore a solution to the wetting phe-

nomenon and the problem with two droplets merging is also addressed. Kaneda

et al. [70] proposed a method for generating realistic animations of water droplets

that meander down a transparent surface based on the work presented in [71, 72].

This work is useful for applications such as drive simulators and animation of wa-

ter droplets on a windshield. The main difference between this work and previous

work is the modelling of obstacles that move against water droplets on a surface,

for example a windshield wiper. The droplet is represented by a single particle

system and modelled as a sphere. The contact angle between the droplet and the

surface is also taken into account.

Jonsson [68] proposed a physically plausible model using normals of the bump

map surface in the computation of water droplet flow based on the model presented

in [71]. Solid spheres are used to model the droplets, where each droplet is a particle

system. Jonsson assumed that the external force that affects the water droplet flow

is due to gravity, while the internal force is due to the resistance. The direction

of the internal force is opposite the direction of movement and is computed by

applying the Gram Schmidt orthogonalization algorithm [102] to orthogonalise the

external force against the unit length normal vector, which is retrieved at every

point from the bump map.


Fournier et al. [54] presented a model oriented towards an efficient and visually-

satisfying simulation of a droplet moving down a surface. The efficiency arises from

the separation between the shape and the motion of the droplet. The aim was to

simulate the shape and motion of large liquid droplets travelling down a surface

when it is affected by surface roughness, adhesion, gravity and friction forces. The

surface is defined by a mesh of triangles. A “neighbourhood” graph is built at the

beginning of the simulation so that each triangle is linked to adjacent triangles.

The neighbourhood graph is used to identify to which triangle the droplet moves

and during the simulation it is known exactly in which triangle a droplet is located.

A droplet might traverse several triangles between two time steps. The motion is

computed over each individual triangle to ensure the droplet is properly affected

by the deviations on the surface it has traversed. The gravity and friction forces

are assumed to be constant over a triangle for simplicity, and the friction force is

modelled as a linear viscous force with a constant negative factor due to surface

roughness. The shape of a droplet is characterized by a small set of properties,

for example, volume conservation and surface tension. A droplet will fall from

the surface if the component of the droplet acceleration force that is normal to

the surface is larger than the adhesion force of the droplet. The motion of the

droplet is generated by a particle system, with the droplet represented by a single

particle [117].

Computational fluid dynamics has been successfully applied to simulate realis-

tic animation of fluids. Chen [26] presented a disturbance model to simulate water

flow using the Navier-Stokes equations. Foster [51, 52] and Enright [36] used this

approach to develop liquid surfaces and to simulate complex liquid motion. Losas-

son [87] simulated water on a refined grid, such as an octree structure instead of a

regular grid to capture more surface details using the Navier-Stokes equations. In

the model presented in this paper we chose not to use this approach to calculate

the motion of the droplet because of the computation expense of this method,

which would require solution on each element in the leaf surface mesh.


6.3 Leaf surface model

As mentioned above, before any simulation of the water droplet movement on a

leaf surface can be simulated, it is necessary to construct a “virtual” leaf surface.

In previous work by the authors [104,105] we have introduced a new surface fitting

interpolation method that combines the Clough-Tocher method with radial basis

functions for this purpose. A set of representative data points sampled from a

Frangipani leaf using a laser scanner was used to reconstruct the surface of the

leaf. The surface fitting method was then applied to the laser scanned leaf data

points to reconstruct the surface. However, in order to apply this method to the

leaf data a preprocessing phase was required, which includes the determination of

a new reference plane for the data and the subsequent triangulation for the leaf

surface mesh.


The laser scanner returns the coordinates of points on the leaf surface. These

coordinates may not necessarily coincide with the xy-plane in the data point co-

ordinate system. To overcome this problem, we used a reference plane that is a

least squares fit to these data points and then the coordinate system was rotated

so that the reference plane becomes the xy-plane. These rotations can be achieved

by at first rotating the normal vector of the reference plane about the y-axis into

the yz-plane and then rotating about the x-axis into the xz-plane [104, 105].

6.3.2 Triangulation of the leaf surface

Our surface fitting method is an interpolation based finite element method and

consequently, a triangulation of the leaf surface needs to be constructed. The leaf

data points that represent the surface are numerous. As a consequence, a subset

of the data set is selected to reduce the computational expense for surface fitting,

which is then used to generate the triangulation. This triangulation is generated

using the EasyMesh mesh generator, which is software written in the C language

by Bojan [99]. EasyMesh generates two-dimensional Delaunay triangulations in


general domains. For more details see Oqielat et al. [104, 105]. An example of a

triangulated leaf surface is shown in figure 6.10(c,d).

6.4 Droplet model

The fundamental unit of the model is a triangular element. We now address the

issues of forces on the droplet, a mathematical description of the thin film and the

kinematics of the motion. The triangulation offers many advantages; for example,

the motion and the position of the droplet over each individual triangle are easy

to compute, and the determination of the location of the droplet on the surface at

any time instant is straightforward. Such simulations of droplet movement could

be computationally demanding if thousands of triangles have to be considered, so a

coarser mesh based on a smaller subset of data points is used that is representative

of the major surface features (see figure 6.10 (c-f)).

6.4.1 External and internal forces

We consider in our model that the external force fext that affects the droplet

movement is due to gravity Fg, which does not change over a triangle. The gravi-

tational force is resolved (projected) in the direction of movement (see figure 6.1)

as

dp = Fg −(Fg · N

)N, (6.1)

where N is the unit normal vector and N, dp/‖dp‖ is an orthonormal set of vec-

tors. The unit normal vector of the surface is found by letting S = (x, y, f(x, y))T

be the surface of interest, with tangent vectors Sx = (1, 0, fx)T and Sy = (0, 1, fy)

T

. The normal of the surface is then given by n = Sx × Sy = −fxi − fyj + k, and

the unit normal vector N = n/‖n‖ is

N =(−fx,−fy, 1)√

f 2x + f 2

y + 1. (6.2)

The internal forcef int consists of a resistance force Fr and a friction, or drag force,

Ff . We have adopted the same notation of vectors used by Kaneda et al. [72],


dp

N

fext

Figure 6.1: The direction of movement dp with normal N and gravity fext.

Fournier et al. [54] and Jonsson [68] along with the new vectors denoting the

gravitational force Fg, the triangle edge ℓ = (ℓx, ℓy, ℓz)T , and the droplet position

p = (px, py, pz)T . The resistance force originates from the interfacial tension that

exists between the water droplets and the leaf surface [72, 68], and its direction

is opposite to the direction of movement (dp). This force is modelled using the

degree of affinity as

Fr = −αdp,

where 0 ≤ α ≤ 1 is the affinity, which is set experimentally in advance and assumed

to be constant over each triangle. The degree of affinity depends on the interfacial

tension as it expresses the status of the surface, such as impurities and small

scratches [72]. The friction force Ff is modelled as a linear retarding force with a

constant negative factor kf due to surface roughness [54]:

Ff (t) = −kfv(t),

where kf is the friction coefficient and v(t) is the droplet velocity at time t. The

motion of the water droplet on the surface depends on the external force fext.

When this force exceeds a static critical force (internal force f int), the water

droplet starts to meander down the surface.


6.4.2 Thin-film flow down a slope

Although there is a large literature on modelling the spreading of droplets on

surfaces, a literature search of papers that describe the simulation of droplet motion

on leaf surfaces has found an absence of thin-film theoretic models to approximate

when to stop the droplet motion. The one-dimensional flow of a thin-film of viscous

fluid down a slope of angle α to the horizontal is governed by the following partial

differential equation [78, 98, 101]:

∂h

∂t+

(g sin α

ν

)h2∂h

∂x=

∂

∂x

1

3h3

(g cos α

ν

∂h

∂x− σ

ν

∂3h

∂x3

), (6.3)

where z = h(x, t) describes the film height, the x-axis points down the slope, and

t is time (see figure 6.2). The physical parameters that describe the fluid are the

kinematic viscosity ν and the surface tension σ. The constant g is the acceleration

due to gravity. Equation (6.3) is derived under the assumption that the film is

x

g

z

xN (t)

h(x, t)

α

Figure 6.2: Thin-film flow down a slope.

‘thin’ (a representative height of the fluid h is much less than a typical length L in

the x-direction) and the flow is slow (the Reynolds number Re = vL/ν ≪ 1, where

v is the velocity scale v = gh2/ν). Unless the surface of the leaf is horizontal (or

nearly horizontal), then away from the front of the film (the nose) equation (6.3)

can be approximated by


∂h

∂t+

(g sin α

ν

)h2 ∂h

∂x= 0. (6.4)

By applying the method of characteristics, the general solution of (6.4) is found to

be

h = f

(x − g sin α

νh2t

), (6.5)

where h(x, 0) = f(x) is a function describing the initial profile of the film [4]. Thus,

we have a travelling wave type solution with wave speed gh2 sin α/ν.

For an initial droplet profile with compact support as shown in figure 6.2, we

can denote the nose of the film by xN (t), so that at any time t the film is in contact

with the substrate in the region 0 < x < xN (t). The x-axis points down the line of

steepest descent, which is assumed to be slowly varying per unit distance. As time

evolves the profile near the nose of the droplet will steepen, so that surface tension

becomes important [66]. However, at intermediate to long times, the height of the

main part of the droplet is small, and in fact, h → 0 as t → ∞. Thus, from (6.5),

we have that [66]

h ∼(

ν

g sin α

)x

12

t12

, (6.6)

for large times (away from the nose), regardless of the initial profile f(x). By

coupling conservation of mass with (6.6) we can derive the location of the droplet

front [66] as

xN(t) =

(9Ag sin α

4ν

) 13

t13 , (6.7)

where A is the surface area of the thin-film given by

A =

∫ xN (t)

0

h(x, t)dx,

and sin α is computed as

sin α =Fg · dp

‖Fg‖‖dp‖,

where dp is the direction of movement given by (6.1).

As mentioned before, the leaf surface is represented by a mesh of triangles

across which the droplet moves. To implement the thin-film concept in our model,


we compute the height of the thin-film given in equation (6.6) over the known

(computed) droplet path on each triangle to determine the height of the thin-film

along its polygonal path. If this height is less than a set tolerance ǫ the droplet

movement is stopped, otherwise it will continue to move to the triangle edge. More

details on this algorithm are given in the next section.

6.4.3 Motion of a droplet over the leaf surface

We first develop a single droplet model. It offers many advantages in terms of

flexibility and generality; for instance, it will make the droplet movement straight-

forward to control and it will be easy to add more droplets to the animation at a

later stage.

Newton’s second law F = ma is used to determine the features of the motion,

so that the droplet is specified according to position p, acceleration a, velocity v

and mass m. The forces acting on the droplet movement (given in section 6.4.1)

are then taken into consideration to derive the model:

mdv

dt= mdp − kfv(t) − αdp, (6.8)

where αdp is the resistance force and kfv(t) is the frictional force due to air. One

way to estimate the parameter kf is to use Stokes’s law for a resistance for a

sphere moving through air of radius r = 0.001(m), which has the same volume as

the droplet used in our simulations, to give kfv ∼ O(10)−8(kg.m/s2). The mass

m of the droplet is assumed to be constant.

In our model, the droplet moves down the virtual leaf surface defined as a mesh

of triangles, which offers the benefit in that the equation of motion is simplified

for an individual triangle.

At the beginning of the simulation, a droplet rolls on the virtual leaf structure.

We specify the initial time t0, the initial velocity v0, initial position p0, the transit

time of the droplet, which is accumulated as the droplet moves from one triangle to

another and the time frame (total specified transit time). We determine the initial

triangle from which the droplet commences to move using the Matlab command

tsearch. Next, we determine the direction of movement using equation (6.1) and


then allow the droplet to move to the next triangle. The time taken for the

droplet to move to the next element is calculated and the accumulated transit

time is stored.

Suppose that the droplet enters the kth triangle at time t = tk (see figure 6.3).

The velocity and the position of the droplet are then computed respectively using

equations (6.9) and (6.10). Denote the droplet transit time for the kth triangle as

tf . The time interval [tk, tk+tf ] is discretised into Nt divisions using ∆t = tf/Nt for

the purposes of calculating the thin-film height and visualising the droplet motion.

As the droplet traces a path across the triangle it can be located on the leaf surface

at the time instant td = tk + i∆t, i = 1, · · · , Nt using p(td). Next, the location

t = tk

p0(tk), v0(tk)

t = tk + tf

p(tk + tf), v(tk + tf)dk

p

Nk

kth triangle

Figure 6.3: The droplet movement within the kth triangle.

of the droplet front xN (td) is computed using equation (6.7). The height of the

thin-film h is evaluated by substituting xN(td) into equation (6.6). Finally, the

droplet is moved to the next triangle provided the accumulated transit time is less

than the time frame and the height of the thin-film is above a specified tolerance,

here taken as h > ǫ = 10−5m.

The motion of the droplet is computed over each triangle and the equation for

velocity and position of the water droplet at any time t are derived from equation

(6.8) as follows:

v(t) = −m

kfdk

p +

(m

kfdk

p + v0(tk)

)exp(tkf/m), (6.9)


p(t) = p0(tk) −(

m

kf

dkp

)t +

m

kf

(m

kf

dkp + v0(tk)

)exp(tkf/m) − 1

, (6.10)

where v0(tk), p0(tk) and dkp are respectively the initial velocity, the initial position

and the direction of movement of the droplet at the time tk when it enters the kth

triangle, see figure 6.3, and we have defined dkp = (1 − α/m)dk

p. When the droplet

enters the kth triangle at time tk, we directly computed the transit time tf , the

exit time td = tk + tf and the exit position p(td) as well as the velocity v(td) at this

time, the transit time tf is found by intersecting the droplet path using equation

(6.10) with each triangle edge using a Newton algorithm. We now explain this

strategy in the following paragraphs.

Each triangle edge has three components (ℓx, ℓy, ℓz)T that are given in standard

parametric form by:

ℓx(τ) = aix + τ(ajx − aix),

ℓy(τ) = aiy + τ(ajy − aiy),

ℓz(τ) = aiz + τ(ajz − aiz),

where the parameter 0 ≤ τ ≤ 1, (aix, aiy, aiz)T and (ajx, ajy, ajz)

T represent the

coordinates of the two vertices for the triangle edge. The position vector p given

by equation (6.10) also has three components:

px(t) = p0x(tk) −(

m

kf

dkpx

)t +

m

kf

(m

kf

dkpx + vx(tk)

)exp(tkf/m) − 1

,

py(t) = p0y(tk) −(

m

kfdk

py

)t +

m

kf

(m

kfdk

py + vy(tk)

)exp(tkf/m) − 1

,

pz(t) = p0z(tk) −(

m

kfdk

pz

)t +

m

kf

(m

kfdk

pz + vz(tk)

)exp(tkf/m) − 1

,

where (p0x(tk), p0y(tk), p0z(tk))T , (vx(tk), vy(tk), vz(tk))

T and(dk

px, dkpy, d

kpz

)Tare, re-

spectively, the initial position, initial velocity and direction of droplet movement

for the kth triangle. We now determine the intersection point (if it exists) between

p(t) and each of the triangle edge vectors using Newton’s method. Define the three


coordinate functions:

f1(t, τ) = px(t) − ℓx(τ) = 0,

f2(t, τ) = py(t) − ℓy(τ) = 0,

f3(t, τ) = pz(t) − ℓz(τ) = 0,

as functions of the independent variables t and τ . Together, we then have a system

of three nonlinear equations that must be solved for t and τ . Here the Newton

method has been applied to the reduced system F (t, τ) = 0 where F = (f1, f2)T ;

f3 is used to validate the solution. At each iteration, tn and τn are updated

according to Newton’s method as tn+1 = tn + δt, τn+1 = τn + δτ , where (δt, δτ)T

= −J−1(tn, τn)F (tn, τn), and J(t, τ) is the Jacobian matrix of F . The iterations

are terminated once ‖F n+1‖2 ≤ τr‖F 0‖2 + τa. For all of the droplet simulations

performed here we have used the initial approximations t = 0.1 and τ = 0.5;

the relative tolerance was chosen as τr = 10−8 and the absolute tolerance was

τa = 10−7. These parameters provided convergence within eight iterations in most

cases. We systematically solve this nonlinear system for each triangle edge until

the intersection point is found. This intersection point must satisfy the physical

requirement that t > 0 and 0 ≤ τ ≤ 1. If this does not occur, or Newton’s method

fails to converge, we proceed to the next edge and repeat the iterative process.

Note that, the point with the minimum time t among all of the intersection points

is chosen as tf .

Once the intersection point is located via the converged solution (tf , τf), the

droplet path using equation (6.10) can be traced across the kth triangle by gradually

incrementing tk until td is reached. In order to proceed to the next triangle we move

slightly past the intersection point by allowing tk to be incremented to just beyond

td. Then, the Matlab command tsearch was used to identify the new triangle into

which the droplet moves. If no such triangle can be located, the droplet is deemed

to have left the leaf surface.

As mentioned before, the droplet has an initial velocity v0(tk) from equation

(6.9) when it moves along the leaf surface from one triangle to another. To ensure

the droplet is adhered to the surface we project this velocity onto the surface in


the direction of movement using v0 =(v0 · dp

)dp = v0 −

(v0 ·N

)N , where N is the

unit normal vector given in equation (6.2). Moreover, this initial speed is updated

when the droplet arrives at the next element.

The procedure for simulating the droplet flow on the surface is summarised in

the following algorithm:

Algorithm 1: Simulating the Flow of Droplet on a Leaf Surface

INPUT: Mesh of triangles (virtual leaf surface), initial position p0, initial velocity v0, initial

time t0, degree of affinity α, friction coefficient kf and gravity force Fg = −(0, 0, 9.8)T .

Step 1: Place the droplet at some specified point on the leaf surface.

Step 2: Initialize the transit time of the droplet, which is accumulated as the droplet moves

from one triangle to another.

Step 3: Determine the triangle in which the droplet is placed using the Matlab command tsearch.

Step 4: Determine the direction of movement dkp using equation (6.1) for the kth triangle.

Step 5: Compute the velocity equation (6.9) of the droplet and then the displacement equation

(6.10) of the droplet at the time tk.

Step 6: Calculate the transit time tf required for the droplet to move to the next triangle by

intersecting the displacement equation with each side of the triangle edges using Newton’s

method.

Step 7: Discretise the time interval [tk, tk+tf ] using ∆t = tf/Nt; td = tk+i∆t, i = 1, · · · , Nt.

Step 8: Find the rate of spreading of the thin-film xN (td) along the path in the kth triangle

using equation (6.7).

Step 9: Evaluate the height of the thin-film h by substituting xN (td) in equation (6.6).

Step 10: Move the droplet to the next triangle, provided the transit time is less than the time

frame and the height of the thin-film h > ǫ.

Step 11: Update the accumulative time, calculate the initial velocity and the new position of

the droplet at t = tk + tf using steps 6 and 7.

Step 12: Locate the triangle to which the droplet now moves.

Step 13: Repeat steps 4 through 12 for the duration of the animation or until the droplet falls

from the leaf surface.


6.5 Experimental procedure

To illustrate the power of this simple droplet modelling approach, and also to

validate the model, a series of water droplet experiments were performed on a

freshly cut Frangipani leaf. Initially, six artificial dots were marked on the leaf

surface (see figure 6.4(c)) so that they were clearly visible on all captured images.

These six points were used as reference points for the droplet movement on the leaf

surface. The sonic digitizer device shown in figure 6.4(d) was used to measure the

locations of these six points along with the series of leaf boundary points including

the end points of the vein depicted in figure 6.5(a,b). The sonic digitiser used here

was a model GP 12-XL, which is nowadays known as Freepoint 3D [84]. This device

was manufactured by the Science Accessories Division of the GTCO Corporation

(GTCO Calcomp); for further details see [84]. This device captures the x-, y- and

z-coordinates of each data point relative to a defined frame of reference in a data

file stored on the acquisition computer. Four additional data points on the string

attached to the clamp holding the leaf were also recorded. These points were used

to determine the direction of gravity with respect to the reference plane of the

leaf surface, because the string is assumed to be aligned with the direction of the

z-axis (refer figures 6.4(a) and 6.5(a)). A syringe was used to measure the droplet

mass and two different masses of 0.1 and 0.2 grams were used in our experiment.

A video camera recorded the path that the droplet traversed on the leaf surface

and the transit time of the droplet also was recorded.

Two different leaf orientations were chosen to simulate the droplet movement

shown in figures 6.4(a,b) and 6.5(a,b). The second orientation was chosen at a

steeper angle than the first. The droplet of mass 0.2 grams was used for the first

orientation while the droplet of mass 0.1 grams was used for the second orienta-

tion. In fact, the experiment showed that the droplet of mass of 0.1 grams moves

very slowly, and in some instances does not move on the leaf surface for the first

orientation because the leaf was positioned very close to horizontal. The same

sized droplet does, however, move on the surface of the second orientation. The

purpose of choosing two different orientations was to test if the droplet path would

change if the orientation was altered.


(a) (b)

(c) (d)

Figure 6.4: (a) exhibits the first orientation of the leaf, (b) shows the second orientation

of the leaf, (c) shows the six dots captured using the sonic digitizer and (d) depicts the sonic

digitizer device.


The laser scanner was used to capture the leaf surface points shown in figure

6.5(c) for the reconstruction of the virtual leaf surface. It was then necessary to

transform this more detailed leaf data point set with the points recorded by the

sonic digitizer shown in figure 6.5(a,b). This transformation process required that

the laser scanner data points shown in figure 6.5(c) be rotated to bring them in

line with the leaf position that we have in the experiment (again refer to figure

6.5(a,b)). We now outline the steps carried out to achieve this transformation,

where we now refer to the set of data points that were captured using the sonic

digitizer as data set 1 and the set of data points that were captured using the laser

scanner as data set 2.

Transformation Process

1. Determine the reference plane for each data set using the strategy outlined in

§6.3.1 (see figures 6.5(d,e)). Rotate the axes such that the z-axis is perpen-

dicular to the reference plane using the strategy outlined in §6.3.1 for both

data sets (see figure 6.5(f)).

2. Find the minimum z for each data set and subtract it from the z values such

that each set has zero as the minimum point, shown in figure 6.6(a).

3. The end points of both data sets (the leaf tail) represented by the circle

points shown in figure 6.6(a) were used to bring both data sets together. If

(xp, yp)T is the coordinate of one end of the vein of the leaf, change the origin

such that (0, 0)T becomes the end point of the leaf vein.

4. Project both sets of points onto the xy-plane as shown in figure 6.6(c). Mea-

sure the angle between the veins and then rotate the axis so that the veins

coincide.

5. Scale the x, y and z coordinates such that the two images coincide as exhib-

ited in figure 6.6(d).

These transformations are all reversible; they may be applied to the coordinates of

the vertical string so that the direction of the gravitational field can be expressed

in either set of reference plane coordinates.


(a) (b)

(c) (d)

(e) (f)

Figure 6.5: (a) shows the boundary points of the leaf, the string points and the six dots for

the first orientation; (b) shows the the second orientation of the data; (c) shows the leaf surface

points that were captured using the scanner; (d) depicts the leaf surface points after rotation

to the reference plane and its normal; (e) exhibits the sonic digitizer leaf boundary points after

rotation to the reference plane and its normal; (f) shows both data sets in the same reference

plane.


(a) (b)

(c) (d)

(e) (f)

Figure 6.6: (a) and (b) show the transformation of both data sets into the xy−plane; (c)

is the projection of both data sets into the xy−plane; (d) is the rotation of the data to become

coincident; (e) depicts the inverse rotation of both data sets into the original position that we

have in the experiment, where data set 1 is represented by circles while data set 2 is represented

by dots; (f) exhibits the final rotation of the first orientation data set and the string.


After the final representation of the leaf data set has been produced, we started

simulating the droplet movement on the virtual leaf surfaces. All of our simula-

tions were performed in Matlab version 7.4 on a 3 GHz pentium 4 processor. The

triangulations shown in figures 6.10(c) and (d) have been used for these simula-

tions. Our model, as we can see from figures 6.7-6.9(b,d,f), captured the motion

of the droplet on the surface quite well when compared to the motions that were

produced in the experiments shown in figures 6.7-6.9(a,c,e). We remark that the

viewing angle is slightly different between the experimental and simulation results.

However, this is the best viewpoint chosen from the perspective of the data visu-

alization software. Overall it appears that the simulation results exhibit close to

linear behaviour for the flow paths, except near the leaf vein. In the experiments,

however, it can be seen that the droplet paths were slightly curved. Figure 6.7

shows the experimental results compared with the droplet simulations for the first

leaf orientation. Observe in figure 6.7 (a-b) that when the droplet was initially

positioned on the lower side of the leaf that the droplet moved parallel to the leaf

vein because the external force due to gravity dominated the internal forces on the

droplet. However, in 6.7 (c-d) when the droplet is positioned on the high side of

the leaf it moves across the surface until reaching the leaf vein, at which stage it

continued to move along the vein due to its surface characteristics being conducive

to flow. The behaviour of the droplet in figure 6.7(c) is similar to that shown in

figure 6.7(a), however in this case the droplet was close to the leaf edge and, as

expected, eventually fell from the surface.

The situation is somewhat different for the second orientation exhibited in figure

6.8, which is positioned much steeper than the first orientation. In particular, we

focus on the behaviour of the droplet depicted in figures 6.8(a-b) and 6.8(c-d).

When the droplet is placed near the upper edge of the leaf, refer to figure 6.8(a-b)

we can see that the velocity of the droplet is large enough to enable it to pass

over the vein and continue across the surface until it reaches the lower edge of

the leaf, at which time it falls from the surface. However, in figure 6.8(c-d) the

droplet velocity is not large enough to enable it to immediately pass over the vein.

Instead, it meanders along the vein before the gravitational force pulls it to leave

the vein and continue moving towards the lower edge of the leaf.


(a) (b)

(c) (d)

(e) (f)

Figure 6.7: (a,c,e) show the droplet movement across the leaf surface from three different

starting positions for the fist orientation. (b,d,f) exhibit the corresponding droplet movement

generated by the model for the three different starting locations shown in (a,c,e).


(a) (b)

(c) (d)

(e) (f)

Figure 6.8: (a,c,e) show the droplet movement across the leaf surface from three different

starting positions for the second orientation. (b,d,f) exhibit the corresponding droplet movement

generated by the model for the three different starting locations shown in (a,c,e).


(a) (b)

(c) (d)

(e) (f)

Figure 6.9: The figures show a comparison of the thin-film model results against the exper-

imental data.


One notes from figure 6.7(c) that when the leaf orientation is close to horizontal

that the droplet, after reaching the leaf vein, continues to move along the vein. A

plausible explanation for this is that the leaf vein has properties different to the leaf

surface properties, for example, the surface tension and the viscosity along the leaf

vein are different to the leaf surface and this has an impact on the droplet velocity.

In order to capture this movement along the leaf vein, we have modified the velocity

in our model when the droplet reaches this vein to be a linear combination of the

droplet velocity v0 when it reached the vein, together with some imposed velocity

vn resolved as (see figure 6.7(d));

vvein = αv0 + βvn (6.11)

where 0 < α < 1 and β = 1/‖v0‖. Another approach to capture the movement

along the vein is achieved by decreasing the surface tension and the resistance of

the droplet along this vein.

The procedure we used for simulating the droplet flow along the leaf vein is

summarised in the following algorithm:

Algorithm 2: Simulating the flow of a droplet along the leaf vein

INPUT: Mesh of triangles (virtual leaf surface), initial position p0, initial velocity v0, initial

time t0, degree of affinity α, friction coefficient kf and gravity force Fg = −(0, 0, 9.8)T .

Step 1: Find the data points along the vein that coincide with the triangle vertices.

Step 2: Find the triangle that the droplet will move into.

Step 3: Determine if this triangle has any points in common with the vein points.

Step 4: If it has common points, update the velocity to be a linear combination of the droplet

velocity when it reaches the vein together with some imposed velocity vn resolved as shown

in equation (6.11). Otherwise, do not modify the velocity and continue.

The result of applying this algorithm to both leaf orientations can be seen in

figures 6.7(d) and 6.8(d). One observes from these figures that the droplet motion

is more realistic once it reaches and continues along the vein. Without applying

this algorithm, the droplet would cross the vein and continues moving until it

reaches the leaf boundary, which represents unrealistic droplet motion.


(a) (b)

(c) (d)

(e) (f)

Figure 6.10: (a) and (b) exhibit the six dots on the final transformed first and second

orientation data sets; (c) and (e) represent the triangulation and the refined triangulation re-

spectively of the first orientation data set; (d) and (f) represent the triangulation and the refined

triangulation respectively of the second orientation data set.


(a) (b)

(c) (d)

Figure 6.11: Each of these figures show two paths of the same droplet on the refined tri-

angulation. One represents the path on the unrefined triangulation, given in figures 6.10 (c,d),

while the other represents the path on the refined triangulation, given in figures 6.10 (e,f).


As mentioned above, the droplet starts to move down the inclined leaf sur-

face, and eventually stops at some stage. Figures 6.9(b,d,f) show comparisons

of the droplet movement of our model against the experiments shown in figures

6.9(a,c,e). By controlling the height of the thin-film in our model we obtained

similar movements to the those depicted in the experiments.

To test if the droplet movement is affected by the triangulation of the leaf

surface, we have refined the triangulation in both orientations by dividing each

triangle into three subtriangles that have their common vertex the centroid of the

divided triangle as shown in the figures 6.10(e,f). Figure 6.11 shows the path of

the droplet on the refined mesh compared to the original mesh. One can see from

this figure that the droplet paths using the unrefined and refined triangulations,

plotted on the refined triangulation, are indistinguishable. Clearly the droplet

motion appears unaffected by the mesh refinement, offering very little change in

the direction of movement. We conclude that the motion of the droplet on this

particular leaf appears unaffected by refining the triangulation and therefore the

coarser resolution can be used to produce acceptable results. This is an important

finding because using a refined grid is more computationally demanding.

As mentioned in §6.4.1, the droplet model contains some parameters such as

friction and the resistance coefficient that can be used for calibration. By changing

these parameters we can control the droplet movement, or simulate the motion of

a pesticide droplet, or nutrient droplet. These movements can be controlled also

by changing the height of the thin-film discussed in §6.4.2. These ideas will be

pursued in future research.

6.6 Conclusions and future research

The work presented in this paper describes a model for a water droplet moving

down a leaf surface. The flexibility of the model offers the user an understanding

of how a droplet moves on a leaf surface and how small changes in the dominating

factors produce different droplet motions. A new idea based on using thin-film

theory has been used to develop a stopping criterion for the droplet as it moves

on the surface. Overall the model produces a good representation of the droplet


behaviour.

The research described here provides a basis on which future studies can be

built. For example, the model may be extended to generate not only realistic

movements of a droplet on the leaf surface, but it can also be extended to produce

a more physically correct simulation by involving more of the dominating factors

and forces that affect the droplet movement. The differences in the nature of leaf

surfaces can be included in the model by studying the behaviour of the droplet

movement on different leaf surfaces. It can be also extended to study the paths of

many droplets of not only water, but also droplets of pesticide moving and colliding

on the surface. Knowledge of this path is important for many applications, such

as the simulation of a pesticide application to plant surfaces [60, 118]. In the

future the model may be used to determine the effectiveness of a treatment, and

then to develop certain pesticides that have the ability to protect leaves for longer

periods of time. Similar models may treat moisture precipitation and energy uptake

through photosynthesis enabled by ray tracing techniques.

Future work will also see the development of more realistic mathematical mod-

els for the spreading and sliding of liquid drops on inclined leaf surfaces.

Wetting effects, merging, spraying, evaporation and adhesion of the droplets

have not been implemented in this model. We can include these phenomena by

including some of the dominating parameters. Spraying and adhesion could be

also included based on the work presented in [49, 50, 140].

6.7 Acknowledgments

This paper was carried out thanks to funding from the School of Mathematical

Sciences. We thank Mark Barry and Mark Dwyer from the Queensland Univer-

sity of Technology HPC centre for many helpful discussions concerning the droplet

visualizations shown in figures 6.7(b,d,f), 6.8(b,d,f) and 6.9(b,d,f). We also ac-

knowledge Dr Jim Hanan from University of Queensland for allowing the use of

the equipment to perform the droplet experiments. Finally, we acknowledge the

insightful comments of the reviewers that have improved the presentation of the

paper.

Chapter 7

Summary and Discussion

The central aim for the research undertaken in this thesis was the development of

a model for simulating water droplet movement on a leaf surface and to compare

the model behavior with experimental observations. A series of five papers [14,

104, 105,106,135] has been presented, in this thesis, to explain systematically the

way in which this modelling work has been realised. The leaf model presented

here appears to provide an excellent representation of the leaf, which is essential in

the context of simulating movement of a water droplet on the surface. A droplet

model is verified and calibrated using experimental measurement of water droplets

traversing a freshly cut Frangipani leaf; the results are promising and appear to

capture reality quite well.

The research methodology that enabled the main objective of the research

to be achieved saw an in-depth investigation of important issues concerning the

virtual leaf. This included the appropriate selection of the surface fitting methods,

triangulation of the surface, and the estimation of the required gradients of the

CT triangular elements for the purpose of leaf surface reconstruction. The forces

that affect the droplet movement were identified to be gravity, friction and surface

resistance. An innovation of the model was the use of thin-film theory to develop

a stopping criterion for the droplet as it moves on the surface. The kinematics

of the motion were derived and finally, verification and calibration of the droplet

164

Summary and Discussion 165

model was made based on experimental measurement.

The objectives of this research programme were listed in chapter 1. In this

section, an examination of the objectives and a demonstration of how these were

achieved is described.

• Survey existing methods for surface fitting and propose new tech-

niques for modelling the leaf surface

In chapter 2, three different mathematical techniques for surface fitting that al-

lows the user to construct accurate leaf surface representations from scattered

data sets were examined. A new mathematical surface fitting technique based

on using a multiquadric radial basis function (RBF) to estimate the gradients of

the Clough-Tocher (CT) triangle together with a hybrid CT-RBF methodology,

has been successfully applied and compared with other interpolation methods and

shown to produce an accurate leaf surface representation. The new method allows

the user to construct the leaf surface from a set of scattered data points captured

from a laser scanner. The hybrid method is an interpolation finite element method

based on a triangulation of the domain, which enables a piecewise cubic surface

with a continuous gradient to be obtained if the function values and the gradients

are given at the vertices of the triangular elements, as well as at the midpoints of

the edges of the triangulation. In the application described here these gradients

need to be estimated and two methods were proposed for this purpose.

The first method uses a set of nearest neighbours to generate approximate di-

rectional derivatives using a truncated multivariable Taylor series expansion. This

procedure enables an overdetermined linear system to be constructed that can

be solved in the least squares sense to extract the required gradient approxima-

tion. Truncated first, second and third order Taylor expansions were used for the

gradient estimates.

The directional derivatives at the edge midpoints, which are the scalar prod-

uct of the gradient with the unit normal to the edge were determined by either

estimating the gradients at the edge midpoints, or by taking the average of the

gradients at the two vertices associated with the same edge. The latter method

implies that three less gradient estimates are required for each CT element, which


represents a considerable computational saving.

The second method to estimate the gradients was the hybrid CT-RBF method,

which is based on using the multiquadratic RBF to estimate the gradients at each

of the vertices and edge midpoints of the CT triangle. As mentioned before, the

hybrid method is based on a triangulation of the domain. The triangulation is

generated by selecting a subset of points from the complete leaf data set, and

these points form the vertices of the triangular mesh elements that are used for

the CT method. We considered two hybrid implementations to obtain the required

gradients of the CT triangle, which we referred to as local and global strategies.

In the global hybrid strategy, we used the triangulation points to construct

a global multiquadratic RBF interpolant. For the local hybrid strategy, only a

local subset of points from the complete data set is used to construct a local

multiquadratic RBF interpolant for each triangle. This local subset of points

represents the closest points to each of the vertices and edge midpoints for the CT

triangle. Due to the poor conditioning for a wide variety of width parameter values

of the linear system for either the global or local RBF interpolant, it was necessary

to apply the truncated singular value decomposition to solve the linear system.

The coefficients of the local and global RBF systems were then used to construct,

respectively, the local and global gradient estimates for all CT triangular elements

in the mesh.

In both strategies, the width parameter associated with the multiquadratic

RBF was estimated either, locally or globally based on the algorithm given by

Rippa [121]. This parameter has a large influence on the quality of the approxima-

tion of the RBF interpolant. The Rippa algorithm was based on minimising a cost

function that represents the error between the interpolating radial basis function

and the given function from which the data vector was sampled. In the global

strategy, the triangulation points are used to apply the algorithm to produce one

global value of the width parameter that is used for all CT elements in the mesh;

while in the local strategy the algorithm was applied to each local subset of points

that were used to construct the local RBF to determine a local estimate of the

width parameter that was then used for each CT element.

The accuracy of these surface fitting methods was demonstrated by applying


them to a data set and six test functions taken from Franke [55]. The quality of

the approximation of these methods was measured by computing the root mean

square error (RMS), which represents the error between the interpolation function

and the function from which the data vector was sampled. It was found that

the RMS error obtained for the truncated third order Taylor series method was

slightly better than that produced using a truncated second order Taylor series.

Furthermore, the second order Taylor series method offers little improvement in

RMS over first order Taylor series. In fact, there was only a slight improvement in

the gradient estimation using the third order approximation over the second order

approximation, as well as using a second order approximation over first order. A

plausible explanation for this finding is related to the numbers and selection of the

neighbouring data points. Moreover, it was found that estimating the gradients at

the vertices and midpoints of the triangles produces a slightly more accurate RMS

error than taking the average of the gradients at the edge midpoints.

For the hybrid method, the RMS produced when using the global and local hy-

brid strategies are similar and almost as good as the case where the exact gradient

of the test functions is used. Estimating the parameter c locally is more compu-

tationally costly than estimating c globally because each time the local RBF is

constructed a new value of c must be calculated. Moreover, we observed, from

profiling our codes in Matlab, that most of the computational time was spent in

solving either the least square problem or the RBF problem, via the pseudoinverse

or TSVD. Thus, the global hybrid CT-RBF method was the most computationally

competitive of all methods tested, followed by the local CT-RBF method strategy

and then the truncated Taylor series method.

In conclusion, a comparison of the different surface fitting methods for the

Franke [55] data highlights that the hybrid CT-RBF method produces a marginally

more accurate surface representation for the CT method than the Taylor series

approach. This result was carried to the next thesis objective where the suitability

of the hybrid surface fitting strategy was explored for a real leaf data set sampled

from Frangipani and Anthurium leaves.

In chapters 2 and 3, the hybrid method was applied to laser scanned Frangipani

and Anthurium leaves. In order to apply the method to the leaf data a prepro-


cessing phase was needed, which included the determination of a new reference

plane for the data and then the triangulation of the leaf surface. This reference

plane was constructed using a plane fitted to the leaf data in the least squares

sense, and then the coordinate system was rotated so that the reference plane be-

came the xy-plane. The triangulation of the leaf surface is generated using the

EasyMesh generator, which generates a 2D Delaunay and constrained Delaunay

triangulation for a general domain. The computational expense for surface fitting

is reduced by selecting only a subset of the entire leaf data to generate the triangu-

lation of the surface. This selection of a subset of the data also has the advantage

of avoiding undesirably shaped triangles. Three different meshes were selected by

refining the triangulation points to test the accuracy of the hybrid method. This

enables more accurate surface representations to be generated, which provided a

good assessment of the new hybrid method as the mesh was refined. As expected,

it was found that a more accurate surface representation was obtained when the

number of triangular elements increases. The other data points from the leaf data

set that remained after selecting the triangulation points, were used to measure

the quality of the approximation of the hybrid method (either the local and global

variant) using the RMS error and the maximum error associated with the surface

fit in relation to the maximum variation in z. It was observed that the local hy-

brid method produced more accurate RMS values and maximum errors than using

the global hybrid method for all three mesh refinements. Furthermore, the RMS

error and the maximum errors decrease when the number of triangular elements

increase, resulting in a more accurate surface representation, as expected, and this

provides an excellent validation for the hybrid methodology for obtaining the leaf

surface representation.

As mentioned before, a Taylor series expansion was implemented to estimate

the gradients of the CT triangle. Note that the gradient estimate obtained using

truncated third order Taylor series was similar to that produced by truncating

to second or first order. This motivated us to derive an error bound (for the case

n = 2) associated with these types of gradient estimate to assess the accuracy of the

CT method. This bound contains in its denominator the smallest nonzero singular

value of the constructed least squares matrix and in its nominator hi, which is the


distance from the point of interest and any of the cloud of neighbouring points

used for estimating the gradient. Numerical investigations have been made of the

theoretical bounds of the errors in the least squares gradient estimation from the

scattered data values. It was noted that the smallest singular value of the least

squares matrix produces the poor error bound estimates because it depends on hi,

a detailed description is given in [14]. Therefore, this bound was modified using

the smallest singular value of the reduced form of the least squares matrix given

in [13] rather than using the smallest singular value of the least squares matrix.

This modification produces a better error bound that led to the derivation of a

tighter bound on the gradient estimates. The tighter error bound was subsequently

generalised to order n least squares and weighted least squares gradient estimates,

refer to [135] for more details. These bounds were assessed by performing two

sets of numerical experiments. First a set of results are described that confirm

the error estimates and asymptotic behaviour of the error as the test points move

towards the data point in question. Then, a contrasting situation is described with

scattered data points taken from Franke [55]. The results show that the tighter

error bound is much better than the initial derived bound.

In summary, the leaf surface representation described here is suitable for models

that determine water droplet paths along a leaf surface before they falls from or

come to a standstill on the surface, which was our next objective.

• Determination of the path of a droplet on the surface of a virtual

leaf

The leaf surface model described in our first objective forms the basis for the

droplet model development. In Chapter 6, we proposed a new model for generating

a realistic movement of a water droplet traversing a virtual Frangipani leaf surface.

A new idea based on using thin-film theory was proposed to develop a stopping

criterion for the droplet as it moves on the surface. The complete droplet model

has been successfully implemented and shown to capture the droplet behavior quite

well.

The virtual leaf surface is divided into a mesh of triangles and the motion

of the droplet is computed over each triangle. This triangulation offered many


advantages; because the motion and the position of the droplet over each individual

triangle is easy to compute, and the determination of the location of the droplet

on the surface at any time instant is straightforward. Since the scattered data that

represents the leaf surface are numerous; the simulation of droplet movement over

the surface could be computationally demanding if thousands of triangles have to

be considered, so a coarser mesh based on a smaller subset of data points is used

that is representative of the major surface features.

To model the exact droplet dynamics is extremely challenging and as a conse-

quence some simplification is necessary. However, the motion of the droplet in our

model is assumed to be affected by two forces, namely an internal force and an

external force. The external force is due to gravity, which does not change over a

triangle and is resolved in the direction of droplet movement. The internal force

consists of a resistance force and a friction force. The resistance force originates

from the interfacial tension that exists between the water droplet and the leaf sur-

face and its direction is opposite to the direction of droplet movement, dp. The

friction force is modeled as a linear retarding force with a constant negative factor

due to surface roughness. The motion of the droplet on the surface depends on

these forces, when the external force exceeds the internal force the droplet starts

to meander down the leaf surface.

Newton’s second law is used to determine the features of the droplet motion.

The motion of the droplet is computed over each triangle and the equation for

velocity and position of the water droplet at any time t are derived. When the

droplet enters a triangle at time t, the exit position and the exit time as well as

the velocity at this time were computed, with the position and time found by

intersecting the droplet path with each triangle edge using a Newton algorithm.

The innovation of our model is the use of thin film theory to estimate when

the droplet movement should stop. The thin-film concept is implemented in our

droplet model by computing the height of the thin-film over the known polygonal

droplet path over each triangle. In order to compute the film height, the front of

the film XN(t) needs to be computed at time t. If the film height is less than a set

tolerance ǫ the droplet movement is stopped, otherwise it will continue to move

to the triangle edge. As a result, the model shows that the droplet stays on the


surface, or it leaves the surface depending on the chosen droplet parameters.

An experimental verification of the water droplet model for a Frangipani leaf

is presented. A number of water droplet experiments were performed on a real

Frangipani leaf. Initially, five reference points for the droplet movement on the

leaf were marked. The sonic digitizer device was used to measure the location of

these five dots as well as the leaf boundary points. This device returns the x, y

and z-coordinates of each data point. Four additional data points on a weighted

string attached to the metal clamp holding the leaf were measured to determine

the direction of gravity with respect to the reference plane of the leaf surface. The

laser scanner was used to capture the leaf surface points for the reconstruction of

the virtual leaf surface. However, the scanner could not record the leaf data points

while it was held on the clamp because both the reference plane and the clamp

were made of the metal. Thus, we scanned the leaf in a different plane and then

applied a rotation to the laser scanner data points to bring them in line with the

points recorded by the sonic digitizer (see Appendix B for full details). After that,

the corresponding points from the scanned points to the five dots were found to

use them in our model as starting points of the droplets.

To test if the path of the same droplet would change if the leaf orientation

was changed, two different leaf orientations were chosen to simulate the droplet

movement; one with a steeper angle than the other. As a result, the droplet path

was changed by altering the leaf orientation. Two different masses of 0.1 and 0.2

grams for the droplet were also used in the experiment. The droplet of mass 0.2

grams was used for the first orientation, while the droplet of mass 0.1 grams was

used for the second orientation. The experiment showed that the smaller droplet

moves slowly on the leaf surface in the second orientation because the leaf was

positioned close to horizontal. However, the same mass droplet does not move on

the surface of the first orientation.

Simulation of the droplet movement on the virtual leaf surface was commenced

once the final representation of the leaf data set had been produced. Our model

captured the motion of the droplet on the surface quite well when compared to the

motions that were observed in the experiments.

In our experiment, for the second orientation, which is positioned much steeper


than the first orientation, it was observed that when the droplet is placed near the

upper edge of the leaf we can see that the velocity of the droplet is large enough

to enable it to pass over the vein and continue across the surface until it reaches

the lower edge of the leaf, at which time it falls from the surface.

For the first leaf orientation, it was observed that when the droplet was initially

positioned on the lower side of the leaf the droplet moved parallel to the leaf vein

because the external force due to gravity dominated the internal forces on the

droplet. However, when the droplet is positioned on the high side of the leaf it

moves across the surface until reaching the leaf vein, along which it continues to

move. A reasonable explanation for this is that the leaf vein has properties different

to the leaf surface characteristics, for example, the surface tension and the viscosity

along the leaf vein are different to the leaf surface and this has an impact on the

droplet movement.

In order to produce and capture realistic movement along the leaf vein, the

velocity in the model was modified when the droplet reaches this vein. At this

point it was decided to use a linear combination of the droplet velocity when it

reaches the vein, together with some imposed velocity resolved in the direction of

the vein. One observes, from applying this modification to both leaf orientations,

that the droplet motion is more realistic once it reaches and continues to move

along the vein and this provides extra evidence of the capabilities of our model.

Without applying this correction it was found that the droplet would cross the

vein and continue to move until it reaches the leaf boundary, which represents

unrealistic droplet motion. As mentioned above, the droplet starts to move down

the inclined leaf surface and eventually stops at some stage. By estimating the

height of the thin-film in our model we obtained similar movements to the those

observed in the experiments.

The triangulation in both orientations was refined by dividing each triangle

into three subtriangles that have their common vertex the centroid of the divided

triangle, to test if the droplet movement is affected by the triangulation. The

droplet motion appears unaffected by the mesh refinement, offering very little

change in the direction of movement. This is an important finding, because using

a refined grid is more computationally demanding.


In conclusion, the model captured reality quite well. It was observed using the

model that the droplet moves along the leaf vein if the leaf surface is horizontal,

or close to horizontal; while on other occasions the droplet moves and then falls

from the surface, or it stays on the surface. By changing the model parameters

(such as the friction and the resistance coefficients) we can control the droplet

movement. The inclusion of the thin-film concept enables the droplet to meander

down an inclined leaf surface following its contours and possibly come to rest on

the surface. The flexibility of the model offers the user with an understanding of

how a droplet moves on a leaf surface and how small changes in the dominating

factors produce different droplet motions.

The inclusion of a thin-film concept enables the motion of the droplet to be

stopped at a point where the height of the thin-film along the polygonal path is

less than some specified tolerance. As a result, we observe using our model that

if the leaf surface is horizontal, or close to horizontal, the droplet moves along the

leaf vein; on other occasions the droplet moves and then falls from the surface.

7.1 Directions for Future Research

7.1.1 Droplet Modelling

In this thesis, a new model for generating a realistic movement of a water droplet

on a virtual leaf surface is developed. This model provides a basis on which future

studies can be built. For example, the droplet model may be extended to produce a

more physically correct simulation of the droplet movement by incorporating more

of the dominating factors and forces that affect the movement. The differences in

the nature of leaf surfaces can be included in the model by studying the behaviour

of the droplet movement on different leaf surfaces.

It may be possible to simulate the motion of a pesticide droplet on the leaf

surface by changing the model parameters (such as friction, surface tension and the

resistance coefficient). Knowledge of this path is important for many applications,

such as the simulation of a pesticide application to plant surfaces [118,60,33]. In the

future the model may help with the evaluation of different pesticide compounds and


can be used to determine the effectiveness of a treatment, and to develop certain

pesticides that have the ability to protect leaves for longer periods of time. Similar

models may treat moisture precipitation and energy uptake through photosynthesis

enabled by ray tracing techniques.

The model can be also extended to study the paths of many droplets of not

only water, but also droplets of pesticide moving and colliding on the surface.

Wetting effects, merging, spraying, evaporation and adhesion of the droplets

have not been implemented in this model. We could account for these phenomena

by including some of the dominating parameters into our model. Spraying and

adhesion could be also included based on the work presented in [33, 46, 47, 48, 49,

50, 140, 141].

7.1.2 Surface Fitting

As was mentioned in chapters 2 and 3, the CT method requires derivative estimates

at the vertices and midpoints of the elements for its evaluation and we proposed

a new hybrid approach for surface fitting that is based on using a multiquadric

radial basis function (RBF), given in equation 2.19, to estimate the gradient at

the vertices and midpoints of the Clough-Tocher triangle.

Another possibility for future research is to estimate the gradients for the CT

triangle by including a linear polynomial term p(x) to the RBF as follows

S(x) = p(x) +

N∑

i=1

aiR (‖x − xi‖) , x ∈ R2. (7.1)

In this case some conditions would be imposed on the “pure RBF” coefficients ai

to remove the extra degrees of freedom introduced by the polynomial part. As a

result of this procedure one might expect an improvement in the estimation of the

directional derivatives.

Furthermore, as was mentioned in chapters 2 and 3, a preprocessing step is

required to apply the hybrid method to the leaf data sets, which includes the

determination of a new reference plane for the leaf data. In this work a linear least

square plane to the leaf data was constructed. This process will perform well except


when the leaf is nearly vertical. In this case it may be possible to compute the

reference plane more robustly via orthogonal distance regression using the Eckart-

Young-Mirsky theorem, which enables the computation of the plane minimising

the sum of squares of orthogonal distances to a cloud of points in 3-D using the

SVD.

Appendices

Appendix A

This appendix was reproduced from pages 203 to 206 in Lancaster book [80].

Cardinal Basis Functions of the Clough-Tocher Interpolant

The cardinal basis functions bi(x, y), ci(x, y), di(x, y) and ej(x, y), i =

1, 2, 3 needed for the Clough-Tocher interpolant are listed below;note that for each subtriangle, a different set of basis functions is

required. The twelve pieces of independent information about thefunction(namely, the three data points and nine partial first deriva-

tives), each corresponding to a basis function.

b1(ξ, η) =1

3+

√3

2η +

−√

32 η3 in T1;

− 14 ξ3 − 3

√

38 ξ2η −

√

38 η3 in T2, b1(P1) = 1;

14ξ3 − 3

√

38 ξ2η −

√

38 η3 in T3

b2(ξ, η) =1

3− 3

4ξ −

√3

4η +

14ξ3 +

√

34 η3 in T1;

916ξ3 + 9

√

316 ξ2η + 9

16ξη2 +√

316 η3 in T2, b2(P2) = 1;

516ξ3 − 3

√

316 ξ2η + 9

16ξη2 +√

316 η3 in T3

b3(ξ, η) =1

3+

3

4ξ −

√3

4η +

−14 ξ3 +

√

34 η3 in T1;

−516 ξ3 − 3

√

316 ξ2η − 9

16ξη2 +√

316 η3 in T2, b3(P3) = 1;

−916 ξ3 + 9

√

316 ξ2η − 9

16ξη2 +√

316 η3 in T3

c1(ξ, η) =1

4ξ +

√3

2ξη +

34ξη2 in T1;

−516 ξ3 − 5

√

38 ξ2η − 3

16ξη2 in T2,∂c1

∂z1

(P1) = 1;

−516 ξ3 + 5

√

38 ξ2η − 3

16ξη2 in T3

176

Appendices 177

c2(ξ, η) = −1

8ξ +

√3

8η +

3

8ξ2 −

√3

4ξη − 3

8η2

+

− 18ξ3 +

√(3)

4 ξ2η − 3√

(3)

8 η3η in T1;

− 932ξ3 +

3√

(3)

32 ξ2η + 1532ξη2 + 3

√

332 η3 in T2,

∂c2

∂z2

(P2) = 1;

1132 ξ3 − 17

√(3)

32 ξ2η + 1532ξη2 + 3

√

332 η3 in T3

c3(ξ, η) = −1

8ξ −

√3

8η − 3

8ξ2 −

√3

4ξη +

3

8η2

+

− 18ξ3 −

√(3)

4 ξ2η +3√

(3)

8 η3η in T1;

1132 ξ3 +

17√

(3)

32 ξ2η + 1532ξη2 − 3

√

332 η3 in T2,

∂c3

∂z3

(P2) = 1;

− 932ξ3 − 3

√(3)

32 ξ2η + 1532ξη2 − 3

√

332 η3 in T3

d1(ξ, η) = −7√

3

81− 13

36η +

√3

18η2 +

1736η3 in T1;

41144η3 + 1

8√

3ξ3 + 3

16ξ2η in T2,∂d1

∂w1(P2) = 1;

41144η3 − 1

8√

3ξ3 + 3

16ξ2η in T3

d2(ξ, η) = −7√

3

81+

13√

3

72ξ +

13

72η +

3

24ξ2 +

1

12ξη +

√3

72η2

+

−√

38 ξ3 − 1

4 ξ2η −√

(3)

12 ξη2 − 1172η3 in T1;

− 17√

396 ξ3 − 17

32ξ2η − 17√

396 ξη2 − 17

288η3 in T2,∂d2

∂w2(P2) = 1;

− 13√

396 ξ3 − 5

32ξ2η − 17√

396 ξη2 − 17

288η3 in T3

d3(ξ, η) = −7√

3

81− 13

√3

72ξ +

13

72η +

√3

24ξ2 − 1

12ξη +

√3

72η2

+

3√

324 ξ3 − 1

4ξ2η +

√(3)

12 ξη2 − 1172η3 in T1;

13√

396 ξ3 − 5

32 ξ2η + 17√

396 ξη2 − 17

288η3 in T2,∂d3

∂w3

(P2) = 1;

17√

396 ξ3 − 17

32 ξ2η + 17√

396 ξη2 − 17

288η3 in T3

e1(ξ, η) = −4√

3

81−

√3

9ξ − 1

9η − 2

3ξη +

2√

3

9η2

+

59η3 −

√

33 ξη2 in T1;

− 736η3 + 5

√

312 ξ3 + 9

4ξ2η + 5√

312 ξη2 in T2,

∂e1

∂n4(P4) = 1;

− 736η3 +

√

312 ξ3 − 3

4ξ2η + 5√

312 ξη2 in T3

Appendices 178

e2(ξ, η) = −4√

3

81+

√3

9ξ − 1

9η +

2

3ξη +

2√

3

9η2

+

59η3 +

√

33 ξη2 in T1;

− 312η3 − 7

36η3 − 34ξ2η − 5

√

312 ξη2 in T2,

∂e2

∂n5

(P5) = 1;

− 5√

312 ξ3 − 7

36η3 + 94ξ2η − 5

√

312 ξη2 in T3

e3(ξ, η) = −4√

3

81+

2

9η +

√3

3ξ2 −

√3

9η2

+

ξ2η − + 139 η3 in T1;

−√

33 ξ3 − 1

2ξ2η + 118η3 in T2,

∂e3

∂n6(P6) = 1;

√

33 ξ3 − 1

2ξ2η + 118η3 in T3

Appendices 179

Appendix B

In this appendix, the steps which were carried out to achieve the

transformation that we performed, in the experimental verification

of the water droplet model, on the laser scanner data points to bring

them in line with the points recorded by the sonic digitizer are out-

lined. The set of data points that were captured using the sonic

digitizer are refereed to as data set 1 and the set of data points that

were captured using the laser scanner are refereed to as data set 2.

• Transformation Process

1. Determine the reference planes using the strategy outlined

in § for both data sets (refer figures 6.5(d,e)).

2. Rotate the two data sets using the rotation matrix that

rotates the unit normal vector of the respective reference

planes about the x-axis and then about the y-axis to align

its normal with the z-axis, so that each reference plane has

normal (0, 0, 1)T .

3. Rotate both data sets to have the same reference coordi-

nate system is exhibited in figure 6.5(f). Observe that the

data sets are clustered on different scales and in different

positions.

4. Bring the two data sets down to lie in the xy-plane by sub-

tracting (0, 0, m)T from data set 1, where m is the minimum

z-coordinate value of data set 1. The process puts the points

from data set 1 with the least value of z onto the xy-plane.

Appendices 180

This process is repeated for data set 2 where the results can

be seen in figure 6.6(a).

5. The end points of both data sets (leaf tail) represented by

the circle points shown in figure 6.6(a) were used to bring

both data sets together as follows: let u1 = (x1,y1, 0)T and

u2 = (x2,y2, 0)T where x1, y1 and x2, y2 represent the x-

and y- coordinates of the end points of data set 1 and data

set 2 respectively (figure 6.6(a)).

6. Subtract u1 from data set 1 and u2 from data set 2 so that

the end point of data set 1 and data set 2 coincide, becoming

respectively, (0, 0, z1)T and (0, 0, z2)

T , so now both data sets

have the same xy end points with different z-coordinates.

These end points can be seen in figure 6.6(b) as blue circles.

7. Project both sets of data points into the xy-plane by using

the first two coordinates (xy-coordinate) of the other end

points of data set 1 and data set 2 as follow: let r1 =

(x1,y1, 0)T and r2 = (x2,y2, 0)T where x1, y1 and x2, y2

represent the x- and y- coordinates of the other end points

of data set 1 and data set 2 respectively. Determine the

vectors v1 = r1 − u1 and v2 = r2 − u2 in the xy-plane and

then determine the angle between these two vectors, such

that the two data set when projected into the xy-plane are

as shown in figure 6.6(c).

8. Use the same angles determined in the previous step to ro-

tate data set 2 to data set 1 where in this case the two

Appendices 181

vectors v1 and v2 coincide as shown in figure6.6(d)).

9. Apply the inverse transformation of the previous steps for

data set 1 to data set 2, so we bring data set 2 to correspond

with data set 1 (see figure 6.6(e)). Now, to obtain the final

representation of data set 2, as shown in the experiment (see

figure 6.4(a)), the string needs to be parallel to the z−axis,

which represents the gravity vector, which can be done as

follows

10. Find the rotation matrix that rotates the string points to

align them with the z-axis and then apply this rotation ma-

trix to data set 1 to ensure that both data sets coincide with

the original orientation shown in figure 6.4(a,b), having the

same gravity vector (see figure 6.6(f)). In this case we ob-

tained the final position of the first orientation of the data

as well as the five points on this orientation is obtained, see

figure 6.10(a). To obtain the second orientation of the data

in figure 6.5 as well as the five points, the same strategy was

applied to the second orientation data set.

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