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STATISTICAL METHODS IN ANALYZING THE SHAPE OF

MAXILLARY DENTAL ARCHES FOR DENTAL

APPLICATIONS

NORLI ANIDA BINTI ABDULLAH

FACULTY OF SCIENCE

UNIVERSITY OF MALAYA

KUALA LUMPUR

2016

STATISTICAL METHODS IN ANALYZING THE SHAPE OF

MAXILLARY DENTAL ARCHES FOR DENTAL APPLICATIONS

NORLI ANIDA BINTI ABDULLAH

THESIS SUBMITTED IN FULFILLMENT

OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

FACULTY OF SCIENCE


KUALA LUMPUR

2016

ii


ORIGINAL LITERARY WORK DECLARATION

Name of Candidate: Norli Anida Binti Abdullah (I.C No: 851121-14-6336)

Registration/Matric No: SHB090010

Name of Degree: Doctor of Philosophy

Title of Project Paper/Research Report/Dissertation/Thesis (“this Work”):

STATISTICAL METHODS IN ANALYZING THE SHAPE OF MAXILLARY

DENTAL ARCHES FOR DENTAL APPLICATIONS

Field of Study:

Applied Statistics

I do solemnly and sincerely declare that:

(1) I am the sole author/writer of this Work;

(2) This Work is original;

(3) Any use of any work in which copyright exists was done by way of fair

dealing and for permitted purposes and any excerpt or extract from, or

reference to or reproduction of any copyright work has been disclosed

expressly and sufficiently and the title of the Work and its authorship have

been acknowledged in this Work;

(4) I do not have any actual knowledge nor do I ought reasonably to know that

the making of this work constitutes an infringement of any copyright work;

(5) I hereby assign all and every rights in the copyright to this Work to the

University of Malaya (“UM”), who henceforth shall be owner of the

copyright in this Work and that any reproduction or use in any form or by any

means whatsoever is prohibited without the written consent of UM having

been first had and obtained;

(6) I am fully aware that if in the course of making this Work I have infringed

any copyright whether intentionally or otherwise, I may be subject to legal

action or any other action as may be determined by UM.

Candidate’s Signature Date:

Subscribed and solemnly declared before,

Witness’s Signature Date:

Name:

Designation:

iii

ABSTRACT

This study aimed to propose shape features and statistical shape models to

develop a novel shape discrimination procedure for the maxillary dental arch with

important applications in dentistry. Standardized digital images of randomly selected

dental casts were obtained and the image calibration and registration were attended to

enable comparison of shape of the dental arches. A collective teeth positions from the

digital images were proposed as a novel shape feature of the dental arch. Each tooth

position is established from origin defined from stable anatomical landmarks. The mean

shape category obtained from clustering method and the probability distribution of each

shape category was further investigated to provide better statistical inference for the

shape models of the dental arch. A modified COVRATIO statistics which incorporates

the problem of small sample size and minimal model assumptions was then proposed as

a discrimination method of shape and compared to the linear discrimination method.

The proposed shape discrimination method was then used to determine suitable arch

shape and indicate natural teeth positions. The results from this study show that

multivariate normal and multivariate complex normal shape models together with the

use of the proposed discrimination method can be used to discriminate shape of the

dental arch. Consequently guides to determining suitable impression trays and

predicting teeth positions for the edentulous patients (patients with all teeth missing) are

provided. Verification of the proposed guides show that 91.42% of the sample studied

indicates appropriate fitting to the resultant impression trays, and the original teeth

positions (with an average error of 0.95 mm for each tooth position) were adequately

estimated by 80% of the arches studied. The presented statistical methods may be

beneficial in assisting inexperienced dentists and dental laboratory technicians to choose

the most appropriate impression tray and to determine natural teeth positions for the

Malaysian population.

iv

ABSTRAK

Kajian ini mencadangkan ciri-ciri serta model bentuk statistik untuk

menghasilkan prosedur diskriminasi yang baru bagi bentuk arkus pergigian pada rahang

atas dengan aplikasi penting dalam bidang pergigian. Imej digital yang seragam dari

acuan pergigian dipilih secara rawak dan penentukuran serta padanan imej dilakukan

untuk membolehkan perbandingan bentuk arkus pergigian. Kedudukan gigi secara

kolektif dari imej-imej digital dicadangkan sebagai ciri-ciri bentuk arkus pergigian yang

baru. Setiap kedudukan gigi ditentukan daripada penanda anatomi yang stabil. Nilai

purata setiap kategori bentuk dianggarkan daripada kaedah kelompok dan taburan

kebarangkalian setiap kategori bentuk seterusnya disiasat untuk memberikan inferens

statistik yang lebih baik untuk model bentuk arkus pergigian. Statistik COVRATIO

yang diubahsuai untuk disesuaikan dengan masalah saiz sampel yang kecil dan andaian

model yang minimum kemudiannya dicadangkan sebagai kaedah diskriminasi bentuk

serta dibandingkan dengan fungsi diskriminasi linear. Kaedah diskriminasi yang

dicadangkan ini kemudiannya digunakan untuk menentukan bentuk arkus dan

kedudukan gigi asal. Hasil daripada kajian ini menunjukkan bahawa model bentuk

multivariat normal dan multivariat kompleks normal dengan kombinasi kaedah

diskriminasi yang dicadangkan boleh digunakan untuk membezakan bentuk arkus

pergigian. Ini seterusnya membolehkan dua panduan dicadangkan dalam pemilihan

ceper impresi yang sesuai dan meramalkan kedudukan gigi untuk pesakit edentulus

(pesakit dengan ketiadaan gigi). Kajian pengesahan untuk panduan-panduan yang

dicadangkan menunjukkan bahawa 91.42% daripada sampel yang dikaji menunjukkan

pemilihan ceper impresi yang sesuai, dan 80% daripada arkus pergigian yang dikaji

boleh menganggarkan kedudukan gigi asal (dengan ralat purata 0.95 mm untuk setiap

posisi gigi). Kaedah statistik yang dicadangkan boleh dimanfaatkan dalam membantu

v

doktor gigi dan juruteknik makmal pergigian yang tidak berpengalaman untuk memilih

ceper impresi yang sesuai dan memudahkan anggaran kedudukan gigi asal untuk

penduduk Malaysia.

vi

To my honey, Hafiz

To my awesome buddies, Iqbal & Yati

vii

ACKNOWLEDGEMENTS

Praise to the Almighty, for giving me the strength to complete this work and

blessed me with loving people around me.

My greatest gratitude goes to my respected supervisors Assoc. Prof. Dr. Omar

Mohd Rijal and Assoc. Prof. Dr. Zakiah Mohd Isa for their guidance throughout my

study. Their supervision has taught me to appreciate knowledge by heart and never

giveup on your dreams.

Special thanks to Assoc. Prof. Dr. Yong Zulina, Dr. Ali Zaid, Dr. Mamun,

Professor Dr. Hanif, Professor Dr. Imon, Professor Dr. Rao and Professor Dr. Sahar for

their willingness to read through my thesis and suggest necessary improvements. To my

friends Iqbal, Yati, Omar, Adia, Adzhar, Rany, Dela, Siti, Zanariah, Faizol, and

Hafrizal, Kuna, Katz, Laili, Maz and Meksu, thank you for always being there, listening

and motivating me throughout these years – you guys are the best. Not forgetting the

staff at the Center of Foundation Study in Science, Institute of Mathematical Science

and Dean Office of Faculty of Science for their assistance right up to the completion of

this thesis.

My heartfelt appreciation goes to my beloved husband, for his continuous love

and the shoulder that I always look for to cry on. To mak, abah, mama, papa, my

siblings, and my favourite uncle Pak itam, thank you for your continuous prayers which

have driven me to complete this thesis. Not forgetting my two boys, Hamza and Aqeel,

that would never fail to make me smile everyday.

To my late sister, Maria, I miss you so much and may Allah place you in a

garden of paradise with the righteous ones.

viii

TABLE OF CONTENTS

ABSTRACT iii

ABSTRAK iv

ACKNOWLEDGEMENT vii

TABLE OF CONTENTS viii

LIST OF FIGURES xv

LIST OF TABLES xx

LIST OF SYMBOLS AND ABBREVIATIONS xxv

LIST OF APPENDICES xxvii

CHAPTER 1: INTRODUCTION 1

1.1 General Introduction: Shape Analysis 1

1.2 Relevance of Shape in Dentistry 2

1.3 Issues Related to Shape Analysis of the Dental Arch in Dentistry 3

1.3.1 Discriminating Shape of the Dental Arch 3

1.3.2 Shape Feature of the Dental Arch 3

1.3.3 Statistical Shape Model of the Dental Arch 4

1.3.4 Issues with Stock Impression Trays 5

1.3.5 Issues with Rehabilitating the Edentulous Patients 5

1.4 Objectives of the Study 6

1.5 Thesis Outline 6

ix

CHAPTER 2: LITERATURE REVIEW 9

2.1 Shape Analysis from Digital Images: Image Acquisition and Storing 9

2.2 Quantitative Description of the Dental Shape from 2D Images 12

2.2.1 Multivariate Morphometrics 12

2.2.2 Boundary Morphometrics 13

2.3 Discrete Fourier Transform (DFT) 18

2.3.1 Derivation of DFT 19

2.3.2 Application of DFT in 2D Shape Analysis 20

2.4 Shape Classification from Digital Images 22

2.4.1 Shape Alignment: Comparing two or more Arch Shapes 22

2.4.2 Arch Shape Classification 24

2.5 Statistical Shape Model of the Dental Arches 26

2.6 Discrimination of Shape 33

2.7 Multivariate Normal Distribution and its Tests 34

2.7.1 Mardia’s Multivariate Skewness and Kurtosis Tests 35

2.7.2 Doornik and Hansen Omnibus Test 36

2.7.3 Royston Test 38

2.7.4 Henze-Zirkler Test 40

2.8 Multivariate Complex Normal Distribution (MVCN) 41

2.8.1 The Univariate and Multivariate Complex Random Variables and

Distributions

42

2.8.2 Properties of the MVCN Distribution 43

2.8.3 Parameter Estimation and Hypothesis testing of the MVCN

Distribution

44

2.9 Missing Values Analysis 47

2.9.1 Data Augmentation (DA) Algorithm 47

x

2.9.1.1 Definition 47

2.9.1.2 Application of DA in the Missing Values Problem 49

2.9.2 Expectation maximization (EM) Algorithm 50

2.9.2.1 Definition 50

2.9.2.2 EM for Regular Exponential Families in Missing

Values Problem

51

CHAPTER 3: EXPLORATORY DATA ANALYSIS OF SHAPE

FEATURE AND INVESTIGATION OF CATEGORIES OF THE

DENTAL ARCH SHAPE

54

3.1 Introduction 54

3.2 Data Collection 54

3.2.1 Selection of Samples 55

3.2.2 Dental Impression and Stone Cast Making 55

3.2.3 Cast Preparation 56

3.2.4 Image Acquisition, Shape Alignment and Calibration of

Measurements

57

3.3 Shape Feature of the Dental Arch 59

3.4 Properties of the Shape Feature 62

3.4.1 Approximation of Circular Normal to Linear Normal

Distribution

64

3.4.2 Results on Approximation of Circular Variables of the Shape

Feature

66

3.5 Dental Arch Symmetry and Dimension Reduction 69

3.5.1 Test of Arch Symmetry 70

3.5.2 Results for Test of Symmetry 72

xi

3.6 Categories of Dental Arch Shape 74

3.6.1 Clustering Method 75

3.6.2 Definitive Number of Clusters 81

3.6.2.1 Principal Component Analysis 81

3.6.2.2 Dunn’s Validity Index 83

3.6.3 Results on Categories of Dental Arch Shape 84

3.7 Discussion 86

CHAPTER 4. SHAPE MODEL AND DISCRIMINATION OF THE

DENTAL ARCH: AS A GUIDE IN DETERMINING APPROPRIATE

IMPRESSION TRAY FOR ORAL DIAGNOSIS AND TREATMENT

PLANNING

89

4.1 Introduction 89

4.2 Shape Models of Maxillary Dental Arch 90

4.2.1 A Simulation Study on Performance of Multivariate Normality

Tests for Small Sample Size

90

4.2.1.1 Type I Error Rates 91

4.2.1.2 Power of Test 95

4.2.1.3 Summary of Simulation Results for Testing

Multivariate Normality

103

4.2.2 Multivariate Normal Distribution of Categories of Shape 104

4.2.3 Test of Separation between the Shape Models 109

4.2.3.1 Test for Equality of Covariance Matrices 109

4.2.3.2 Investigating Separation of Mean Shape using the

Hotelling T2 test

110

4.2.4 Verification of the Arch Shape Models 111

xii

4.3 Application of Shape Models to Impression Tray Design 113

4.3.1 Fabricating Three Impression Trays 113

4.3.2 Verification of the Fabricated Impression Trays 114

4.4 Discrimination of the Dental Arch Shape 117

4.4.1 Proposed )( iCOVRATIO as Discrimination Method 118

4.4.2 Simulation Study for Comparing Performance of LDF and

)( iCOVRATIO

120

4.4.3 Results for Discrimination of Dental Arch Shape 124

4.5 Investigating Dental Arch Shape with Missing Tooth: A Simulation

Study on Performance of Data Augmentation and Expectation

Maximization

125

4.6 Application of Shape Discrimination 128

4.6.1 A Proposed Guide for Determining Appropriate Impression Tray

for Patients (Without and with Missing Tooth)

128

4.6.2 Verification of the Proposed Guide for Determining Appropriate

Impression Trays

129

4.7 Discussion 130

CHAPTER 5. SHAPE MODEL AND DISCRIMINATION OF THE

DENTAL ARCH: AS A GUIDE IN ESTIMATING NATURAL TEETH

POSITIONS ON COMPLETE DENTURES FOR THE EDENTULOUS

132

5.1 Introduction 132

5.2 Shape model of the dental arch using Fourier Descriptor (FD) 133

5.2.1 The Ability of FD in representing Dental Arch Shape 133

5.2.2 Selecting Number of FD Terms as Shape Feature 136

5.2.3 Categories of Arch Shape using FD Shape Feature 139

xiii

5.2.4 Probability Distribution of Shape Categories using FD 141

5.2.5 Test of Separation between MVCN Shape Models 148

5.2.5.1 The Proposed Hotelling T2 Test for MVCN

Distribution

149

5.2.5.2 Simulation Study on Performance of the Hotelling T2

for MVCN Distribution

150

5.2.5.2.1 Type I Error Rates 151

5.2.5.2.2 Power of Test 153

5.2.5.3 Results on Test of Separation between MVCN Shape

Models of Dental Arch

155

5.3 Discrimination of Shape for MVCN Model 156

5.3.1 LDF for MVCN Distribution 157

5.3.2 Proposed MVCNiCOVRATIO )( for MVCN Distribution as

Discrimination Method

158

5.3.3 Simulation Study on Performance of LDF and

MVCNiCOVRATIO )(

159

5.3.4 Results for Shape Discrimination of Dental Arch using FD 164

5.4 Linking Anatomical Landmarks to the MVCN Shape Models of the

Dental Arch on Edentulous Arch

165

5.4.1 Shape model of Anatomical Landmarks 166

5.4.2 Verification of the Anatomical Landmarks Model 168

5.5 Application of Shape Discrimination for MVCN model 170

5.5.1 A Proposed Guide to Teeth Positioning on

Complete Dentures

170

5.5.2 Verification of the Proposed Teeth Positioning Guide for the

Edentulous

174

xiv

5.6 Discussion 177

CHAPTER 6: CONCLUDING REMARKS 178

6.1 Conclusion 178

6.2 Limitations of the study 180

6.3 Direction for Future Research 181

6.3.1 Exploring other Applications of Shape Model and

Shape Discrimination Procedure in Dental Problems

181

6.3.2 Extension to Mandibular and 3D Shape of the Dental Arches 181

6.3.3 Multivariate Complex Normal with Relation Matrix 182

6.3.4 Comparing Shape Discrimination using COVRATIO and

Bayesian Methods

182

6.3.5 Regression Ideas for Shape Models and Discrimination 183

REFERENCES 184

LIST OF PUBLICATIONS AND PAPERS PRESENTED 198

APPENDICES 199

xv

LIST OF FIGURES

Figure 2.1: Maxillary dental cast of a dentate subject 9

Figure 2.2: Each pixel in a digital image can be indicated by a Cartesian

coordinate with origin (1,1) at the left top of the picture.

10

Figure 2.3: Cartesian axes were established using teeth as landmarks. 11

Figure 2.4: Common linear measurements used to represent the dental arch. 13

Figure 2.5: Bezier curve defined by four control points of which has starting

point ),,( 00 yx directional points ),( 11 yx and ),( 22 yx and end point

),( 33 yx .

16

Figure 2.6: Two copies of the same shape of a hand. The picture on the right

was rotated roughly at 45 anti clockwise from the vertical axis.

24

Figure 3.1: Dental impression made in an impression tray. 55

Figure 3.2: Dental plaster was poured onto the impression to obtain the

stone dental cast.

56

Figure 3.3: Method used to make the base of the stone cast parallel to the

occlusal plane.

57

Figure 3.4: Two metal rulers positioned on a plane parallel to the occlusal

plane enable measurements to be calibrated. Shape alignment is

achieved by the creation of the Cartesian plane define from

anatomical landmarks which were marked with crosses.

58

Figure 3.5: Selected points on the teeth used to represent the dental arch

shape

59

Figure 3.6: Computation of angle and distance of the central incisor tooth

from the geometrical Cartesian origin made on the digital

images.

62

xvi

Figure 3.7: Differences of linear and circular measurement. 63

Figure 3.8: The arithmetic mean points the wrong way. 63

Figure 3.9: QQ circular normal plot for RRRR wwww 4321 and ,, . 68

Figure 3.10: Raw data plot for RRRR wwww 4321 and ,, . 68

Figure 3.11: Chi square plot for testing 16-variate normal distribution. 72

Figure 3.12: Dendrograms showing number of clusters at different cut-off

levels obtained from complete linkage method.

79

Figure 3.13: The pyramid of cluster boxes where each cluster is represented

by the vector of means.

80

Figure 3.14: Investigation of separation of clusters using the first two

principal components indicates the existence of three distinct

groups of dental arches in the sample studied.

82

Figure 3.15: Mean shapes of )3,2,1( kGk 85

Figure 3.16: Boxplot of each shape feature for )3,2,1( kGk 85

Figure 4.1: Examples of different sets of generated MVN data for p = 2. 93

Figure 4.2: Example of generated multivariate t distribution for df = 10 and

83

38Σ .

97

Figure 4.3: Example of generated )87,36(U data for p = 2. 97

Figure 4.4: Example of a multivariate lognormal distribution generated for

0μ and 2IΣ .

98

Figure 4.5: Empirical power for MS, MK, DH, HZ and Royston test statistic

against multivariate t distribution.

99


against multivariate t distribution.

100

xvii


against lognormal distribution.

102

Figure 4.8: Mean location of a particular tooth and its variations. 107

Figure 4.9: Mean shape and variation of 1G . 107



Figure 4.12: (a) An example of commercial impression tray with space

around the tray to allow for variation of arch sizes and adequate

thickness of alginate impression material. (b) Mean shape v

was indicated by solid line. The broken line indicates 5mm

added to the perimeter of the arch shape. (c) The light cure

acrylic resin tray.

114

Figure 4.13: Some of the mal-aligned arches in the 40 samples used for

verification of the three fabricated trays.

116

Figure 4.14: A small ruler was placed near the teeth (for image calibration)

while capturing the 2D image of the patient’s dental arch by

using an intraoral camera.

129

Figure 5.1: An example of a particular dental cast which shows the hamular

notches (HN) and incisive papilla (IP) that were used to establish

the Cartesian axes. Twenty one selected points as illustrated in

the diagram were used as the shape boundary.

134

Figure 5.2: The ability of 2, 3, 4, 5, 6, 19, 20 and all 21 FD terms in

representing the dental arch shape.

135

Figure 5.3: The magnitude plot for 2 casts as an example. The first 6 and the

last 2 terms shows higher contribution in representing the

boundary of the arch shape.

137

xviii

Figure 5.4: The Procrustes distances between the original 21 points and their

estimated positions using q-FD terms gradually levels off at q=8.

138

Figure 5.5: Plot of the original arch shape oix and the shape boundary q

ix

approximated using q = 8 FD terms.

139

Figure 5.6: Mean shapes of 21 ˆ,ˆ AA and 3A using 8 FD as shape feature. 141

Figure 5.7: Variation of the k-th point from its mean. 146

Figure 5.8: Mean shape and variation for shape model )1(A . 147



Figure 5.11: Examples of the generated complex normal data for 4p and

.4p

153

Figure 5.12: Examples of the generated complex normal data for different

values of 1θ and

2θ .

154

Figure 5.13: Examples of the generated complex normal data for 4p and

.4p

163

Figure 5.14: Edentulous cast showing the incisive papilla (IP) and hamular

notches (HN).

165

Figure 5.15: Brief outline of steps in the construction of complete dentures. 170

Figure 5.16: Proposed guide for construction of complete dentures. 172

Figure 5.17: Smallest Procrustes distance PD = 9.7520 mm between the

original and estimated teeth position. Red dots indicate the

estimated teeth position (sample N110).

175

Figure 5.18: An average Procrustes distance PD = 17.4664 mm, between the



176

xix

Figure 5.19: Largest Procrustes distance PD = 26.1236 mm, between the



176

xx

LIST OF TABLES

Table 2.1: Summary of literature for shape feature and shape model of the

dental arch.

29

Table 3.1: The sample correlation coefficient r, and the test statistic for

1;0 H to test the reliability of measurements taken at three

different times. The lower and upper critical values are -2.0141

and 2.0141 respectively when α = 0.05 was used.

60

Table 3.2: Mean, standard deviation (SD), minimum and maximum values

of the angular measurements (in degrees).

61

Table 3.3: Mean, standard deviation (SD), minimum and maximum values

of the length measurements (in mm).

61

Table 3.4: The test statistic for the Watson U2 and Kuiper’s tests and

estimated concentration parameter κ. All the variables show

non-significant results and indicate normality.

69

Table 3.5: Mean and variance of angular variables using circular and linear

statistic.

69

Table 3.6: Results of Kolmogorov-Smirnov (KS) goodness-of-fit test for

testing normality of residuals. Critical value of KS statistic used

is 0.1743.

73

Table 3.7: Test statistic for the Hotelling one-sample T2 test. 73

Table 3.8: Number of clusters formed using the available linkage methods

(M1 to M7) at selected percentage of similarity levels.

78

Table 3.9: Validation of number of clusters using Dunn’s index. 83

Table 3.10: Ethnic and gender homogeneity for each cluster. 84

Table 3.11: The anterior and posterior width and length for each of the shape

xxi

category. 86

Table 4.1: Empirical Type I error rate (in percentage) against the MVN

distribution with dimension p = 2 for different sets of

parameters.

94



parameters.

94



parameters.

95

Table 4.4: The proportion of observations satisfying

)())(())(( 21 pii kk vvSvv in each shape category .kG

Arbitrary values of 0.7 and 0.6 ,5.0 were used giving

3441.7)5.0(28 , 8.3505)6.0(2

8 and 9.5245)7.0(28 .

104

Table 4.5: The T values for Kolmogorov-Smirnov test with sample size

,111 n .14,22 32 nn

105

Table 4.6: Test statistic to investigate multivariate normality using Doornik

and Hansen (DH) test. Lower and upper 2.5% critical values are

6.9076 and 28.8453 respectively.

105

Table 4.7: Test statistic to investigate multivariate normality using Henze-

Zirkler (HZ) test.

105

Table 4.8: Test statistic to investigate multivariate normality using

Royston’s test.

106

Table 4.9: Hotelling 2T test for comparing two multivariate means. 111

Table 4.10: Doornik and Hansen (DH) test to investigate multivariate

xxii

normality of the shape categories using 122 casts. Lower and

upper 2.5% critical values are 6.9077 and 28.8454 respectively.

112

Table 4.11: Henze-Zirkler (HZ) test to investigate multivariate normality of

the shape categories using 122 casts.

112

Table 4.12: Royston’s test to investigate multivariate normality of the shape

categories using 122 casts.

112

Table 4.13: p-value of the Hotelling two sample 2T test for comparing 3

clusters using 47 casts and 122 casts.

113

Table 4.14: Amount of plasticine thickness in the impression tray and

number of cast which fit the fabricated trays C1, C2 and C3

when impression of 47 control samples was taken.

115

Table 4.15: Amount of plasticine thickness in the impression tray and

number of cast which fit the fabricated trays C1, C2 and C3

when impression of 40 test samples was taken.

116

Table 4.16: Misclassification probability for LDF and )( iCOVRATIO when

different sample size, dimension, mean vector and covariance

matrices were considered.

123

Table 4.17: Misclassification probability when 47 control casts were re-

assigned into either one of the population of the shape model

using the LDF and )( iCOVRATIO .

124

Table 4.18: The misclassification probability using DA and EM for missing

values imputation when the number of missing values 20 n .

127


values imputation when the number of missing values 40 n .

127


xxiii

values imputation when the number of missing values 60 n . 127

Table 4.21: Plasticine thickness in the impression tray and number of casts

(n=35) which fit the fabricated trays C1, C2 and C3.

130

Table 5.1: Mean, standard deviation (SD) and range of the FD terms 140

Table 5.2: Test statistic to investigate bivariate normality of each

ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c using HZ test in the 3 shape categories. LCV

and UCV are the abbreviation for lower and upper critical value

respectively.

142


ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c using DH test in the 3 shape categories. LCV

and UCV are the abbreviation for lower and upper critical value

respectively.

143


ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c using Royston’s test in the 3 shape categories.

The critical values were denoted in the bracket as (Lower 2.5% ,

Upper 2.5%).

143

Table 5.5: Empirical Type I error rate against equal means for different sets

of parameters.

152

Table 5.6: Empirical power of test against unequal means for different sets

of parameters.

155

Table 5.7: Hotelling 2T test for comparing two MVCN mean. 156

Table 5.8: Misclassification probability of LDF and MVCN

iCOVRATIO )( for

xxiv

MVCN model when different sample size, dimension, mean

vector and covariance matrices were considered.

161

Table 5.9: Misclassification probability when 47 control casts were re-

assigned into either one of the population of the shape model

using LDF and MVCNiCOVRATIO )( .

164

Table 5.10: The T2 test for comparing two MVCN mean for anatomical

landmarks.

166

Table 5.11: Mean, standard deviation (SD) and range of the FD terms for the

anatomical landmarks.

167

Table 5.12: Misclassification probability when 47 control casts was

discriminated using the anatomical landmarks ),,( 321 iiii LLLL

of the dental arch.

169

Table 5.13: Misclassification probabilities when 40 test casts were

discriminated using anatomical landmarks ),,( 321 iiii LLLL and

teeth location iii

i aaa 821ˆ , ,ˆ ,ˆˆ A .

169

Table 5.14: Group membership when the anatomical landmarks of the 35

test cast was assigned using MVCNiCOVRATIO )( .

175

Table 5.15: The number and percentage of casts according to the Procrustes

distance (PD) intervals indicating sum squared of difference

between the estimated and original teeth position.

175

xxv

LIST OF SYMBOLS AND ABBREVIATIONS

AHC Agglomerative hierarchical clustering

ANOVA Analysis of variance

c.d.f Cumulative distribution function

CI Confidence interval

DA Data augmentation

d.f Degrees of freedom

DFT Discrete Fourier Transform

DH Doornik and Hansen

EFF Elliptical Fourier function

EM Expectation maximization

FD Fourier descriptor

FS Fourier series

HN Hamular notches

HZ Henze-Zirkler

i.i.d Independent and identically

IP Incisive papilla

KS Kolmogorov-Smirnov

LDF` Linear discrimination function

m.l.e Maximum likelihood estimates

mm Millimeters

MS Mardia’s skewness

MK Mardia’s kurtosis

MVN Multivariate normal

MVCN Multivariate complex normal

xxvi

PC Principal component

PD Procrustes distance

SD Standard deviation

SW Shapiro-Wilk

xxvii

LIST OF APPENDICES

Page

Appendix A Ethics approval 198

Appendix B Examples of Research Data for

),,,,,,,,( 44332211

RRRRRRRRR lwlwlwlwv iii aaa 821ˆ , ,ˆ ,ˆˆ A and

),,( 321 LLLL

199

Appendix C Summary of works on dental arch symmetry 201

Appendix D MATLAB Program for Determine Appropriate Impression

Tray for Dentate Patients

205

Appendix E MATLAB program for Estimating Natural Teeth Positions

on Complete Dentures for the Edentulous

209

1

CHAPTER 1: INTRODUCTION

1.1 General Introduction: Shape Analysis

The word shape is commonly used to refer to the external form of an object.

Describing, comparing, classifying, recognizing and discriminating the shape of natural

and man-made objects are of interest in many disciplines. In physical anthropology,

forensic anthropology and archaeology, investigation of bodily appearance of living and

dead humans and animals was carried out to study the evolution of primates

(O’Higgins, 2000), identifying body shape or structure of an athlete according to a

particular type of sports (Maas, 1974), to differentiate skeletal of the apes (White, 1960)

and shape of the human head (Lacko et al., 2015). Other fields of research that involve

the study of shape are biology (classification of plant leaves according to their species,

discrimination of honey bee based on wing shape and species identification of fish)

(Bruno et al., 2008; Yu et al., 2014; Charistos et al., 2014), medicine (detecting

osteoporosis from dental radiographs, predicting 3D surface model of knee from 2D

image of knee joint for computer-aided knee surgery and discrimination of liver for

diagnosis of liver cirrhosis) (Allen et al., 2007; Tsai et al., 2015; Uetani et al., 2015),

security (fingerprints recognition) (Dass & Li, 2009) and computer vision analysis

(automatic trademark image search system to ensure uniqueness of each new registered

trademark) (Eakins et al., 2003).

Advances in technology particularly in image acquisition techniques have

facilitated the collection of data for shape information. The information was then used

to model the object of interest. A popular approach for shape modelling is using

statistical shape model, which finds the mean shape and shape variations of the training

2

data from the principal component analysis (Allen et al., 2007; Lacko et al., 2015; Tsai

et al., 2015). The shape models were usually incorporated into computers and used as a

model-based tool to consistently predict, compare or recognize shapes by iteratively

deforming the initial shape set from the shape model to fit the new shape feature (also

known as an active shape model).

Another approach for shape modelling is the Procrustes analysis, which analyzes

the distribution of a random sample of a set of shape feature that must be optimally

superimposed so that all shapes are comparable. This approach is usually used to

describe differences of shape or how shape is related to size (O’higgins and Dryden,

1993; Sampson, 1983; Dryden & Mardia, 1998).

The development of statistical shape model is essentially dependent on how the

shape feature was quantified. Quantitative description of an object is called

morphometrics. Basically, an object can be seen as a set of measurements (multivariate

morphometrics) or a boundary (boundary morphometrics). The multivariate

morphometrics though considered a traditional approach is, however, still commonly

used and useful for exploring the main characteristics of the shape and also for

discriminating shape (Lee et al., 2011; Scanavini et al., 2012; Uetani et al., 2015).

1.2 Relevance of Shape in Dentistry

The development of shape analysis in dentistry is not as extensive as in

biomedical engineering and other associated fields. However, works related on arch

shape analysis are increasingly sophisticated in the following applications:

In orthodontics, information about normal arches are used in designing and

manufacturing preformed arch wires to correct arches with irregularities in tooth

positions and in the maintenance phase of the corrected teeth in the new position

(Kairalla et al., 2014; Lee et al., 2011). In restorative dentistry, successful complete

3

dentures requires that the artificial teeth be placed where the natural teeth were, so that

natural function and aesthetics are restored.

Orthodontic and prosthetic treatments are completed on casts made from

impressions of the dental arches. The selection of stock trays to be used by dentists

when making impressions for the casts will be facilitated if the tray design and size are

as close as possible to the patient’s dental arch (Wiland, 1971; Yergin et al., 2001).

In forensic dentistry, bite mark patterns found either on the victim of an assault

or homicide, or inflicted by a deceased victim on the assailant may be compared with

the dental arch shape and dentition of either victim or assailant for the purpose of

identifying or exclusion of suspect or victim (Blackwell et al., 2007; Bush et al., 2011).

1.3 Issues Related to Shape Analysis of the Dental Arch in Dentistry

From the review of the literature, a few issues related to the analysis of dental arch

shape were identified as follows:

1.3.1 Discriminating Shape of the Dental Arch

This is important when choosing suitable available pre-formed arch wire and

stock impression tray for a particular patient (Yergin et al., 2001; Beale, 2007; Lee et

al., 2011), and discriminating an assailant’s identity through the bite marks from a few

suspects (Bush 2011; Sheets et al., 2013). To date, shape discrimination based on

statistical methods has not been used to address these applications relevant to dentistry.

1.3.2 Shape Feature of the Dental Arch

Collective shape features such as linear distances and angles between teeth,

ratios of two distances, perimeter, area and points on the teeth were typically used to

describe dental arch shape and size. The shape features were obtained from manipulated

pixels of digital image that correspond to an established Cartesian coordinate system.

4

However, some of these shape features together with the methods of analysis may be

inferior, as they only indicate size, but not the shape of the arch (Maurice & Kula, 1998;

Nojima et al., 2001; Šlaj et al., 2003; Tong et al., 2012). Moreover, as most of the

proposed Cartesian coordinate axes were established from teeth, the shape feature and

reconstruction of shape may not be possible if the teeth are lost (Kasai et al., 1995;

Triviño et al., 2008; Mikami et al., 2010; Lee et al., 2011). Hence, applications related

to the shape of the dental arch may be restrictive.

1.3.3 Statistical Shape Model of the Dental Arch

A statistical shape model should include a mean shape and variability (Hufnagel,

2001). Most studies on dental arches only determined the “average” shape of the arch.

Information on how the sample arches deviated from the established average arch shape

was not mentioned (Arai & Will, 2011; Ferrario et al., 1994; Lee et al., 2011; McKee &

Molnar, 1988; Raberin et al., 1993).

Only three studies considered the shape variation (Sampson, 1983; Wu et al.,

2012; Elhabian & Farag, 2014). These shape models were defined as a single ideal arch

shape, with relatively large variations from their respective mean shapes. Other studies

have however, showed that more than one category of arch shape exists when possible

cluster of arches were investigated or assigned to a specific arch form (Lee et al., 2011;

Nakatsuka et al., 2011; Triviño et al., 2008; Zia et al., 2009). Clinical experience shows

that a single impression tray would not fit all patients’ arches. The existence of a single-

ideal arch shape model in a particular population is therefore unrealistic and categories

of shape model should be proposed instead.

5

1.3.4 Issues with Stock Impression Trays

Stock impression trays are available in various sizes and shapes to accommodate

different mouths. In many instances the trays require modification as they do not

provide for variations found in patients’ mouths (Beale, 2007; Bomberg et al., 1985). It

was shown that some commercially available stock impression trays may only be

suitable for a particular patient population (Yergin et al., 2001).

There also appears to be no scientific basis for the design of stock impression trays

(Yergin et al., 2001). Specifications for the commercialization of stock impression trays

were made without any reference to the average dimensions and variability of the

human dental arch size and shape. Further, a guide using statistical methods to assign a

suitable tray for a particular patient has never been proposed.

1.3.5 Issues with Rehabilitating the Edentulous Patients

Treating the edentulous patient may be difficult and require years of clinical

observation and experience, especially when pre-extraction records (diagnostic casts or

photographs) are not available to be used as guides for indicating the positions of the

natural teeth (Bissasu, 1992; Faigenblum & Sharma, 2007). With the loss of teeth,

alveolar bone resorption follows and this further complicates the process of

redeveloping the natural arch shape and determining the position of the natural teeth.

Many studies have provided guides to the size of the artificial teeth, based on

observations of the size of the natural teeth and facial dimensions (Abdullah, 2002; Isa

et al., 2010). However, guides for reconstruction of arch shape and precise location of

each tooth in the edentulous arch remains a valid endeavour (Chu, 2007). A guide using

statistical methods has never been proposed to assist inexperienced dentists and dental

laboratory technicians in locating the natural teeth positions.

6

1.5 Objectives of the Study

The aim of this study was to develop a statistical shape model and subsequently

a discrimination procedure of shape for the maxillary dental arch with respect to:

a. Designing stock impression trays and selecting suitable impression trays for

patients.

b. Reconstructing of arch shape and predicting natural teeth positions for

edentulous patients.

To achieve these aims, the following objectives were set:

1. To propose novel shape features derived from stable anatomical landmarks

to enable reconstruction of the arch shape even when all teeth are lost.

2. To investigate the properties of the shape features of the dental arch,

particularly arch symmetry and shape categories as means of shape variation.

3. To develop statistical categories of shape model of dental arches by

investigating the probability distribution of each shape category and its

distinction.

4. To propose a shape discrimination procedure with minimal model

assumptions.

5. To propose guides in assisting inexperienced dentists and dental laboratory

technicians to choose the most appropriate stock tray for dentate or partially

edentulous patients and to estimate natural teeth positions for edentulous

patients.

1.6 Thesis Outline

This thesis is organized as follows:

Chapter 2 is a review of the literature on quantitative description, shape

classification and statistical shape model of the dental arch from 2D digital images.

7

Tests on multivariate normal distribution, introduction to multivariate complex normal

distribution and COVRATIO statistics are also reviewed to provide possible statistical

shape model of the dental arch and discrimination method for shape. An introduction to

the analysis of missing values is provided for a missing tooth, so that a complete shape

feature of the dental arch can still be obtained.

Chapter 3 reviews the properties of the shape feature, particularly categories of

dental arch shape. A novel shape feature v which indicate the teeth positions by a

collective of distances and angles from digital images was proposed and exploratory

data analysis of the shape feature was carried out. The linear property of the angular

variables used as shape feature was investigated. Further, a test of dental arch

symmetry reduced the dimensions of the shape feature. Exploration on categories of

dental arch shape was then carried out using the agglomerative hierarchical clustering

method. Investigation of the best linkage method to provide realistic clusters of arch

shape was carried out. These clusters represent shape categories and were then verified

using principal component analysis and the Dunn’s validity index.

A particular probability distribution of the dental arch as statistical shape model

should explain the mean shape and variation in each shape category. Chapter 4 proposes

statistical shape models of dental arches by investigating the probability distribution of

each shape category. A simulation study comparing tests of multivariate normal

distribution which performs best when the data mimics the shape feature and the sample

population was carried out. Three multivariate normal distributions define the shape

model of the dental arch. Test casts were used to validate the existence of the MVN

shape models. Application of the shape models was demonstrated in the design of three

impression trays. Subsequently, discrimination procedure for the shape of the maxillary

dental arch using the established statistical shape models was developed. A modified

COVRATIO statistics was proposed as a discrimination method of shape and compared

8

to the linear discrimination method. Using this knowledge, a guide for determining

appropriate impression tray for patients (without and with a missing tooth) was

proposed to facilitate the procedures of oral diagnosis and treatment planning.

Chapter 5 proposes a more precise shape feature that provides sufficient detailed

information about teeth positions on the dental arch using the Fourier descriptor (FD).

The ability of 8 FD terms in estimating all teeth positions was demonstrated. Using

these FDs, the 3 categories of shape established from v were verified and tested for

multivariate complex normality as its shape model. A hypothesis testing for two sample

means from MVCN based on the Hotelling 2T test was derived and employed to

confirm the existence of the 3 MVCN shape models. Then, a shape discrimination

procedure for the maxillary dental arch using the established statistical shape models for

MVCN was developed using the modified COVRATIO statistics. Further, 3 anatomical

landmarks which remain stable in the edentulous arch were linked to the 3 categories of

MVCN shape models. The application of the knowledge about the shape discrimination

for MVCN model together with the relation between the MVCN shape model and

anatomical landmarks was demonstrated in a guide for estimation of natural teeth

positions on complete dentures. Verification of the proposed guide for estimation of

natural teeth positions was carried out by comparing the original teeth positions of

dentate patients with the estimated teeth positions using the proposed guide.

Chapter 6 concludes the study. Limitations of the study and possible future

directions were also presented.

9

CHAPTER 2: LITERATURE REVIEW

2.1 Shape Analysis from Digital Images: Image Acquisition and Storing

The shape of the dental arch is usually represented by the curve composite

structure of the natural dentition. The arch curve is commonly indicated by a series of

landmarks on the arches in some biologically-meaningful way (Dryden & Mardia,

1998).

To analyze the dental arch of a particular patient, a replica of his oral structure

was cast in dental stone or plaster (Figure 2.1). An image of the dental cast was then

acquired and stored for analysis. Two common approaches may be used to acquire the

2-dimensional digital image of the dental cast. The dental cast was either:

a. Photographed using a digital camera (Henrikson et al., 2001; Lestrel et al.,

2004; Mikami et al., 2010) or

b. Scanned using a scanner (Lee et al., 2011; Taner et al., 2004; Wellens, 2007).

Rulers or calibration sheets were included when casts were being photographed or

scanned to ensure careful control of magnification.

Figure 2.1: Maxillary dental cast of a dentate subject.

10

Each digital image consisting of numerical presentations and images are stored

in an array of real or complex numbers and the elements of such a digital array are

called image elements, picture elements or pixels (Gonzalez & Woods, 2002). Each

pixel can be referred as a Cartesian coordinate. Establishment of the Cartesian axes and

subsequently its origin may be carried out as follows:

a. The origin of the Cartesian axes is usually taken as the upper left of the image,

as illustrated in Figure 2.2 (Henrikson et al., 2001; Lestrel et al., 2004; Taner et

al., 2004).

b. Landmarks on the dental arch may be used to establish the x and y Cartesian

axes and subsequently the origin (Kasai et al., 1995; Lee et al., 2011; Mikami et

al., 2010; Triviño et al., 2008). For instance, a line joining the cusp tip of the

molar teeth was regarded as the x axis and a perpendicular line to the x axis

which passes through the center of two central incisor teeth was regarded as the

y axis (Figure 2.3).

)3,3()2,3()1,3(

)3,2()2,2()1,2(

)3,1()2,1()1,1(

Figure 2.2: Each pixel in a digital image can be indicated by a Cartesian coordinate

with origin (1,1) at the left top of the picture. Source: www.ashleymills.com.

http://www.ashleymills.com/

11

Figure 2.3: Cartesian axes established using teeth as landmarks.

Source: Kasai et al. (1995).

The pixel coordinate of the selected landmarks representing the dental arch with

respect to the established Cartesian axes will be subsequently transformed to the actual

coordinate with the scale calibrated from rulers or calibration sheet. This procedure was

usually carried out using custom-made program in any available programming language

such as Delphi (CodeGear, Scotts Valley, Calif) and available digitised software such as

DigitizeIt 1.5.7 (I. Bormann, Bormisoft, Germany) (Lee et al., 2011; Wellens, 2007).

With the actual coordinate of the arch shape, shape feature such as linear

distances between landmarks and angles between two landmarks can be easily obtained

by manipulating the pixel coordinates (Henrikson et al., 2001). Other studies on the

other hand used the coordinates as the shape feature (Triviño et al., 2008; Sampson,

1983). The literature on quantitative description of the dental arch shape is reviewed in

the following section.

12

2.2 Quantitative Description of the Dental Arch Shape from 2D Images

The following subsections review multivariate and boundary morphometrics

approaches to dental arch size and shape.

2.2.1. Multivariate Morphometrics

Extraction of several features of the arch shape such as linear distances between

landmarks, angles between two landmarks, ratios of two linear distances, perimeter and

area of the dental arch measured from the coordinates of the landmarks were typically

used to describe dental arch shape and size. Among the distances used were tooth to

tooth distance, such as the intercanine distance and intermolar distance and the length of

the perpendicular line that passes through the midline of the central incisors (Burris &

Harris, 2000; Hao et al., 2000; Lee et al., 2011). Ratios of these linear measurements

were also used to provide a general idea of the arch shape (Ferrario et al., 1993; Nojima

et al., 2001; Raberin et al., 1993).

Besides distances, angular measurements such as the distolateral angle at the

intersection of lines formed by a cusp tip of a tooth and the midpalatal raphe, perimeter

and area of the arch were also collectively used to indicate arch shape (Figure 2.4)

(Burris & Harris, 2000; Cassidy et al., 1998; Nakatsuka et al., 2011; Scanavini et al.,

2012). These j-th measurements were then defined as a (1 x j) vector and analyzed

using multivariate analysis. This approach is known as the “multivariate

morphometrics”.

Several studies have used the above features collectively to describe shape;

however they were not analyzed simultaneously using a multivariate approach (Maurice

& Kula, 1998; Nojima et al., 2001; Šlaj et al., 2003; Tong et al., 2012). These studies

may therefore only indicate size, but not the shape of the arch. It is known that

distances, ratios, and angles obtained from landmarks may not be the best method to

13

represent arch shape as they only represent the general form of the arch and only

capture certain features of shape. However, this approach may utilize available

multivariate techniques such as cluster analysis, which has successfully unravelled the

problems of shape classification and is still commonly used in biological research

(Costa & Cesar, 2009; Dryden & Mardia, 1998).

Figure 2.4: Common linear measurements used to represent the dental arch.

Source: Nakatsuka, et al. (2011)

2.2.2 Boundary Morphometrics

Instead of extracting certain features about shape, the coordinates of the selected

landmarks were used to indicate the boundary of the shape. These boundary coordinates

were then fitted to the following mathematical curves:

i. Polynomial Function

The n-th order of a polynomial function is given as

nn xaxaxaaxf 2

210)( , (2.1)

where naaa ,,, 10 are constant coefficients and n is a non-negative integer

which indicates the order of the function. The 2-nd order polynomial

function was used to predict the shape of the anterior (frontal) arch in an

edentulous patient (Preti et al., 1986). The 3-rd, 4-th and 6-th order

polynomial functions on the other hand, were shown to follow the shape

14

of the whole arch (Shrestha, 2013; McKee & Molnar, 1988). The

coefficients may describe the characteristics of arch shape. For the 4-th

order polynomial, the odd numbered terms ( 31 and aa ) define the left-

right asymmetry of the arch and the even numbered terms measure the

taperedness ( 2a ) and squareness ( 4a ) (Burris & Harris, 2000; Lu, 1966).

Higher degree of polynomials were shown to provide more precise

representations of the arch whereby they may capture any off-aligned

teeth on the arch and give a wavy, rather than smooth polynomial

(Triviño et al., 2008).

ii. Conic Section

Conic section is a general function which comprises of a family

of the simplest plane curves which include circle, ellipse, parabola or

hyperbola given as

022 FEyDxCyBxyAx , (2.2)

where FBA ,,, are the constant coefficients.

The eccentricity, e, of a conic section (a measure of how far it

deviates from being circular) determines the types of conics, whereby if

e=0 corresponds to circle, e < 1 being ellipses, those with e=1 being the

parabolas and e > 1 being hyperbolas (Sampson, 1981). This measure

was used to quantify the shape of the conics, specific for each dental arch

(de la Cruz et al., 1995).

The limitation of this function in quantifying arch shape is that it

provides a symmetry curve, whereas naturally, the arch shape may be

asymmetrical (Henrikson et al., 2001).

15

iii. Cubic Spline

The cubic spline is a spline constructed from piecewise of the 3rd

order polynomials which passes through a series of points called knots. This

curve will insert a touch of individuality to each arch shape as the curve is

forced to pass through the knots, therefore generating different curve

configurations and does not necessarily produce a symmetric curve. This

curve appears to be an ideal means for representing dental arch form as it

may more adequately reflect the actual shape of the arch (BeGole, 1980;

BeGole & Lyew, 1998).

iv. Beta Functions

An empirical function derived from beta function based on two

parameters: arch length, D and molar width, W, was proposed to indicate

the shape of the arch, given as:

8.08.0

2

1

2

10314.3

W

x

W

xDy . (2.3)

Since the beta function is symmetric, therefore this function will

inherently produce a symmetrical curve, which may not always be the

case in all arch shapes (AlHarbi et al., 2008; Braun et al., 1998).

http://mathworld.wolfram.com/Spline.html

http://mathworld.wolfram.com/Polynomial.html

https://en.wikipedia.org/wiki/Symmetric_function

16

v. Bezier Cubic

A cubic Bezier curve is defined by four control points of which has

starting point ),( 00 yx , directional points ),( 11 yx and ),( 22 yx and end point

),( 33 yx . The Bezier curve is given as:

)}(),({)( tytxtR , (2.4)

where 001

2

120

3

2103 )()2()33()( xtxxtxxxtxxxxtx and

001

2

120

3

2103 )()2()33()( ytyytyyytyyyyty , are the

cubic equations defined in the interval 10 t . As increasing values for t

are supplied to the equations, the curve starts at ),( 00 yx going

towards ),( 11 yx and arrives at ),( 33 yx from the direction of ),( 22 yx (Figure

2.5) .

If the Bezier curve is fitted to a set of points representing the arch shape,

it will only pass through the first and last points. This is a shortcoming since

the other points will only provide directional information and do not lie on

the curve. Henceforth, the Bezier curve may only be an approximation of the

dental arch shape, but not an accurate shape.

Figure 2.5: Bezier curve defined by four control points of which has starting point

),,( 00 yx directional points ),( 11 yx and ),( 22 yx and end point ),( 33 yx .

Source: http://math.fullerton.edu

y

x

http://math.fullerton.edu/

17

vi. Fourier Series

A Fourier series (FS) represents a function as an infinite sum of sines and

cosines signals of various amplitudes and frequencies. The FS of periodic

continuous-time function with frequency 0 and period 0

0

2

T is given by

tuctuba

tf u

u

u 0

1

00 sincos

2)(

, (2.5)

where the Fourier coefficients are:

0

)(2

00

Tdttf

Ta , (2.6)

0

00

cos)(2

Tu dttutf

Tb , (2.7)

0

00

sin)(2

Tu dttutf

Tc , (2.8)

for u =1, 2, …. (Oppenheim et al., 1983).

The FS may be decomposed into separate components or harmonics. If

the limit of (2.5) is changed, it can be simplified to the finite form

N

u

u

N

u

u tuctuba

tf

1

0

1

00 sincos

2)( , (2.9)

where N is the maximum harmonic number. It is clear that the term 0a

corresponds to the average value of the original function along the period

],[ , and ua and ub are coefficients of the sine and cosine functions

respectively (Costa & Cesar, 2009). Therefore, the amplitude for the u-th

harmonic (u=1,…,n) may be obtained as

22uuu cbA , (2.10)

which can be considered as a measure of the influence (or weight) that each

harmonic has on the curve. The first 4 Fourier harmonics were shown to well

reproduce the maxillary dental arch (Kasai et al., 1995; Mikami et al., 2010).

18

Another study used all 14 Fourier harmonics and concluded that FS

precisely express the form and size of dental arches, compared to using the

fourth-grade polynomial function (Valenzuela et al., 2002).

vii. Elliptical Fourier Function

The elliptical Fourier function (EFF) was developed from the

conventional FS which defines x and y points on a curve in 2 dimensions as

separate functions of a third variable t as follows:

N

n

n

N

n

n ntcntbAtx

11

0 sincos)( , (2.11)

and

N

n

n

N

n

n ntentdBty

11

0 sincos)( , (2.12)

respectively, where the estimated coefficients are given in algebraic form,

instead of integral (Kuhl & Giardina, 1982).

The EFF was shown to fit the dental arch (Lestrel et al., 2004; Lestrel,

2008). An increase in the number of harmonic gives better representation of the

dental arch shape. With 24 harmonics, a small mean residual of 0.10 mm was

obtained between observed data and the fitted data using EFF.

Although representation of the dental arch using FS is also favourably

satisfactory, the EFF was claimed easier to imply, in the sense that its

coefficients are generated algebraically, instead of the integral solutions. This

makes computations simpler and faster.

2.3 Discrete Fourier Transform (DFT)

The DFT is another approach which is based on the FS and is increasingly

important in the study of shape. It has been widely used for vision and pattern

19

recognition applications but was never been applied for representing and modelling the

human dental arch.

The following section briefly explains the fundamental theory on DFT and its

application in 2D shape analysis.

2.3.1 Derivation of DFT

Using the Euler formula ,sincosexp ji the FS for periodic continuous-

time signal in (2.5) can be expressed in complex form of

u

u tjuatf 0exp)( , (2.13)

where

0

00

exp)(1

Tu dttjutf

Ta . (2.14)

are the coefficients of the FS for periodic continuous-time function (Kreyszig, 2007;

Oppenheim et al., 1983).

In the case when function f(t) is a discrete-time function, the FS for periodic

continuous-time function in (2.13) and (2.14) can be extended to discrete-time function

(Oppenheim et al., 1983). The FS for periodic discrete-time function s(k) at period

0

2

N is defined as follows

, 2

exp

exp)( 0

Nu

u

Nu

u

N

kjua

kjuaks

(2.15)

where

Nk

uN

kjuks

Na

2exp)(

1. (2.16)

Consequently, a one period (finite-duration) signal can be constructed from the periodic

signal in (2.15) by letting

20

.10 interval theoutside ,0)( N-nks (2.17)

The coefficients of FS at the interval of one period is given by

1,...,1,0for 2

exp)(1

1

0

NuN

kjuks

Na

N

k

u

. (2.18)

The set of complex coefficients ua defined in (2.18) comprise the DFT of )(ks

(Oppenheim et al., 1983).

Using (2.15), the original finite-duration signal can be recovered from its DFT

by

2

exp )(

1

0

N

u

uN

kjuaks

, (2.19)

1,...,1,0for Nk .

2.3.2 Application of DFT in 2D Shape Analysis

The basic underlying idea of the DFT in 2D shape analysis is by letting the 2D

data points of a boundary be transformed as a 1 dimensional data of

)()()( kjykxks , (2.20)

for k = 0, 1,…, N-1 and )]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx are the anti-clockwise

sequence of N coordinate points on the xy plane.

The complex coefficients ua in (2.19) are also known as the Fourier descriptors

(FD) of a boundary in shape analysis. Let )(')(' ujyux denotes the complex form of .ua

Rewrite (2.18) as

21

.2

sin)(2

cos)()(')('

,2

sin2

cos)()(')('

, 2

exp)(1

1

0

1

0

1

0

1

0

N

k

N

k

N

k

N

k

u

N

kuksj

N

kuksujyux

N

kuj

N

kuksujyux

N

kjuks

Na

(2.21)

From (2.21), the real and imaginary parts of ua are

, )('2

sin)()('

and )('2

cos)()('

1

0

1

0

uyN

kuksuy

uxN

kuksux

N

k

N

k

(2.22)

respectively. The physical interpretation of ua can be explained by writing (2.19) as

a

b

barrbaN

uka

N

ukjuy

N

ukux

N

ukjj

N

uka

N

ukjaks

N

u

au

N

u

N

u

u

N

u

u

u

1

221

0

1

0

1

0

1

0

tan and

where,cossincos ,)2

(cos

,)2

sin()(')2

cos()('

,)2

sin()2

cos(

,)2

exp( )(

(2.23)

where

22 )(')(' uyuxau , (2.24)

and

)('

)('tan 1

ux

uyua encode the amplitude and phase of )(ks respectively (Costa &

Cesar, 2009).

Since the amplitude ua (also known as the FD or DFT of )(ks ) also expresses the

size implicating the synthesized signal, we can evaluate how ua affects the boundary.

The ua may be regarded as quantification of contribution or weight in representing the

boundary of the arch shape (Mikami et al., 2010). Therefore, any u-th term may be

selected and used as an approximation of the boundary as follows

22

terms}{

2exp)()(ˆ

thuUN

ukjuaks

. (2.25)

The larger the total number of FD terms becomes, the closer the shape of the

boundary will be to the original one. One of the advantages of using FD is that it is a

reversible linear transformation which retains all the information in the original

boundary of the object (Keyes & Winstanley, 1999). This important feature is desirable

when precise shape and its boundary point are required.

Computation of DFT was usually done using fast Fourier transform, which uses

the Butterfly algorithm (Cooley & Tukey, 1965) for removing redundancies in the DFT

calculation, therefore allowing faster computation time (Costa & Cesar, 2009).

2.4 Shape Classification from Digital Images

Many applications involving 2D shape analysis require comparing, matching

and classifying the quantified shapes. The first step in achieving these tasks requires

alignment or registration between two or more associated shapes from the images, so

that post variation may be eliminated and the shapes are comparable, therefore yielding

meaningful results. This section explores how dental arch shape is aligned from digital

images and classified.

2.4.1 Shape Alignment: Comparing two or more Arch Shapes

Shape alignment of the dental arch shape may be carried out in two different

ways:

1. Indirect Shape Alignment by Establishing Cartesian Coordinate System

The shape of the arch was indirectly aligned when establishment of

Cartesian coordinate system was carried out (Kasai et al., 1995; Lee et al.,

2011; Taner et al., 2004) (see Section 2.2). Any measurements or

23

coordinates collected with respect to this origin are comparable with each

other.

2. Procrustes Alignment

If the standardizations above is not done, coordinates representing each

arch shape may be aligned using the Procrustes transformation (Banabilh et

al., 2006; Schaefer et al., 2006).

Let the 2D coordinate )]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx of two

dental casts be denoted in )2( N matrix 1x and 2

x . Alignment of two

shapes 1x and 2

x can be carried out by moving the points 2x relative to points

1x until their residual sum of squares

N

r

rrrr

1

1212 )()'( xxxx . (2.26)

is minimal (Mardia et al., 1979). Movement of 2X relative to points 1

X

through rotation and translation can be carried out by

' 1*2bxAx rr , (2.27)

where Nr ,...,1 , *2rx is the points after transformation, 'A is a Procrustes rotation

of )( pp orthogonal matrix and b is the translation factor. Therefore, the

goodness of fit measure (or superimposition) of 2 shape boundaries 1x and 2

x

can be obtained by solving

n

r

rrrrR

1

12122 )()'(min bAxxbAxxbA,

. (2.28)

The estimates of A and b are found by least squares estimation method given as

'ˆ VUA , (2.29)

and

12ˆ xA'xb , (2.30)

24

respectively, where V and U are the orthogonal )( pp matrices and Γ is a

diagonal matrix of non-negative elements obtained using the singular value

decomposition theorem by writing

'21UVX'X . (2.31)

The shape alignment of 2x relative to points 1

x can be done by

substituting the estimates of A and b in (2.27).

3. Bookstein’s Alignment

Bookstein’s shape alignment is the earliest and the simplest form of

shape alignment. It uses geometry shape with linearized spaces (Bookstein,

1986). It was defined in such a way that the first 2 landmarks for all

configurations were set to (a,0) and (b,0) as baseline, where a and b are

arbitrary constants. An example of the baseline was (0,0) and (1,0) in order

to align the coordinate boundary of the dental arch (Sampson, 1983).

Once the baseline has been defined, the remaining coordinates were

translated, rotated and rescaled according to the baseline. Bookstein admits

that there is a problem when using this approach for shape alignment. This

method works well only if the variations of the landmarks are small.

Figure 2.6: Two copies of the same shape of a hand. The picture on the right was

rotated roughly at 45 anti clockwise from the vertical axis. Source: Stegmann and

Gomez (2002).

25

2.4.2 Arch Shape Classification

Once the shapes of a particular interest were aligned, the arch shape

classification was carried out in two approaches according to how the shape was

quantified:

A. Shape Classification using Multivariate Morphometrics

Arch shape categories may be investigated using the k-means

clustering method from which the ratio of arch width and length were

used to quantify the arch shape (Raberin et al., 1993). Two to 8 shape

categories were considered and the homogenous sizes within each

classification were examined. Five categories of dental arch were

deemed appropriate after employing the analysis of variance and the

mean values of ratios of arch length and width in each category were

calculated. Then, the polynomial function was fitted to these mean

values to indicate 5 shape (and size) models of the dental arch.

Other clustering methods besides k-means were also used to

explore possible categories of the dental arch. The partitioning around

medoids method formed 3 categories of arch shape: narrow, middle and

wide (Lee et al., 2011).

The agglomerative hierarchical clustering method together with

analysis of variance (ANOVA) shows that the 4 categories of arch shape

(and size) were significantly different (Nakatsuka et al., 2011). The

mean shape of these shape categories were then indicated by the average

of linear and angular measurements made from the casts of the

corresponding categories.

26

B. Shape Classification using Boundary Morphometrics

Three categories of shape, determined by factor analysis were established

by McKee and Molnar (1988). Mean slopes (of a series of third degree

polynomial fitted on the arch) for each shape category were graphically

illustrated to indicate the 3 mean shapes. Arch shape variation in each arch

category was summarized heuristically as having steeper slope or flared at a

particular region of the arch.

Besides using factor analysis, dental arches were divided into 3

categories of sizes as defined from fixed constants of polynomial equation

(Preti et al., 1986). They were categorized as small, medium and large. The

disadvantage of this kind of arch category is that, they are purely varied to

the arch size only, with similar polynomial shape.

2.5 Statistical Shape Model of the Dental Arches

A summary of the literature which involves the study of the dental arch shape is

presented in Table 2.1. In general, the shape of the dental arch was described as either a

single-ideal or in categories of arch shape, according to the application of the study.

Most of the works only provided respective mean arch shapes. However, there appears

to be little work done to define the probability distribution of arch shape in order to

demonstrate the variation of the shape from its mean shape.

Only three studies provided the shape distribution of the dental arches. Work

carried out by Sampson (1983) defined shape model of the dental arch with mean and

variation using Bingham distribution from four coefficients of the conic arch. The

density of the four-variate Bingham distribution is given as

ΛγYYγΔ

ΛYYΛΛY

2124

1

14

2

21

1

214

4

- of tionparameriza , exp2

1

exp2

1);(

j jTT

jj

T

K

K

(2.32)

27

where TECBA ,,,Y denotes the coefficients of conic section

0)( 22 EyCyBxyxxA , (2.33)

which passes through (0,0) and (1,0) in the yx, plane and TΓΔΓΛ

21 is the spectral

decomposition of the symmetric matrix Λ where 41 ,, Γ is the matrix of

eigenvectors and )( 41 diagΔ are the corresponding eigenvalues (Sampson,

1983).

The appropriateness of conic arch for the Bingham distribution was assessed by

investigating the asymptotic properties of )4,...,1( jTj Yγ whereby they should be

asymptotically normally distributed as j 4 . Further test of the adequacy of the

Bingham model was carried out after obtaining the approximate maximum likelihood

estimators of the distribution.

Graphical characteristic of the distribution model for the arch shape was

examined using the modal arch, as a reflection of the average shape in the sample. The

50% and 95% confidence region for the population model arch was constructed as an

envelope to show variations in the model.

Wu et al. (2012) on the other hand built the statistical shape model using

principal component (PC) analysis to study the arch shape variations. The vector

TMzMyMxzyx DDDDD ],,,,,,[ 111 X of selected n points on the gingival contour of each

individual tooth in 3 dimensions where M = Ln and L is the number of teeth were used

as shape feature of the dental arch. All X were then represented as a weighted linear

combination of the PCs and the mean shape was given as

K

iii pb

1XX , (2.34)

where ip is the unit eigenvector corresponding to the eigenvalues i of the covariance

matrix, and ib are the weights that define the shape parameters of a deformable model

that are linearly independent and follow a normal distribution of a null mean and

28

variance i . It was found that 96% of the total amount variation present in the training

set can be described by the first 25 PCs and 68% by the first 3 PCs which in turn gives

3 standard deviations from X . The same approach was carried out by Elhabian and

Farag (2014) for jaw reconstruction.

These studies have defined a single ideal arch shape model, with relatively large

variations from the respective mean shape. This type of shape model does not allow

shape to be discriminated. Additionally, clinical experience shows that a single

impression tray would not fit all patients’ arches. The existence of a single-ideal arch

shape model in a particular population is therefore invalid and categories of shape

model should be proposed instead.

Table 2.1 summarizes the literature for shape feature and shape model of the

dental arches whereby in general mean shape were demonstrated. However, information

regarding the variations from the mean shapes established is lacking.

29

Table 2.1: Summary of literature for shape feature and shape model of the dental arch

Authors Applications of

the study

How shape of the

dental arches was

described

The shape

feature used

Mean shape

demonstrated

Variability

model

demonstrated

Shape

distribution

demonstrated

Sampson

(1981)

Orthodontic: evaluate

changes before and

after treatment

Single ‘ideal’ shape

of conic arcs

coefficient of

conic arcs

1 mean shape Variability is

measured as

concentration

parameter Λ of

the Bingham

distribution

The coefficients

of conic arcs are

distributed as the

Bingham

distribution

Preti et al.

(1986)

Prosthetic: prediction

of arch shape and teeth

position for edentulous

patients

3 types of shapes For arch shape:

coefficient of

parabolic

3 means of arch

shape

No No

Raberin et al.

(1993)

Orthodontic::

Classification of arch

forms for easier

clinical practice

5 forms of arch

shape

Ratios of arch

width and

length.

5 mean shapes No No

Kasai et al.

(1995)

Not stated. Investigate

suitability of Fourier

harmonic in describing

dental arch form and

estimate contribution

of genetic factors to

observed variability

Single ideal shape of

Fourier series

Fourier

coefficient

No. each arch was

fitted to FS

Not related Not related

30

BeGole and

Lyew (1998)

Orthodontic:

evaluate changes

before and after

treatment

Single ‘ideal’

shape of conic

spline

Coefficients of

conic spline

No No No

Braun et al.

(1998)

Orthodontic:

treatment planning

Single ‘ideal’

shape of beta

function

Coefficients of

beta function

1 Mean of each

coefficient of beta

function

No No

Ferrario et al.

(1999)

Orthodontic:

treatment planning

2 types of shape Linear distances 2 Mean of each

linear distances

No No

Burris and

Harris (2000)

Orthodontic:

treatment planning

2 types of shape Linear distances

(arch width and

length), arch

parameter,

arch area,

4th order polynomial

curve fitted to 16

dental landmarks

2 Mean of each

linear distances

No No

Henrikson et

al. (2001)

Orthodontic:

evaluate changes

before and after

treatment


of Sampson (1981)

conic arc

Coefficients of

conics

No No No

Nojima et al.

(2001)

Orthodontic:

Fabricate preformed

arch wire

3 types of shapes Linear distance:

inter canine and

molar, length from

origin to intercanine

and intermolar lines

Ratio: canine and

molar arch width

and length ratio

3 Mean of each

linear distances

No No

31

Yergin et al.

(2001)

Develop an

automated technique

for the selection of

an appropriate tray

for the patient

Not related –

matching segmented

tray and arch for

matching.

2D pixel coordinate Not related Not related Not related

Gu et al.

(2002)

Orthodontic:

Fabricate preformed

arch wire

3 types of shapes

grouped by vector

quantization, VQ

2D pixel coordinates 3 typical shapes

extracted from VQ

No No

Lestrel et al.

(2004)

Orthodontic:

treatment planning


of elliptical fourier

function, EFF

Coefficients of EFF 1 Mean of each coeff

of EFF

No No

Taner et al.

(2004)

Orthodontic:

evaluate changes

before and after

treatment


of Bezier arch curve

Coefficients of

Bezier

No No No

Wellens

(2007)

Orthodontic:

evaluate changes

before and after

treatment


of hyperbolic cosine

curve, HCC

Coefficients of HCC No No No

Triviño et al.

(2008)

Orthodontic:

Fabricate preformed

arch wire

8 types of shapes by

assigning arches to

specified values of

coefficient in

ascending order (all

possible values).

Coefficient of 6

degree polynomial

8 mean shapes No No

32

Mikami et al.

(2010)

Clinical purpose

related to studies of

morphological of the

dental arch

4 categories of

shapes, preliminary

classified visually,

then, Fourier series

were fitted to each of

the category and

essential factors that

affect the arch shape

categories were

investigated using

the amplitude of

different harmonics

Coefficient of FS 4 mean shapes

obtained from 4

coefficient

(maxillary) and 6

coefficients

(mandibular)

No No

Bush et al.

(2011)

Forensic: investigate

uniqueness of human

dentition for

bitemark analysis

Not related – finding

matches between

samples using

Procrustes distance

2D and 3D pixel

coordinate from 14

selected landmarks

on the teeth

Not related Not related Not related

Lee et al.

(2011)

Orthodontic:

Fabricate preformed

arch wire

3 types of shapes Linear distances 3 mean shapes No No

Nakatsuka et

al. (2011)

Orthodontic:

treatment planning

4 types of shapes

obtained from cluster

analysis

Linear and angular

variables

4 mean shapes Variation in each

cluster was described

using PCA.

No

Wu et al.

(2012)

Prosthetic: contour

reconstruction for

the partial edntulous


of a collective B-

spline curve from

contour of all teeth

3D pixel coordinates

from the B-spline

curve

1 mean shape from

training set

Eigenvector and

weight that define the

shape parameter from

training set

No

Elharbian and

Farag (2014)

Surgery: jaw

reconstruction


of a collective thin-

plate spline from

contour of all teeth

3D pixel coordinates

from thin-plate

spline

1 mean shape from

training set

Eigenvector and

weight that define the

shape parameter from

training set

No

33

2.6 Discrimination of Shape

Another important goal in the analysis of shape is to discriminate categories or

classes of shapes to enable classification of a new shape. A literature survey shows no

work on dental arch shape discrimination has been done, however work on

discrimination of animal morphology and human anatomy are increasingly

sophisticated, especially in discriminating the normal and abnormal shape of human

anatomy for medical diagnosis (Mukherjee et al., 2013; Uetani et al., 2015; Higashuira

et al. 2012; Pilgram et al., 2006).

Most of the existing studies represented the shape variability of normal and

abnormal anatomy using statistical shape model which was constructed by the principle

component analysis. Then, discrimination of normal and diseased anatomy shape were

carried out using support vector machine, whereby a hyperplane in a hyper dimensional

space was constructed based on the two closest points of two convex hull from training

examples of the respective categories (Bennett and Bredensteiner, 2000). However, the

major drawback of support vector machine are the high training time complexity,

extensive memory requirements and limited to binary classification (Kang et al., 2015;

Yu et al.,2010). Multiclass SVM has been developed and due to the complexity of the

method, its applications have been integrated with image features such as colour or

grayscale histogram and texture features to improve discrimination (Hu et al. 2014;

Zhang and Wu 2012).

Uetani et al. (2015) on the other hand did not use the support vector machine,

but used the linear discriminant function for discriminating shape of normal and

abnormal livers. The weight of the discriminant function was determined from the

learning data by the method of least squares. A relatively good classification on

abnormal livers was obtained using this method, with 84% classification accuracy.

34

2.7 Multivariate Normal Distribution and Its Tests

The multivariate normal distribution is useful in practical for two reasons. First,

it serves as a natural population model in many natural phenomena as many real

problems fall naturally within the framework of normal theory. Secondly, it serves as an

approximate sampling distribution for many multivariate statistics due to a central limit

property, regardless of the form of the parent population (Johnson & Wichern, 1992).

Shape analysis using multivariate morphometrics often involve multivariate

response data. Many of the analyses often necessitate procedures such as discriminant

analysis, multivariate ANOVA and multivariate regression which assume multivariate

normality (MVN). However, establishing a multivariate normal distribution is relatively

difficult, especially when the dimension of the data gets higher. More than 50 tests of

MVN were proposed in the literature for detecting departures from MVN. In general,

four categories of MVN test were identified:

1. Procedure based on graphical plots and correlation coefficient

2. Goodness-of-fit test which is an extension from the univariate method

3. Tests based on measure of skewness and kurtosis

4. Consistent procedures based on the empirical characteristic function.

Although a large number of MVN tests exist in the literature, they are seldom

used by practicing statisticians (Looney, 1995). This is due to the lack of readily

available software to conduct such tests and the reluctance to use the procedures

because of the little information about the quality and the power of the procedures. Most

work done in evaluating the power of MVN tests were carried out when developing new

tests and comparing them to several other tests with limited range. Often, the

simulations only consider the situation for data with dimension of p = 2 (bivariate

normal) and some studies simply considered data with components that were identically

and independently distributed from a common univariate distribution.

35

Mecklin and Mundfrom (2003) examined the power of 8 promising tests of

MVN via an extensive simulation study. In their study, realistic data were simulated

with small sample size and higher dimension up to p = 5 were considered using

different multivariate distribution ranging from multivariate normal to severe departures

from normality. Though the study showed that the Henze-Zirkler is the best procedure

to test MVN, it was concluded that no single method is sufficient for testing the MVN.

It was suggested that multiple approaches of MVN tests should be carried out and it is

best to start with simpler tests such as the univariate Kolmogorov-Smirnov, ellipsoid

test and graphical approach (for example the chi square plot). Following this, it is

recommended to use the multivariate measures of skewness and kurtosis by Mardia

(1970) and the Henze and Zirkler’s empirical characteristic function approach (Mecklin

& Mundfrom, 2003).

The following subsections review some of the basic and advance tests of

multivariate normal distribution which was recommended by many studies.

2.7.1 Mardia’s Multivariate Skewness and Kurtosis Test

The most commonly used test for MVN test is the Mardia’s multivariate

skewness and kurtosis test (Mardia, 1970). The test statistic for Mardia’s test of

skewness and kurtosis are

6

,1 pnb

A , (2.35)

and

,/)2(8

)2(,2

npp

ppbB

p

(2.36)

respectively, where ,)()'(1

3

1 1

1

2,1

n

i

p

j

iipn

b xxSxx

,)()'(1

2

1

1,2

n

i

iipn

b xxSxx p = 4

is the dimension of ix , n = 47 and S is the sample covariance matrix. It is known that

36

A is asymptotically a chi-square probability distributed with 6

)2)(1( ppp degrees

of freedom and B has a standard normal distribution (Mecklin & Mundfrom, 2003). If

the test rejects the null hypothesis, then ix is not following MVN distribution.

MVN tests based on measures of skewness and kurtosis have been demonstrated

to have relatively low power towards rejecting the alternative hypothesis (Horswell &

Looney, 1992; Mecklin & Mundfrom, 2004). Simulation studies by Horswell and

Looney (1992) and Mecklin and Mundfrom (2004) confirm the relatively low power of

these tests. The tests have no power against the multivariate t distribution, which is a

mild deviation from normality and symmetric.

Nevertheless, the power of Mardia’s skewness and kurtosis tests is higher when

compared to Small (1980) and Srivastava (1984) tests, which are also based on the

measure of skewness and kurtosis. The power of performance for Mardia’s skewness is

considerably high, at almost 100% power of rejecting MVN as the sample size increases

(Mecklin & Mundfrom, 2004). This shows that Mardia’s tests are relatively good

indicators of multivariate normal distribution. However other tests should be supplied to

support the finding.

2.7.2 Doornik and Hansen Omnibus test

To improve upon power of test based on skewness and kurtosis, some authors

have attempted to combine measures of skewness and kurtosis into a single òmnibus'

test statistic. Mardia and Foster (1983) derived six omnibus statistics; however it was

found that this statistic lacked power (Horswell & Looney, 1992).

Doornik and Hansen (2008) proposed an extension of the omnibus univariate

test based on skewness and kurtosis by Shenton and Bowman (1977). Simulation study

demonstrated that this test possessed good power properties and was suggested to be

37

used when Mardia’s test of multivariate skewness and kurtosis was considered as a

supportive result.

Let ),,(' 1 nxxX be a )( np matrix of n observations on a p-dimensional vector

with sample mean )( 11

nn xxX and covariance )(1XXS n where

),,( 1 XxXxX n

. The observations were transformed into

XVHHΛR1/2

, (2.37)

where ),,(' 1 nRR R is a )( np matrix of n observations on a p-dimensional vector, V is

a matrix such that pIVV 1 with elements on the diagonal:

2/12/111 ,, ppSSdiag V , (2.38)

VSVC is a correlation matrix, Λ is the matrix with eigenvalues of C on the diagonal, H is

the corresponding eigenvectors such that pIHH and CHHΛ .

The univariate skewness and kurtosis of each transformed n-vector of

observations are given as 2/3

2

31

m

mb and

22

42

m

mb respectively, where

n

i

ii RRnm

1

1 )( and )( 11

nRRnR .

The Doornik-Hansen multivariate omnibus test statistic is

)2( ~ 22211 pZZZZEp

, (2.39)

where ),,( 1111 pzzZ

and ),,( 2212 pzzZ

are the transformation for skewness 1b

and kurtosis 2b to standard normal. Each transformed n-vector of observations for

skewness are:

])1(log[ 2/121 yyz , (2.40)

38

where

2/12

1)2)(6(2

)3)(1)(1(

n

nnby

,

2/12 )log(

1

, 2/12 )1(21 and

)9)(7)(5)(2(

)3)(1)(7027(3 2

nnnn

nnnn . As for kurtosis, its transformation is given as

2/13/1

2 )9(9

11

2

z , (2.41)

where kbb 2)1( 12 , cba 1 , 12

)3131137)(7)(5( 23

nnnnnk

6

)52)(7)(5)(7( 2

nnnnnc ,

6

)7027)(7)(5)(2( 2

nnnnna and

)415)(1)(3( 2 nnnn .

2.7.3 Royston Test

The Shapiro and Wilk (1965) test was shown to be among the most powerful

tests for detecting departures from univariate normality (Srivastava & Hui, 1987).

Royston (1983) proposed an extension of the Shapiro-Wilk (SW) goodness of fit test to

MVN distribution. A revision of Royston (1992) MVN test for approximation of

coefficients requires the calculation of the SW test to correct the specification of the null

distribution. The revised version of the Royston’s MVN test was shown to have good

Type I error control and power against rejecting non MVN distributions (Farrell et al.,

2007).

Suppose ),,( 1 pxxX is the p-variate random vector to be examined for

multivariate normality consisting of j-th random sample ),,( 1 njjj xx x of n cases. Let

nyy 1 be an ordered sample jx of univariate data. The SW test is defined as

2

2

)(

)(

yy

yaW

i

ii

, (2.42)

39

where ),,( 1 naa a is such that ii yan 2/1)1( is the best linear unbiased estimate of the

standard deviation of the iy , assuming normality (Royston, 1983). A revised

approximation of a is given as

ii ma ~~ 2/1 , (2.43)

for , where )/()(~ 14

38

1 nimi , is the normal cumulative

distribution function (c.d.f), )~2~21/()~2~2~~( 21

221

2 nnnn aammmm , naa ~~

1 , 12~~

naa ,

5432 706056.2434685.4071190.2147981.02221157.0~ xxxxxca nn

543211 582663.3682633.5752461.1293762.0042981.0~ xxxxxca nn

nx , and ii mc ~)~~(12 mm (Royston, 1992).

The W statistic could be transformed to an approximately standard normal

variate, z. A simpler normalize transformation for the W statistic is given as

wz , (2.44)

where 32 0006714.00205054.039978.05440.0 nnn , )1ln(ln Ww ,

n459.0273.2 , and )0020322.0062767.077857.03822.1exp( 32 nnn , for

114 n . As for 200012 n , 32 0038915.0083751.031082.05861.1 xxx ,

)1ln( Ww , )0030302.0082676.04803.0exp( 2xx and nx ln (Royston, 1992).

Suppose pzz ,,1 of the transformed W statistic have been obtained as above,

with ),...,1( pjz j correspond to the j-th component from ),,( 1 pxxX . The Royston’s

MVN test based on the Shapiro-Wilk is defined as

eGH , (2.45)

)5for ( 2,,3 nni

40

where pkG

p

j

j

1

,

2

1 )(2

1

jj zk , is the normal c.d.f, cp

pe

)1(1 ,

)( 2 ppcc ij is an estimate for the average correlation among the jk ,

),( jiij kkcorrc is the correlation matrix of k. Approximation of ijc , is given as

)1(1ˆv

cij , (2.46)

where is the correlation coefficient, 715.0,5 ,

32 0018034.0015124.021364.0 xxv and nx ln . The Royston’s MVN test H was

shown to approximately distributed as chi-square distribution with e degrees of

freedom (Royston, 1983).

2.7.4 Henze-Zirkler Test

The Henze-Zirkler (HZ) test is recommended as a formal test of MVN (Mecklin

& Mundfrom, 2003). The HZ test is based on a nonnegative functional distance

that measures the distance between two distribution functions

dtttQtPQPd )(|)(ˆ)(ˆ|),( 2

, (2.47)

where the characteristic function of the multivariate normal distribution )(ˆ tP and the

empirical characteristic function )(ˆ tQ are the Fourier transformations of P and Q

respectively, is the weight or kernel function of ),0( 2

pp IN and β is the smoothing

parameter (Henze & Zirkler, 1990).

The closed form for the Henze-Zirkler statistic is

,)21(||||)1(2

exp1

)1(2

||||2

exp1

1

2/2

22

22/2

1,

2

2,

n

i

p

i

p

n

ji

jin

Yn

YYn

T

(2.48)

41

where )4/(1)4/(1

4

12

2

1

p

p

np

is the smoothing parameter, )()'(|||| 12ddSdd

jiji YY

and )()'(|||| 12ddSdd

iiiY . The limiting distribution of ,nT is a lognormal

distribution with the following mean and variance:

22

4

2

2

2/2

)21(2

)2(

211)21(1)]([

ppppTE p , (2.49)

,)(2

)2(

)(2

31)(4

)21(4

)2(3

)21(2

21)21(2)41(2)]([

2

84

2/

42

8

22

4

22/2

w

pp

w

pw

ppppTVar

p

pp

(2.50)

where )31)(1()( 32 w (Henze & Zirkler, 1990). Rejection of the null hypothesis

indicates non-normality distribution.

2.8 Multivariate Complex Normal Distribution (MVCN)

The complex random variables often appear as actual data in areas such as in

geophysics, circular analysis, communications, signal processing and shape analysis.

The random variables come in pairs of the form x + jy, where x is the real part and y is

the imaginary part. The introduction of a complex-number-based theory can often lower

the dimension involved and subsequently simplify mathematical models.

Modelling complex random variables has been indirectly applied in the field of

physics, which examine the arrangement of atoms in solids. However Wooding (1956)

and Goodman (1963) are among the first who explicitly introduced the multivariate

distributions of complex random variables which is also called the “multivariate

complex normal distribution”. Although work to develop the theory of the multivariate

complex normal distribution has been carried out, little applications of the distribution

can be found in the literature (Pannu et al., 2003).

42

The following section introduces the complex random variables and complex

normal distribution together with their properties. Then, a review on the parameter

estimation and test of separation between two vectors from multivariate complex

normal distribution is presented.

2.8.1 The Univariate and Multivariate Complex Random Variables and

Distributions

Let U and V be real random variables. The random variable iVUX where

X is taking values in C (field of complex numbers), may be regarded as a univariate

complex normal random variable by the following definition:

Definition 1. A univariate complex random variable is a complex random variable

iVUX such that the distribution of T

V

UX

has a bivariate normal distribution

(Anderson et al., 1995, p. 5; Giri, 2003, p. 86).

Using the above definition, the univariate complex random variable of complex

random variable X may be investigated by proving that T

V

UX

has bivariate normal

distribution.

The p-variate complex random vector on the other hand is defined as follows:

Definition 2. Let kXX be a p-dimensional vector, where kX for pk ,...,1 is a

complex random variable such that the real 2p-vector ),( kk VUY has a 2p-variate

normal distribution (Andersen et al., 1995; Giri, 2003, p. 86).

Now, we consider the definition of the univariate complex normal distribution

which is given as follows:

Definition 3. The univariate complex random variable X is distributed as univariate

complex normal distribution ),( 2CN with its probability density function given as

43

,)var()var(

)()(exp

))var()(var(

1

)var(

*))((exp

)var(

1

*))((exp

1)(

22

22

VU

vu

VU

X

xx

X

xxxf X

(2.51)

where j is the mean of x and *)( x is the conjugate and transpose of

)( x (Andersen et al., 1995, p. 20; Giri, 2003, p. 86).

The definition of the multivariate complex normal (MVCN) distribution on the

other hand is as follows:

Definition 4. A complex random p-vector kXX with values in Cp has a multivariate

complex normal distribution if, for b Cp , each Xb* has a univariate complex normal

distribution (Giri, 2003, p. 91).

The probability density function of the MVCN distribution is given as

(Andersen, et al., 1995, p. 26; Giri, 2003, p. 86):

.)()(exp)det()( 1*1θxHθxHxX pf , (2.52)

where θ is the mean vector, H is a Hermitian matrix which correspond to a real matrix

being symmetry and *)( θx is the conjugate and transpose of ).( θx We write

),(~ HθX pMVCN (Andersen et al., 1995).

2.8.2 Properties of the MVCN Distribution

There is a close relationship between real and complex normal distribution

established by the isomorphism, : pp 2RC . For ),( ~ HθX pMVCN , it holds that

HθX

2

1, 2~ pN

kVkU

, (2.53)

44

where ,,,1 pk (Andersen et al., 1995, p. 25). The distribution of the pp complex

random matrix XXW* follows a complex Wishart distribution with parameters H and

n , denoted by ),(~ nCWp HW and it holds that

),,2

1(~ 2 nW p HW (2.54)

for the real Wishart distribution (Andersen et al., 1995, pg. 40).

Further, most of the properties of the multivariate complex normal distribution are

relatively similar to the real multivariate normal distribution. Let pCc , it holds that

),*,*(~ HccθccX pMVCN (2.55)

and

).,(~ 2121 nnCWp HWW (2.56)

(Andersen et al., 1995, p. 23 & 43).

2.8.3 Parameter Estimation of the MVCN distribution

The parameters θ and H can be estimated using the maximum likelihood

method analogous to real multivariate normal distribution (Goodman, 1963). Let

)( kXX be a p-dimensional random vector, where kX for pk ,,2,1 is a complex

random variable kkk jVUX . Therefore, for n independent realisation X is called an

)( pn complex random matrix given as

npnn

p

p

XXX

XXX

XXX

21

22221

11211

X . (2.57)

45

The likelihood function of ),(~ HθX pMVCN in (2.52) can be written as

. )()(exp)det(

)()(exp)det(

)()(exp)det(

.)()(exp)det(

)()(

1

1*

1*1

1

1*

1

1

1

1*1

1

n

i

ii

nnp

nn

p

p

n

i

ii

p

n

i

ifL

θxHθxH

θxHθxH

θxHθxH

θxHθxH

Hθ,;xHθ,X;

(2.58)

The term

n

j

jj

1

1* )()( θxHθx can be simplified using identity in Mardia et al. (1979)

(p. 97, 456) therefore yielding.

*1

1

*1

1

1* ))(( tr ))((tr )()( θxθxHxxxxHθxHθx

n

n

i

ii

n

i

ii . (2.59)

Let

n

i

iin

1

*))(( θxθxM and substitute (2.59) in (2.58) gives

. ))(( tr tr exp)det(

))(( tr )(tr exp)det()(

*11

*11

θxθxHMHH

θxθxHMHHHθ,X;

nn

nnL

nnp

nnp

(2.60)

Therefore, the log likelihood function of )(xXf can be written as

.))(( tr tr )det(loglog

))(( tr tr exp)det(log )(

*11

*11

θxθxHMHH

θxθxHMHHHθ,X;

nnnnp

nnl nnp

(2.61)

The maximum likelihood estimate (m.l.e) of the mean θ and variance matrix H

for the MVCN distribution is that value of the parameter which maximizes the

likelihood of the given observation. This value may be obtained by differentiation and

since the log likelihood )( Hθ,X;l is at a maximum when )( Hθ,X;L at a maximum, the

equation 0

θ

l and 0

H

l is solved for θ and H respectively.

Consider in (2.61). Using matrix differentiation theorems,

)(1θxH

θ

nl . (2.62)

)( Hθ,X;l

46

Solving 0

θ

l , 0)ˆ(1 θxHn . Therefore, the m.l.e of θ is xθ ˆ .

Define new parameter 1 HV . Equation (2.61) becomes

*))(( tr tr loglog )( θxθxVVMVHθ,X; nnnnpl . (2.63)

To calculate V

l , the right terms of (2.63) can be written as (Mardia et al, 1979, p. 104

and 481)

,)(22

)))(((2))((2

)))((())((222

))(())((2 )(2 )(2 )(

**

**

**

UU

θxθxMHθxθxMH

θxθxMHθxθxMH

θxθxθxθxMMHHHθ,X;

Diagn

Diagn

Diagn

DiagnDiagnDiagnl

(2.64)

where *))(( θxθxMHU . Solving 0

V

l,

.)ˆ)(ˆ(ˆ

0)ˆ)(ˆ(ˆ

0

*θxθxMH

θxθxMH

U

(2.65)

Since xθ ˆ , therefore the m.l.e of H is

MH ˆ . (2.66)

Analogous to the real multivariate normal distribution, the unbiased estimator for θ and

H are xθ ˆ and

n

i

iin

1

*))((1

1ˆ xxxxMH respectively.

2.9 Missing Values Analysis

Missing values is a common problem in data analysis. Deleting a particular

sample when one of the variables is unavailable seems to be the most common way to

handle missing data. A few deletion procedures were proposed such as the list wise and

pair wise deletion. However, this procedure is unfavorable particularly when the sample

size is small and a single sample deletion is crucial as it may affect the biased estimates,

statistical model and consequently the power of statistics (Barzi & Woodward, 2004).

47

Simple imputation approach seems to be the alternative whereby the missing

values of a particular variable will be imputed using the mean or mode of the respective

observable variable. Another sophisticated imputation approach is by generating

estimates of the missing values using a predicted probability distribution or regression

model from the knowledge of the observable data.

The following sub-section reviews some of the model-based imputation methods

for the analysis of missing values.

2.9.1 Data Augmentation (DA) Algorithm

2.9.1.1 Definition

The data augmentation algorithm was first proposed by Taner and Wong (1987).

The theory behind this method was derived from Gibbs sampling; a well known form of

Markov chain Monte Carlo which creates pseudorandom draws from a particular

probability distribution.

Suppose z is a random vector with two sub-vectors; ),( vuz , and its joint

distribution )(zP is not easily simulated or intractable., however, their conditional

distributions )|()|( vugvuP and )|()|( uvhuvP are. Let

,,,,,,,

,,,

)()()(

2

)(

2

)(

1

)(

1

)()(

2

)(

1

)(

00

0

t

n

t

n

tttt

t

n

ttt

vuvuvu

zzzZ

(2.67)

be a sample of size 0n simulated from a distribution that approximates the target

distribution )(zP at iteration t . The data augmentation algorithm updates this sample in

two steps.

Step 1: Generate the sub-vector

)1()1(2

)1(1

)1(

0,,, t

nttt uuuU , (2.68)

independently for 0,,2,1 ni from )|(~ )()1( ti

ti vugu .

Step 2: Generate the other sub-vector

48

)1()1(2

)1(1

)1(

0,,, t

nttt vvvV , (2.69)

independent and identically (i.i.d) sampled from the average of the conditionals

:)|( )1( tiuvh

0

1

)1(

0

)1( )|(1

)|(

n

i

ti

t uvhn

Uvh . (2.70)

Consequently completes the new sample

)1()1()1(2

)1(2

)1(1

)1(1

)1(

00,,,,,, t

nt

nttttt vuvuvuZ . (2.71)

Using functional analysis, the distribution of )(tZ will converge to )(zP as t

(Tanner & Wong, 1987).

The advantage of using the data augmentation is that the convergence does not

require a large value of 0n . With 10 n , data augmentation reduces to a special case of

the Gibbs sampler

,,,,|~

,,,,|~

,,,,|~

)1(

1

)1(

3

)1(

2

)()1(

)()(

3

)1(

2

)(

2

)1(

2

)()(

3

)(

2

)(

1

)1(

1

000

0

0

t

n

ttt

n

t

n

t

n

tttt

t

n

tttt

ZZZZPZ

ZZZZPZ

ZZZZPZ

(2.72)

with the random vector ),( vuz partitioned into two sub-vectors u and v .

2.9.1.2 Application of DA in the Missing Values Problem.

In many incomplete-data problems, the observed-data posterior distribution

)|( obsYP are sometimes difficult to determine and therefore the data from )|( obsYP

cannot be easily simulated. However, if obsY is augmented by an assumed value m isY , the

resulting complete-data posterior ),|( misobs YYP may be used to find its true posterior

when the data augmentation algorithm is adopted as follows:

49

Step 1 (Imputation step): Given a current guess of a parameter )(t , draw

independent 0n values of m isY

)1()1(2

)1(1

)1(

0,,, t

misnt

mist

mist

miss yyyY , (2.73)

generated from the conditional predictive distribution of m isY

),|(~ )()1( tobsmis

tmis YYPY . (2.74)

Step 2 (Posterior step): Draw new values of

)1()1(2

)1(1

)1(

0,,, t

nttt , (2.75)

sampled from the conditional distribution of obsY and )1( tmisY

),|(~ )1()1( tmisobs

t YYP . (2.76)

Repeating steps (1) – (2) from the starting value )0( for a value of t that

is relatively large yields a stochastic sequence

,2,1:, )()( tY tmiss

t , (2.77)

which possesses )|,( obsmis YYP as its stationary distribution (Schafer, 2010).

Consequently, the sub-sequences ,2,1:)( tt and ,2,1:)( tY tmiss have

)|( obsYP and )|( obsmiss YYP as their respective stationary distributions. A

stopping criterion can be introduced to determine the sufficient number of t .

The process of repeating steps (1) – (2) will continue until )(t and

)1( t

converge to a certain pre-specified tolerance. An extension for the multivariate

observation can be done using the same idea as above.

2.9.2 Expectation Maximization (EM) Algorithm

2.9.2.1 Definition

The EM algorithm is an iterative method for approximating the maximum

likelihood estimates (m.l.e) of parameters. It uses the idea of interdependence between

50

missing data m isY and parameters, say, . Since m isY holds relevant information for

estimating and may help in finding likely values of m isY , therefore is it also

possible to estimate in the presence of observable data obsY .

The scheme to estimate using this idea is as follows:

1. Replace the m isY (denoted as )(t

misY ) using arbitrary initial estimate of

(denoted as )(t ), where ,1t .

2. will be re-estimated (denoted as )1( t ) based on obsY and

)(tY

3. Step 1 is repeated iteratively until )(t converge to a specified

tolerance

Dempster et al. (1977) generalized the above idea for broader class of problems such as

in applications for cluster and truncated data, finite mixture models, factor analysis,

variance component estimation and missing value problems. The generalized EM

involves 2 steps as follows:

1. the Expectation or E-step, which require the estimation of )(t for the

complete data, when obsY is given.

2. the Maximization or M-step, where )1( t is estimated by maximum

likelihood similar to as though the estimated complete data were the obsY .

The EM algorithm was shown to be a reliable method whereby it is consistently

converging to a stationary point of the observed-data log likelihood (Dempster et al.,

1977; Wu, 1983). The application of EM is commonly used in computer vision and

machine learning for data clustering and analyzing data with missing values (Barzi &

Woodward, 2004; Bishop & Nasrabadi, 2006).

51

2.9.2.2 EM for Regular Exponential Families in Missing Values Problem

If the complete-data probability model has the regular exponential family, the

complete-data log likelihood on n i.i.d (possibly multivariate) observations

nyyyY ,,, 21 may be written as

cngYTYl T )()()()|( , (2.78)

where

Ts )(),(),()( 21 , (2.79)

is the canonical form of the parameter ,

Tsss YTYTYTYT )(),(),()( , (2.80)

is a (1 x s) vector of complete-data sufficient statistics and c is a constant. The

sufficient statistics when (2.68) holds, exist in an additive form of

n

i

ijj yhYT

1

)()( , (2.81)

for some function jh and j=1,2,…s.

The E-step estimates the )(YT jby finding

)(,|)(

tobsj YYTE . (2.82)

In many of the models in the form of (2.68), the expectations )(,|)(

tobsj YYTE can be

obtained in closed form and E-step will be computationally straightforward.

An example of i.i.d observations nyyyY ,,, 21 from a univariate normal

distribution with mean and variance , so that , is the unknown parameter,

is presented to illustrate the E-step. Let the first 1n components of the data vector Y are

observed, and the remaining 10 nnn are missing at random. The sufficient statistics

for , are given by:

52

Tn

i

i

n

i

iT yyTTYT

1

2

1

21 ,,),()( . (2.83)

The E-step estimates the 1T and 2T by finding

0

111

1

1

1

1

,| nyyyEYTE

n

i

i

n

ni

i

n

i

iobs

, (2.84)

and

)( ,| 20

1

2

1

2

1

22

1

1

1

nyyyEYTE

n

i

i

n

ni

i

n

i

iobs , (2.85)

respectively (Schafer, 2010).

The M-step determines )1( t by replacing the estimates from E-step into the

complete-data m.l.e as the solution of

tYTE |,)( , (2.86)

where t is the realized value of the vector )(YT .

It is known that the m.l.e for parameter from univariate normal distribution are

n

iiyny

1

1 and

n

i i

n

ii

n

ii ynynyyn

1

2

1

21

1

221 )( . The M-step

of i.i.d observations from univariate normal distributions may be obtained by inserting

the expected sufficient statistics in (2.74) and (2.75) into the expressions for the

complete-data m.l.e and yields

)(0

1

1)1( 1 tn

ii

tnyn , (2.87)

2

)(

10

222)(0

)(

10

21)1( 11

tn

ii

ttn

ii

tnynnnyn , (2.88)

Imputation of missing values using EM usually uses )1( t . Since the calculation

of )1( t does not depend on

)1( t , therefore it may be possibly extended to the case

of multivariate normal.

53

It can be shown that the observed-data likelihood )|( obsYL for normal

distribution is just a complete-data likelihood based on 1

,,, 21 nobs yyyY and the

observed-data m.l.e for and are

1

1

11

n

iiobs yny and

1

1

2211 )(

n

iobsi yyn

respectively. Although the observed-data m.l.e exist in closed form, one can also

compute them using the EM algorithm. Nevertheless, EM is very useful when dealing

with complicated functions whereby obtaining their maximum likelihood estimates of

parameters in statistical models appear to be difficult.

54

CHAPTER 3: EXPLORATORY DATA ANALYSIS OF SHAPE FEATURE AND

INVESTIGATION OF CATEGORIES OF THE DENTAL ARCH SHAPE

3.1 Introduction

Categories of dental arch shape may exist due to ethnicity, hereditary and

environmental factors. This may necessitate different considerations in dental treatment

planning. To categorize dental arches, a novel shape descriptor indicating the variation

of tooth position was derived from digital images of maxillary dental casts. Initial tests

on control samples of 47 dental casts showed that, firstly, angular measures can be

linearly approximated, and secondly, the maxillary arch is symmetrical. Then, clustering

methods, principal component analysis and cluster validity index were carried out to

show possible groups of dental arch shapes.

3.2 Data Collection

One hundred and twenty-two Malaysian dentate subjects (45 males and 77

females) of 74 Malay, 36 Chinese and 12 Indian ethnic groups aged between 19 and 48

years were randomly selected from University of Malaya students and dental personnel

attached to the Faculty of Dentistry University of Malaya and patients receiving dental

treatment in the out patients clinic at the University Malaya Medical Centre.

Impressions of the patients’ dental arches were made to obtain their dental casts.

Obtaining a reasonably large sample size is time consuming as subject inclusion

criteria were imposed. Forty seven dental casts were collected from March 1st until

November 30th 2010 with the assistance of a dentist from the Department of Restorative

Dentistry, Faculty of Dentistry, University of Malaya. These casts were used as control

samples to investigate symmetry of the dental arches and possible clusters or categories

55

of arch shape as well as to establish the existence of multivariate normal shape model.

Another set of test sample consisting of 75 dental casts was collected from October 1st

2011 until March 30th 2012 for verification studies needed in several steps of this study.

The study was approved by the Faculty of Dentistry, University of Malaya Medical

Ethics Committee (MEC no: DF PD1106/0089(L), dated 20 December 2011).

3.2.1 Selection of Samples

Certain inclusion criteria were set to select control samples for the study. The subjects

must have:

i) Well aligned maxillary and mandibular teeth with minimal attrition

ii) Angle's Class I dental relationship

iii) No history of orthodontic treatment, anterior restoration or fixed dental

prosthesis in the maxilla or mandible.

iv) No facial asymmetry.

The inclusion criteria (i) was not strictly imposed for the remaining 75 subjects.

3.2.2 Dental Impression and Stone Cast Making

An impression of the dental and oral structures in the mouth of each patient was

made using irreversible hydrocolloid (Duplast fast set alginate impression material;

Dentsply Dental Co Ltd, Tianjin, China) (Figure 3.1). Impressions were then cast using

type III dental stone (Moldano; Heraeus Kulzer GmbH, Hanau, Germany) (Figure 3.2).

Then, each cast was numbered and labelled to identify the subjects to whom the casts

belonged to.

56

Figure 3.1: Dental impression made in an impression tray.

Source: http://www.firstbtob.com.

Figure 3.2: Dental plaster was poured onto the impression to obtain the stone dental

cast. Source: www.howtomakevampireteeth.com/.

3.2.3 Cast Preparation

To avoid image variation that may resulted from differences in camera to cast

distance, standardization of each cast was made. The occlusal plane of each cast was

made parallel to the base of the cast by ensuring that the incisal edges, canine tips and

mesiopalatal cusps of the first molars touched the surface of the table in the most stable

position. Then, a geometric compass was positioned on a flat surface of a table with one

of its arm resting on and parallel to the flat surface. Adhesive tape was used to stabilize

the compass on the flat surface.

A line was then inscribed on the base of the cast using a pencil attached to the

compass so that it is parallel to the occlusal plane. The base of the cast was then

http://www.firstbtob.com/

http://www.howtomakevampireteeth.com/

57

trimmed following this line to obtain an occlusal plane which is parallel to the

horizontal (Figure 3.3). This was done to standardize the height of each cast so that

image variation that may be resulted from differences in camera to cast distance can be

avoided.

Figure 3.3: Method used to make the base of the stone cast parallel to the occlusal

plane

3.2.4 Image Acquisition, Shape Alignment and Calibration of Measurements

Correspondences of the dental arch shapes were ensured by identifying

landmarks on the digital images. Anatomical landmarks are defined as prominent points

that match between organisms in some biologically-meaningful way (Dryden & Mardia,

1998). The incisive papilla and the hamular notches were identified as the common

anatomical landmarks as they have been shown to be relatively stable, even after the

loss of teeth (Grave & Becker, 1987; Nikola et al., 2005). They were marked on each

cast using a 0.5 mm pencil lead with a cross sign (x) (Figure 3.4).

Digital images of the casts were then captured by a high resolution digital

camera (Nikon D70s; Nikon Corp, Tokyo, Japan). The camera to object distance was

fixed at approximately 50 centimeters to ensure distortion-free images. Two metal

rulers fixed perpendicular to each other and positioned on a plane parallel to the

occlusal plane were used as frames for calibration of the measurements (Figure 3.4).

These images were then saved as JPEG files.

58

Figure 3.4: Two metal rulers positioned on a plane parallel to the occlusal plane

enabled measurements to be calibrated. Shape alignment is achieved by the creation of

the Cartesian plane defined from anatomical landmarks which were marked with

crosses.

Digital images of these casts were then imported to a program developed in

MATLAB software (version R2009b, The MathWorks Inc., USA). In order to align the

arch shapes, the line joining the two hamular notches was used geometrically as the

Cartesian x-axis, while the line perpendicular to the x-axis passing through the incisive

papilla was defined as the y-axis. The point where both axes meet was defined as the

origin of the coordinates.

Shape alignment of the dental arch shape is achieved by the creation of these

Cartesian coordinate axes on the digital images (Figure 3.4). In this way, the dental

arches were comparable to one another.

Hamular notches

Incisive papilla

59

Figure 3.5: Selected points on the teeth used to represent the dental arch shape.

3.3 Shape feature of the Dental Arch

The midpoint of the anterior teeth and distobuccal cusp of the first molars were

selected (Figure 3.5). The j-th point may be represented by the length of a line joining

the cusp tips of a tooth to the origin (jl ) and an angle (

jw ) with respect to the

horizontal axis. The arch shape for a given dental cast was described using the shape

feature vector

),,,,,,,,,,,,,,,( 4433221144332211RRRRRRRRLLLLLLLL lwlwlwlwlwlwlwlwq , (3.1)

where for example ),( 11LL lw and ),( 11

RR lw represent the point on the left central incisor and

right central incisor respectively, in reference to the Cartesian origin established from

the digital image of the dental cast.

Let (𝑂𝑥, 𝑂𝑦) be the coordinate of the origin and ),( 11LL yx be the coordinate of the

first landmark on the left side of the arch. The angle Lw1 and distance Ll1 of this

landmark from the origin (in pixels coordinate) was obtained by

x

y

Ox

Oy

1

11tan and

21

21 )()( yx OyOx respectively (Figure 3.6). Then, a standard image processing

technique was applied whereby a calibration method converts all measurements in terms

of pixel number into millimeters (mm). The same procedure was used to find the

60

distances and angles of the other teeth. To simplify the above tasks for measuring q , a

program was developed in MATLAB software.

Measurements were repeated three times by the same investigator and a pilot

study using 47 control samples for test-retest reliability using Pearson’s correlation

shows that the measurements are reliable as the measurements taken in different times

show high correlation, r (Table 3.1). Further, the hypothesis testing on the population

correlation coefficient 1:0 H demonstrates consistency of the data (Murphy &

Davidshofer, 1991). The data were assumed to follow normal distribution since the

variation of teeth position and human arch shape may be regarded as natural

phenomena. In the later sections, this matter will be established.

Table 3.1: The sample correlation coefficient r, and the test statistic for 1;0 H to test

the reliability of measurements taken at three different times. The lower and upper

critical values are -2.0141 and 2.0141 respectively when α =0.05 was used.

Times of

measurements

Variable

First and second

times

First and third

times

Second and third

times

r Test

statistic r

Test

statistic r

Test

statistic Rw1 0.9715 -0.8065 0.9644 -0.9037 0.9884 -0.5115

Rl1 0.9687 -0.8454 0.9423 -1.1558 0.9582 -0.9801 Lw1 0.9709 -0.8145 0.9653 -0.8918 0.9883 -0.5135

Ll1 0.9636 -0.9132 0.9328 -1.2512 0.9551 -1.0167 Rw2 0.9689 -0.8435 0.9617 -0.9369 0.9842 -0.5986

Rl2 0.9641 -0.9073 0.9323 -1.2552 0.9498 -1.0759

Lw2 0.9634 -0.9158 0.9583 -0.9790 0.9861 -0.5621

Ll2 0.9677 -0.8595 0.9353 -1.2268 0.9537 -1.0328

Rw3 0.9737 -0.7745 0.9589 -0.9722 0.9776 -0.7147

Rl3 0.9664 -0.8775 0.9307 -1.2711 0.9470 -1.1070

Lw3 0.9730 -0.7841 0.9602 -0.9560 0.9834 -0.6140

Ll3 0.9666 -0.8745 0.9305 -1.2730 0.9479 -1.0974

Rw4 0.9909 -0.4524 0.9526 -1.0455 0.9575 -0.9883

Rl4 0.9671 -0.8675 0.9446 -1.1318 0.9667 -0.8734

Lw4 0.9899 -0.4788 0.9632 -0.9178 0.9703 -0.8240

Ll4 0.9669 -0.8697 0.9577 -0.9865 0.9749 -0.7556

61

The average measurement was used as the final measure. The notations for the

points together with their summary statistics for jw and jl where j = 1, ..4 are tabulated

in Table 3.2 and Table 3.3, respectively.

Table 3.2: Mean, standard deviation (SD), minimum and maximum values of the

angular measurements (in degrees).

Variable Mean SD Min Max

Right central incisor ( Rw1) 85.51 2.14 80.68

89.39

Left central incisor ( Lw1) 85.66 2.07 81.28

89.79

Right lateral incisor ( Rw2) 77.26 2.32 72.33

82.47

Left lateral incisor ( Lw2) 77.31 2.16 73.05

82.27

Right canine ( Rw3) 69.62 2.48 64.90

75.27

Left canine ( Lw3) 69.62 2.47 63.86

75.60

Right distobuccal cusp of first

molar ( Rw4)

38.55

4.43

30.10 47.99

Left distobuccal cusp of first

molar ( Lw4)

38.63

4.54

30.20 47.21

Table 3.3: Mean, standard deviation (SD), minimum and maximum values of the length

measurements (in mm).

Variable Mean SD Min Max

Right central incisor ( Rl1 ) 56.20 3.33 49.20

62.66

Left central incisor ( Ll1 ) 56.27 3.22 50.02

63.03

Right lateral incisor ( Rl2) 54.03 3.03 48.70

60.42

Left lateral incisor ( Ll2) 54.06 3.14 48.54

60.30

Right canine ( Rl3) 50.56 2.88 44.94

56.82

Left canine ( Ll3) 50.79

2.87

45.09

56.41

Right distobuccal cusp of

first molar ( Rl4)

35.41

2.31

30.51

40.71

Left distobuccal cusp of first

molar ( Ll4)

35.78

2.84

31.66

43.04

62

Figure 3.6: Computation of angle and distance of the central incisor tooth from the

geometrical Cartesian origin.

3.4 Properties of the Shape Feature

Shape feature ),,,,,,,,,,,,,,,( 4433221144332211RRRRRRRRLLLLLLLL lwlwlwlwlwlwlwlwq was

proposed to represent the shape of human dental arches and teeth location. Obviously,

there would be a problem in analyzing these measures as the nature topology of jw and

jl are different. However, if the angles recorded in the range ]360 ,0(

degree were

analyzed using the linear metric, then the directions o f 360 are close to the

opposite end-points. This made them near neighbours in a circular metric but

maximally distant in linear metric (Figure 3.7). Further, if we apply the angles

using the conventional linear techniques, it can lead to paradoxes; for example, the

arithmetic mean (ordinary mean on the straight line) of the angles 1 and

359 is

180 whereas by geometrical (mean direction of circular data), the mean has to be

0 (Figure 3.8).

63

Figure 3.7: Differences of linear and circular measurement.

Figure 3.8: The arithmetic mean points the wrong way.

A distribution that is often used to describe the physical properties of circular

data is the von Mises distribution. As a continuous probability distribution, the von

Mises is analogous to the normal distribution for linear data and has some similar

characteristics with the normal distribution. Thus, the von Mises is also known as

the circular normal distribution.

Let be the circular random variable with circular normal distribution,

denoted by ),( 0 CN . The probability density function of is

)}cos(exp{)(2

1),;( 0

0

0

I

g , (3.2)

where )20( 00 is the mean direction, )0( is known as the

concentration parameter and )(0 I denotes the modified Bessel function of order

zero.

Linear:

Circular:

Circular:

64

Note that in the context of circular statistics, 0 does not give an average

value of n

n ...1 which is commonly used for linear data. However, the

interpretation of mean direction is similar to the arithmetic mean, and is sometimes

called as the “ preferred direction”.

The concentration parameter (also known as the variance for the angular

variable) measures the departure of observations in a circle. For large value of ,

the observations became very concentrated and are close around the mean direction,

0 . If approaches 0, the distribution tends to converge to a uniform

distribution (Fisher, 1993, p. 49).

3.4.1 Approximation of Circular Normal to Linear Normal Distribution

The restriction that jw

varies between 0 degree and 90 degrees is the first

suggestion it may be regarded as a linear variable. The work by Jammalamadaka and

Sengupta (2001) has shown that a circular random variable θ from a circular normal

distribution with mean direction 0 and concentration parameter , can be

approximated to a linear normal distribution with mean direction 0 and variance

1

when , and can be written as

)1

,(),(~ 00

NwCNw ,

(3.3)

Parameter 0 is estimated using the maximum likelihood using

,0,0 2)/(tan

0 )/(tan

0,0 )/(tan

ˆ

1

1

1

0

CSCS

CCS

CSCS

(3.4)

and the estimate of concentration parameter is approximated by

65

,85.0 34

1

85.00.53 1

43.039.14.0

53.0 6

52

ˆ

23

5

3

RRRR

RR

R

RR

RR

(3.5)

where

n

i iC1cos ,

n

i iS1sin and 22 )/()/( nSnCR is the mean

resultant length (Fisher, 1993, p. 88; Jammalamadaka & Sengupta, 2001, pp. 85-86).

A QQ plot for circular normal distribution which assesses the goodness-of-fit of

sample to the circular normal distribution is one of the graphical tools used to

verify the circular normal distribution. The plotting is done as follows:

1. The quantiles nqq ,,1 of the corresponding best fitted circular normal

distribution, )ˆ,ˆ( CN as estimated in equation (3.4) and (3.5), are

calculated.

2. Then, the sample quantiles

)ˆ(2

1sin iiz , (3.6)

are obtained and rearranged in ascending order.

3. Plot the pair wise of circular normal theoretical and sample quantiles as:

(sin(1

2

1q ),

1z ), . . . , (sin(nq

2

1),

nz ). (3.7)

Tests such as Watson U2 and Kuiper tests also provide another alternative to test

the goodness of fit of the data to the circular normal distribution. The Watsons U2 test

(Fisher, 1993; Mardia & Jupp, 2000) performs a goodness-of-fit test against a

specified distribution, either uniform or circular normal. The cumulative frequency

values niFz ii ,,1),ˆ(ˆ were rearranged into increasing order as

)()1( nzz . The Watson 2U test statistic calculates the mean square deviation

for the fitted distribution given as

66

n

znnizUn

i

i12

1

2

1)2/()12(

2

1

2

)(

2

, (3.8)

where

n

i i nzz1 )( / . If the deviation is too high (resulting in a high U and a low

probability) then the null hypothesis that the data fit the chosen distribution is

rejected. The critical values of this test can be found in Fisher (1993, p. 230).

Kuiper’s Test (Fisher, 1993; Mardia & Jupp, 2000) takes the alternative

approach of directly comparing the distribution of the data to the desired

distribution, either uniform or circular normal. The Kuiper’s statistic is based on the

largest vertical deviations above and below the diagonal line (representing the

desired distribution) given as

)/24.0155.0( 2/12/1 nnVV n , (3.9)

where DDVn ,

2,,

2 ,/)1(max , /max

)()1(

n

iii znizDzniD

,, and )()1( n are the re-arranged i in increasing order )()1( n . The critical

value for the statistic V was given in Fisher (1993). Greater deviation gives high value

of V and low probability, leads to rejection of the null hypothesis that the data fit

the distribution.

3.4.2 Results on Approximation of Circular Variables of the Shape Feature

In this study, the Oriana 2 software (Kovach Computing Services, 1994-2003)

was used to obtain the QQ circular normal plot, calculating the parameters of circular

normal distribution and performing the hypotheses testing for the circular data.

Evidence of the circular normal distribution is illustrated with the QQ circular

normal plot. Figure 3.9 suggests that RRRR wwww 4321 ,,, are following circular normal

distribution with sample quantiles for the entire plots lie on the straight line

obtained from quantiles of theoretical von Mises distribution. The corresponding

67

four landmarks on the left side of the arch gave similar results.

Further confirmation of circular normal distribution was shown from the Watson

U 2 and Kuiper tests. Results from Table 3.4 also showed that all the circular variables

follow the circular normal distribution with a large value of κ. The raw data plot

illustrated in Figure 3.10 clearly shown that RRRR wwww 4321 ,,, were concentrated

around the circular mean (indicated as the bold line) and indicates that the

concentration parameter κ may be large enough to be approximated to linear

variables.

Comparison of mean and variance for LRLRLRLR wwwwwwww 44332211 and ,,,,,, using

circular and linear statistic in Table 3.5 gives similar results indicating that they may be

regarded as linear variables.

(a) Rw1

(b) Rw2

(c) Rw3

(d) Rw4

Figure 3.9: QQ circular normal plot for RRRR wwww 4321 and ,,

68

(a) Rw1

(b) Rw2

(c) Rw3

(d) Rw4

Figure 3.10: Raw data plot for RRRR wwww 4321 and ,,

Table 3.4: The test statistic for the Watson U2 and Kuiper’s tests and estimated

concentration parameter κ. All the variables show non-significant results and indicate

normality.

Test Watson U2 Kuiper

Critical value

Variable

Lower 2.5% = -0.117

Upper 2.5% = 0.117

Lower 2.5% = -1.747

Upper 2.5% = 1.747 Rw1 0.065 1.141 729.812

Lw1 0.031 0.810 779.770

Rw2 0.056 1.184 621.983

Lw2 0.026 0.791 721.246

Rw3 0.067 1.324 546.268

Lw3 0.053 1.034 550.745

Rw4 0.035 0.815 171.063

Lw4 0.100 1.283 163.182

69

Table 3.5: Mean and variance of angular variables using circular and linear statistic.

Statistic Variable Linear statistic Circular statistic

Mean Rw1 85.5090 85.5090

Lw1 85.6593 85.6590

Rw2 77.2581 77.2580

Lw2 77.3090 77.3090

Rw3 69.6219 69.6220

Lw3 69.6192 69.6190

Rw4 38.5546 38.5540

Lw4 38.6323 38.6320

Variance Rw1 0.0803 0.0785

Lw1 0.0751 0.0735

Rw2 0.0942 0.0921

Lw2 0.0812 0.0794

Rw3 0.1073 0.1049

Lw3 0.1064 0.1040

Rw4 0.3431 0.3349

Lw4 0.3597 0.3511

3.5 Dental Arch Symmetry and Dimension Reduction

In the previous section, the angular components in shape feature q have been

shown to have similar properties with linear data. Further analysis of q may now be

incorporated using the analysis of linear statistic. The shape feature q using eight

landmarks is clearly a statistical multivariate problem with dimension 16p .

Dimension reduction will be carried out by investigating the symmetry of the dental

arch. The symmetry of the arch would in turn imply that the four selected teeth is a

reasonable way of representing arch shape.

By definition, symmetry of the dental arch means that identical teeth are facing

each other around the midline, and is studied by comparing the corresponding shape

feature on the left and the right sides of the arch. A summary of the work examining

dental arch symmetry in Appendix C shows ambiguous results. The dental arch is said

to be symmetrical when the shape feature representing the teeth position on one side of

the arch was a mirror image of the other-side (Ferrario et al., 1993; Shrestha &

Bhattarai, 2009; Sprowls et al., 2008; Tong et al., 2012). On the other hand, a certain

70

degree of morphological asymmetry was shown to be present, with smaller asymmetry

degree in Class I compared to Class II and Class III arches (Cassidy et al., 1998;

Scanavini et al., 2012). Further, depending on the measurements used to represent the

dental arches, some of the tests investigated arch symmetry based on size only. It is

therefore of interest to investigate the symmetry of the dental arch based on arch size

and shape simultaneously, using the shape feature q .

3.5.1 Test of Arch Symmetry

Let ),,,,,,,( 44332211

LLLLLLLLL lwlwlwlwv represent the four landmarks on the left

side of a given arch. Similarly let, ),,,,,,,( 44332211

RRRRRRRRR lwlwlwlwv be the

corresponding landmarks on the right side of the same arch. For purposes of notation,

let L

iv represent L

v for the i-th dental cast. Similarly, let R

iv be the corresponding

vector for the right side of the cast. Without loss of generality, if ),(11LL lw and ),(

11RR lw

represent the left central incisor and right central incisor respectively, symmetry of the

arch is implied if RL ll 11

and RL ww 11

. This implication is strengthening if similar

equalities are true for the other three teeth. In total, symmetry of the arch is regarded as

confirmed if 0xAT where

71

1100000000000000

0011000000000000

0000110000000000

0000001100000000

0000000011000000

0000000000110000

0000000000001100

0000000000000011

TA and

R

L

R

L

R

L

R

L

R

L

R

L

R

L

R

L

l

l

w

w

l

l

w

w

l

l

w

w

l

l

w

w

4

4

4

4

3

3

3

3

2

2

2

2

1

1

1

1

x .

If x can be shown to be multivariate normal such that ),(~ Σμx pN , then a test for

symmetry is equivalent to a test on the hypothesis 0μA T.

Since x is a 16 dimensional vector with sample size of 47, evidence of

multivariate normality was shown using the Kolmogorov-Smirnov test (Dudewicz &

Mishra, 1988) to investigate univariate normality for each element of x and the Chi-

square plot.

Let 47,,2,1, ii

T

i xAy where T

A is a )( qp matrix and denote the

normality of the x vectors as 47,,2,1),,(~ iNqi Σμx . Therefore, ) ,(~ yy Σμy pi N

where μAμTy and AΣAΣ y

T (Mardia et al., 1979).

Define nn

yyyy 21

1 and Tr

n

r

r yyyyS 1

y as the unbiased

estimators of yy and Σμ and )

1,(~ yy Σ0μyn

N p . Let

),(~ yy Σ0μyu pNn , (3.10)

and

yy ,1~ ΣSW nWp, (3.11)

72

follow the Wishart distribution.

The test statistic for the one sample Hotelling T2 test statistic under

0:0 μATH is given as

,)1(

)()())(1(

)()()1(

)1(

1

y

1

y

y

1

y

12

ySy

μAySμAy

μySμy

uWu

T

TTT

T

y

T

nn

nn

nnn

nT

(3.12)

such that )(,2

)(

)1(~ pnpF

pn

pnT

where p =8 and n = 47.

3.5.2 Results for Test of Symmetry

Kolmogorov-Smirnov tests (Dudewicz & Mishra, 1988) showed that each

element of TRLRLRLRLRLRLRLRL llwwllwwllwwllww ),,,,,,,,,,,,,,,( 4444333322221111x is univariate normal

(Table 3.6). Further evidence of multivariate normality of x is indicated in the Chi-

square plot (Figure 3.11).

Figure 3.11: Chi square plot for testing 16-variate normal distribution.

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Chi-square quartiles (theoretical)

Squ

are

d d

ista

nce

(O

bse

rvati

on)

73

Table 3.6: Results of Kolmogorov-Smirnov (KS) goodness-of-fit test for testing

normality of residuals. Critical value of KS statistic used is 0.1743.

Variables KS test statistic

Lw1 0.0973

Rw1 0.067

Ll1 0.0691 Rl1 0.0722 Lw2 0.0826

Rw2 0.0639

Ll2 0.0650

Rl2 0.0869

Lw3 0.0928

Rw3 0.0849

Ll3 0.1028

Rl3 0.0635

Lw4 0.0729

Rw4 0.0922

Ll4 0.0790

Rl4 0.1373

Table 3.7 illustrates the result of the Hotelling T2 test. The test appears to accept

0H at 5% significant level which gives evidence of maxillary arch symmetry.

Therefore, using landmarks on one side of the arch only is a reasonable way of

representing arch shape.

Table 3.7: Test statistic for the Hotelling one-sample T2 test

Critical value T2statistics

Lower 2.5% = 2.4547

Upper 2.5% = 23.9492 8.3441

74

3.6 Categories of Dental Arch Shape

Studies on arch size and shape of people from different ethnic groups in

different populations have shown varying results. Significant variations in a specific

population such as in Italian Caucasian and Chilean, American blacks and whites and

among the Southern Chinese were found, while no differences could be found between

the Caucasian and Japanese, and Caucasian and non- Caucasians (Burris & Harris,

2000; Ferrario et al., 1999; Ling & Wong, 2009; Nojima et al., 2001; Radmer &

Johnson, 2009).

When investigating arch size and shape for disparities, some studies have at the

outset segregated the arches according to ethnic groups and then compared the average

values measured in each group using univariate two sample t-test or other appropriate

discriminant analysis tests (Ferrario et al., 1999; Ling & Wong, 2009; Nojima et al.,

2001). It was concluded that the arches were discriminated according to ethnic groups

when the statistical tests on individual variables show significant differences. Other

studies used the multivariate approach, whereby the arches were grouped using cluster

analysis and subsequently the ethnic and gender homogeneity for each cluster

established was investigated (Hao et al., 2000; Raberin et al., 1993). Though many

studies have made an attempt to find groups of dental arches and relate them to ethnicity

and arch shape, the uniqueness of the groups was not thoroughly verified (Hao et al.,

2000; Lee et al., 2011; Preti et al., 1986).

Malaysia is a multi-racial country with citizens comprising of three major ethnic

groups; the Malays, Chinese and Indians. In 2010, the Malays formed 50.1% of the

population, the Chinese 22.5% and the Indians 6.7%. The rest of the population in the

country is made up of other indigenous groups and non Malaysian citizens (Malaysia,

2012). It is advantageous to study differences in arch size and shape of the ethnic

75

population, although intermarriages do occur in Malaysia (4.6% of the population)

(Nagaraj, 2009).

This section explores the existence of groups of dental arches applied on

),,,,,,,( 44332211RRRRRRRRR

i lwlwlwlwv , i = 1,..,47 which in turn validates that the v -vectors

is a reliable shape feature for the dental arch.

3.6.1 Clustering Method

An indicator of arch shape variation is the existence of groups or clusters of the

v -vectors. Agglomerative hierarchical clustering (AHC) is the most commonly used

and efficient method in describing pattern similarities and differences in the data to

reveal characteristics given in homogenous groups or clusters. A measure of distance or

separation between the i-th dental cast and the j-th dental cast is as follows (Anderberg,

1973):

244

244

211

211 )l(l+)w(w++)l(l+)w(w=j)d(i, jijijiji . (3.13)

The whole set of distances may be expressed in the matrix:

.

0121

03,23,1

02,1

0i0,

)nd(n,)d(n,)d(n,

)d()d(

)d(

=j),D(

(3.14)

The first step in the AHC method was to merge two dental casts with the

smallest ),( jid value between them into the same group. The distance between this

new group and the remaining original dental casts was then redefined using one of the

76

following methods (Anderberg, 1973; Everitt et al., 2001; McKenzie & Goldman,

1998):

M1) Average linkage method; ),( jid = average distance between all pairs of

dental casts in cluster-i and cluster-j.

M2) Centroid linkage method; ),( jid = distance between centroid of cluster-

i and cluster-j.

M3) Complete linkage method;

),( jid = maximum distance between dental

casts in cluster-i and cluster-j.

M4) McQuitty’s linkage method; ),( jid = average distance from cluster-i to

other cluster (e.g., cluster-l) and cluster-j to cluster-l.

M5) Median linkage method; ),( jid = median distance between all pairs of

dental casts in cluster-i and cluster-j.

M6) Single linkage method; ),( jid = minimum distance between dental casts

in cluster-i and cluster-j.

M7) Ward’s linkage method; ),( jid = minimum error sum of squares of

cluster-i and cluster-j.

A summary of clusters formed from the control samples using methods M1 to

M7 with selected similarity values are shown in Table 3.8. The problem of misfit of

impression trays with patients’ mouths suggests that there should be at least 2 clusters

of dental arches. Further, previous studies discussing morphology of dental arches

reported about 3 to 5 different shapes based on geometrical representations (de la Cruz

et al., 1995; Raberin et al., 1993). It is seen that only with the complete linkage method

was there a realistic number of groups or clusters of dental arch shapes (2 to 6) for a

similarity level of up to 50%.

Using the linkage method, the set of distances after the merger would be found

in the matrix ),,1( jiD . The next step in the AHC method was to combine two dental

77

casts (or two groups of dental casts) with the smallest ),( jid obtained from the matrix

),,1( jiD . This process of combining two groups and merging their characteristics was

repeated until all dental casts were placed in one large group. The result of this

hierarchical cluster analysis is shown graphically in a dendrogram (Figure 3.12), where

all the samples are listed and the level of similarity showing how any two clusters were

joined are indicated. In Figure 3.12, the horizontal axis indicates the position of the

dental casts (relative to each other) whilst the height of the vertical axis is a measure of

the disparity among the casts. The similarity level at the m-th merger of clusters is

defined as (McKenzie & Goldman, 1998):

100% x 0,in max

,in min1

j)i,D(j)d(i,

j)i,D(mj)d(i,=S(m) . (3.15)

A large )(mS value suggests dental cast-i and dental cast-j (or two groups of dental

casts) have similar arch shape. The number of clusters may be chosen by selecting the

height of the vertical axis which represents the cut-off point.

The means of ),,,,,,,( 44332211

RRRRRRRRR

i lwlwlwlwv were denoted as

),,,,,,,()( 44332211

RRRRRRRR lwlwlwlwk v where kGk ,,1 and

kG is the number of group

for a given similarity value, may also be graphically represented by a box containing the

means (Figure 3.13).

78

Table 3.8: Number of clusters formed using the available linkage methods (M1 to M7)

at selected percentage of similarity levels.

Linkage

method

Similarity

Level (dij)

M1 M2 M3 M4 M5 M6 M7

1% 1 1 2 1 1 1 5

5% 1 1 2 1 1 1 5

10% 1 1 2 1 1 1 5

15% 1 1 2 1 1 1 5

20% 1 1 2 1 1 1 5

25% 1 1 3 1 1 1 5

30% 1 1 3 1 1 1 6

35% 1 1 3 1 1 1 8

40% 1 1 4 2 1 1 8

45% 2 1 5 2 1 1 8

50% 2 1 5 3 1 1 10

55% 2 1 6 4 1 1 10

60% 5 1 8 5 1 1 10

65% 7 1 9 7 2 1 11

79

Number of

clusters

Similarity

level

Dendrogram

2

1% - 20%

3

21% - 38%

4

39% - 44%

5

45% - 50%

Figure 3.12: Dendrograms showing number of clusters at different cut-off levels

obtained using complete linkage method.

4523371446313428423629413541644303826212027223233134011172518152431924743121039659871

0.00

33.33

66.67

100.00

Observations

Similarity

DendrogramComplete Linkage, Euclidean Distance

4523371446313428423629413541644303826212027223233134011172518152431924743121039659871

0.00

33.33

66.67

100.00

Observations

Similarity


4523371446313428423629413541644303826212027223233134011172518152431924743121039659871

0.00

33.33

66.67

100.00

Observations

Similarity


4523371446313428423629413541644303826212027223233134011172518152431924743121039659871

0.00

33.33

66.67

100.00

Observations

Similarity


80

2 clusters

(1 - 20%)

3 clusters

(21 - 38%)

4 clusters

(39 - 44%)

5 clusters

(45 - 50%)

Figure 3.13: The pyramid of cluster boxes where each cluster is represented by the

vector of means

83.16

57.30

74.85

54.91

67.27

51.49

36.82

36.43

87.30

59.51

79.43

57.13

72.21

53.30

43.76

36.97

85.55

53.55

77.07

51.61

69.15

48.35

36.11

33.91

84.75

54.80

76.33

52.71

68.53

49.39

36.35

34.75

87.30

59.51

79.43

57.13

72.21

53.30

43.76

36.97

83.16

57.30

74.85

54.91

67.27

51.49

36.82

36.43

84.66

51.76

75.84

49.74

67.55

46.32

32.69

32.70

87.30

59.51

79.43

57.13

72.21

53.30

43.76

36.97

86.05

54.57

77.78

52.68

70.07

49.50

38.06

34.60

83.16

57.30

74.85

54.91

67.27

51.49

36.82

36.43

84.66

51.76

75.84

49.74

67.55

46.32

32.69

32.70

84.59

61.83

76.92

59.75

70.21

55.87

42.50

38.80

86.05

54.57

77.78

52.68

70.07

49.50

38.06

34.60

88.38

58.58

80.44

56.08

73.00

52.27

44.26

36.24

81

3.6.2 Definitive Number of Clusters

The definitive number of clusters is determined when each group resulting from

a chosen cut-off point is well separated, rather than overlapping with another group

(Everitt et al., 2001). To assess the uniqueness of the formed clusters, a 2D graphical

method using the first two principal components of v vectors and the validity indices

were carried out.

3.6.2.1 Principal Component Analysis

Let Tk kkkk )()( )()( vvvvS be the sample covariance of clusters

),,1( gkGk resulting from a chosen cut-off point where g is the number of clusters and

)(kv are v vectors from kG . Let kn

S1

1

be an estimate of the cluster population, say

kΣ . The spectral decomposition of the estimate of kΣ is given by T

Q ΛQ where

)|||( 821 qqqQ and ),,,(diag 821 Λ are the matrices of eigenvectors and

eigenvalues respectively (Anderson, 2003).

Henceforth, each v vector in )(kv is represented by its first and second principal

components, namely i

Tvq1

and i

Tvq2

. The two principal components of all )(kv was then

plotted in a 2D scatter plot. The same procedure was repeated for different cut-off

points, where 3 and 4 clusters resulted.

The plots are shown in Figure 3.14(a) – 3.14(c) whereby the 2 principal

components explained about 75% of the variation of )(kv . Figure 3.14(a) and Figure

3.14(b) shows 2 and 3 distinct clusters respectively. In Figure 3.14(c), overlapping of

members is seen in two of the 4 clusters formed. They were not clearly separated and

may be considered as one cluster. Therefore, the sample studied strongly suggests the

existence of 3 distinct groups of dental arches.

82

(a)

(b)

(c)

Figure 3.14: Investigation of separation of clusters using the first two principal

components indicates the existence of three distinct groups of dental arches.

-120 -100 -80 -60 -40 -20 0 20 40-150

-100

-50

0

50

100

150

1st PC

2nd

PC

G1

G2

-150 -100 -50 0 50-60

-40

-20

0

20

40

60

80

100

120

1st PC

2nd

PC

G1

G2

G3

-150 -100 -50 0 50-200

-150

-100

-50

0

50

100

150

1st PC

2nd

PC

G1

G2

G3

G4

83

3.6.2.2 Dunn’s Validity Index

Apart from the principal components, the Dunn’s validity index was also used to

establish the definitive number of clusters formed. The Dunn’s validity index validates

the separation and compactness of the groups which resulted from a chosen cut-off

point and is defined as

,))((max

))(),((minmin

1

11

k

mkD

gk

gmgk v

vv (3.16)

where )(kv are v vectors from kG and where g is the number of clusters. Further,

),(max))((,

jiG

dkkji

vvvvv

and ),(max))(),((,

jiGG

dmkmjki

vvvvvv

are the intra-cluster

diameter and inter-cluster linkage respectively and ),( jid vv is the Euclidean distance

between iv and jv (Bolshakova & Azuaje, 2003). The optimal number of clusters is

suggested by the maximum value of D.

Table 3.9 gives the Dunn’s validity index for all possible number of clusters

which suggests 3 groups as an optimum number of clusters. The 3 clusters, denoted as

kG (k= 1, 2, 3), have group sizes 11, 22 and 14 respectively. Table 3.10 shows the

percentage of each cluster membership according to gender and ethnicity. There is a

mixture of gender and ethnicity which further suggests that this information may be

derived by well defined groups of arch shape using the shape feature v . The v -vector

has successfully captured the shape of the dental arch and may be regarded as a reliable

shape feature for the dental arch.

Table 3.9: Validation of number of clusters using Dunn’s index.

Number of clusters, g Dunn’s Index, D

2 Clusters 1.2561

3 Clusters 1.2977 (max)

4 Clusters 1.1064

5 Clusters 1.0365

6 Clusters 1.1163

84

Table 3.10: Ethnic and gender homogeneity for each cluster.

Cluster

Total

membership

(percentage)

Ethnic

Percentage

of

ethnicity

Gender Percentage

of gender

1G 11 (23.40%)

Malay (n = 6) 54.54% Male (n = 2) 18.18%

Chinese (n = 5) 45.45% Female (n = 9) 81.81%

2G 22 (46.81%)

Malay(n = 18) 81.81% Male (n = 5) 22.72%

Chinese (n = 4) 18.18% Female (n = 17) 77.27%

3G 14 (29.79%)

Malay (n = 10) 71.42% Male (n = 7) 50%

Chinese (n = 4) 28.57% Female (n = 7) 50%

3.6.3 Results on Categories of Dental Arch Shape

Three categories of arch shape (1G ,

2G and 3G ) have been established from

meticulous handling of the clustering methods. Their respective means are

)1(v (83.16, 57.30, 74.85,54.91,67.27,51.49,36.82,36.43) ,

)91.33,11.36,35.48,15.69,61.51,07.77,55.53,55.85()2( v ,

)97.36,76.43,30.53,21.72,13.57,43.79,51.59,30.87()3( v .

Interpolation of the shape feature using means of the 4 teeth used in this study with the

mirror image of the other half of the arch gives the general shape of the dental arch. The

shapes of the arch for kG (k= 1, 2, 3), illustrated in a 2-dimensional is shown in Figure

3.15. The shape of 2G and 1G are almost similar, however 2G is smaller than 1G and 3G

is medium sized. 3G also differs from 1G and 2G as it has more taper in the frontal arch.

Figure 3.16 further shows separation of the three categories of shape when each shape

feature was considered separately. For the purpose of comparison of arch dimension

with other studies, the mean anterior and posterior widths and lengths for the arches in

each of the shape category are shown in Table 3.11.

85

Figure 3.15: Mean shapes of kG (k= 1, 2, 3).

Figure 3.16: Boxplot of each shape feature for kG (k= 1, 2, 3).

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

60

70

width (mm)

leng

th (

mm

)

G1

G2

G3

theta1 length1 theta2 length2 theta3 length3 theta4 length4

30

40

50

60

70

80

90

Valu

es

G1

G2

G3

86

Table 3.11: The anterior and posterior width and length for each of the shape category.

Shape

category

Length (mm) Width (mm)

Anterior Posterior Total Anterior Posterior

1G 9.5 25.5 57.0 40.0 58.0

2G 8.5 27.5 59.0 48.0 63.0

3G 9.0 27.0 65.0 36.5 60.5

3.7 Discussion

A novel shape feature ),,,,,,,( 44332211RRRRRRRRR

i lwlwlwlwv of the dental arch was

derived from digital images. Precautions were taken when digitizing the dental casts,

whereby the base of the cast were standardized, distortion of the images were avoided

or minimized, calibration of the measurements were made each time the shape feature

was measured and the shape features were aligned to enable comparison between arch

shape. The shape features were accurately measured, as shown by the small standard

deviation values of the measurements (Table 3.2 and Table 3.3).

Nakatsuka et al. (2001) also used a combination of linear and angular variables

as shape feature which were obtained from different landmarks. However, unlike this

study, the properties of shape feature were not carefully investigated, particularly the

treatment of the angular variables. This study on the other hand addressed this issue and

found that the angular variables may be regarded as a linear measured vector therefore

enabling investigation of arch symmetry using the usual linear statistic (Table 3.5).

The symmetry of the v -vectors was also investigated. Most studies which

investigated arch symmetry have commonly used univariate tests such as two sample t

test and paired t test for normally distributed data, and Wilcoxon sum rank and signed

rank tests for non-normal data (Cassidy et al., 1998; Maurice & Kula, 1998; Shrestha &

Bhattarai, 2009; Šlaj et al., 2003; Tong et al., 2012). However, it should be noted that

the use of two sample t test may not be appropriate since variables depicting shape

features on the left and right sides of the arch are clearly dependent. Further, the use of

87

several univariate tests on each variable individually makes the overall error rate greater

than error rate per test (also known as the Bonferroni inequality). In this study, all the

variables indicating the shape feature were tested simultaneously using the paired

Hotelling T2 together with thorough check on its assumptions to ensure that the decision

made about the null hypothesis was reliable (Table 3.6 and Figure 3.11). Investigation

of arch symmetry in turn allowed reduction of dimension of the shape feature.

Clustering methods using the proposed shape feature yields 3 categories of

dental arch shape (Figure 3.15). Although the mean of the shape feature for each shape

category were illustrated, this study also reported on the arch width and length to enable

comparison with other studies (Table 3.11). The first shape category can be described as

being widest on the posterior and anterior widths with medium arch length. The second

shape category has medium posterior and anterior widths and the shortest arch length.

On the other hand, the third shape category has narrowest posterior and anterior widths

with the longest arch length.

Differences in clustering analyses may contribute to the different results

obtained. Lee et al. (2011) and Raberin et al. (1993) used the partitional clustering

method that requires the user to specify the number of clusters, which was set from a

subjective viewpoint. Nakatsuka et al. (2011) on the other hand employed similar

hierarchical clustering methods used in this study. However the choice of Ward linkage

as distance measure for merging the clusters was not properly justified.

The strength of this study in investigating categories of the dental arch shape

was on the proper statistical analysis in each step of the clustering analysis technique

used. Further verification of the groups established also ensured that the groups are

deemed meaningful (Figure 3.14 and Table 3.9). This chapter considers the categories

of arch shape without the knowledge on how the variation of shape in each category

88

deviates from its mean shape. Three shape models will be established in the following

chapter to describe the variation in each of the shape category.

89

CHAPTER 4. SHAPE MODELS OF THE DENTAL ARCH: AS A GUIDE IN

DETERMINING APPROPRIATE IMPRESSION TRAYS FOR ORAL

DIAGNOSIS AND TREATMENT PLANNING

4.1 Introduction

A clustering method applied on ),,,,,,,( 44332211

RRRRRRRRR

i lwlwlwlwv , i = 1,..,47 in

Section 3.7 has established 3 categories of the v -vectors denoted as )3,2,1)(( kkv .

Three shape models can be defined if the mean and covariance of )(kv can be derived

(Hufnagel, 2011). The probability distribution of v-vectors, if exists, may in turn allow

more accurate inferences to be made.

The shape models were derived in the following procedure. Firstly, the

performance of several multivariate normality tests for small sample size and high

dimensional data were investigated before each shape category is tested for multivariate

normality. The existence of 3 meaningful shape models were tested to investigate if the

three MVN models possess significant differences in shape. To ensure that the shape

models are valid, verification study was carried out using 75 test samples. The

knowledge about these shape models were then used in the design of 3 impression trays.

A modified COVRATIO statistics which incorporates the problem of small

sample size and minimal model assumptions was proposed as a discrimination method

of shape and compared to the linear discrimination method. Using this knowledge, a

guide for determining appropriate impression tray for a patient (without and with a

missing tooth) was proposed.

90

4.2 Shape Models of Maxillary Dental Arch

Since there was evidence of normality distribution for the shape feature (section

3.5.2), three shape models can be obtained if each of the shape categories )3,2,1( kGk

can be shown to have a multivariate normal distribution and possess significant

differences in shape. However, the sample size of each category kG )3,2,1( k was

small and possesses relatively high dimension. Some multivariate normal tests may

have failed to reject normality. The following subsection investigated the performance

of several multivariate normality tests in the case of small sample size and high

dimension, followed by the test of normality for the three shape categories.

4.2.1 A Simulation Study on Performance of Multivariate Normality Tests for

Small Sample Size

In real life situations, particularly in shape analysis, smaller sample sizes relative

to numbers of landmarks selected are likely to occur. Therefore, the assumption of

multivariate normality and the validity of the MVN tests are most critical. The sample

size for category 1G ,

2G and 3G were 11, 22 and 14 respectively. Before performing

multivariate normality tests on )3,2,1)(( kkv , the effect of sample size as small as

10n and higher dimension up to 8p which mimic the sample size and dimension

of the v-vectors from kG was investigated. Five multivariate normal (MVN) tests

which possess high power from the literature were considered (see section 2.6.1 to

2.6.4):

1. Mardia’s skewness (MS)

2. Mardia’s kurtosis (MK)

3. Doornik and Hansen (DH)

4. Henze-Zirkler (HZ)

91

5. Royston

The null hypothesis of the above tests was, 0H : Data set follow a MVN

distribution, ),( ΣμpN , and the performance of the tests was evaluated according to:

1. Type I error rates against the MVN distribution

2. Power of test against non-MVN distribution

4.2.1.1 Type I Error Rates

To estimate the Type I error rates for MS, MK, DH, HZ and Royston tests, the

following steps were taken:

Step 1. A random sample from ),(~ Σμx pN with n = 10 and p = 2,4,8 was generated

using R random number generator function. The number of parameters to be

fixed is )2( pp . To reduce the number of parameters to be fixed, let μ be a p

zeros vector and A be the eigenvector of Σ . Therefore,

),(~ D0Ax pN , (4.1)

where

pd

d

00

0

0

001

AAΣD is the canonical form of Σ and ),,( 1 pdd d

is the corresponding eigenvalues (Mardia et al., 1979, p. 62 & p. 214). Now, the

number of parameters is reduced to p . The parameter ),,( 1 pdd d is fixed such

that the covariance

pppp

p

p

21

22221

11211

Σ demonstrates various possible data

dispersion as follows:

92

Set A: pIΣ

Set B: The diagonal and off-diagonal elements of Σ are 8 and 0

respectively.

Set C: The diagonal and off-diagonal elements of Σ are 8 and 3

respectively.

Set D: The diagonal and off-diagonal elements of Σ are 8 and 5

respectively.

Step 2. Obtain the p-value from the test statistic of MS, MK, DH, HZ and Royston

tests using the generated data as described in step 1 (Korkmaz et al., 2015; Aho,

2015).

Step 3. The above process of getting the p-value was repeated for s = 10000 times.

Step 4. The proportion of 10000 samples for which the test rejects the null hypothesis

0H at 05.0 significance level was calculated. This is equivalent to

calculating the probability of making Type I error:

trueis |reject 00 HHP . (4.2)

The corresponding error rate was then obtained.

Step 5. Similar steps as above were repeated to calculate the Type I error rates for

sample size n = 20 and 50.

Examples of generated data using the above set of parameters for p = 2 are

illustrated in Figure 4.1 and the above steps were also repeated for p = 4 and 8.

93

Example of generated MVN data using

parameter set A. The data are expected to

distributed around 0 with covariance

10

01Σ .


parameter set B. The data are expected to


80

08Σ .


parameter set C. The data are expected to


83

38.


parameter set D. The data are expected to


.85

58

Figure 4.1: Examples of different sets of generated MVN data for p = 2.

x y

z

94

Table 4.1: Empirical Type I error rate (in percentage) against the MVN distribution

with dimension p = 2 for different sets of parameters.

n MVN test Set A Set B Set C Set D

10 Mardia’s skewness (MS) 0.35 0.35 0.51 0.38

Mardia’s kurtosis (MK) 0.00 0.00 0.00 0.00

Doornik & Hansen (DH) 4.32 4.18 4.15 4.51

Henze-Zirkler (HZ) 3.40 3.32 3.66 3.61

Royston 5.79 5.32 5.80 5.67




Henze-Zirkler (HZ) 4.49 4.26 4.20 4.51

Royston 6.30 6.20 6.26 6.20




Henze-Zirkler (HZ) 4.80 4.64 4.95 4.86

Royston 6.50 6.40 6.46 6.40







Henze-Zirkler (HZ) 3.17 2.88 2.96 3.43

Royston 5.23 5.14 5.98 5.49




Henze-Zirkler (HZ) 4.60 4.50 4.53 4.40

Royston 7.18 6.63 7.39 7.35




Henze-Zirkler (HZ) 5.28 5.28 5.03 5.05

Royston 7.11 6.90 7.15 7.25

95







Henze-Zirkler (HZ) 6.39 6.72 6.55 6.62

Royston 4.96 5.12 6.26 6.36




Henze-Zirkler (HZ) 6.02 5.85 5.85 5.95

Royston 7.92 7.80 8.95 8.90




Henze-Zirkler (HZ) 5.06 5.01 5.58 5.32

Royston 7.38 7.80 9.00 9.07

Table 4.1, Table 4.2 and Table 4.3 show the empirical Type I error rates against

the generated multivariate normal data. Since the 0H is true, each test should reject

the0H at about the nominal rate of 5%. MS and MK tests are far below from reaching

5% when sample size is small. With increasing n, MS test appear to show reliable

performance, however it breaks down as p gets higher. MK test on the other hand is

reaching 5% as the p and n gets higher. DH test appears to be the best performer when

different sets of parameters were considered, whereby it consistently possess close to

5% Type I error rate, followed by HZ test with higher variation compared to DH. The

estimates for the Royston were also relatively good in most of the cases, with a slight

higher error from the nominal rate. In a particular case where n is small and p is high

(Table 4.3), DH test perform best followed by Royston and HZ tests.

4.2.1.2 Power of Test

To estimate the power of tests between MS, MK, DH, HZ and Royston tests, the

following procedure was carried out:

Step 1. Alternative distributions which reflect non-normality of n = 10 were generated

96

using R random number generator functions. The distributions considered are:

(a) Multivariate t distribution with 10 degrees of freedom (d.f) and covariance

pppp

p

p

21

22221

11211

Σ , where the diagonal and off-diagonal elements of Σ

are 8 and 3 respectively, for p = 2, 4, 8, and 16 (Genz and Azzalini, 2015).

This distribution represents very mild departure from MVN distribution.

Figure 4.2 shows an example of generated data from multivariate t

distribution.

(b) Uniform distribution ),( baU with minimum value a and maximum value b

were set as 36 and 87 respectively. The properties of ),( baU are symmetric

and having lower kurtosis than the MVN distribution. Figure 4.3 shows an

example of generated data from multivariate uniform distribution.

(c) Multivariate lognormal distribution with parameters 0μ and pIΣ . The

property of this distribution is highly skewed and represents a drastic

departure from MVN distribution (Boogaart et al., 2015). Figure 4.4 shows

an example of generated data from multivariate lognormal distribution.

Step 2. Obtain the p-value from the test statistic of the MS, MK, DH, HZ

and Royston tests using the generated data as described in step 1 (Korkmaz et

al., 2015; Aho, 2015).

Step 3. The above process of getting the p-value was repeated for s = 10000 times.

Step 4. The proportion of 10000 samples for which the test rejects the null hypothesis

0H at 05.0 significance level was calculated. This is equivalent to calculating the

probability of not making a Type II error:

false is |reject 1 00 HHP . (4.3)

97

The corresponding error rate in percent was then obtained as the power of the

test.

Step 5. Similar steps as above was repeated to calculate the power of tests for

n = 20 and 50.

Figure 4.2: Example of generated multivariate t distribution for d.f = 10 and

83

38Σ .

Figure 4.3: Example of generated )87,36(U data for p = 2.

98

Figure 4.4: Example of a multivariate lognormal distribution generated for 0μ and

2IΣ .

Figure 4.5 shows the power estimates for the MVN test against multivariate t

distribution. For small sample size of n = 10 the rate of rejection for all tests was very

low and generally below 20% due to mild differences between multivariate t and normal

distribution. As n and p increases, Royston and MS tests perform best with power of test

around 70%. This result supports the findings of Farrell et al. (2007) with an added

knowledge on how the tests behave when an extremely small sample size with higher

dimension data set is considered.

99

(a) n = 10

(a) n = 20

(c) n = 50

Figure 4.5: Empirical power for MS, MK, DH, HZ and Royston test statistic against

multivariate t distribution.

2 3 4 5 6 7 8

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston

Power of test against MV t distribution with n=10

0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston

Power of test against MV t distribution with n=20

0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston

Power of test against MV t distribution with n = 50

100

Results for the empirical power for the 5 MVN tests against multivariate

uniform distribution are given in Figure 4.6. As one would expect, MS had virtually no

power against symmetric alternatives whereby the power of the test is consistently 0%

for all cases. Other tests show low power of rejecting multivariate uniform distribution

when n is small. The Royston test produced the best result especially as n and p

increases. At n = 50, the power of Royston, DH and MK tests were nearly 100%, except

for HZ. Our results for the Royston and DH tests differ noticeably from those in Farrell

et al. (2007). Specifically, when n = 50, a decrease of DH and Royston’s power can be

evident in Farrell et al. (2007). This may be due to the differences of parameter used to

generate the multivariate uniform data set, which was not given in Farrell et al. (2007).

(a) n = 10


multivariate uniform distribution.

2 3 4 5 6 7 8

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston

Power of test against MV uniform distribution with n=10

101

(b) n = 20

(c) n = 50

Figure 4.6, continued.

Figure 4.7 illustrates the results of the power of MVN tests against the

lognormal distribution. The multivariate lognormal is a heavily skewed distribution that

is a drastic departure from normality. Here, the power of the tests of MVN was expected

to be very high, close to 100%. For small sample size, Royston’s had the power of

nearly 100%, while the power of other tests dipped as low as 0% when p increases.

When n = 10 and 20, the MK and MS tests decreased as p increases. This may

be due to higher dimension data relative to the sample size which resulted in singularity

of the covariance matrices. This reason is supported as MK and MS showed consistent

rate of rejection which are close to 100% when the sample size greatly increases to n =

0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston


0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston


102

50. Results in this study are similar to Mecklin and Mundfrom (2003) and Farrell et al.

(2007), where as n increased to 50, the power of all procedures was at least 99.9%.

(a) n = 10

(b) n = 20


lognormal distribution.

2 3 4 5 6 7 8

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston

Power of test against MV lognormal distribution with n=10

0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston


103

(c) n = 50


4.2.1.3 Summary of Simulation Results for Testing Multivariate Normality

The above simulation study considers 5 MVN distribution tests which were

recommended by many studies. Results for n = 50 in this study were mostly in

agreement with studies by Farrell et al. (2007) although the error and power rates may

have slight differences. However, results for n = 10 and 20 cannot be compared as, to

the best of our knowledge, no studies have considered these cases in the literature.

This study showed that for small sample size, DH is the best performer when

only Type I error is considered. HZ and Royston tests on the other hand perform well in

the sense that their Type I error rates are consistently small in all considered cases.

Further, their power of estimates against severe departure of non-normality is

considerably high, ranging from 70% to 100%. It is therefore suggested that multiple

tests should be carried out when testing for multivariate normal distribution, especially

when dealing with data from small sample size and higher dimension. Some tests may

not be working well due to non-singularity of the sample covariance matrix.

0 2 4 6 8 10 12 14 16

Dimension, p

0

20

40

60

80

100

Pow

er

of

test

MS

MK

DH

HZ

Royston


104

4.2.2 Multivariate Normal Distribution of Categories of Shape

The MVN contour and univariate Kolmogorov-Smirnov (KS) tests were initially

carried out to investigate the multivariate normality of each shape category. The 50%

probability ellipsoid appears to capture half of the observations (Table 4.4). The 60%

ellipsoid and 70% ellipsoid show an increasing number of observations being captured.

This result may be considered as weak evidence of multivariate normality in view of the

small sample sizes. On the other hand, each of the univariate variables was shown to be

normally distributed using the KS test (Table 4.5). These results strongly suggest the

existence of multivariate normality of )(kv for each shape category )3,2,1( kGk.

Since DH, HZ and Royston tests managed to perform well in the simulation

study for the case of small n and high p (Table 4.3 and Figure 4.7), these tests were

employed to further confirm the multivariate normality for each shape category. Table

4.6, Table 4.7 and Table 4.8 show evidence of multivariate normality for each kG .

Table 4.4: The proportion of observations satisfying )())(())(( 21 pii kk vvSvv in

each shape category kG . Arbitrary values of 0.7 and 0.6 ,5.0 were used giving

3441.7)5.0(28 , 8.3505)6.0(2

8 and 9.5245)7.0(28 .

Shape category

1G 1G

1G

5.0 11

5= 0.4545

22

11=0.5000

14

6= 0.4285

6.0 11

6= 0.5455

22

12=0.5455

14

7= 0.5000

7.0 11

11= 1.0000

22

15=0.6818

14

9= 0.6429

105

Table 4.5: The T values for KS test with sample size .14,22,11 321 nnn

Shape category 1G 2G

3G

Critical value at 5%

sig. level

Variables

0.3912 0.2809 0.3489

1w 0.1800 0.1392 0.1787

1l 0.0946 0.1070 0.1425

2w 0.1457 0.0926 0.2213

2l 0.2298 0.1178 0.1534

3w 0.1642 0.1360 0.1927

3l

0.1927 0.1663 0.1400

4w 0.2293 0.1259 0.1709

4l 0.0980 0.1380 0.1683

Table 4.6: Test statistic to investigate multivariate normality using Doornik and Hansen

(DH) test. Lower and upper 2.5% critical values are 6.9076 and 28.8453 respectively.

Category DH Test statistic

1G 27.2909

2G 10.1494

3G 23.5899

Table 4.7: Test statistic to investigate multivariate normality using Henze-Zirkler (HZ)

test.

Category Critical Value HZ Test statistic

1G Lower 2.5% = 0. 8186

Upper 2.5% = 0. 9614 0.9357

2G Lower 2.5% = 0.8499

Upper 2.5% = 0.9795 0.8870

3G Lower 2.5% = 0.8300

Upper 2.5% = 0.9683 0.9181

106

Table 4.8: Test statistic to investigate multivariate normality using Royston’s test.

Category Critical Value Royston’s Test

statistic

1G Lower 2.5% = 1.0944

Upper 2.5% = 13.9102 3.4711

2G Lower 2.5% = 0.7789

Upper 2.5% = 12.6017 3.1804

3G Lower 2.5% = 0.2140

Upper 2.5% = 9.3334 3.6874

Therefore, the three fitted categories of shape models are given as

3,2,1 ),(~)(: kkMVNk kk Svv , (4.4)

where the means are:

36.43), 36.82, 51.49, 67.27, 54.91, 74.85, 57.30, 83.16,()1( v

33.91), 36.11, 48.35, 69.15, 51.61, 77.07, 53.55, 85.55,()2( v

36.97) 43.76, 53.30, 72.21, 57.13, 79.43, 59.51, 87.30,()3( v , and the shape variability

can be explained by the estimated covariance matrices

47.4.......

40.398.7......

12.110.195.1.....

85.189.387.057.2....

20.127.188.188.004.2...

19.021.257.078.147.024.2..

52.059.005.258.021.238.092.2.

15.037.159.037.139.005.233.001.2

1S

,

33.3.......

98.243.9......

23.239.412.4.....

51.091.363.101.3....

84.100.478.365.173.3...

15.090.220.153.227.134.2..

26.137.375.370.184.353.162.4.

21.083.158.006.278.095.104.178.1

2

S

and

.

73.2.......

52.133.6......

03.204.011.4.....

37.195.101.264.2....

99.123.089.320.206.4...

64.108.251.296.274.254.3..

68.157.085.304.297.354.283.4.

72.152.191.279.210.350.399.270.3

3

S

107

Using the above results, the variation of a particular tooth may be illustrated as

Figure 4.8. The mean shape and variation for 1G ,

2G and 3G are shown in Figure 4.9,

Figure 4.10 and Figure 4.11 respectively.

Figure 4.8: Mean location of a particular tooth and its variations.

Figure 4.9: Mean shape and variation of

1G .

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

60

70

width (mm)

leng

th (

mm

)

where,

𝐴 = (��𝑗 , ��𝑗)

𝐵 = (��𝑗 + 𝜎𝑤𝑗 , 𝑙�� − 𝜎𝑙𝑗

)

𝐶 = (��𝑗 + 𝜎𝑤𝑗 , 𝑙�� + 𝜎𝑙𝑗

)

𝐷 = (��𝑗 − 𝜎𝑤𝑗 , 𝑙��+𝜎𝑗)

𝐸 = ( ��𝑗 − 𝜎𝑤𝑗 , 𝑙�� − 𝜎𝑗)

E

C

D B

𝑙��

��𝑗

A

Width (mm)

(mm

Length (mm)

(mm

108


2G .


3G .

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

60

70

width (mm)

leng

th (

mm

)

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

60

70

width (mm)

leng

th (

mm

)

109

4.2.3 Test of Separation between the Shape Models

The existence of 3 meaningful MVN shape models can be obtained if these

models are unique and possess significant differences in shape. The Hotelling two

sample T2 test (Johnson & Wichern, 1992) was used to compare mean vectors from two

populations. However since the sample size of each category kG (k=1, 2, 3) is

relatively small, further assumptions have to be met, that is, both populations are MVN

with equal covariance matrix (Johnson & Wichern, 1992). The equality of covariance

matrices was imposed to reduce the number of parameters in the model and

consequently helped in reducing the error of the T2 test. Since the normality assumption

was investigated earlier, the latter was further investigated.

4.2.3.1 Test for Equality of Covariance Matrices

This section investigated the equality of covariance matrices using the Box test

(Stevens, 2001). Let g be the number of shape category, N be the total sample size, kn

and 1 kk nr be the sample size and degrees of freedom of the k-th shape category,

respectively. Let the null hypothesis for the three MVN covariance matrices be

3210 : H . (4.5)

The Box’s M test statistic is defined as

g

k

kkpooled rgNM1

||ln||ln)( SS , (4.6)

where,

g

k

kkpooled

gN

n

1

)1( SS is the sample covariance matrix for the pooled data,

1

))()())(()((

k

T

kn

kkkk vvvvS is the corresponding covariance matrix for category-k and

p is the dimension of the v-vector.

110

When the sample size 20kn , MF is approximately distributed as an F

distribution with r and 0r degrees of freedom, where rrrCF 0/1 ,

2

)1)(1(

gppr ,

2

0

0

2

CC

rr

,

g

k

k gNrg

ppC

1

22

0 )/(1/1)1(6

)2)(1(,

g

k

k gNrgp

ppC

1

22

2

)/(1/1)1)(1(6

132. The test statistic was found to be 1.004

which is between the lower and upper 2.5% critical value, namely

3580.1004.16982.0 , indicating the covariance matrices are equal.

Therefore, the three categories of fitted shape models may be now written as

3,2,1 ),(~)(: kkMVNk pooledk Svv , (4.7)

where the common shape variability is

41.3.......

75.119.8......

92.136.263.3.....

26.133.338.080.2....

74.113.238.334.045.3...

37.050.203.049.209.067.2..

21.191.139.334.051.306.030.4.

64.063.144.012.245.043.231.040.2

pooledS . (4.8)

4.2.3.2 Investigating Separation of Mean Shape using the Hotelling T2 test

Let the null hypothesis for two mean vectors be 210 : μμ H . The Hotelling two

sample test statistic with 21 is

)()'( 21121

21

212 vvSvv

poolednn

nnT , (4.9)

such that )1(,

21

212

211

)2(~ pnnpF

pnn

pnnT

. Table 4.9 clearly shows rejection of

0H and

indicates meaningful existence of 3 MVN models with significant differences in shape.

111

Table 4.9: Hotelling 2T test for comparing two multivariate means.

Mean vector Critical value HotellingT2 statistics

)2()1( vv Lower 2.5% =2.6179

Upper 2.5% =28.7177 75.0389

)3()1( vv

Lower 2.5% =2.8213

Upper 2.5% =35.9355

113.8008

)3()2( vv

Lower 2.5% =2.5714

Upper 2.5% =27.2745

124.0771

4.2.4 Verification of the Arch Shape Models

The 3 MVN shape models were established using a relatively small sample of 47

casts. To ensure that the shape models are valid, verification study were carried out after

obtaining another 75 test samples.

The shape features v from 47 control casts and 75 test samples were pooled

together and denoted as iv where .122,,1i Using these v -vectors, 3 categories of

shape (denoted as 3,2,1 ,122 lGl ) together with its mean (denoted as 3,2,1 ,)( 122 kkv )

were obtained from AHC with complete linkage and the multivariate normal

distribution and equality of the covariance matrices of 3,2,1 ,122 lGl were investigated.

Then, the 3,2,1 ,)( 122 llv were compared to 3,2,1 ),( kkv from the 47 control casts.

If the hypothesis testing under lklkH for )()(: 122

0 vv fails and reject the 0H

lk for , these would imply the validity of the 3 shape models established using the 47

control sample.

The 3 shape categories 3,2,1 ,122 kGk , with group sizes 26, 40 and 56

respectively, followed the MVN distribution when tested using DH, HZ and Royston

normality tests (Table 4.10, Table 4.11 and Table 4.12). The test statistic for Box test

(Stevens, 2001) was found to be 1.1499 which is between the lower and upper 2.5%

critical value, namely 0.7001< 1.1499<1.353, indicating the covariance matrices

112

122

3

122

2

122

1 ΣΣΣ are equal. Table 4.13 shows the p-values of Hotelling two sample 2T

test when comparing the means of shape using 47 and 122 samples. This result verifies

the validity of the 3 shape models established using the 47 control sample.

Table 4.10: Doornik and Hansen (DH) test to investigate multivariate normality of the

shape categories using 122 casts. Lower and upper 2.5% critical values are 6.9077 and

28.8454 respectively.

Category DH Test statistic 122

1G 14.9283

122

2G 15.5736

122

3G 13.1403

Table 4.11: Henze-Zirkler (HZ) test to investigate multivariate normality of the shape

categories using 122 casts.

Category Critical Value HZ Test statistic

122

1G Lower 2.5% = 0.8570

Upper 2.5% = 0.9830 0.9138

122

2G Lower 2.5% = 0.8740

Upper 2.5% = 0.9910 0.9819

122

3G Lower 2.5% = 0.8850

Upper 2.5% = 0.9960 0.9505

Table 4.12: Royston’s test to investigate multivariate normality of the shape categories

using 122 casts.

Category Critical Value Royston’s Test

statistic

122

1G Lower 2.5% = 1.1698

Upper 2.5% = 14.1979 1.4984

122

2G Lower 2.5% = 0.9001

Upper 2.5% = 13.1270 8.5142

122

3G Lower 2.5% = 0.7636

Upper 2.5% = 12.5331 6.088

113

Table 4.13: p-value of the Hotelling two sample 2T test for comparing 3 clusters using

47 casts and 122 casts.

Mean vectors for k-th cluster )1(v )2(v )3(v

122)1(v 0.9046 0 0

122)2(v 0 0.4137 0

122)3(v 0 0 0.6956

4.3 Application of Shape Models to Impression Tray Design

4.3.1 Fabricating Three Impression Trays

This section presents the application of the shape models of the dental arch

defined by MVN distributions in designing three impression trays for the population

studied. Existing stock trays were generally U-shaped and the dimensions include space

for an adequate thickness of the impression material all around the dental arch (Figure

4.12(a)). To provide impression trays which would be suitable for the population

studied, three trays with mean dimensions derived from each of the three shape models

were produced. Each tray was fabricated based on the knowledge of the corresponding

mean shape models together with the maximum deviation obtained from common

covariance matrix to indicate the variability of the arch shape.

In the k-th category of arch shape, select the i-th dental cast and let d(i), i=1,.., kn

denote the distance between the two hamular notches. Further, denote the average of

these distances by kd . On the Cartesian x-coordinate, the points 0,5.0 kd and

0,5.0 kd represent the position of the two hamular notches. Using the vertical line

passing through the origin as the y-axis, the four selected teeth can be represented by

),( 11 lw , ),( 22 lw , ),( 33 lw and ),( 44 lw where for example, 1w represent the average of 1w

for all casts in a given category. The remaining four points on the other side of the arch

were similarly derived due to the symmetry of the arch (see Section 3.5). Straight lines

114

joining the eight teeth positions and the two hamular notches give good approximation

to the arch shape.

An additional 5 mm was added to the outside perimeter on both sides of each

shape, thereby increasing the transverse distance of the simulated casts by 10 mm. The

5 mm increase in width on either side of the arch was added taking into account a

maximum standard deviation of 3 mm, and an allowance of 2 mm for the impression

material to be used in the stock tray (Hatrick et al., 2003; McCabe & Walls, 1998)

(Figure 4.12(b)). Light cure material (Kemdent Works, Wiltshire, UK) was then used to

make three impression trays for each shape model and were labelled as C1, C2 and C3

(Figure 4.12(c)).

(a) (b) (c)

Figure 4.12: (a) An example of a stock impression tray with space around the tray to

allow for variation of arch sizes and adequate thickness of alginate impression material.

(b) Mean shape v was indicated by solid line. The broken lines indicate 5mm added to

the perimeter of the arch shape. (c) The light cure acrylic resin tray.

4.3.2 Verification of the Fabricated Impression Trays

The fabricated trays C1, C2 and C3 should be verified by inspecting whether or

not they were constructed according to the designed measurements. To confirm this, the

membership of the 47 control casts which belong to each of the 3 shape models were

tracked from the dendrogram (see Figure 3.12 in section (3.6.1)). All casts which made

up the shape model )1(v should fit tray C1. Similarly, casts which belong from shape

models )2(v and )3(v should fit tray C2 and C3 respectively.

A particular cast is said to fit the corresponding tray adequately if the amount of

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

5mm

115

impression material between the inside edge of the tray to the buccal surfaces of the

teeth is at least 2 mm (Hatrick et al., 2003; McCabe & Walls, 1998). Plasticine was used

as a simple method (instead of using impression material), to examine the space for the

material impression in the fabricated tray. This was done by first loading a tray with

plasticine and a cast was then placed vertically into the tray. When placing the cast

vertically into the loaded plasticine tray, the distance of the tray to the labial surface of

the central incisors on the cast was ensured to be about 3 mm. Once properly seated,

the cast was removed gently from the plasticine, to ensure minimal distortion of the

plasticine. Finally, a minimum of 2 mm amount of the plasticine between the inside

edge of the tray to the buccal surfaces of the teeth was ensured by a dentist to confirm

that the tray is suitable for the cast.

Table 4.14 shows that the minimum amount of plasticine in the impression tray

for all casts using its corresponding fabricated tray was more than 2 mm, indicating all

47 casts fit their corresponding trays. This result shows that the trays were constructed

according to its corresponding shape model and therefore verifies the fabricated trays.

Table 4.14: Amount of plasticine thickness in the impression tray and number of cast

which fit the fabricated trays C1, C2 and C3 when impression of 47 control samples

was taken.

Tray

Material

thickness(mm)

C1 C2 C3

<2 0 0 0

2-4 4 14 9

4-9 7 8 5

>9 0 0 0

Number (percentage) of

casts which fit the tray 11 (23.4%) 22 (46.8%) 14 (29.8%)

Further, the ability of the fabricated impression trays to fit the Malaysian

population was carried out by investigating the fit of the trays on 40 dental casts which

were randomly chosen from the 75 test casts (the remaining 35 test casts was used for

116

another verification study in the later part of this chapter). The impression of each test

cast was taken using the same procedure as above, and the thickness of the plasticine in

the impression tray was examined. In a situation when the condition was not satisfied,

the procedure was repeated using the other 2 trays. The trays were considered suitable

when the cast could go into a tray, regardless of whether the teeth were in good

alignment or not (Figure 4.13). The result illustrated in Table 4.15 show that all 40 test

casts could fit in at least one of the fabricated trays.

Table 4.15: Amount of plasticine thickness in the impression tray and number of cast

which fit the fabricated trays C1, C2 and C3 when impression of 40 test samples was

taken.

Tray

Material

thickness(mm)

C1 C2 C3

<2 0 0 0

2-4 1 7 6

4-9 2 12 10

>9 0 0 2



Figure 4.13: Two of the mal-aligned arches in the 40 samples used for verification of

the three fabricated trays.

117

4.4 Discrimination of the Dental Arch Shape

Three impression trays were fabricated using the three MVN shape models

)(kv in equation (4.7). An immediate question is how to discriminate which tray is

suitable for a particular patients’ arch shape. This section aims at developing a

discrimination procedure for the shape of the maxillary dental arch using the established

statistical shape models.

By definition, discrimination is an act of assigning object to categories or classes

(Costa & Cesar, 2009). The PCA may provide information for discrimination as it

explains the variability of the data that quantifies shape differences. However the main

limitation of the use of PCA is that it does not consider class separability since there is

no mention of class label of the shape. Therefore, assigning new objects to a particular

class may be difficult to accomplish.

The nearest neighbours method is non-parametric and the simplest

discrimination approach. It basically identifies the sample in a set of N samples that is

closest to a new object and takes its class. However, the performance of this method is

generally inferior to the likelihood function method as it does not supply information

about the probability density function of the class (Costa & Cesar, 2009).

Since the probability distribution of the shape has been identified as MVN with

equal covariance matrices, the discriminant function, in particular the linear

discrimination function (LDF) can be used (Uetani et al., 2015). Let )47,1( iiv be the

i-th control cast and )(vkf be the probability density function of the k-th population.

The LDF is given as:

118

jkpooledpooled

jkjk

jk

jk

j

kkj

jkjkjk

jjkkjk

jjjkkk

jjvp

kkp

ff

f

fh

SSvvSvvvSvv

vSvvSvvSvvSv

vSvvSvvSvvSv

vvSvS

vvSvvS

vv

v

vv

)).()(())'()((2

1))'()((

)()'(2

1)()'(

2

1)'()'(

)()'(2

1)'()()'(

2

1)'(

))(())'((2

1||log

2

1)2log(

2

))(())'((2

1||log

2

1)2log(

2

)(log)(log

)(

)(log)(

11

1111

1111

1

1

(4.10)

where jk .The discriminant rule for the 2 populations is as follows (Johnson &

Wichern, 1992; Mardia et al., 1979)

if 0)( vkjh , then kv ,

if 0)( vkjh , then jv , and (4.11)

The Gaussian and equality of covariance matrices assumptions for the

discriminant function are particularly restrictive and sometimes difficult to satisfy. If

violated, it may reduce the performance of discriminant analysis. The following section

therefore proposes COVRATIO as an alternative discrimination method which uses the

determinant ratio of two covariance matrices that are not necessarily from Gaussian

distribution. Subsequently, comparison between the LDF and )( iCOVRATIO was carried

out via simulation study.

4.4.1 Proposed )( iCOVRATIO as Discrimination Method

The COVRATIO procedure dates back to Belsley et al. (1980). They proposed a

numerical statistic to identify the presence of influential observation in linear regression

models. This numerical statistic is based on the determinantal ratio given as:

119

COV

COV iCOVRATIO i

, (4.12)

where COV is the determinant of covariance matrix for full data set and ( )iCOV is

that for the reduced data set by excluding the thi observation. If the ratio is close to 1,

then there is no significant difference between them. In other words, the thi observation

is consistent with the other observations. Alternatively, if the value of ( ) 1iCOVRATIO

is close to or larger than a derived cut off point, then it indicates that the thi observation

is a candidate of an outlier. This procedure has been extended and employed for other

models such as the simple linear functional relationship model and circular regression

model (Uddameri et al., 2014; Ghapor et al., 2014; Ibrahim et al., 2013; Abuzaid et al.,

2008).

In this study, the idea of COVRATIO statistics for outlier detection was extended

for the purpose of discrimination. Let 1Σ and 2Σ be the pp covariance matrix for

population 1 and 2 respectively, where each of the population has a known

probability distribution. The determinant of 1Σ and 2Σ , denoted as 1Σ and

2Σ gives

the descriptive measures of multivariate variability (Peña and Rodrıguez, 2003). Next,

instead of using reduced observation as formulated by Belsley et al. (1980), define the

ratio of covariance determinant for 1 as:

Σ

Σ

1

)(1

)(1

i

iCOVRATIO

, (4.13)

where Σ)(1 i

is the determinant of covariance matrix )(1 iΣ which is estimated by adding

new thi observation in 1 .

120

The same thi observation will be added in 2 and its

)(2 iCOVRATIO is obtained

using

Σ

Σ

2

)(2

)(2

i

iCOVRATIO

. (4.14)

Smaller value of )( iCOVRATIO (which is close to 1) indicates no variation when new

observation was included in a particular population. In other words, the thi observation

is consistent with the other observations and therefore belongs to the population.

Therefore, the discrimination rule using )( iCOVRATIO for 2 populations is given

as:

(4.15)

The above result can be generalized for g populations, given as:

(4.16)

4.4.2 Simulation Study for Comparing Performance of LDF and )( iCOVRATIO

The performance of LDF and )( iCOVRATIO was examined via simulation study.

Random samples from two MVN populations were generated and re-assigned to their

corresponding population using LDF and )( iCOVRATIO

. The misclassification

Allocate new thi observation in 1 if

)(2)(1 ii COVRATIOCOVRATIO ,

Else, allocate to 2

Allocate new thi observation in

j if )()( ihij COVRATIOCOVRATIO ,

where ghj ,,2,1, and hj .

121

probabilities using these discrimination methods were then compared. This procedure

was done as follows:

Step 1: Two sets of random samples with sample size n from two unequal p-variate

normal distributions; 1111 ,~: ΣμX pN and 2222 ,~: ΣμX pN were generated

using R random number generator.

Step 2: Generate another two random samples A and B which were drawn from 1 and

2 respectively.

Step 3: Calculate the LDF in equation (4.10) using random sample A and B , denoted

as )(12 Ah and )(12 Bh respectively.

Step 4: Calculate )()(1 AiCOVRATIO , )()(1 BiCOVRATIO , )()(2 AiCOVRATIO and

)()(2 BiCOVRATIO using equation (4.13) and (4.14), by adding random sample A and

B in both sets of random samples generated from 1 and 2 in step 1.

Step 5: Discrimination rule for LDF and )( iCOVRATIO (in equations (4.11) and (4.15))

were used to determine the membership of A and B .

Step 6: The above steps were repeated for s = 10000 times.

Step 7: The proportion of A and B which does not assign to their corresponding true

population when discriminated using LDF and )( iCOVRATIO may be regarded as the

misclassification probability.

Table 4.16 shows the performance of LDF and )( iCOVRATIO when different values

of n, p, μ and Σ were considered. In general, the misclassification probability for LDF

and )( iCOVRATIO reduces when a particular value of

2μ was chosen to indicate larger

separation between 2 and

1 . The dispersion of the data in a particular population also

122

dictates the misclassification probability, whereby larger value of Σ may result in higher

chances of a sample to be assigned to an incorrect population. Increasing value of n

gives slightly higher misclassification probability. On the other hand, increasing value

of p gives smaller misclassification probability. However at 8p , the LDF breaks

down and shows extremely high misclassification probability.

Overall, the )( iCOVRATIO method gives smaller probability misclassification as

compared to the LDF in all cases, indicating better performance of discrimination. The

proposed )( iCOVRATIO is theoretically preferable, because unlike the LDF, it does not

require Gaussian and equality of covariance matrices assumptions, which may be

difficult to attain. It also uses the raw data instead of descriptive statistics when

discriminating therefore retaining the information about the population. Moreover, due

to its simplicity, the i)(COVRATIO can be employed when discriminating more than 2

groups. The drawback of )( iCOVRATIO

method is that it is computationally more

expensive compared to LDF, however with technology advancement in available

statistical software, such computation is straightforward and incredibly fast to compute.

123

Table 4.16: Misclassification probability for LDF and )( iCOVRATIO when different

sample size, dimension, mean vector and covariance matrices were considered.

Parameters 1 2

LDF )( iCOVRATIO

LDF )( iCOVRATIO

10

01

, 0

0

,2,20

21

1

21

ΣΣ

μ

pnn

2

22μ 0.0741 0.0622 0.0672 0.0577

3

32μ 0.0141 0.0049 0.0149 0.0046

4

42μ 0.0015 0.0001 0.0013 0

10

01

4

4,

0

0

,2,20

1

21

21

Σ

μμ

pnn

5.10

05.12Σ 0.0022 0.0005 0.0093 0.0005

20

022Σ 0.0026 0.0012 0.0203 0.0021

5.20

05.22Σ 0.0031 0.0022

0.0301

0.0070

30

032Σ 0.0031 0.0031 0.0460 0.0114

10

01

2

2,

0

0 ,2

21

21

ΣΣ

μμp

1021 nn 0.0640 0.0530 0.0634 0.0504

2021 nn 0.0741 0.0622 0.0672 0.0577

5021 nn 0.0820 0.0663 0.0799 0.0636

100

0

10

001

,

2

2

,

0

0

,20

21

21

21

ΣΣ

μμ

nn

2p 0.0741 0.0622 0.0672 0.0577

4p 0.0177 0.0087 0.0163 0.0104

6p 0.0040 0.0018 0.0050 0.0015

8p 0.4000 0 0.9000 0

124

4.4.3 Results for Discrimination of Dental Arch Shape

The membership of the 47 control samples which belong to each of the 3 shape

models were tracked from the dendrogram (see Figure 3.12 in section (3.6.1)). The LDF

and )( iCOVRATIO were employed to re-assign these casts into one of 3 populations of

the shape models. Their misclassification probability was calculated to ensure that these

methods are deemed good for shape discrimination.

Table 4.17 shows the misclassification probabilities when the samples were re-

assigned to *

2

*

1 , or *

3 using the LDF and )( iCOVRATIO

. A relatively low

misclassification probabilities for LDF were obtained, however )( iCOVRATIO

outperformed the LDF method by giving zero misclassification probabilities. These

results demonstrate the ability of the )( iCOVRATIO to discriminate the dental arch shape

correctly and can be regarded as a better discrimination method.

Table 4.17: Misclassification probability when 47 dental casts were re-assigned into

either one of the population of the shape model using the LDF and )( iCOVRATIO .

Discrimination

True

population

*1 *

2 *3

LDF )( iCOVRATIO LDF )( iCOVRATIO

LDF )( iCOVRATIO

1 - - 0.18 0 0.18 0

2 0.09 0 - - 0.09 0

3 0.21 0 0.07 0 - -

125

4.5 Investigating Dental Arch Shape with Missing Tooth: A Simulation Study on

Performance of Data Augmentation and Expectation Maximization

In usual clinical practice, a dentist will choose an impression tray that to his

observation may best fit the patient’s arch. If the resultant cast is not adequate for

restoration to be constructed upon it, a custom impression tray will have to be

constructed from this cast. The construction of custom tray is time consuming, incur

more expenses and suitable for only one particular patient (Patil et al., 2008).

Shape discrimination using )( iCOVRATIO may provide a good guide (see Section

4.4.3) to assign a suitable tray for a particular patient; however, this is only possible if

the teeth used as shape descriptor v are available. Since missing tooth occurs for a

variety of reasons, assigning a suitable tray to a particular shape of dental arch may be

problematic if a tooth is lost.

This section considers the data augmentation (DA) and expectation

maximization (EM) to impute any missing values. The performance of the imputation

method was evaluated if a particular method has lower misclassification probability

when discriminated into its true shape category. The effect of sample size n, number of

missing value 0n and specific missing tooth was investigated through the following

steps:

Step 1. A random sample of n = 10 which mimics one of the three shape models was

generated from

3,2,1 ),(~)(: kkMVNk pooledk Svv (4.17)

where the parameters were given in section (4.2.2). Consider k = 1 and let the

data denoted as

126

44332211

2424232322222121

1414131312121111

,,,,,,,

,,,,,,,

,,,,,,,

nnnnnnnn lwlwlwlw

lwlwlwlw

lwlwlwlw

, (4.18)

Step 2. Missing values of size 20 n were introduced into the generated data at the j-th

tooth ( 4,3,2,1j ). Since one tooth was defined as a pair wise variable,

therefore the missing values for the j-th tooth and true population 1 are denoted

as follow

44332211

3434333332323131

2424232322222121

1414131312121111

,,,,,,,

,,,,,,,

,,,,,,,

,,,,,,,

nnnnnnnn

NANA

NANA

lwlwlwlw

lwlwlwlw

lwlwlwlw

lwlwlwlw

, (4.19)

Step 3. The estimated values from DA (section 2.9.1) and EM (section 2.9.2) were

imputed to the missing values.

Step 4. The sample with imputed values from DA and EM were then assigned to the

population using the )( iCOVRATIO in section (4.4.1).

Step 5. If any samples with imputed DA or EM were not correctly classified to their

true population, the number of misclassification for the first simulation, s is

counted as 11 misn , else 01 misn . The initial value was arbitrarily taken as 30

with 50 iterations and the stopping criterion for EM and DA was set as 0.1.

Step 6. The above process of counting ),,1( sinmis

i was repeated for s = 10000 and

the misclassification probability using EM and DA was calculated as

s

n

icationmisclassifP

s

i

misi

1)( . (4.20)

Step 7. The above steps were repeated to calculate )( icationmisclassifP as a performance

127

indicator of the imputation method for pair wise cases of 6,4,20 n and

50,20,10n respectively.

Table 4.18: The misclassification probability using DA and EM for missing values

imputation when the number of missing values 20 n .

n

j-th tooth

10 20 50

DA EM DA EM DA EM

j=1 (central incisor tooth) 0.4809 0.2495 0.4965 0.2321 0.4917 0.2167

j=2 (lateral incisor tooth) 0.5099 0.4159 0.4979 0.4045 0.4961 0.4090

j=3 (canine tooth) 0.6010 0.4765 0.5804 0.4627 0.5821 0.4717

j=4 (1st molar tooth) 0.3219 0.2313 0.3210 0.2231 0.3227 0.2177



n

j-th tooh

10 20 50

DA EM DA EM DA EM



j=3 (canine tooth) 0.8176 0.7130 0.8259 0.7118 0.8317 0.7152

j=4 (1st molar tooth) 0.5354 0.4130 0.5345 0.3879 0.5452 0.3990



n

j-th tooh

10 20 50

DA EM DA EM DA EM



j=3 (canine tooth) 0.8977 0.8324 0.9234 0.8426 0.9243 0.8467

j=4 (1st molar tooth) 0.6465 0.5358 0.6842 0.5310 0.6953 0.5330

Table 4.18 to Table 4.20 illustrate the results. The misclassification probability

increases when the number of missing values increase, with EM performs best in all

cases, particularly for larger n. This simulation study also shows that the

misclassification probabilities are highest when canine tooth was missing, followed by

lateral incisor, central incisor and molar teeth. This indicates the hierarchy of the teeth

and their weight in determining the dental arch shape.

128

4.6 Application of Shape Discrimination

4.6.1 A Proposed Guide for Determining Appropriate Impression Tray for

Patients (Without and With Missing Tooth).

This section proposed a guide for determining appropriate fabricated impression

tray for clinical use. The method is as follows:

1) An intraoral camera connected to a PC is inserted into a patient’s mouth. A

small ruler is placed near the teeth while capturing the dental arch (Figure

4.14).

2) The image is directly transferred to the PC.

3) A program developed in MATLAB software is used to import and also to

calibrate the image, obtain the shape descriptor of the patient’s dental arch,

and employ the )( iCOVRATIO to assign a shape model or tray which best fits

the patient’s arch shape (sections 3.3 and 4.4.1).

4) If a missing tooth is detected, the estimation of the missing values using EM

will be imputed into the shape descriptor (section 4.5). However, there may

be 0.2177 to 0.601 probabilities (depending on which tooth is missing - see

Table 4.18) that the tray assigned does not fit the patient, and minor

modifications to the tray may be required.

129

Figure 4.14: A small ruler was placed near the teeth (for image calibration) while

capturing the 2D image of the patient’s dental arch by using an intraoral camera.

4.6.2 Verification of the Proposed Guide for Determining Appropriate Impression

Tray

The proposed method for determining appropriate impression tray was verified

to ensure that the guide is satisfactory for use on patients. The v-vector shape feature of

the remaining 35 test samples were obtained and discriminated using )( iCOVRATIO to

assign each test cast to any one of the 3 fabricated trays (or shape models). Then, the

impression of each test cast was made using its corresponding tray and the thickness of

the plasticine (see section 4.3.2) in the impression was used to indicate the fit of the tray

to cast. Table 4.21 shows the percentage of the cast that fit the assigned tray.

130

Table 4.21: Plasticine thickness in the impression and number of casts (n=35) which fit

the fabricated trays C1, C2 and C3.

Tray

Space(mm)

C1 C2 C3

<2 0 0 3

2-4 5 16 4

4-9 1 4 1

>9 0 0 1



* Three casts (8.58%) could not go into any of the assigned tray.

4.8 Discussion

Three shape models resulting from the clustering method were used to construct

three impression trays. In the verification study, 91.42% of an independent random

sample of 35 casts could fit into at least one of the three impression trays (Table 4.21).

The remainder 8.58% were regarded as outliers, being too long or too wide for them to

go into any of the trays. Although only four points were used to define the arch shape,

the ability of the impression trays to capture 91.42% of the test dental casts showed that

as far as impression tray design is concerned, the MVN shape model is adequate. This

is inspite of some of the test casts had teeth which were not well aligned in the dental

arch.

A simulation study on missing tooth shows that the selected 4 teeth used to

describe the arch shape may be arguable, and that all 18 teeth should represent the arch

shape of the maxillary arch for a more accurate arch shape representation. If all teeth

were used, the MVN shape model will clearly fail because of dimensionality problems.

Other techniques should be introduced for example using Fourier descriptor to represent

the maxillary arch shape. However, a more complex probability shape model needs to

be defined. Nevertheless, the v-vector as shape descriptor is capable of capturing the

general shape of the dental arch and assigning arch shape to its correct population.

131

The value of the study in this chapter was: (1) the establishment of 3 categories

of arch shape with mean and variation for each shape category; (2) verification that the

shape models and trays produces can accommodate adequately randomly selected casts.

(3) Proposal of a guide to select suitable impression tray for a particular patient. In the

next chapter, a more precise arch shape representation of the dental arches is considered

for rehabilitating the edentulous patients.

132

CHAPTER 5. SHAPE MODELS OF THE DENTAL ARCH: AS A GUIDE IN

ESTIMATING NATURAL TEETH POSITIONS ON COMPLETE DENTURES

FOR THE EDENTULOUS

5.1 Introduction

The knowledge of teeth positions on the dental arch and the categories of arch

shape are important factors in restoring aesthetics and function of the edentulous

patient. Limitations to the shape feature ),,,( 4,43,312,21,1 lwlwlwlwv proposed in the

previous chapter may be that only 4 teeth were considered and were regarded to be

symmetrical. It may be argued that 4 selected teeth on the standardized digital images

of the dental casts could be considered as insufficient with respect to representing shape.

However, increasing the number of teeth would create problems with dimensions and

proof of existence of the multivariate normal distribution is extremely difficult.

This chapter investigates the ability of Fourier descriptors (FD) to represent all

maxillary teeth as alternative shape models. To avoid the curse of dimensionality, a

relatively small number of FD terms are used to enable the formulation of a new shape

model. Using these FDs, the 3 categories of shape established from v were verified, and

tested for multivariate complex normality as its shape model. A hypothesis testing for

two sample means from MVCN based on the Hotelling 2T test was derived and

employed to confirm the existence of shape models that possess significant differences

in shape.

133

A shape discrimination procedure for the maxillary dental arch using the

established statistical shape models for MVCN was developed using the modified

COVRATIO statistics. Then, three anatomical landmarks which remain in edentulous

arch were linked to the 3 categories of MVCN shape models. A guide for estimation of

natural teeth positions on complete dentures for the edentulous patient was then

proposed. Test casts were used to verify the proposed guide whereby the estimated teeth

positions using the MVCN shape model will be compared to the original teeth positions.

5.2 Shape model of the dental arch using Fourier Descriptor (FD)

5.2.1 The Ability of Fourier Descriptor (FD) in Representing Dental Arch Shape

The same control samples of 47 standardized digital images of dental casts were

used. Twenty one points representing the dental arch shape were rearranged in an anti-

clockwise sequence starting from the origin and denoted as

)]1(),1([)],...,1(),1([)],0(),0([ NyNxyxyx (Figure 5.1). Each coordinate pair can be treated

as a complex number so that )()()( kjykxks , where k = 0, 1,…, N-1 and N=21.

The discrete Fourier transform of )(ks is

2

exp)(1

1

0

N

k

uN

kjuks

Na

, (5.1)

1,...,1,0for Nu (Oppenheim et al., 1983). The complex coefficients ua are known as

the Fourier descriptors (FD) of the boundary. The set of FD for each dental cast was

denoted as ],,,[ 110iN

iii aaa A , 47,,1i where

iu

iu

iu jdca , (5.2)

134

The inverse Fourier transform of these coefficients restores )(ks where

1,,1,0 Nk , as given by

1

0

2exp )(

N

u

uN

kjuaks

. (5.3)

A pilot study was carried out to investigate the ability of FD terms to

approximate the shape boundary. The results are illustrated in Figure 5.2. It can be seen

that the boundaries of the arch shape are closer to the original as the number of selected

FD terms increase. An approximation of boundary using all 21 FD terms gives exact

original boundary of the dental arch shape.

Figure 5.1: An example of a particular dental cast which shows the hamular notches

(HN) and incisive papilla (IP) that were used to establish the Cartesian axes. Twenty

one selected points as illustrated in the diagram were used as the shape boundary.

135

(a) Plot of the original and

approximation of boundaries using

2 FD terms

(b) Plot of the original and


3 FD terms

(c) Plot of the original and


4 FD terms

(d) Plot of the original and


5 FD terms

Figure 5.2: The ability of 2, 3, 4, 5, 6, 19, 20 and all 21 FD terms in representing the

dental arch shape.

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

136

(e) Plot of the original and


6 FD terms

(f) Plot of the original and


19 FD terms

(g) Plot of the original and


20 FD terms

(h) Plot of the original and


all 21 FD terms


5.2.2 Selecting Number of FD Terms as Shape Feature

Each of the amplitude of ua given by

22uuu dca , (5.4)

indicates the size implicating the synthesized signal and may be regarded as

quantification of contribution or weight in representing the boundary of the arch shape

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

137

(Mikami et al., 2010). Larger amplitudes account for global shape, and small ones carry

fine detail of a boundary (Gonzalez & Woods, 2002). An example of magnitude plot of

ua , were plotted in Figure 5.3. The first 6 and the last 2 terms show higher contribution

in representing the boundary of the arch shape.

(a) Sample N022

(b) Sample N006

Figure 5.3: The magnitude plot for 2 casts as an example. The first 6 and the last 2

terms shows higher contribution in representing the boundary of the arch shape.

The q largest ua values were selected from the magnitude plot and its

corresponding ua were re-substituted in (5.3), whilst the remaining terms were

ignored. The 21 points 21,,1, rqrx obtained by using q of

ua terms where Nq ,

have to be compared with the corresponding 21 points from the original boundary,

21,,1, rorx when all ua terms were considered. A measure of similarity between

these two shapes will show the appropriateness of the choice of q as the shape feature of

the dental arch.

0 5 10 15 20 250

100

200

300

400

500

600

700

u

Am

plitu

de

, |a

u|

0 5 10 15 20 250

100

200

300

400

500

600

700

u

Am

plitu

de

, |a

u|

138

Since the images are aligned, the Procrustes distance

PD

N

r

qr

or

qr

or

1

)()'( xxxx , (5.5)

can be used as a similarity measure (Zhang & Lu, 2001; Stegmann and Gomez, 2002).

The value of q that minimizes PD will be regarded as the optimal choice of q.

The ability of the 8 FD terms to represent the 21 points of the teeth position is

illustrated in Figure 5.4. The Procrustes distance decreases rapidly when q changes from

2 to 8 and gradually levels off for increasing q. The 8-FD terms used were

i

q

ii aaa ˆ , ,ˆ ,ˆˆ21 A , (5.6)

where iu

iu

iu djca ˆˆˆ and qu ,,1 , may be used as an approximation of shape boundary.

The 8 selected FD are from the first 6 and the last 2 terms (Figure 5.3).

Figure 5.4: The Procrustes distances between the original 21 points and their estimated

positions using q-FD terms gradually levels off at q=8.

Figure 5.5 illustrates 2 cases of the original 21 points of the dental arch overlaid

with the 21 points derived from the 8-FD terms.

0 5 10 15 20 250

5

10

15

20

25

30

35

q-FD terms

Pro

cru

ste

s d

ista

nce

(m

m)

Cast number N006

Cast number N022

q=8

139

(a) Sample N022

(b) Sample N006

Figure 5.5: Plot of the original arch shape oix and the shape boundary q

ix approximated

using q = 8 FD terms.

5.2.3 Categories of Arch Shape Using FD Shape feature

Three categories of arch shape (1G ,

2G and 3G ) established in section 3.6.3 may

be used to define categories of shape when 8 FD was used to represent the shape feature

of the dental arch. Let each of the membership in the corresponding category

represented as ,ˆ , ,ˆ ,ˆˆ21

i

q

ii

i aaa A where 11,,1i for the first category, 33,,12 i

for the second category, and 47,,34 i for the third category with )3,2,1(ˆ ggA as

the respective means.

Table 5.1 shows descriptive statistics for the total casts and casts segregated

according to each shape category. Smaller values of standard deviation can be seen in

each category as compared to the total casts. This result supports the existence of three

categories of shape. The mean shapes for the 3 shape categories using FD,

iii aaa 821 ˆ , ,ˆ ,ˆˆ A as shape feature are illustrated in Figure 5.6. Shape feature using FD

provides additional information of all locations of teeth and the hamular notches instead

of only 8 teeth locations provided when the shape of the dental arch was represented by

),,,,,,,( 44332211RRRRRRRRR

i lwlwlwlwv .

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

-40 -30 -20 -10 0 10 20 30 40-10

0

10

20

30

40

50

60

70

width(mm)

leng

th(m

m)

Approx boundary

Original boundary

140

Table 5.1: Mean, standard deviation (SD), and range of the FD terms.

FD

terms 1a 2a 3a 4a 5a 6a 7a 8a

Total

casts Mean

-0.23

+61.69i

-0.32

-58.69i

0.02

-7.59i

0.02

-9.03i

0.03

-4.50i

0.05

-2.56i

0.01

+6.24i

0.02

+4.94i

SD 6.47 2.76 1.42 1.03 0.73 0.52 1.03 1.73

Min -9.75

+51.34i

-2.61

-63.84i

-0.86

-10.48i

-1.21

-11.37i

-0.33

-5.84i

-0.45

-3.51i

-0.83

+4.24i

-1.41

+0.91i

Max 9.25

+72.16i

2.96

-53.80i

0.64

-4.33i

0.71

-6.91i

0.82

-2.78i

0.80

-1.30i

1.23+

8.15i

1.26+

8.41i

Shape

category

1

Mean 2.98

+63.57i

-1.65

-59.41i

-0.14

-7.72i

-0.15

-9.43i

0.06

-4.55i

0.17

-2.67i

-0.01

+5.89i

0.11

+4.58i

SD 5.22 2.20 1.83 0.88 0.63 0.53 0.89 1.62

Min -2.07

+56.67i

-2.61

-63.5i

-0.86

-10.48i

-1.21

-10.77i

-0.28

-5.47i

0.01

-3.23i

-0.45

+4.72i

-0.99

+3.12i

Max 9.25

+68.68i

-0.16

-56.01i

0.31

-4.33i

0.26

-8.07i

0.33

-3.49i

0.34

-1.30i

0.46

+7.63i

0.98

+8.41i

Shape

category

2

Mean -0.65

+57.6i

-0.48

-57.6i

0.040

-7.22i

0.04

-8.35i

0.00

-4.27i

0.08

-2.45i

0.06

+6.74i

0.10

+5.57i

SD 4.83 2.40 1.30 0.63 0.80 0.47 0.98 1.26

Min -9.75

+51.34i

-2.23

-63.84i

-0.56

-9.59i

-0.34

-9.70i

-0.33

-5.71i

-0.43

-3.25i

-0.83

+5.45i

-1.41

+3.71i

Max 4.76

+64.03i

2.27

-53.8i

0.61

-5.02i

0.57

-6.91i

0.46

-2.78i

0.80

-1.57i

1.23

+8.15i

1.26

+8.03i

Shape

category

3

Mean -2.08

+66.58i

0.98

-59.83i

0.12

-8.07i

0.12

-9.77i

0.050

-4.81i

-0.10

-2.65i

-0.030

+5.73i

-0.17

+4.24i

SD 4.50 2.58 1.16 0.98 0.57 0.55 0.87 2.15

Min -7.22

+61.11i

-2.51

-62.7i

-0.31

-9.66i

-0.34

-11.37i

-0.26

-5.84i

-0.45

-3.51i

-0.53

+4.24i

-0.87

+0.91i

Max 4.08

+72.16i

2.96

-56.55i

0.64

-5.97i

0.71

-8.78i

0.82

-4.07i

0.27

-1.95i

0.58

+7.21i

0.40

+7.23i

141

Figure 5.6: Mean shapes of 21 ˆ,ˆ AA and 3A using 8 FD as shape feature.

5.2.4 Probability Distribution of Shape Categories using FD

For completeness, variation of each of the shape category must be stated and this

is done by seeking the probability distribution of the iq

iii aaa ˆ , ,ˆ ,ˆˆ

21 A . The existence

of mean and variation of the dental arch shape will consequently provide the statistical

shape model. From (5.2), iA be can be written as

iii jdcA ˆˆˆ , (5.7)

where )ˆ,,ˆ,ˆ(ˆ21

iq

iiTi ccc c , )ˆ,,ˆ,ˆ(ˆ

21iq

iiTi ddd d and 11,,1i for the first shape category,

34,,12 i for the second category, and 49,,35i for the third category. Now, for the

convenience of notation, let )(ˆ gA (g = 1, 2, 3) represent the k-the shape category. The

-40 -30 -20 -10 0 10 20 30 40

0

10

20

30

40

50

60

70

width (mm)

leng

th (

mm

)

A(1)

A(2)

A(3)

142

complex random variable in (5.7) is a univariate complex normal random variable if the

distribution of each ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c is a bivariate normal distribution (see Definition

1 in section 2.7.1). Further, dc ˆˆ j will then have the multivariate complex normal

distribution (see Definition 4 in section 2.7.1).

Table 5.2: Test statistic to investigate bivariate normality of each ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c

using HZ test in the 3 shape categories. LCV and UCV are the abbreviation for lower

and upper critical value respectively.

FD term

Shape category 1

2.5% LCV = 0.1403

2.5% UCV = 0.7352

Shape category 2

2.5% LCV = 0.1786

2.5% UCV = 0.8568

Shape category 3

2.5% LCV = 0.1531

2.5% UCV = 0.7784

Tdc 11ˆˆ 0.4072 0.5170 0.2277

Tdc22

ˆˆ 0.3208 0.5936 0.2922

Tdc 33ˆˆ 0.3692 0.2945 0.3634

Tdc 44ˆˆ 0.4929 0.7338 0.4376

Tdc 55ˆˆ 0.1847 0.3428 0.5068

Tdc 66ˆˆ 0.4918 0.3273 0.3212

Tdc 77ˆˆ 0.2401 0.5191 0.3006

Tdc 88ˆˆ 0.5462 0.2975 0.6890

143


ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c

using DH test in the 3 shape categories. LCV and UCV are the abbreviation for lower

and upper critical value respectively.

FD term

Shape category 1

2.5% LCV = 0.4844

2.5% UCV = 11.1433

Shape category 2

2.5% LCV = 0.4844

2.5% UCV = 11.1433

Shape category 3

2.5% LCV = 0.4844

2.5% UCV = 11.1433

Tdc 11ˆˆ 2.9185 9.5717 1.8439

Tdc22

ˆˆ 3.6049 11.5220 3.7941

Tdc 33ˆˆ 1.8408 1.7985 1.9049

Tdc 44ˆˆ 7.4605 8.8383 6.4315

Tdc 55ˆˆ 4.8990 3.0334 4.3175

Tdc 66ˆˆ 6.0771 6.2587 0.1502

Tdc 77ˆˆ 5.0529 4.7464 0.4338

Tdc 88ˆˆ 8.0012 3.4041 7.6285


ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c

using Royston’s test in the 3 shape categories. The critical values were denoted in the

bracket as (Lower 2.5% , Upper 2.5%).

FD term

Shape category 1

Shape category 2

Shape category 3

Critical value Test

statistic Critical value

Test

statistic Critical value

Test

statistic

Tdc 11ˆˆ

(0.0503 , 7.3702)

0.5543

(0.0458 , 7.2788)

1.3450

(0.0539 , 7.4424)

0.0331

Tdc22

ˆˆ (0.0506 , 7.3776)

1.8940 (0.0507 , 7.3786) 4.8374 (0.0526 , 7.4165) 2.2363

Tdc 33ˆˆ

(0.0506 , 7.3777)

0.3441 (0.0506 , 7.3777) 0.1963 (0.0508 , 7.3803) 0.5318

Tdc 44ˆˆ

(0.0519 , 7.4029)

5.9471 (0.0506 , 7.3770) 3.8766 (0.0506 , 7.3778) 4.5656

Tdc 55ˆˆ

(0.0506 , 7.3778)

1.0985 (0.0506 , 7.3778) 1.9266 (0.0427 , 7.2115) 5.0150

Tdc 66ˆˆ

(0.0533 , 7.4303)

5.8561 (0.0504 , 7.3738) 0.3820 (0.0499 , 7.3625) 1.3295

Tdc 77ˆˆ

(0.0472 , 7.3076)

0.4836 (0.0507 , 7.3788) 2.1720 (0.0512 , 7.3887) 0.0134

Tdc 88ˆˆ (0.0506 , 7.3778) 6.0304 (0.0506 , 7.3778) 0.5082 (0.0507 , 7.3786) 2.9674

144

Table 5.2 to Table 5.4 strongly suggest that each ˆ

ˆ,,

ˆ

ˆ,

ˆ

ˆ

2

2

1

1

q

q

d

c

d

c

d

c has a

bivariate normal when making use of HZ, DH and Royston’s tests for normality.

Except for DH test on

2

2

ˆ

ˆ

d

c in shape category 2, and Royston’s test on

1

1

ˆ

ˆ

d

c and

7

7

ˆ

ˆ

d

c in

shape category 3 which appear to reject the null hypothesis. Nevertheless, if 1%

significance level is considered, the test is likely to show bivariate normality. Therefore,

dc ˆˆ j will have the multivariate complex normal probability distribution

ggpCN Hθ ,8 where 3 ,2 ,1g are the corresponding shape category.

The three fitted shape models are given as

3,2,1 ,ˆ~)(ˆ:

gMVCNg g

gg MAA , (5.8)

where the means are given as

), 4.5777i0.1148 5.8928i,0.0124- 2.6657i,-0.1717 4.5459i,-0.0583

9.4307i,-0.1546- 7.7225i,-0.1445- 59.4081i,-1.6451- 63.5671i,2.9753(ˆ 1

A (5.9)

),5.5652i0.0980 6.7442i,0.0559 2.4500i,-0.0793 4.2743i,-0.0037

8.3538i,-0.0374 7.2191,-0.0425 57.5987i,-0.4808- 57.6496i,-0.6514(ˆ 2

A (5.10)

).i2385.40.1655- i,7306.50.0302- i,6505.2i,-0.10328052.40.0519

9.7729i,-0.1155 8.0748i,0.1246 i,8310.590.9793 i,5773.66-2.0787(ˆ 3

A (5.11)

and shape variability can be explained by the estimated Hermitian covariance matrices

whereby the element in the i-th row and j-th column is equal to the complex conjugate

of the respective element for all indices i and j, given as

2.61.......

0.17i + 0.680.79......

0.00i - 0.04-0.01i + 0.29 0.28.....

0.14i + 0.17-0.00i - 0.360.02i - 0.210.40....

0.10i + 0.080.11i - 0.310.04i + 0.270.12i - 0.280.76...

0.23i + 0.59- 0.24i - 0.670.12i - 0.720.15i - 0.730.36i - 1.083.36..

0.36i - 0.05-0.57i -0.200.09i - 0.030.37i - 0.510.08i + 0.680.35i - 0.764.85.

1.97i - 2.34-1.22i + 3.19- 0.51i + 1.36-1.34i + 1.83-0.27i + 2.77-0.68i + 5.67-0.92i + 4.79-27.32

1M

, (5.12)

145

1.57.......

0.18i + 0.61 0.95......

0.13i + 0.15- 0.09i + 0.12 0.21.....

0.04- 0.02i + 0.430.01i - 0.15 0.64....

0.17i + 0.06-0.00i - 0.210.10i - 0.16 0.04i - 0.280.39...

0.03i - 0.59- 0.17i + 0.310.01i - 0.320.03i + 0.730.15i + 0.491.67..

0.15i + 1.06-0.36i + 1.09-0.13i + 0.04-0.17i + 0.05- 0.11i - 0.080.03i - 0.495.78.

1.70i - 0.59-1.23i - 2.71-0.50i + 1.03-0.15i - 2.32-1.97-1.24i + 3.61-3.87i + 2.86-23.28

2M

, (5.13)

4.63.......

0.27i + 0.840.76......

0.19i + 0.24-0.13i + 0.02- 0.30.....

0.07i + 0.07- 0.06i + 0.090.02i - 0.19 0.32....

0.01i - 0.580.07i + 0.06-0.04i + 0.200.00i + 0.260.96...

0.10i + 1.69-0.13i + 0.17-0.12i + 0.360.12i + 0.370.07i + 0.22 1.34..

0.00i + 2.61- 0.08i + 0.94-0.45i + 0.33-0.39i + 0.31-0.66i + 0.14-0.85i + 1.14 6.64.

5.26i - 1.91-0.78i - 1.02- 0.01i + 0.12-0.18i - 0.38-1.22i - 1.60-1.08i + 1.36-3.75i + 5.04- 20.27

3M

. (5.14)

The mean shape and shape variability of the arch shape model )(ˆ gA from the

MVCN distribution can be illustrated. The inverse DFT of gg jg dcA ˆˆ)(ˆ restores

),()()( kjykxks ,20 , ,0 k and the Cartesian coordinate pair

,)20(),20(,,)0(),0( yxyx were plotted as the g-th mean shape.

The variance of )(ˆ gA that can be obtained from the diagonal elements of

Hermitian covariance matrix gM , was used to illustrate the the variation of each shape

model from its mean shape without considering the covariances. Since )(ˆ gA may also

be written as i

u

i

u

i

u djca ˆˆˆ , where 8,...,1u , and 11,,1i indicates the first shape

category, 34,,12i for the second category, and 49,,35i for the third category,

consequently its mean, variance and standard error may also be written as i

u

i

u

i

u djca ˆˆˆ

, )ˆvar()ˆvar()ˆvar( i

u

i

u

i

u dca (Andersen, et al., 1995, p. 7) and

)ˆvar()ˆvar()ˆ( iu

iu

iu dcase , (5.15)

respectively. From (5.15), the standard error for iuc and

iud may be difficult to obtain.

Therefore, we define the standard error for iuc and

iud as

146

2

)ˆ()ˆ()ˆ(

i

ui

u

i

u

asedsecse . (5.16)

The shape variation of the dental arch shape may be represented by looking at the

variation of each mean of the u-th FD term, ua , and finding the following points

)ˆ(ˆ)ˆ(ˆ i

u

i

u

i

u

i

u

i

u dsedjcsecQ , (5.17)

)ˆ(ˆ)ˆ(ˆ i

u

i

u

i

u

i

u

i

u dsedjcsecR , (5.18)

)ˆ(ˆ)ˆ(ˆ i

u

i

u

i

u

i

u

i

u dsedjcsecS , (5.19)

)ˆ(ˆ)ˆ(ˆ i

u

i

u

i

u

i

u

i

u dsedjcsecT . (5.20)

The inverse DFT of the )8,,1( and ,, uTSRQ i

u

i

u

i

u

i

u yields and ,, i

k

i

k

i

k SRQ

)20,,0( kT i

k . The variation of the k-th point may be illustrated in Figure 5.7. The

mean shape and shape variability of the arch shape model 3.2.1),(ˆ ggA are illustrated

in Figures 5.8, Figures 5.9 and Figures 5.10 respectively.

ka

Sk

Tk

Rk

Qk

Figure 5.7: Variation of the k-th point from its mean.

* *

* *

Width (mm)

(mm

Length (mm)

(mm

147

Figure 5.8: Mean shape and variation for shape model )1(A .


-40 -30 -20 -10 0 10 20 30 40

-10

0

10

20

30

40

50

60

width (mm)

leng

th (

mm

)

-40 -30 -20 -10 0 10 20 30 40

-10

0

10

20

30

40

50

60

width (mm)

leng

th (

mm

)

148


5.2.5 Test of Separation between MVCN Shape Models

Three distinct MVCN shape models exist if these models possess significant

differences in shape. A review of the literature shows that only one sample test

concerning the complex mean of MVCN distribution has been considered (Giri, 1965;

Khatri, 1965; Andersen et al., 1995). The aim of this section was to develop hypothesis

testing for two sample means from MVCN based on the Hotelling 2T test by using the

established relationship of MVCN to the multivariate real normal distribution. The

proposed test was employed for confirmation of the existence of the 3 MVCN shape

models.

-40 -30 -20 -10 0 10 20 30 40

-10

0

10

20

30

40

50

60

width (mm)

leng

th (

mm

)

149

5.2.5.1 The Proposed Hotelling T2 test for MVCN Distribution

Let 1 1 1~ ,pCNX θ H and 2 2 2~ ,pCNX θ H denotes the first and

second shape category respectively. Let 1 21gn

g g g ggn

X X X X and

*

1

1

1

gnr r

g g g g grgn

M X X X X be the unbiased estimators of gθ and 1,2g g H

respectively and 1 1 1 2

1 2

1 1

2pooled

n n

n n

M MM as an estimate of pooled Hermitian

covariance matrix H , where *

denotes the conjugate and transpose of .

A relationship with the multivariate real normal distribution from equation

(2.53) gives 1 2 1 1

1~ ,

2pN

X θ H and

}{

2

1],[~][ 2222 HθX pN respectively.

Consequently,

1 2 2 1 11 2

1 1~ ,

2 2pN

n n

X X θ θ H , (5.21)

1 2 1 2 21 2

1 1~ , ,

2 2pN

n n

X X θ θ 0 H (5.22)

and let

12

1 2 1 2 21 2

1 1~ , ,

2 2pN

n n

u X X θ θ 0 H (5.23)

where H denotes the pooled Hermitian covariance matrix. From equation (2.56),

1 1 2 2 1 21 1 ~ , 2 ,pn n CW n n M M H (5.24)

and can also be written as

1 1 2 2 2 1 22 1 1 ~ { }, 2 ,pn n W n n W M M H (5.25)

since equation (2.54) applies.

150

The test statistic for the Hotelling 2T test statistic under 1 2H 0:θ θ 0 can be

derived similar to the real case:

1

2 *

1 2

12 * 1

1 2 1 21 2

12

1 2 1 21 2

2

1 12

2 2

1 1

2 2

Tn n

n n

n n

pooled

Wu u

X X θ θ M

X X θ θ

1* 1

1 2 1 21 2

1 1 1

2 2 2n n

pooledX X M X X

* 11 21 2 1 2

1 2

.n n

n n

pooledX X M X X (5.26)

and the distribution of 2 T is given as

1 2

2 1 22 ,( 2) (2 1)

1 2

( 2)2~ .

( 2) (2 1)p n n p

n n pT F

n n p

(5.27)

5.2.5.2 Simulation Study on Performance of the Hotelling T2 for MVCN

Distribution

Simulation study were carried out using R software to investigate the performance of

the proposed test under the null hypothesis, 0H : 1 2θ θ and was evaluated according to:

1. Type I error rates against equal means

2. Power of test against unequal means

151

5.2.5.2.1 Type I Error Rates

To estimate the Type I error rates, the following steps were taken:

Step 1: Two random samples of p -variate complex normal distributions with size n

was drawn from 1 1 1~ ,pCNX θ H and 2 2 2~ ,pCNX θ H and generated by using R

package cmvnorm (Hankin, 2015), where 1 2θ θ and 1 2H H .

Step 2: Calculate the test statistic in equation (5.26) and find its corresponding p -

value using equation (5.27).

Step 3: The above steps of obtaining the p -value were repeated for s = 10000 times.

Step 4: The proportion of 10,000 samples for which the test rejects the 0H at

05.0 was calculated. This is equivalent to calculating the probability of making

Type I error, 0 0reject | is trueP H H . The corresponding error rate in percentage was

then obtained.

The examples of the generated complex normal data for 4p and 8p with

equal means are illustrated in Figure 5.11. Since the 0H is true, the test statistic should

reject the 0H at about the nominal rate of 5%. However, Table 5.5 shows high values of

Type I error rate, which gets higher as p increases. A correction factor imposed to the

test statistic 2T may be required to decrease the chance of the test getting rejected.

152

Table 5.5: Empirical Type I error rate against equal means for different sets of

parameters

Parameters

Without

Correction

Factor

With

Correction Factor

Type I Error 5 %

Type I

Error 5 %

Type I

Error 10 %

1 2

1 2

4,

(H ) (H )

1

,

1

0 0

1 0θ θ .

1

0

p

diag diag

i

i

i

i

20n 20.99 % 4.20 % 8.59 %

30n 20.14 % 4.45 % 8.97 %

50n 18.45 % 4.58 % 9.34 %

100n 18.10 % 4.80 % 9.65 %

250n 18.45 % 4.95 % 9.36 %

500n 17. 38 % 5.16 % 10.12%

1

2

30,

(H )

(H )

1

.

1

n

diag

diag

1 2

2,

θ θ

0 0

1 0

p

i

i

8.66 % 4.29 % 9.44 %

1 2

4,

θ θ

0 0

1 0

1

0

p

i

i

i

i

20.14 % 4.45 % 8.97 %

1 2

8,

θ θ

1 2

2

2

1 2

1 2

2

2

1 2

p

i

i

i

i

i

i

i

i

48.35 % 4.75 % 9.94 %

153

(a) 4p (b) 8p

Figure 5.11: Examples of the generated complex normal data for 4p and 8p .

Note that this test was derived using the complex and real normal relationships

which involves transformation of p complex dimension to 2 p (see equation (2.53)).

Therefore, the best guesstimate of the correction factor is two times what it should be

and the test statistic with correction factor is given as

*2 11 21 2 1 2

1 2

2 .pooled

n nT

n n

X X M X X (5.28)

Simulation study was repeated after correction factor was imposed on the test

statistic 2T . The Type I error rates for different sets of parameters are close to the

nominal rates when 5% and 10% significance level were considered, especially for a

large sample size.

5.2.5.2.1 Power of Test

To estimate the power of test, the following procedure was carried out:

Step 1: Two random samples of size n from two p -variate complex normal

distributions were generated similar to Step 1 and Step 2 in Section 4.1, with 1 2θ θ .

Step 2: Calculate the test statistic in equation (5.28) with correction factor imposed,

and find its corresponding p -value using equation (5.27).

Step 3: The above steps of obtaining the p -value were repeated for s = 10000 times.

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

2.5

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

154

Step 4: The proportion of 10,000 samples for which the test rejects the null

hypothesis 0H at 0.05 significance level was calculated. This is equivalent to

calculating the probability of not making a Type II error,

false) is |reject (1 00 HHP .

The corresponding error rate in percentage was

then obtained as the power of the test.

The examples of the generated complex normal data for 4p with unequal

means are illustrated in Figure 5.12. Here, the test was expected to reject the 0H . Table

5.6 shows the power estimates when different sets of mean and covariance matrix were

considered, representing data with mild to complete separation. As expected, the power

of test gets higher as the differences between the means gets bigger and as the value of

covariance matrices gets smaller. With obvious separation of the means and smallest

values of covariance matrices, the power of the test is nearly 100%.

Figure 5.12: Examples of generated complex normal data

for Different Values of 1θ and

2θ .

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

-1 0 1 2 3 4 5-1

0

1

2

3

4

5

155

Table 5.6: Empirical power of test against unequal means for different sets of

parameters.

Parameters Power of

test

1

1 2

4, 30,

0 0

1 0θ ,

1

0

(H ) (H )

1

1

p n

i

i

i

i

diag diag

2

0.1 0.1

1.1 0.1θ

1.1 0.1

0.1 1.1

i

i

i

i

88.90 %

2

0.3 0.3

1.3 0.3θ

1.3 0.3

0.3 1.3

i

i

i

i

96.41 %

2

0.1 0.1

1.1 0.1θ

1.1 0.1

0.1 1.1

i

i

i

i

99.90 %

2

1 1

2 1θ

2 2

1 2

i

i

i

i

100 %

1

4, 30,

0 0

1 0θ ,

1

0

p n

i

i

i

i

2

0.3 0.3

1.3 0.3θ

1.3 0.3

0.3 1.3

i

i

i

i

1

2

(H )

(H )

0.5

0.5

diag

diag

99.54 %

1

2

(H )

(H )

1

1

diag

diag

96.41 %

1

2

(H )

(H )

1.5

1.5

diag

diag

89.88 %

5.2.5.3 Results on Test of Separation between MVCN Shape Models of Dental Arch

Let

g

gMVCNg MAA ,ˆ~)(ˆ8

denotes the g-th shape model. Estimate the

pooled Hermitian covariance matrix 2

)1()1(

21

2111

nn

nnpooled

MMM and use equations (5.28)

and (5.27) to test for mean separation between two MVCN means. The result is shown

in Table 5.7.

156

Only 1 2ˆ Â A and 2 3ˆ Â A appear to reject 0H at 5% significance level, which

indicates separation of shape category 2 from the other two shape categories. 1 3ˆ Â A on

the other hand appear to accept the 0H , however they are significantly separated when

10% significant level is considered. This result confirms the existence of the three

MVCN shape models.

Table 5.7: Hotelling T2 test for comparing two MVCN mean.


21 ˆˆ AA Lower 2.5% = 11.2264

Upper 2.5% = 85.6021 97.66

21 ˆˆ AA Lower 2.5% = 14.7208

Upper 2.5% = 187.5004 16.17

Lower 2.5% = 10.6120

Upper 2.5% = 74.1745 135.47

5.3 Discrimination of Shape for MVCN Model

This section developed shape discrimination procedure for the maxillary dental

arch using the established statistical shape models for MVCN. Since the probability

distribution of the shape has been identified as MVCN, the linear discrimination

function (LDF) for MVCN may be derived and used as discrimination method.

However, the Gaussian assumptions for the discriminant function are particularly

restrictive and sometimes difficult to satisfy if the sample size for the shape category is

small. If violated, it may reduce the performance of discriminant analysis. An

alternative discrimination method will be proposed which uses the determinant ratio of

two covariance matrices that are not necessarily from Gaussian distribution.

Subsequently, comparison between the LDF and )( iCOVRATIO will be carried out via

simulation study.

157

5.3.1 LDF for MVCN Distribution

Let x be the landmarks of the i-th object and the probability density function for

the g-th population ( g ) from the MVCN distribution be

)()(exp)det()( 1*1gggg

p

gf θxHθxHxX

, (5.29)

where gθ and gH are the mean and Hermitian covariance matrix respectively. The

LDF for MVCN is given as

, ,)()()()(

)()()()()det()det(

)()()det(log

)()()det(log

)( log)( log

)(

)(log)(

1*1*

1*1*

1*

1*

HHHθxHθxθxHθx

θxHθxθxHθxHH

θxHθxH

θxHθxH

xx

x

xx

kgkkgg

kkkgggkg

kkkk

gggg

kg

k

gMVCN

gk

p

p

ff

f

fh

(5.30)

where the covariance matrices are equal and kg . The unbiased estimator of

estimators of θ and H are given as

1ˆ 21 gn

ggggn

xxxθ and

2

1 2

)1(

gg

g

N

nHH

respectively (Goodman, 1963).

Let 1x . The )(1 xf from equation (5.29) is maximized because the

exponent is minimized and )()( 21 xx ff (Mardia et al., 1979). Consequently, it is

expected to obtain 0)(12 xMVCNh (equation (5.30)). On the contrary, if 2x , then it

is expected to have 0)(12 xMVCNh . The discrimination rule for two CN populations

1

and 2 can be therefore generalized as:

158

If 0)(12 xMVCNh , then 1x and,

if 0)(12 xMVCNh , then 2x . (5.31)

Similar to the real MVN case, the LDF for 2 populations in equation (5.31) can be

extended for 3 populations as follows (Johnson and Whicern 1992; Mardia et al. 1979):

if 0)(12 xMVCNh and 0)(13 xMVCNh , then 1x ,

if 0)(12 xMVCNh and 0)(23 xMVCNh , then 2x , (5.32)

if 0)(13 xMVCNh and 0)(23 xMVCNh , then 3x .

5.3.2 Proposed MVCN

iCOVRATIO )( for MVCN Distribution as Discrimination Method

The COVRATIO statistics for the MVCN model will be formulated using the

same idea proposed in section (4.4.1) for the purpose of discrimination. Let 1H and 2H

be the Hermitian matrix which corresponds to a real matrix being symmetry for

population 1 and 2 respectively, where each of the population has a known

probability distribution. Instead of using reduced observation as formulated by Belsley

et al. (1980), define the ratio of covariance determinant for 1 as:

H

H

1

)(1

)(1

iMVCNiCOVRATIO

, (5.33)

where H)(1 i

is the determinant of Hermitian covariance matrix )(1 iH which is

estimated by adding new thi observation in 1 and

1H is the determinant of 1H .

The same thi observation will be added in 2 and its

MVCNiCOVRATIO )(2 is obtained

using

159

H

H

2

)(2

)(2

iMVCNiCOVRATIO

. (5.34)

Smaller value of MVCN

iCOVRATIO )( (which is close to 1) indicates no variation when new

observation was included in a particular population. In other words, the thi observation

is consistent with the other observations and therefore belongs to the population.

Therefore, the discrimination rule using MVCN

iCOVRATIO )( for 2 MVCN

populations is given as:

(5.35)

The above result can be generalized for g populations, given as:

(5.36)

5.3.3 Simulation Study on Performance of LDF and MVCN

iCOVRATIO )(

A simulation study was designed to examine the performance of LDF and

)( iCOVRATIO for CN model. Random samples from two CN populations were

generated and re-assigned to their corresponding population using LDF and

i)(COVRATIO . The misclassification probabilities using these discrimination methods

were then compared. This procedure was carried out as follows:

Step 1: Two sets of complex random samples with sample size n from two unequal p-

variate MVCN distributions 1111 ,~: HθX pMVCN and

Allocate new thi observation in 1 if

MVCNi

MVCNi COVRATIOCOVRATIO )(2)(1 ,

Else, allocate to 2

Allocate new thi observation in

j if MVCN

ihMVCN

ij COVRATIOCOVRATIO )()( ,

where ghj ,,2,1, and hj .

160

),(~: 2222 HθX pMVCN were generated using R package cmvnorm

(Hankin, 2015). The parameters for 1 when 4p were set as

i

i

i

i

0

1

01

00

1θ and

5.0

5.0

)( 1 Hdiag ,

where )( 1Hdiag denotes the diagonal matrix with its off-diagonal

elements are 0.

Step 2: Generate another two complex random samples A and B which were drawn

from 1 and

2 respectively.

Step 3: Calculate the LDF for MVCN in equation (5.30) using random sample A and

B , denoted as )(12 AMVCNh and )(12 BMVCNh respectively.

Step 4: Calculate )()(1 AMVCN

iCOVRATIO , )()(1 BMVCN

iCOVRATIO , )()(2 AMVCN

iCOVRATIO and

)()(2 BMVCN

iCOVRATIO using equation (5.33) and (5.44), by adding random sample A

and B in both sets of random samples generated from 1 and 2 in step 1.

Step 5: Discrimination rule for LDF (in equation (5.31)) and MVCN

iCOVRATIO )( (in

equations (5.35) and (5.36)) were used to determine the membership of A and B .

Step 6: The above steps were repeated for s = 10000 times.

Step 7: The proportion of A and B which does not assign to their corresponding true

population when discriminated using LDF and MVCN

iCOVRATIO )( may be regarded as

the misclassification probability.

161

Table 5.8: Misclassification probability of LDF and MVCN

iCOVRATIO )( for MVCN

model when different sample size, dimension, mean vector and covariance matrices

were considered.

Parameters

1 2

LDF MVCN

iCOVRATIO )(

LDF MVCN

iCOVRATIO )(

,30,4 np

)(

)(

2

1

H

H

diag

diag

i

i

i

i

8.18.0

8.08.1

8.08.1

8.08.0

2θ 0.0140 0.0092 0.0122 0.0110

i

i

i

i

21

22

12

11

2θ 0.0026 0.0011 0.0030

0.0013

i

i

i

i

2.22.1

2.22.2

2.12.2

2.12.1

2θ 0. 0005 0 0. 0004 0

,30,4 np

i

i

i

i

21

22

12

11

2θ ,

3.0

3.0

)( 2 Hdiag 0.0022 0.0003 0.0001 0

5.0

5.0

)( 2 Hdiag 0.0026 0.0011 0.0030

0.0013

7.0

7.0

)( 2 Hdiag 0.0034 0.0012 0.0083 0.0042

,4p

i

i

i

i

21

22

12

11

2θ ,

)(

)(

2

1

H

H

diag

diag

20n 0.0211 0.0123 0.0203 0.0125

30n 0.0026

0.0011 0.0030 0.0013

50n 0.0024 0.0009 0.0015 0.0008

162

Table 5.8: Continue.

,30n

)(

)(

2

1

H

H

diag

diag

2p ,

i

i

01

001θ ,

i

i

12

112θ

0.0222 0.0147

0.0224 0.0148

4p ,

i

i

i

i

21

22

12

11

2θ

0.0026 0.0011 0.0030

0.0013

8p ,

i

i

i

i

i

i

i

i

0

1

2

3

03

02

01

00

1θ ,

i

i

i

i

i

i

i

i

21

22

23

24

4

3

2

1

2θ

0 0

0.0001 0

163

The examples of the generated complex normal data for p = 4 and p = 8 with unequal

means are illustrated in Figure 5.13. Table 5.8 shows the performance of LDF and

MVCNiCOVRATIO )( for MVCN model when different values of n, p, μ and Σ were

considered. Overall, the misclassification probabilities for LDF and i)(COVRATIO in

all cases are less than 0.05. However, the misclassification probability reduces when a

particular value of 2θ was chosen to indicate larger separation between

2 and 1 .

The dispersion of the data in a particular population also dictates the misclassification

probability, whereby larger value of H may results in higher chances of a sample to be

assigned to an incorrect population. Increasing value of n and p gives smaller

misclassification probability.

The MVCN

iCOVRATIO )( method gives smaller probability of misclassification as

compared to LDF in all cases, indicating better performance of shape discrimination.

The )( iCOVRATIO is theoretically desirable when discriminating shape of the dental

arch, since differences of shapes may be barely noticeable and the discriminatory

information may not necessarily in the mean shape, but rather in shape variability.

Further, the )( iCOVRATIO can be employed when discriminating k groups of shape and

without any assumptions on the distribution of the shape.

Figure 5.13: Examples of the generated complex normal data for p = 4 and p = 8.

164

5.3.4 Results for Shape Discrimination of Dental Arch using FD

The membership of the 47 control samples which belong to each of the 3 shape

models of the dental arch using FD iq

iii aaa ˆ , ,ˆ ,ˆˆ

21 A as shape feature were tracked

from the dendrogram (see Figure 3.12 in section (3.6.1)). The LDF and MVCN

iCOVRATIO )(

were employed to re-assign these casts into one of 3 populations of the MVCN shape

models. Their misclassification probability was investigated to ensure that these

methods are deemed good for shape discrimination.

Table 5.9 shows the misclassification probabilities when the samples were re-

assigned to *

2

*

1 , or *

3 using LDF and MVCN

iCOVRATIO )( . The results for LDF show

relatively small misclassification probabilities except for the discrimination of 1 from

3 . However this is expected since the test of separation between these 2 shape models

was significantly separated when 10% significant level is considered (see Table (5.7)).

MVCNiCOVRATIO )( outperformed the LDF method by having zero misclassification

probabilities. These results demonstrate the ability of the MVCN

iCOVRATIO )( to discriminate

the dental arch shape correctly as a better discrimination method.

Table 5.9: Misclassification probability when the 47 dental casts were re-assigned into

either one of the population of the shape model using the LDF and MVCN

iCOVRATIO )( .

Discrimina-

tion

True

population

*1 *

2 *3

LDF MVCN

iCOVRATIO )( LDF MVCN

iCOVRATIO )( LDF MVCN

iCOVRATIO )(

1 - - 0.18 0 0.18 0

2 0.09 0 - - 0.09 0

3 0.21 0 0.07 0 - -

165

5.4 Linking Anatomical Landmarks to the MVCN Shape Models on Edentulous

Arch

The only landmarks that a dentist may obtain from an edentulous patient are the

incisive papilla and hamular notches (Figure 5.14). Investigation on these landmarks in

dentate arches which correspond to MVCN shape category models may provide

indicator for determining the shape category or shape model for the edentulous arches.

Consequently, setting up of artificial teeth for edentulous arch may be carried out

according to the assigned MVCN shape category model. This knowledge may provide a

guide for the inexperienced dentists and dental technicians when setting up teeth on

complete dentures.

Since each of the shape categories follows the MVCN, the corresponding shape

category using the anatomical landmarks as shape feature may have the same MVCN

distribution. Confirmation of group separation for the same 47 casts (using only the

incisive papilla and hamular notches, without taking into account the teeth) was carried

out and referred to as the landmark models. The casts were then assigned into either one

of 3 populations of the landmark models using the proposed MVCN

iCOVRATIO )( . Their

misclassification probabilities were also investigated for verification of the landmark

models.

Figure 5.14: Edentulous cast showing the incisive papilla (IP) and hamular notches

(HN).

IP

HN

166

5.4.1 Shape Model of Anatomical Landmarks

Let the set of FD of anatomical landmarks; the right and left hamular notches

and incisive papilla, for the i-th control cast be denoted as ),,( 321 iiii LLLL , where

47 ,...,1i . The membership of the casts were identified according to groups

established and the vector of the anatomical landmarks were denoted by

ggg LLLg 321 ,,)( L . Since each of the shape categories are following the MVCN, the

corresponding shape category when using the anatomical landmarks as shape feature

may have the same MVCN distribution. Therefore, let ) ,(~)( 3

landmark

g

gMVCNg MLL ,

where gL and

landmark

g M are the estimator of gθ and gH (g = 1,2,3). Confirmation of

group separation using the anatomical landmarks will be investigated using test derived

in (5.28).

Table 5.10 indicates separation of shape categories for )(gL where .3,2,1g

Table 5.11 on the other hand shows descriptive statistics for the total casts and casts

segregated according to each shape category. Smaller values of standard deviation can

be seen in each category as compared to the total casts. These further support the

existence of three categories of shape using anatomical landmarks.

Table 5.10: The T2 test for comparing two MVCN mean for anatomical landmarks.


21LL

Lower 2.5% = 1.4035

Upper 2.5% = 21.0661 26.7173

31LL

Lower 2.5% = 1.4738

Upper 2.5% = 24.6937 52.6035

32LL

Lower 2.5% = 1.3868

Upper 2.5% = 20.2874 94.3139

167

Table 5.11: Mean, standard deviation (SD) and range of the FD terms for the

anatomical landmarks.

FD

terms 1L 2L 3L

Total casts Mean -0.07 + 4.73i 7.45 - 4.33i -0.78 - 0.44i

SD 0.57 0.45 0.47

Min 0 + 4.02i 6.60 - 3.77i -0.23 - 0.07i

Max 0.33 + 5.44i 8.21 - 4.73i -1.52 - 0.71i

Shape

category 1 Mean 0.25 + 4.72i 7.46 - 4.29i -0.92 - 0.26i

SD 0.42 0.38 0.49

Min -0.04 + 4.47i 6.79 – 4.14i -0.26 - 0.12i

Max 0.85 + 5.03i 7.94 - 50i -1.55 - 0.43i

Shape

category 2 Mean -0.11 + 4.54i 7.27 - 4.25i -0.59 - 0.32i

SD 0.52 0.46 0.32

Min -0.12 + 4.03i 6.60 - 3.87i -0.23 - 0.071i

Max -0.82 + 5.13i 7.88 - 4.92i -0.99- 0.50i

Shape

category 3 Mean -0.25 + 5.10i 7.71 - 4.49i -0.98 - 0.78i

SD 0.51 0.32 0.39

Min -0.64 + 4.68i 7.44 – 4.10i -0.48 - 0.38i

Max 0.33 + 5.44i 8.21 - 4.73i -1.52 - 0.71i

The above results consequently allow the three fitted landmarks models be labelled as

3,2,1 ,~)(: 3 gMVCNg landmarkg

gg MLL , (5.37)

where the means are given as

)0.2408i - 0.8090- 4.2612i, - 7.4342 4.7012i, + 0.2515(1 L , (5.38)

)0.3231i - 0.5906- 4.2528i, - 7.2656 4.5401i, + -0.1192(2 L , (5.39)

)0.7243i - 0.9210- 4.4183i, - 7.5993 5.0657i, + -0.1738(3 L , (5.40)

168

and shape variability can be explained by the estimated Hermitian covariance matrices

given as

0.2463..

0.0259i + 0.0369-0.1387 .

0.0479i + 0.1028-0.0178i - 0.0217-0.1556

1

landmarkM , (5.41)

0.0993..

0.0190i + 0.0158 0.2156.

0.0211i + 0.0554-0.0102i + 0.0490-0.26526

2

landmarkM , (5.42)

0.1318..

0.0321i + 0.0363- 0.2333.

0.0607i + 0.0832- 0.0019i + 0.0063 0.2886

3

landmarkM . (5.43)

5.4.2 Verification of the Anatomical Landmark Models

A verification study was carried out to confirm the ability of the anatomical

landmarks in indicating the shape category. This is important to enable estimation of

teeth positions for the edentulous patients. The membership of iii

i aaa 821ˆ , ,ˆ ,ˆˆ A from

the 47 control samples which belong to each of the 3 shape models 3,2,1 , )(ˆ: ggg A ,

were tracked from the dendrogram (See Figure 3.12 in section (3.6.1)). The

MVCNiCOVRATIO )( was employed to re-assign these casts to one of 3 populations of )(ˆ gA

using the landmarks ),,,( 321 iiii LLLL where 47 ,...,1i . Their misclassification

probability was then investigated to verify that the landmarks can be used to

discriminate shape category of the dental arch.

Then, shape discrimination using shape feature teeth location, iii

i aaa 821ˆ , ,ˆ ,ˆˆ A

and anatomical landmarks, ),,( 321 iiii LLLL was carried out for comparison.

Discrimination using iii

i aaa 821ˆ , ,ˆ ,ˆˆ A was regarded as the true population. Similar

classification of population is expected using both shape features to further verify the

ability of the anatomical landmarks in indicating the shape category.

169

Table 5.12 and Table 5.13 show the misclassification probability of the 47

control casts and 40 test casts respectively, that were re-assigned to 21, or 3 . Low

misclassification probabilities were obtained in each of the true shape population.

These results support the ability of the ),,( 321 iiii LLLL in discriminating the categories

of arch shape correctly.

Table 5.12: Misclassification probability when 47 control casts was discriminated using

the anatomical landmarks ),,( 321 iiii LLLL of the dental arch.

Assigned using

)( iCOVRATIO

True population

*1 *

2 *3

1 0 0.0909 0

2 0.1818 0 0

3 0.0714 0.0714 0

Table 5.13: Misclassification probabilities when 40 test casts were discriminated using

anatomical landmarks ),,( 321 iiii LLLL and teeth location iii

i aaa 821ˆ , ,ˆ ,ˆˆ A .

Assigned using ),,( 321 iii LLL

True population:

using iii

i aaa 821ˆ , ,ˆ ,ˆˆ A

*1 *

2 *3

1 0 0.14 0

2 0 0 0.17

3 0 0.03 0

5.5 Application of Shape Discrimination for MVCN model

5.5.1 A Proposed Guide to Teeth Positioning on Complete Dentures

The conventional construction of complete dentures requires at least five clinical

appointments or visits. In the first visit, the patient is examined, primary impressions are

made and diagnosic casts are poured from the impressions. In the second visit, final

impressions are made using custom tray from which final casts are poured. In the third

visit, jaw relationship registrations are made for transferring necessary information from

the patient to construct the dentures. In the fourth visit, the trial denture is evaluated in

170

the patient’s mouth and on the articulator for aesthetics and occlusion. In the fifth visit,

the final dentures are inserted into the patient’s mouth amd delivered to the patient.

The schematic diagram below (Table 5.15) is a brief outline of the steps

performed by the dentist and technician in the construction of complete dentures

according to clinical visits.

Number

of clinical

visit

Dentist (clinic) Technician (laboratory)

1st visit - Examination

- Making primary impression

- Making primary cast

- Fabrication of custom tray from the

primary cast

Figure 5.15: Brief outline of steps in the construction of complete dentures.

171

2nd visit - Taking final impression using

custom

tray

- Making final cast

- Fabrication of denture baseplate

and

wax rim

3rd visit - Checking patient’s upper and

lower arch relationship

- Selection of artificial teeth

- Artificial teeth positioning

4th visit - Trial denture visit

- Modification on dentures

5th visit - Final denture and follow up


172

Previous sections have established 3 categories of shape models and they can be

linked using the anatomical landmarks. This section is aimed at using the knowledge to

provide a guide for developing the wax rim and also for teeth positioning on complete

dentures. The proposed guide is as follows:

Number

of clinical

visit

Dentist (clinic) Technician (laboratory)

1st visit - Examination

- Make primary impression

- A 2D image of the edentulous

patient’s arch is captured using an

intraoral camera with a small ruler

attached.

- Make primary cast

- Fabrication of custom tray from the

primary cast

- A program developed in MATLAB

software will be used to import and

calibrate the image, and obtain 3

anatomical landmarks from the

edentulous arch. Consequently, the

landmarks may be used to determine

which shape category the patient

belongs to (section 5.7.2). The

primary wax rim will be developed

using the estimated teeth position

from the corresponding MVCN

model (section 5.5.3).

Figure 5.16: Proposed guide for construction of complete dentures.

173

2nd visit - Make final impression using

custom tray

- Check patient’s upper and

lower arch relationship using

wax rim built on primary casts.

- Make final cast

- Fabrication of final wax rim and

recording jaw relationships using

primary wax rim.

- Selection of artificial teeth

- Artificial teeth positioning will be

carried out using the estimated

teeth positions obtained from the

corresponding MVCN shape

model (section 5.5.3).

3rd visit - Trial denture visit

- Modification on dentures

4th visit - Final denture and follow up


174

There are several advantages of the above proposed guide as compared to the

conventional steps. First, it may reduce one clinical visits. Secondly, shorter clinical

sessions may be needed in the third (trial denture) visit and thirdly, the development of

wax rim, teeth positioning and consequently the whole process of preparing the final

dentures is facilitated.

5.5.2 Verification of the Proposed Teeth Positioning Guide for the Edentulous

The proposed guide for teeth positioning for the edentulous patient was verified

by investigating another 35 selected dentate casts. The MVCN

iCOVRATIO )( in equation (5.33)

together with its discrimination rule established in equation (5.36) was employed using

the anatomical landmarks to determine the membership of shape category. Then, the

teeth position was estimated using the MVCN model corresponding to the assigned

shape category (equation (5.8)). The Procrustes distance (PD) in equation (5.5)

calculates the differences between the original and the estimated teeth position. Table

5.14 shows the membership percentage of the 35 casts in the three shape categories.

Table 5.15 on the other hand indicates that 80% of the 35 casts have 20 mm PD

or sum squared of difference between the 21 estimated and original teeth position. This

signifies that an average squared error for each tooth position is 0.95 mm. Figure 5.17,

Figure 5.18 and Figure 5.19 illustrate the smallest (PD = 9.7520 mm), average (PD =

17.4664 mm) and largest (PD = 26.1236 mm) Procrustes distances respectively, for

comparison of the estimated and original teeth positions.

175

Table 5.14: Group membership when the anatomical landmarks of the 35 test cast was

assigned using MVCN

iCOVRATIO )( .

Population *1 *

2 *3

Number

(Percentage) of

casts assigned

using

)( iCOVRATIO

6

(17.14%)

21

(60.00%)

8

(22.86%)

Table 5.15: The number and percentage of casts according to the Procrustes distance

(PD) intervals indicating sum squared of difference between the estimated and original

teeth position.

PD interval 10 mm )15,10[ mm )20,15[ mm )20,15[ mm 25 mm

Number

(Percentage)

of casts

2

(5.71%)

13

(37.14%)

13

(37.14%)

4

(11.43%)

3

(8.58%)

Figure 5.17: Smallest Procrustes distance PD = 9.7520 mm between the original and

estimated teeth position. Red dots indicate the estimated teeth position (sample N110).

176

Figure 5.18: An average Procrustes distance PD = 17.4664 mm, between the original

and estimated teeth position. Red dots indicate the estimated teeth position (sample

N130).

Figure 5.19: Largest Procrustes distance PD = 26.1236 mm, between the original and

estimated teeth position. Red dots indicate the estimated teeth position (sample N117).

177

5.6 Discussion

The 8 Fourier descriptors provide a more precise arch shape and teeth position

and was capable of closely representing the dental arch shape. The Fourier descriptor

was considered as it is a reversible linear transformation which retains all the

information in the original boundary of the object (Keyes & Winstanley, 1999). This

important feature makes the modelling of arch shape and teeth position using the FDs

easier and more precise. Further, the FDs used in this study were derived from the

origin established from anatomical landmarks which made them valid for use in the

edentulous patient, rather than using teeth which may be lost (Mikami et al., 2010;

Nakatsuka et al., 2011).

Test of separation of two mean vectors from MVCN was derived and the

anatomical landmarks were found capable in discriminating the shape category

established with relatively low misclassification probabilities. The use of shape models

3,2,1 , ,ˆ~)(ˆ

gMVCNg g

g SAA gives the precise locations of the 18 teeth on the

maxillary arch. The proposed guide for estimating teeth positioning on complete

dentures shows that the three categories of shape models may estimate the original teeth

position correctly in 80% of the Malaysian dental arches with an average error margin

of 0.95 mm for each tooth position. This will help the dentist to position the maxillary

artificial teeth as close as possible to the position originally occupied by the natural

teeth.

178

CHAPTER 6: CONCLUDING REMARKS

6.1 Conclusion

The primary goal of this thesis was to propose shape features and statistical

shape models to develop a novel discrimination procedure for the shape of the maxillary

dental arch. This work has been applied to two applications in dentistry, namely in

designing and selection of impression trays and in predicting arch shape and teeth

positions for the edentulous patients.

A review of the literature related to the shape analysis of the dental arch shows

that this study is the first to propose shape discrimination procedure and to demonstrate

its applications in dentistry. The first step in discriminating shape of the dental arch to

enable reconstruction of the arch shape even in edentulous situation was to propose a

shape feature derived from stable anatomical landmarks. Then, the properties of the

shape feature were investigated. The probability distribution of each shape category

which was clustered from the shape feature provides shape variation and was used as

shape model of the dental arch. A new hypothesis testing for two sample means from

multivariate complex normal based on Hotelling T2 test was also derived and employed

to test the distinction of two shape models.

A modified COVRATIO statistics, denoted as )( iCOVRATIO , was then proposed

as a novel shape discrimination method and compared to the linear discrimination

function. Simulation results showed that )( iCOVRATIO

as discrimination method

performs better than the LDF with lower misclassification probabilities in all considered

cases. The )( iCOVRATIO is theoretically desirable when discriminating shape of the

179

dental arch, since differences of the dental arch shapes are barely noticeable and the

discriminatory information may not necessarily be in the mean shape, but rather in

shape variability. Further, the )( iCOVRATIO can be employed when discriminating k

groups of shape and without any assumptions on the distribution of the shape. The

resulting statistical shape discrimination procedure enables the development of two

proposed guides in choosing suitable impression trays and predicting natural teeth

positions for the edentulous. Verification study shows that 91.42% of the test sample

studied indicates appropriate fitting when assigned using the proposed guide. As for the

latter guide, 80% of the studied arches may adequately estimate the original teeth

position, with an average error of 0.95 mm for each tooth position.

The presented statistical shape discrimination procedure therefore may be useful

in assisting inexperienced dentists and dental laboratory technicians to choose the most

appropriate impression tray for the Malaysian population and facilitate the estimation

natural teeth positions for the edentulous. Dental visits may be shortened, and

unnecessary cost of repeated dental procedures may be avoided.

The new contribution of the work can be summarized as follows:

1. Shape features indicating the location of the teeth have been proposed which

allows the reconstruction of the arch shape even when all teeth are lost.

2. Shape models of the dental arch has been proposed and provides not only

presents the mean shape, but also shape variation.

3. A new hypothesis testing for two sample means from multivariate complex

normal based on Hotelling T2 test was derived and employed to test the

distinction of two shape models.

4. A modified COVRATIO statistics, denoted as )( iCOVRATIO , was then proposed

as a novel shape discrimination method.

180

5. A guide in assisting inexperienced dentists and dental laboratory technicians to

choose the most appropriate stock tray for dentate or partially edentulous

patients has been proposed.

6. A guides in assisting inexperienced dentists and dental laboratory technicians to

estimate natural teeth position on the complete dentures for the edentulous

patients has been proposed.

6.2 Limitations of the study

The limitation of this study was the sampling method used to represent the

Malaysian population. Nevertheless, the results of this study may be generalized since

the sample was collected from one of the hospitals in the Klang Valley region in

Malaysia, which is the heart of commercial, business and industrial center of the

country and consists of the 3 major ethnic groups in the population.

The shape feature using angular measurements compared to Fourier descriptor

may be a limitation to the shape model. It may only work for semi-angular measures

and must be approximated to the linear measures. In addition, the shape model should

follow normal distribution. A non-normal distribution of shape model may be explored

in the future research.

It should be noted that the design of the trays, guide in assigning suitable

impression tray and guide in estimating teeth positions on complete dentures are limited

as only two dimensional shapes on the maxillary (upper) arch were considered. Minor

modifications to the impression trays, predicted arch shape or teeth positions may be

necessary to accommodate factors which were not investigated in this study and some

clinical work is needed to further scrutinize the statistical methods to be used in clinical

practices. It has to be pointed out that the palatal shape and depth of maxillary and

mandibular (lower) arches are also important considerations for making stock

181

impression tray and guiding the teeth positioning. This would be the subject of a future

study.

6.3 Direction for Future Research

6.3.1 Exploring other Applications of Shape Model and Shape Discrimination

Procedure in Dental Problems

Two applications were discussed in this study. More potential applications of the

discrimination procedure for the dental arch shape can be extended, namely in

fabricating the arch wire used in orthodontic and matching teeth position from the bite

mark in forensic dentistry. Lee et al. (2011) classified the dental arch forms with the

objective to provide a guide in designing pre-formed orthodontic arch wire forms.

Results from their study may only be used to determine the number of preformed wire

to be designed, since the knowledge of mean shape variability was not provided. The

shape model and discrimination procedure proposed in this study may determine the

elasticity of the arch wire and suitable pre-formed wire that would fit the patient’s arch.

This study may also help to provide evidence to exclude suspects in a judicial

procedure. The bite marks may be first discriminated from the three shape models to

determine which category it belongs to. Then, the teeth positions from all the suspects

are discriminated and those which belong to the same category as the bite marks can be

narrowed as the potential assailant.

6.3.2 Extension to Mandibular and 3D Shape of the Dental Arches

The current study focuses on the 2D digital images of the maxillary dental arch.

This research can be extended to the mandibular arches, whereby the same shape feature

can be derived from stable anatomical landmarks of the mandicular arches, which are

the retromolar pads and lingual frenum (Celebic et al., 1995; Roraff, 1977).

182

A study by Wu et al. (2012) has considered a 3D statistical model of gingival

contours from training dataset for reconstructing missing gingival contours in partially

edentulous patients. Recently, Elhabian & Farag (2014) formulated a statistical model

of teeth surface from training dataset using shape-from-shading approaches to estimate

the 3D teeth shape surface for clinical crowns to aid dentists dealing problems with

irregular teeth and dental pulp. Ideas from these works in obtaining the 3D images of

the dental arch may be incorporated this study to develop 3D shape discrimination

procedure. An extension of the study for 3D images which includes the depth of the

palate of the arches would be beneficial in more accurate design of the stock tray and

teeth positioning on complete denture.

6.3.3 Multivariate Complex Normal with Relation Matrix

The multivariate complex normal distribution considered in this study assumes

zero relation matrix. Piccinboho (1996) showed the importance of the relation matrix to

complete the description of the second-order statistics. Further development of

multivariate complex normal with relation matrix as a shape model may be explored.

6.3.4 Comparing Shape Discrimination using COVRATIO and Bayesian Methods

In real situations, particularly in shape analysis, smaller sample size relative to

numbers of landmarks or variables selected to represent the shape of the subject are

likely to occur. A Bayesian approach mentioned in Mardia et al. (1979) may be an

alternative to deal with this problem. It is an interest to compare the proposed

COVRATIO and the Bayesian methods for shape discrimination.

183

6.3.5 Regression Ideas for Shape Models and Discrimination

Let ),,,,,,,()( 44332211

kkkkkkkk lwlwlwlwk v be the k-th ( 2,1k ) shape category

where ),( k

j

k

j lw represent the j-th selected teeth. Two regression models of the j-th teeth

gives

ˆˆ and ˆˆ 22221111

jjjjjjjj lwlw .

Shape categories )1(v and )2(v are said to be different in shape if 21 ˆˆjj and

21 ˆˆjj for all j. It is an interest to develop an alternative shape model and

discrimination of the dental arch based on this idea and compare with shape model

presented in this study.

184

REFERENCES

Abdullah, M. A. (2002). Inner canthal distance and geometric progression as a predictor

of maxillary central incisor width. The Journal of Prosthetic Dentistry, 88(1),

16-20.

Abuzaid, A. H., Hussin, A. G., & Mohamed, I. B. (2008). Identifying single outlier in

linear circular regression model based on circular distance. Journal of Applied

Probability and Statistics, 3(1), 107–117.

Aho, K. (2015). asbio: A Collection of Statistical Tools for Biologists. R package

version 1.1-5. Retrieve from

http://cran.r-project.org/web/packages/asbio/asbio.pdf

AlHarbi, S., Alkofide, E. A., & AlMadi, A. (2008). Mathematical analyses of dental

arch curvature in normal occlusion. The Angle Orthodontist, 78(2), 281-287.

Allen, P. D., Graham, J., Farnell, D. J., Harrison, E. J., Jacobs, R., Nicopolou-

Karayianni, K., ... & Devlin, H. (2007). Detecting reduced bone mineral density

from dental radiographs using statistical shape models. Information Technology

in Biomedicine, IEEE Transactions on, 11(6), 601-610.

Arai, K., & Will, L. A. (2011). Subjective classification and objective analysis of the

mandibular dental arch form of orthodontic patients. American Journal of

Orthodontics and Dentofacial Orthopedics, 139(4), e315-e321.

Anderberg, M. R. (1973). Cluster analysis for applications. New York, NY: Academic

Press.

Andersen, H. H., Hojbjerre, M., Sorenson, D., & Eriksen, P. S. (1995). Linear and

Graphical Models: For the Multivariate Complex Normal Distribution (Vol.

101). New York, NY: Springer Verlag.

Anderson, T. W. (2003). An introduction to multivariate statistical analysis. New York,

NY: Wiley.

Banabilh, S., Rajion, Z., Samsudin, R., & Singh, G. (2006). Dental arch shape and size

in Malay schoolchildren with Class II malocclusion. Australian Orthodontic

Journal, 22(2), 99.

185

Barzi, F., & Woodward, M. (2004). Imputations of missing values in practice: Results

from imputations of serum cholesterol in 28 cohort studies. American Journal of

Epidemiology, 160(1), 34-45.

Beale, T. (2007). The essential elements of impression tray design. International

Dentistry South Africa, 9(4), 58-60.

BeGole, E. A. (1980). Application of the cubic spline function in the description of

dental arch form. Journal of Dental Research, 59(9), 1549-1556.

BeGole, E. A., & Lyew, R. C. (1998). A new method for analyzing change in dental

arch form. American Journal of Orthodontics and Dentofacial Orthopedics,

113(4), 394-401.

Bennett, K. P., & Bredensteiner, E. J. (2000). Duality and geometry in SVM classifiers.

In P. Langley (Ed.), Proceedings of the 17th International Conference on

Machine Learning (pp. 57-64). San Francisco, CA: Morgan Kaufman.

Bishop, C. M., & Nasrabadi, N. M. (2006). Pattern recognition and machine learning.

New York, NY: Springer.

Bissasu, M. (1992). Copying maxillary anterior natural tooth position in complete

dentures. Journal of Prostethic Dentistry, 67(5), 688-689.

Blackwell, S., Taylor, R., Gordon, I., Ogleby, C., Tanijiri, T., Yoshino, M., Clement, J.

(2007). 3-D imaging and quantitative comparison of human dentitions and

simulated bite marks. International Journal of Legal Medicine, 121(1), 9-17.

Bolshakova, N., & Azuaje, F. (2003). Cluster validation techniques for genome

expression data. Signal Processing, 83(4), 825-833.

Bomberg, T. J., Hatch, R. A., & Hoffman, W. J. (1985). Impression material thickness

in stock and custom trays. Journal of Prostethic Dentistry, 54(2), 170-172.

Boogaart, K. G., Tolosana, and R., Bren, M. (2015). compositions: Compositional data

analysis. R package version 1.40-1. Retrieve from http://cran.r-

project.org/web/packages/compositions/compositions.pdf

Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions.

Statistical Science, 181-222.

186

Braun, S., Hnat, W., Fender, D., & Legan, H. (1998). The form of the human dental

arch. The Angle Orthodontics, 68(1), 29-36.

Bruno, O. M., de Oliveira Plotze, R., Falvo, M., & de Castro, M. (2008). Fractal

dimension applied to plant identification. Information Sciences, 178(12), 2722-

2733.

Burris, B. G., & Harris, E. F. (2000). Maxillary arch size and shape in American blacks

and whites. The Angle Orthodontist, 70(4), 297-302.

Bush, M. A., Bush, P. J., & Sheets, H. D. (2011). Similarity and match rates of the

human dentition in three dimensions: Relevance to bitemark analysis.

International Journal of Legal Medicine, 125(6), 779-784.

Cassidy, K. M., Harris, E. F., Tolley, E. A., & Keim, R. G. (1998). Genetic influence on

dental arch form in orthodontic patients. The Angle Orthodontist, 68(5), 445-

454.

Celebic, A., Valenticcar-Peruzovic, M., Kraljević, K., & Brkić, H. (1995). A study of

the occlusal plane orientation by intra‐oral method (retromolar pad). Journal of

oral rehabilitation, 22(3), 233-236.

Charistos, L., Hatjina, F., Bouga, M., Mladenovic, M., & Maistros, A. D. (2014).

Morphological discrimination of Greek honey bee populations based on

geometric morphometrics analysis of wing Shape. Journal of Apicultural

Science, 58(1), 75-84.

Chu, S. J. (2007). COMMENTARY. A study of dentists’preferred maxillary anterior

tooth width proportions: Comparing the recurring esthetic dental proportion to

other mathematical and naturally occurring proportions. Journal of Esthetic and

Restorative Dentistry, 19(6), 338–339.

Cooley, J. W., & Tukey, J. W. (1965). An algorithm for the machine calculation of

complex Fourier series. Mathematics of Computation, 19(90), 297-301.

Costa, L. F. D., & Cesar, R. M. (2009). Shape analysis and classification: Theory and

practice. Florida, FL: CRC Press.

de la Cruz, A., Sampson, P., Little, R. M., Årtun, J., & Shapiro, P. A. (1995). Long-term

changes in arch form after orthodontic treatment and retention. American

Journal of Orthodontics and Dentofacial Orthopedics, 107(5), 518-530.

187

Dass, S. C., & Li, M. (2009). Hierarchical mixture models for assessing fingerprint

individuality. The Annals of Applied Statistics, 3(4), 1448-1466.

Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from

incomplete data via the EM algorithm. Journal of the Royal Statistical Society.

Series B (Methodological), 39(1), 1-38.

Doornik, J. A., & Hansen, H. (2008). An Omnibus Test for Univariate and Multivariate

Normality. Oxford Bulletin of Economics and Statistics, 70, 927-939.

Dryden, I., & Mardia, K. (1998). Statistical analysis of shape. New York, NY: Wiley.

Dudewicz, E. J., & Mishra, S. N. (1988). Modern Mathematical Statistics: Singapore:

Wiley.

Eakins, J. P., Riley, K. J., & Edwards, J. D. (2003). Shape feature matching for

trademark image retrieval. Proceedings of the 2nd International Conference

on Image and Video Retrieval, 28-38. doi: 10.1007/3-540-45113-7_4

Elhabian, S. Y., & Farag, A. A. (2014). Appearance-based approach for complete

human jaw shape reconstruction. IET Computer Vision, 8(5), 404-418.

Everitt, B. S., Landau, S., & Leese, M. (2001). Cluster analysis. London, England:

Arnold.

Faigenblum, M., & Sharma, P. (2007). Determining replacement teeth position for the

implant-retained prosthesis in the edentulous patient. Alpha Omegan, 100(2),

67-74.

Farrell, P. J., Salibian-Barrera, M., & Naczk, K. (2007). On tests for multivariate

normality and associated simulation studies. Journal of Statistical Computation

and Simulation, 77(12), 1065-1080.

Ferrario, V. F., Sforza, C., Miani, A. J., & Tartaglia, G. (1994). Mathematical definition

of the shape of dental arches in human permanent healthy dentitions. European

Journal of Orthodontics, 16(4), 287-294.

Ferrario, V. F., Sforza, C., Colombo, A., Carvajal, R., Duncan, V., & Palomino, H.

(1999). Dental arch size in healthy human permanent dentitions: ethnic

differences as assessed by discriminant analysis. The International Journal of

Adult Orthodontics and Orthognathic Surgery, 14(2), 153.

188

Ferrario, V. F., Sforza, C., Miani, A., & Serrao, G. (1993). Dental arch asymmetry in

young healthy human subjects evaluated by Euclidean distance matrix analysis.

Archives of Oral Biology, 38(3), 189-194.

Fisher, N. I. (1993). Statistical analysis of circular data. Cambridge, England:

Cambridge University Press.

Genz, A. and Azzalini, A. (2015). mnormt: The Multivariate Normal and t

Distributions. R package version 1.5-3. Retrieve from http://cran.r-

project.org/web/packages/mnormt/mnormt.pdf

Giri, N. C. (2003). Multivariate Statistical Analysis: Revised And Expanded. New York,

NY:Taylor & Francis.

Gonzalez, R. C., & Woods, R. E. (2002). Digital image processing. Englewood Cliffs,

NJ: Prentice Hall.

Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex

Gaussian distribution (an introduction). Annals of Mathematical Statistics, 34,

152-177.

Ghapor A. A., Zubairi, Y. Z., Mamun, A. S. M. A., & Imon, A. H. M. R. (2014). On

detecting outlier in simple linear functional relationship model using

COVRATIO statistics, Pakistan Journal of Statistics, 30(1), 129-142.

Grave, A., & Becker, P. (1987). Evaluation of the incisive papilla as a guide to anterior

tooth position. The Journal of Prosthetic Dentistry, 57(6), 712-714.

Gu, Q. R., Shibata, T., Fujita, K., & Takada, K. (2002). Application of vector

quantization algorithm to dental arch classification in orthodontics practice.

Paper presented at the Proceedings of the 5th Biannual World Automation

Congress Orlando, FL, USA.

Hankin, RKS. cmvnorm: The complex multivariate Gaussian distribution.

R package version 1.0-1, 2015. Retrieve from

http://cran.r-project.org/web/packages/cmvnorm/cmvnorm.pdf.

189

Hannigan, A., & Lynch, C. D. (2013). Statistical methodology in oral and dental

research: pitfalls and recommendations. Journal of Dentistry, 41(5), 385-392.

Hao, J., Ito, R., Kikuchi, N., Kobayashi, S., & Tingshen, Z. (2000). A morphological

study on the dental arch of the Chaoxian. Anthropological Science, 108(2), 215-

222.

Hatrick, C. D., Eakle, W. S., & Bird, W. F. (2003). Dental materials: clinical

applications for dental assistants and dental hygienists. St. Louis, MO: Elsevier

Saunders.

Henrikson, J., Persson, M., & Thilander, B. (2001). Long-term stability of dental arch

form in normal occlusion from 13 to 31 years of age. The European Journal of

Orthodontics, 23(1), 51-61.

Henze, N., & Zirkler, B. (1990). A class of invariant consistent tests for multivariate

normality. Communications in Statistics-Theory and Methods, 19(10), 3595-

3617.

Higashiura, K., Mukherjee, D. P., Okada, T., Yokota, F., Hori, M., Takao, M., ... &

Sato, Y. (2012, October). Disease discrimination based on disease subspace of

organ shape using orthogonal complement of normal subspace. In Information

Science and Service Science and Data Mining (ISSDM). Paper presented at the

6th International Conference on New Trends in, Taipei, Taiwan, 23-25 October

(pp. 453-457). Piscataway, NJ: IEEE.

Horswell, R. L., & Looney, S. W. (1992). A comparison of tests for multivariate

normality that are based on measures of multivariate skewness and kurtosis.

Journal of Statistical Computation and Simulation, 42(1-2), 21-38.

Hu, H., Li, Y., Liu, M., & Liang, W. (2014). Classification of defects in steel strip

surface based on multiclass support vector machine. Multimedia Tools and

Applications, 69(1), 199-216.

Hufnagel, H. (2011). A Probabilistic Framework for Point-Based Shape Modeling in

Medical Image Analysis. Wiesbaden: Vieweg+ Teubner Verlag.

Huttenlocher, D. P., Klanderman, G. A., & Rucklidge, W. J. (1993). Comparing images

using the Hausdorff distance. Pattern Analysis and Machine Intelligence, IEEE

Transactions on Pattern Analysis and Machine Intelligence, 15(9), 850-863.

190

Ibrahim, S., Rambli, A., Hussin, A. G., & Mohamed, I. (2013). Outlier detection in a

circular regression model using COVRATIO statistics. Communications in

Statistiscs-Simulation and Computation, 42(10), 2272-2280.

Isa, Z. M., Tawfiq, O. F., Noor, N. M., Shamsudheen, M. I., & Rijal, O. M. (2010).

Regression methods to investigate the relationship between facial measurements

and widths of the maxillary anterior teeth. The Journal of Prosthetic Dentistry,

103(3), 182-188.

Jammalamadaka, S. R., & Sengupta, A. (2001). Topics in circular statistics (Vol. 5).

River Edge, NJ: World Scientific Publishing.

Johnson, R. W., & Wichern, D. (1992). Applied Multivariate Statistical Analysis:

Englewood Cliffs, NJ: Prentice Hall.

Kairalla, S. A., Scuzzo, G., Triviño, T., Velasco, L., Lombardo, L., & Paranhos, L. R.

(2014). Determining shapes and dimensions of dental arches for the use of

straight-wire arches in lingual technique. Dental press journal of

orthodontics, 19(5), 116-122.

Kang, S., Kang, P., Ko, T., Cho, S., Rhee, S. J., & Yu, K. S. (2015). An efficient and

effective ensemble of support vector machines for anti-diabetic drug failure

prediction. Expert Systems with Applications, 42(9), 4265-4273.

Kasai, K., Richards, L. C., & Clement, G. (1995). Fourier analysis of dental arch

morphology in South Australian twins. Anthropological Science, 103(1), 39-48.

Keyes, L., & Winstanley, A. C. (1999). Fourier descriptors as a general classification

tool for topographic shapes. In .F. Whelan (Ed.), Paper presented at the IMVIP

'99 Proceedings of the Irish Machine Vision and Image Processing Conference,

Dublin City University, Dubin (pp. 193-203). Dublin: Dublin City University.

Kim, J. S. J., Kim, D. K., & Hong, S. J. (2011). Assessment of errors and misused

statistics in dental research. International dental journal, 61(3), 163-167.

Korkmaz, S., Goksuluk, D., Zararsiz, G. (2015). MVN: Multivariate normality test. R

package version 4.0. Retrieve from http://cran.r-

project.org/web/packages/MVN/MVN.pdf

191

Kreyszig, E. (2007). Advanced engineering mathematics. New York, NJ: Wiley.

Kuhl, F. P., & Giardina, C. R. (1982). Elliptic Fourier features of a closed contour.

Computer graphics and image processing, 18(3), 236-258.

Lacko, D., Huysmans, T., Parizel, P. M., De Bruyne, G., Verwulgen, S., Van Hulle, M.

M., & Sijbers, J. (2015). Evaluation of an anthropometric shape model of the

human scalp. Applied Ergonomics, 48, 70-85.

Lee, S. J., Lee, S., Lim, J., Park, H. J., & Wheeler, T. T. (2011). Method to classify

dental arch forms. American Journal of Orthodontics and Dentofacial

Orthopedics, 140(1), 87-96.

Lestrel, P. E. (2008). Fourier descriptors and their applications in biology (2nd ed.).

Cambridge, England: Cambridge University Press.

Lestrel, P. E., Takahashi, O., & Kanazawa, E. (2004). A quantitative approach for

measuring crowding in the dental arch: Fourier descriptors. American Journal of

Orthodontics and Dentofacial Orthopedics, 125(6), 716-725.

Ling, J. Y. K., & Wong, R. W. K. (2009). Dental arch widths of southern Chinese. The

Angle Orthodontist, 79(1), 54-63.

Looney, S. W. (1995). How to use tests for univariate normality to assess multivariate

normality. The American Statistician, 49(1), 64-70.

Lu, K. (1966). An orthogonal analysis of the form, symmetry and asymmetry of the

dental arch. Archives of Oral Biology, 11(11), 1057-1069.

Maas, G. (1974). The Physique of Athletes: An Anthropometric Study of 285 Top

Sportsmen from 14 Sports in a Total of 774 Athletes. Leiden: Leiden University

Press.

Malaysia. (2012). Population projections 2010-2040. Kuala Lumpur: Department of

Statistics Malaysia.

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with

applications. Biometrika, 57(3), 519-530.

192

Mardia, K. V., Bookstein, F. L., & Moreton, I. J. (2000). Statistical assessment of

bilateral symmetry of shapes. Biometrika, 87, 285-300.

Mardia, K. V., & Jupp, P. E. (2000). Directional statistics. New York, NY: Wiley.

Mardia, K. V., & Foster, K. (1983). Omnibus tests of multinormality based on skewness

and kurtosis. Communications in Statistics-Theory and Methods, 12(2), 207-221.

Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate analysis. New York,

NY: Academic Press.

Maurice, T. J., & Kula, K. (1998). Dental arch asymmetry in the mixed dentition. The

Angle Orthodontist, 68(1), 37-44.

McCabe, J. F., & Walls, A. (1998). Applied dental materials. London, England:

Blackwell Science.

McKee, J., & Molnar, S. (1988). Mathematical and descriptive classification of

variations in dental arch shape in an Australian aborigine population. Archives of

Oral Biology, 33(12), 901-906.

McKenzie, J., & Goldman, R. N. (1998). Minitab Handbook for Windows 95 and

Windows NT. Boston, MA: Addison-Wesley.

Mecklin, C. J., & Mundfrom, D. J. (2003). On using asymptotic critical values in testing

for multivariate normality. InterStat. Retrieved from http://interstat.

statjournals.net/ YEAR/2003/articles/0301001.pdf

Mecklin, C. J., & Mundfrom, D. J. (2004). An appraisal and bibliography of tests for

multivariate normality. International Statistical Review, 72(1), 123-138.

Mikami, H., Nakatsuka, M., & Iwai, Y. (2010). Comparison of maxillary and

mandibular dental arch forms by studying Fourier series developed from

mathematically estimated dentitions. Okajimas Folia Anatomica Japonica,

87(3), 85-96.

Mukherjee, D. P., Higashiura, K., Okada, T., Hori, M., Chen, Y. W., Tomiyama, N., &

Sato, Y. (2013). Utilizing disease-specific organ shape components for disease

discrimination: Application to discrimination of chronic liver disease from CT

data. In K. Mori, I. Sakuma, Y. Sato, C. Barillot (Eds.), Proceedings of the 16th

193

International Conference on Medical Image Computing and Computer-Assisted

Intervention, Nagoya, Japan (pp. 235-242). Berlin: Springer Verlag.

Murphy, K. R., & Davidshofer, C. O. (1991). Psychological testing. Englewood Cliffs,

NJ: Prentice Hall.

Nakatsuka, M., Iwai, Y., Huang, S.-T., Huang, H.-C., Kon-I, H., Morishita, A., &

Hsiao, S.-Y. (2011). Cluster Analysis of Maxillary Dental Arch Forms. Taiwan

Journal of Oral Medicine Sciences 27(2), 3-18.

Nagaraj, S. (2009). Intermarrriage in Malaysia. Malaysian Journal of Economic Studies

46, 75-92.

Nikola, P., Čelebić, A., Baučić, M., & Robert, A. (2005). Importance of hamular

distance for calculation of the width of maxillary anterior teeth. Acta

Stomatologica Croatica, 39(3), 291-294.

Nojima, K., McLaughlin, R. P., Isshiki, Y., & Sinclair, P. M. (2001). A comparative

study of Caucasian and Japanese mandibular clinical arch forms. The Angle

Orthodontist, 71(3), 195-200.

Oppenheim, A. V., Willsky, A. S., & Young, I. T. (1983). Signals and systems.

Englewood Cliffs, NJ: Prentice Hall.

O'Higgins, P., & Dryden, I. L. (1993). Sexual dimorphism in hominoids: further studies

of craniofacial shape differences in Pan, Gorilla and Pongo. Journal of Human

Evolution, 24(3), 183-205.

O'Higgins, P. (2000). The study of morphological variation in the hominid fossil record:

Biology, landmarks and geometry. Journal of Anatomy, 197(1), 103-120.

Pannu, N. S., McCoy, A. J., & Read, R. J. (2003). Application of the complex

multivariate normal distribution to crystallographic methods with insights into

multiple isomorphous replacement phasing. Acta Crystallographica Section D:

Biological Crystallography, 59(10), 1801-1808.

Patil, P. S., Chowdhary, R., & Mishra, S. (2008). Comparison of custom trays and stock

trays using polyvinylsiloxane to evaluate linear dimensional accuracy: An in

vitro study. The Journal of Indian Prosthodontic Society, 8(3), 156-161.

Peña, D., & Rodrıguez, J. (2003). Descriptive measures of multivariate scatter and

linear dependence. Journal of Multivariate Analysis, 85(2), 361-374.

194

Picinbono, B. (1996). Second-order complex random vectors and normal

distributions. IEEE Transactions on Signal Processing, 44(10), 2637-2640.

Pilgram, R., Schubert, R., Fritscher, K. D., Zwick, R. H., Schocke, M. F., Trieb, T., &

Pachinger, O. (2006). Shape discrimination of healthy and diseased cardiac

ventricles using medial representation. International Journal of Computer

Assisted Radiology and Surgery, 1(1), 33-38.

Preti, G., Pera, P., & Bassi, F. (1986). Prediction of the shape and size of the maxillary

anterior arch in edentulous patients. Journal of Oral Rehabilitation, 13(2), 115-

125.

Raberin, M., Laumon, B., Martin, J. L., & Brunner, F. (1993). Dimensions and form of

dental arches in subjects with normal occlusions. American Journal of


Radmer, T. W., & Johnson, L. T. (2009). The Correlation of Dental Arch Width and

Ethnicity. Journal of Forensic Identification, 59 (3), 268-274.

Roraff, A. R. (1977). Arranging artificial teeth according to anatomic landmarks.The

Journal of prosthetic dentistry, 38(2), 120-130.

Royston, J. P. (1983). Some techniques for assessing multivarate normality based on the

Shapiro-Wilk W. Applied Statistics, 32(2), 121-133.

Royston, P. (1992). Approximating the Shapiro-Wilk W-Test for non-normality.

Statistics and Computing, 2(3), 117-119.

Sampson, P. D. (1981). Dental arch shape: A statistical analysis using conic sections.

American Journal of Orthodontics, 79(5), 535-548.

Sampson, P. D. (1983). Statistical analysis of arch shape with conic sections.

Biometrics, 39(2), 411-423.

Scanavini, P. E., Paranhos, L. R., Torres, F. C., Vasconcelos, M. H. F., Jóias, R. P., &

Scanavini, M. A. (2012). Evaluation of the dental arch asymmetry in natural

normal occlusion and Class II malocclusion individuals. Dental Press Journal of

Orthodontics,17(1), 125-137.

Schafer, J. L. (2010). Analysis of incomplete multivariate data. Florida: CRC Press.

195

Schaefer, K., Lauc, T., Mitteroecker, P., Gunz, P., & Bookstein, F. L. (2006). Dental

arch asymmetry in an isolated Adriatic community. American Journal of

Physical Anthropology, 129(1), 132-142.

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality

(complete samples). Biometrika, 52, 591-611.

Sheets, D. H., Bush, P. J., & Bush, M. A. (2013). Patterns of Variation and Match Rates

of the Anterior Biting Dentition: Characteristics of a Database of 3D‐Scanned

Dentitions. Journal of forensic sciences, 58(1), 60-68.

Shenton, L. R., & Bowman, K. O. (1977). A bivariate model for the distribution of √b1

and b2. Journal of the American Statistical Association, 72 (357), 206-211.

Shrestha, R. M., & Bhattarai, P. (2009). Dental arch length and arch symmetry analysis

of Nepalese permanent dentition. Journal of Nepal Dental Association, 10(2),

110-114.

Shrestha, R. M. (2013). Polynomial Analysis of Dental Arch Form of Nepalese Adult

Subjects. Orthodontic Journal of Nepal, 3(1), 7-13.

Šlaj, M., Ježina, M. A., Lauc, T., Rajic-Meštrovic, S., & Mikšic, M. (2003).

Longitudinal dental arch changes in the mixed dentition. The Angle

Orthodontist, 73(5), 509-514.

Small, N. J. H. (1980). Marginal skewness and kurtosis in testing multivariate

normality. Applied Statistics, 29, 85-87.

Srivastava, M. S. (1984). A measure of skewness and kurtosis and a graphical method

for assessing multivariate normality. Statistics & Probability Letters, 2(5), 263-

267.

Srivastava, M. S., & Hui, T. K. (1987). On assessing multivariate normality based on

Shapiro-Wilk W statistic. Statistics & Probability Letters, 5(1), 15-18.

Stegmann, M. B., & Gomez, D. D. (March, 2002). A brief introduction to statistical

shape analysis. Retrieved from http://www2.imm.dtu.dk/pubdb/views/

publication_details.php?id=403

196

Stevens, J.P. (2001). Applied MultivariateSstatistics for the Social Sciences. New

Jersey, NJ: Lawrence Erlbaum.

Sprowls, M. W., Ward, R. E., Jamison, P. L., & Hartsfield Jr, J. K. (2008). Dental arch

asymmetry, fluctuating dental asymmetry, and dental crowding: A comparison

of tooth position and tooth size between antimeres. Seminars in Orthodontics,

14(2), 157-165.

Taner, T. U., Ciǧer, S., El, H., Germec, D., & Es, A. (2004). Evaluation of dental arch

width and form changes after orthodontic treatment and retention with a new

computerized method. American Journal of Orthodontics and Dentofacial

Orthopedics, 126(4), 463-474.

Tanner, M. A., & Wong, W. H. (1987). The calculation of posterior distributions by

data augmentation. Journal of the American statistical Association, 82, 528-540.

Tong, H., Kwon, D., Shi, J., Sakai, N., Enciso, R., & Sameshima, G. T. (2012).

Mesiodistal angulation and faciolingual inclination of each whole tooth in 3-

dimensional space in patients with near-normal occlusion. American Journal of


Triviño, T., Siqueira, D. F., & Scanavini, M. A. (2008). A new concept of mandibular

dental arch forms with normal occlusion. American Journal of Orthodontics and

Dentofacial Orthopedics, 133(1), 10. e15-10. e22.

Tsai, T. Y., Li, J. S., Wang, S., Li, P., Kwon, Y. M., & Li, G. (2015). Principal

component analysis in construction of 3D human knee joint models using a

statistical shape model method. Computer methods in biomechanics and

biomedical engineering, 18(7), 721-729.

Uetani, M., Tateyama, T., Kohara, S., Tanaka, H., Han, X. H., Kanasaki, S., ... & Chen,

Y. W. (2015). Statistical Shape Model of the Liver and Its Application to

Computer‐Aided Diagnosis of Liver Cirrhosis. Electrical Engineering in

Japan, 190(4), 37-45

Valenzuela, A. P., Pardo, M. A., & Yezioro, S. (2002). Description of dental arch form

using the Fourier series. The International Journal of Adult Orthodontics and

Orthognathic Surgery, 17(1), 59.

Wellens, H. (2007). A clinical–experimental simulation of changes in intercanine width

associated with the correction of crowding: A pilot study. The European Journal

of Orthodontics, 29(6), 632-638.

197

White, J. E. M. (1960). Teach yourself anthropology. London, England: English

Universities Press.

Wiland, L. (1971). Evaluating the size of dentulous stock trays. The Journal of

Prosthetic Dentistry, 25(3), 317-322.

Wooding, R. A. (1956). The multivariate distribution of complex normal variables.

Biometrika, 43, 112-115.

Wu, C. (1983). On the convergence properties of the EM algorithm. The Annals of

Statistics, 11(1), 95-103.

Wu, T., Liao, W., & Dai, N. (2012). Three-Dimensional statistical model for gingival

contour reconstruction. Biomedical Engineering, IEEE Transactions on, 59(4),

1086-1093.

Yergin, E., Ozturk, C., & Sermet, B. (2001). Image processing techniques for

assessment of dental trays. Proceedings of the 23rd Annual International

Conference of the IEEE Engineering in Medicine and Biology Society 3, 2571-

2573. doi: 10.1109/IEMBS.2001.1017305

Yu, Q., Miche, Y., Sorjamaa, A., Guillen, A., Lendasse, A., & Séverin, E. (2010). OP-

KNN: Method and applications. Advances in Artificial Neural Systems, 2010, 1-

6.

Yu, X., Cao, L., Liu, J., Zhao, B., Shan, X., & Dou, S. (2014). Application of otolith

shape analysis for stock discrimination and species identification of five goby

species (Perciformes: Gobiidae) in the northern Chinese coastal waters. Chinese

Journal of Oceanology and Limnology, 32, 1060-1073.

Zhang, D., & Lu, G. (2001). A comparative study on shape retrieval using Fourier

descriptors with different shape signatures. In Intelligent Multimedia,

Computing and Communications: Technologies and Applications of the

Future: Proceedings of International Conference on Intelligent Multimedia and

Distance Education Fargo, USA (pp. 1-9). New York: John Wiley & Sons.

Zhang, Y., & Wu, L. (2012). Classification of fruits using computer vision and a

multiclass support vector machine. Sensors, 12(9), 12489-12505.

Zia, M., Azad, A. A., & Ahmed, S. (2009). Comparison of distance between maxillary

central incisors and incisive papilla in dentate individuals with different arc

forms. J Ayub Med Coll Abbottabad, 21(4), 125-128.

198

LIST OF PUBLICATIONS AND PAPERS PRESENTED

1. N.A. Abdullah1, O.M. Rijal1, Y.Z. Zubairi2 and Z.M. Isa3 (2015). Hotelling two

sample T2 test for complex normal distribution. Accepted for publication in the

Pakistan Journal of Statistics, (ISI-Cited Publication).

2. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Noor, N. M., & Tawfiq, O. F. (2013,

July). Shape model of the maxillary dental arch using Fourier descriptors with

an application in the rehabilitation for edentulous patient. In Engineering in

Medicine and Biology Society (EMBC), 2013 35th Annual International

Conference of the IEEE (pp. 209-212) (IS-Cited Publication).

3. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Noor, N. M., & Tawfiq, O. F. (2012,

August). A probability distribution of shape for the dental maxillary arch using

digital images. In Engineering in Medicine and Biology Society (EMBC), 2012

Annual International Conference of the IEEE (pp. 5420-5423) (ISI-Cited

Publication).

4. Rijal, O. M., Abdullah, N. A., Isa, Z. M., Davaei, F. A., Noor, N. M., & Tawfiq,

O. F. (2011, August). A novel shape representation of the dental arch and its

applications in some dentistry problems. In Engineering in Medicine and

Biology Society, EMBC, 2011 Annual International Conference of the IEEE (pp.

5092-5095) (ISI-Cited Publication).

5. Isa, Z. M., Tawfiq, O. F., Abdullah, N. A., Noor, N. M., & Rijal, O. M. (2011).

Statistical clustering of maxillary dental arches. Scientific Research and Essays,

6(13), 2710-2719 (SCOPUS-Cited Publication).

6. N.A. Abdullah1, O.M. Rijal1, Y.Z. Zubairi2 , Z.M. Isa3 . COVRATIO Statistics as

discrimination method for complex normal distribution. Sent to Statistical

Methods in Medical Research on October 2015.

199

APPENDIX A: Ethics Approval

200

APPENDIX B: Examples of Research Data for ),,,,,,,( 44332211

RRRRRRRRR lwlwlwlwv , iii aaa 821ˆ , ,ˆ ,ˆˆ A and ),,( 321 LLLL .

Patient

No.

Gender Race Age Shape feature using ),,,,,,,( 44332211

RRRRRRRRR lwlwlwlwv . The selected tooth on the right side of

the arch is represented by the length of a line joining the cusp tips of a tooth to the origin (R

jl )

and an angle (R

jw ) with respect to the horizontal axis.

Rw1 Rl2

Rw2 Rl2

Rw3 Rl3

Rw4 Rl4

N001 Male Malay 21

83.312621 57.944650 74.485064 54.575058 65.099794 51.244767 30.104148 33.004298

N002 Female Malay 27

85.165569 49.202121 75.641731 48.695497 68.162934 44.940678 33.092770 33.263693

N093 Male Chinese 30

89.377146 59.761346 81.352490 57.074108 73.565726 52.713229 47.993159 37.722920

20

1

Patient

No.

Gender Race Age Shape feature using 8 FD terms, iii aaa 821

ˆ , ,ˆ ,ˆˆ A used to represent all teeth on the dental arch.

1a 2a 3a 4a 5a 6a 7a 8a

N001 Male Malay 21 3.156537 +

56.666303i

-2.612131 -

60.720536i

-0.588401 -

4.331291i

0.179727 -

8.073411i

0.103580 -

3.488072i

0.341993 -

1.295228i

0.103206 +

7.633484i

-0.530347 +

4.967460i

N002 Female Malay 27 -2.735498 +

64.028726i

-1.390996 -

59.425994i

0.142704 -

9.589793i

0.528870 -

8.970231i

0.351559 -

5.507708i

0.260832 -

2.708948i

0.605500 +

6.252824i

0.348833 +

6.299513i

N003 Male Chinese 30 -5.047854 +

68.200273i

2.648933 -

58.916754i

0.012492 -

8.904548i

0.076124 -

11.062200i

-0.151649 -

5.378759i

-0.390196 -

3.510903i

0.050893 +

5.386973i

-0.382475 +

4.431804i

Patient

No.

Gender Race Age Shape feature using 3 FD terms, ),,( 321 LLLL depicting 3 anatomical landmarks on the

dental arch.

1L 2L

3L

N001 Male Malay 21

-0.055406 + 4.026401i 6.681682 - 3.877263i -0.274842 - 0.178272i

N002 Female Malay 27

-0.117819 + 4.029429i 6.599693 - 3.864965i -0.360941 - 0.193751i

N003 Male Chinese 30

-0.252726 + 5.027066i 7.790054 - 4.481303i -0.755679 - 0.812290i

20

2

APPENDIX C: Summary of Works on Dental Arch Symmetry

Authors Purpose of

investigation

of arch

symmetry

Subject

(adult or

adolescent)

and results

of the arch

symmetry

Measurements used Size and

shape

demonst-

rated

Image

registration

demonstra-

ted

Method used to assess symmetry

of arch

Lundstrom

(1961)

To investigate

arch symmetry

in genetically

simiar

individuals.

Children aged

13 years old.

Asymmetry

arches.

The center of the arch is

defined from the palatal

raphae line and the

transverse distance

connecting on each side

of the teeth to the center

were obtained.

Size only.

Not using

image.

Distance of the left side of the teeth

to the raphae line was greater than

the right side.

Lavelle &

Plant

(1969)

To provide a

record of the

British

dentition.

Adults aged

18 years.

Symmetry.

Mesiodistal crown

diameters from 2nd molar

to 1st incisor on each side

of the dental arch.

Size only. No Unclear method of symmetry test.

Uses p-value to indicate

insignificance of the statistical test.

Ferrario et

al. (1993)

Patients with

normal arch

and

malocclusion

were evaluated

for the

symmetry.

Adult 20-27

years.

Symmetry,

but not in

maxillary

female arch.

Each patient’s dental arch

is represented by linear

distances between all

possible pairs of

landmarks on the left and

right arch (21 distances

for each side of arch).

Size only. No. Ratio of corresponding linear

distances on the left and right arch

for each patient and the average

were calculated. Then, T1=max

average ratio/ min average ratio

was obtained. Hypothesis testing is

tested on the null hypothesis H0: T1

= 1 (indicating that the size is

similar and therefore symmetry).

20

3

Araujo et

al. (1994)

To investigate

dental

symmetry in

individuals

with normal

dental

occlusions.

Adults

Mean age 22.4

years

Symmetry

dental arches.

(1) Difference between

mesiobuccal cusps of

right and left first molars

and median axis of the

defined grid (molar

transverse asymmetry

using the transferred

palatal raphae).

(2) Molar

anteroposterior

asymmetry using a plane

that passess by the

buccal surfaces of

incisors.

(3) Mandibular midline

deviation to the

transferred palatal

raphae.

Shape

only.

Not using

image.

Student’s t test and chi-square tests

were conducted to check for

statistically signifiacant

asymmetries between right and left

sides of the arch.

Cassidy et

al. (1998)

To investigate

the role

heredity plays

in determining

dental arch

size and shape.

Children aged

10-19 years

old.

Not symmetry.

Linear and angular

measurements on the left

and right arch

(See Fig 3A-D in this

paper).

Size only. No. Paired t-test testing for mu_d=0

Use mixed model ANOVA.

Maurice &

Kula (1998)

Describe the

degree of

asymmetry.

For

orthodontic

treatment and

to evaluate

arch symmetry

0 to 9 years.

Symmetry.

Assymmetry

only present at

mandibular

canine region.

Linear distances from

teeth to medial palatal

plane.

Size only. No. One tailed median signed test

mu(|d|)>2mm. (nonparametric-

using median instead of mean)

Paired t-test indicate significant

differences from 0

20

4

Šlaj et al.

(2003)

Analysze

changes of

arch symmetry

for developing

occlusal

relationship.

For

orthodontic

treatment.

3-13 years

Symmetry.

Assymmetry

only present at

mandibular

canine region.

Arch width and length

of left and right arch

measured from medial

palatal plane.

Size only. No. Two sample t test for independent

sample.

Schaefer et

al. (2006)

To investigate

is ethnic origin

influences

dental arch

symmetry.

Children 7-15

years

Symmetry. No

results/tables

were

displayed.

3D Digitized coordinate

(x,y,z)

Shape and

size.

Yes. In text: The mirror image of each

dental cast was produced using one

side of the arch. Then, the forms of

the original and the mirror were

superimposed using generalized LS

Procrustes.

Modified Hotelling 2T test were

used to test the symmetry of arch

simultaneously (Mardia et al.,

2000).

Sprowls et

al. (2008)

to determine

the

relationship

between dental

arch

asymmetry

and

asymmetry.

11.5 to 48.3

years.

Symmetry.

Distance between

permanent teeth and

MPR (medial palatal

raphae)

On the right and left

side of arch.

Shape

only.

No. Two way ANOVA for testing:

H0_d: tooth position depends on

the presence/absence of directional

asymmetry

Ho_e: tooth position depends on

whether the remaining fluctuation

asymmetry exceeds the within

observer measurement error or not.

Shrestha &

Bhattarai

Of interest to

the

Adult 17-32

years. Symmetry

Linear distance from

(1) midpoint of central

Size only. No. Two sample t-test testing for

independent sample

20

5

(2009) orthodontist

for functional

and aesthetic

value. Also

useful for

odontometric

and

anthropometric

purposes.

in both

maxillary and

mandibular

female and male

arches.

incisor to canine and

(2) midpoint of central

incisor to first molar

cusp teeth

on the left and right

arch.

- Left and right arch is

independent?

Scanavini

et al. (2012)

To verify the

degree of

asymmetry in

Brazilian

dental arches.

For

orthodontic

treatment.

12 to 21 years

subjects

Asymmetry in

the dental arches

was found in all

individuals.

Normal arches.

has smaller

asymmetry

degree

Liner and Angular

measurements of teeth

position in digital

images and protractor

(standardized) on the

left and right arch.

Shape

only.

Not using

digital

image.

ANOVA.

Test of Pearson’s correlation

coefficient, 0 for differences

between left and right side of the

arch.

Tong et al.

(2012)

to obtain the

angulations

and inclination

for all teeth in

near-normal

occlusion.

Adult subjects.

Almost perfect

symmetry

between the

right and left

side

measurements

allowed them to

combine the 2-

side data.

Used 76 CBCT scans

before treatment,

They measured the

mesiodistal and

faciolingual

angulations between

Long axis of upper-

lower teeth.

Shape

only

(tooth

position in

dental

arch).

Yes,

comparable

somehow to

present

study.

Paired t test for normally

distributed data.

And wilcoxon signed rank test for

non-normal data.

Use Bonferoni correction.

206

APPENDIX D:

MATLAB Program for Determine Appropriate Impression Tray for Dentate

Patients

%new cast image to assign to a suitable fabricated tray %select only 4 teeth on the right arch: central incisor, lateral

incisor, %canine and distobuccal of first molar teeth

%% retrieve image

cd c:/; % Change directory [f,d] = uigetfile('*'); % * means can select any file. d=directory

name. f=filename S=imread([d,f]); imshow(S) h = helpdlg('select the desired area and double click to crop the

picture'); uiwait (h); Y=imcrop(S); Y=rgb2gray(Y); figure; imshow(Y) h = helpdlg('select 2 points from the ruler to calibrate the distance

for 1cm'); uiwait (h); [x1,y1]=ginput(1); hold on; plot(x1,y1,'-c+') [x2,y2]=ginput(1); hold on; plot(x2,y2,'-c+')

D1 = (x2-x1)^2 + (y2-y1)^2; oneCm = sqrt(D1);

h = helpdlg('Click on 2 hamular notches'); uiwait (h); [x3,y3]=ginput(1); %Titik hamular notch pertama hold on; plot(x3,y3,'-c+') [x4,y4]=ginput(1); %Titik hamular notch kedua hold on; plot(x4,y4,'-c+') h = helpdlg('Click on the incisive papilla'); uiwait (h); [x5,y5]=ginput(1); %Titik incisive papilla hold on; plot(x5,y5,'-c+')

%------------------------------------- % Dapatkan persamaan garislurus %-------------------------------------

m1 = (y3-y4)/(x3-x4); m2 = -1/m1; c2 = y3 - m1*x3; %Equation for the hamular notch c1 = y5 - m2*x5; % Equation perpendicular to hamular notch line

207

%Dapatkan titik origin (u, w) (in pixels) antara 2 garisan u = (c2-c1)/(m2-m1); w = m2*u + c1; xHamular=[x3 u x4]; yHamular=[y3 w y4]; plot(xHamular,yHamular)

xIncisive=[x5 u]; yIncisive=[y5 w]; plot(xIncisive,yIncisive)

data=double([]); cnt=1; button = questdlg('Click at the point (cusp tips of the teeth) you

wish to measure from the origin. Next point?',... 'Continue Operation','Yes','No','No'); while strcmp(button,'Yes') [x,y]=ginput(1); hold on; plot(x,y,'-c+')

xx=x-u; yy=y-w;

if xx<0 xx=-xx; end

theta=abs(atan(yy/xx))*180/pi; lengthcm=sqrt(xx^2+yy^2)/oneCm; length=lengthcm*10;

data=[data;theta,length];

button = questdlg('Take the next point?',... 'Continue Operation','Yes','No','No'); cnt=cnt+1; if strcmp(button,'No') | cnt >= 20 break end end

save data

X=[data(1,1) data(1,2) data(2,1) data(2,2) data(3,1) data(3,2)

data(4,1) data(4,2)];

%mean and covariance matrix for C1 meanc1=[83.16 57.30 74.85 54.91 67.27 51.49 36.82

36.43];

Sc1=[2.012123598 0.337432009 2.052163747 0.399304037 1.374913377

0.592820423 1.372807356 -0.150785691 0.337432009 2.922243216 0.381430085 2.214038665 0.58930365

2.054974648 0.597471715 0.525553419 2.052163747 0.381430085 2.241715613 0.47811376 1.78768176

0.573042135 2.210770242 0.192481994 0.399304037 2.214038665 0.47811376 2.046860634 0.886332222

1.882684422 1.270886268 1.206468829

208

1.374913377 0.58930365 1.78768176 0.886332222 2.579873342

0.877251632 3.890586272 1.854193936 0.592820423 2.054974648 0.573042135 1.882684422 0.877251632

1.953917399 1.100585987 1.124200586 1.372807356 0.597471715 2.210770242 1.270886268 3.890586272

1.100585987 7.9871065 3.406908715 -0.150785691 0.525553419 0.192481994 1.206468829 1.854193936

1.124200586 3.406908715 4.472910114];


33.91];

Sc2=[1.787781732 1.043992557 1.952824521 0.781918286 2.065771236

0.58693496 1.834273356 -0.214270913 1.043992557 4.62791343 1.538818563 3.845541984 1.700776267 3.75215652

3.377773901 1.25664875 1.952824521 1.538818563 2.342034167 1.274220143 2.53759831

1.201756339 2.900485286 0.153779997 0.781918286 3.845541984 1.274220143 3.737919597 1.659892627

3.782797801 4.007685346 1.841344663 2.065771236 1.700776267 2.53759831 1.659892627 3.019286244

1.639383456 3.91572139 0.514891077 0.58693496 3.75215652 1.201756339 3.782797801 1.639383456

4.129723757 4.398579999 2.23415283 1.834273356 3.377773901 2.900485286 4.007685346 3.91572139

4.398579999 9.438333114 2.989053801 -0.214270913 1.25664875 0.153779997 1.841344663 0.514891077

2.23415283 2.989053801 3.33636598];


36.97];

Sc3=[3.705327047 -2.997999787 3.507847911 -3.104122741

2.792800093 -2.914275848 1.522582331 -1.72004541 -2.997999787 4.835122711 -2.544628522 3.971776525 -2.041828018

3.852192782 0.570477185 1.687803257 3.507847911 -2.544628522 3.545887763 -2.742954926 2.962438686 -

2.51232233 2.087531852 -1.648866261 -3.104122741 3.971776525 -2.742954926 4.06513582 -2.202586106

3.898300992 -0.238788688 1.999575682 2.792800093 -2.041828018 2.962438686 -2.202586106 2.64042465 -

2.013866607 1.95582961 -1.374264035 -2.914275848 3.852192782 -2.51232233 3.898300992 -2.013866607

4.118655357 0.049982414 2.031833986 1.522582331 0.570477185 2.087531852 -0.238788688 1.95582961

0.049982414 6.333348038 -1.524028874 -1.72004541 1.687803257 -1.648866261 1.999575682 -1.374264035

2.031833986 -1.524028874 2.736373079];

%% finding mahalanobis distance:

%pooled sigma Spooled=(11*Sc1 + 22*Sc2 + 14*Sc3)/(47-3); %3groups, n for c_j=11, 22

and 14

XMinusMuC1=double([]); XMinusMuC2=double([]); XMinusMuC3=double([]); Z2C1=double([]); Z2C2=double([]);

209

Z2C3=double([]);

sizeX=size(X,1);

for i=1:sizeX %for C1 xMinusMuC1=X(i,:)-meanc1; XMinusMuC1=[XMinusMuC1; xMinusMuC1];

z2C1=XMinusMuC1(i,:)*inv(Spooled)*transpose(XMinusMuC1(i,:)); Z2C1=[Z2C1; z2C1];

%for C2 xMinusMuC2=X(i,:)-meanc2; XMinusMuC2=[XMinusMuC2; xMinusMuC2];

z2C2=XMinusMuC2(i,:)*inv(Spooled)*transpose(XMinusMuC2(i,:)); Z2C2=[Z2C2; z2C2];

%for C3 xMinusMuC3=X(i,:)-meanc3; XMinusMuC3=[XMinusMuC3; xMinusMuC3];

z2C3=XMinusMuC3(i,:)*inv(Spooled)*transpose(XMinusMuC3(i,:)); Z2C3=[Z2C3; z2C3]; end

mahalnob=[Z2C1 Z2C2 Z2C3];

if Z2C1<Z2C2 && Z2C1<Z2C3 disp ('Assign to tray 1') else if Z2C2<Z2C1 && Z2C2<Z2C3 disp('Assign to tray 2') else if Z2C3<Z2C1 && Z2C3<Z2C1 disp('Assign to tray 3') end end end

210

APPENDIX E:

MATLAB program for Estimating Natural Teeth Positions on Complete Dentures

for the Edentulous

%select MVCN shape model 1, 2 or 3

%% reterieve image cd c:/; % Change directory [f,d] = uigetfile('*'); % * means can select any file. d=directory

name. f=filename S=imread([d,f]); imshow(S) h = helpdlg('select the desired area and double click to crop the

picture'); uiwait (h); Y=imcrop(S); Y=rgb2gray(Y); figure; imshow(Y) h = helpdlg('select 2 points from the ruler to calibrate the distance

for 1cm'); uiwait (h); [x1,y1]=ginput(1); [x2,y2]=ginput(1);

D1 = (x2-x1)^2 + (y2-y1)^2; oneCm = sqrt(D1);

h = helpdlg('Click on 2 hamular notches'); uiwait (h); [x3,y3]=ginput(1); %Titik hamular notch pertama %select on right side

first hold on; plot(x3,y3,'-c+') [x4,y4]=ginput(1); %Titik hamular notch kedua hold on; plot(x4,y4,'-c+') h = helpdlg('Click on the incisive papilla'); uiwait (h); [x5,y5]=ginput(1); %Titik incisive papilla hold on; plot(x5,y5,'-c+') hold on;

%------------------------------------- % Dapatkan persamaan garislurus %-------------------------------------

m1 = (y3-y4)/(x3-x4); m2 = -1/m1; c2 = y3 - m1*x3; %Equation for the hamular notch c1 = y5 - m2*x5; % Equation perpendicular to hamular notch line

%Dapatkan titik origin (u, w) (in pixels) antara 2 garisan

u = (c2-c1)/(m2-m1); w = m2*u + c1;

xHamular=[x3 u x4]; yHamular=[y3 w y4];

211

plot(xHamular,yHamular) hold on;

HamularX=[x3-u x4-u]'/oneCm; yHamularX=[y3-w y4-w]'/oneCm; Hamular=[HamularX yHamularX];

xIncisive=[x5 u]; yIncisive=[y5 w]; plot(xIncisive,yIncisive) hold on;

IP=[x5-u w-y5]/oneCm; landmark=[Hamular;IP];

%% choose mean of assigned shape (AS) category in FD % meanc1 % AS=[2.975253561 63.56708953 % -1.645126476 -59.40807746 % -0.144520751 -7.722531286 % -0.154638083 -9.4306985 % 0.058291401 -4.545900269 % 0.171673126 -2.665689093 % -0.012414048 5.892822351 % 0.114782775 4.577652201];

% meanc2 % AS=[-0.651386156 57.64963381 % -0.480754733 -57.59873011 % 0.04254522 -7.219076249 % 0.037382931 -8.35381279 % 0.003718754 -4.274265505 % 0.079266153 -2.450033353 % 0.055917418 6.744189641 % 0.09803187 5.565180921];

%meanc3 AS=[-2.078679961 66.5773416 0.979342286 -59.83098877 0.12456784 -8.074783287 0.11547679 -9.77286547 0.0518563 -4.805206904 -0.103224426 -2.650462174 -0.030196412 5.730557796 -0.165540068 4.238463245];

as1=AS(:,1)+AS(:,2)*i; %change in complex number zero=zeros(13,1); as=[as1(1:6) ;zero;as1(7:8)]; %make it in 21 points xas=ifft(as); %inverse transform of the 8FD giving 21 coordinates in

complex form

X1=(xas(1:2)*oneCm)+(u+w*i); X2=transpose(((xas(3:20)*oneCm)+(u-w*i))'); X3=(xas(21)*oneCm)+(u+w*i); X=[X1;X2;X3];

scatter(real(X),imag(X),'MarkerEdgeColor','r') scatter(real(X),imag(X),'fill','r')

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