OPTIMIZATION OF NON-UNIFORM RELATIONAL

B-SPLINE SURFACE RECONSTRUCTION USING

GROWING GRID- DIFFERENTIAL EVOLUTION

PRIZA PANDUNATA

UNIVERSITI TEKNOLOGI MALAYSIA

OPTIMIZATION OF NON-UNIFORM RELATIONAL

B-SPLINE SURFACE RECONSTRUCTION USING

GROWING GRID-DIFFERENTIAL EVOLUTION

PRIZA PANDUNATA

A thesis submitted in fulfillment of the

Requirements for the award of the degree of

Master of Science (Computer Science)

Faculty of Computer Science and Information Systems

Universiti Teknologi Malaysia

NOVEMBER 2011

iii

To my beloved wife, mother and father

iv

ACKNOWLEDGEMENTS

In preparing this thesis, I was in contact with many people, researchers, and

academicians. They have contributed towards my understanding and thoughts. In

particular, I wish to express my sincere appreciation to my main thesis supervisor, Prof.

Dr. Siti Mariyam bt Shamsuddin, for encouragement, guidance, and critics. I am also

indebted to Universiti Teknologi Malaysia (UTM) for funding my study.

My fellow Indonesian Students should also be recognized for their support. My

sincere appreciation also extends to all my brothers, colleagues and others who have

provided assistance at various occasions. Their views and tips are useful indeed.

Unfortunately, it is not possible to list all of them in this limited space. I am grateful to all

my family members.

v

ABSTRACT

Computer graphics is a fast growing field as it contributes significantly to the

advancement of modern technology aimed at empowering human and nation wealth

creation. Computer-Aided Design (CAD), Computer-Aided Manufacturing (CAM) and

Computer-Aided Geometric Design (CAGD) are commonly used to reconstruct surfaces

in order to obtain a set of limited and disorganized geometric sample values. The process

of surface reconstruction consists of two main steps: parameterization and surface fitting.

Various solutions have been used in previous studies to reconstruct surfaces such as Non

Uniform Rational B-Spline (NURBS) and B-Spline. However, in recent years, Artificial

Intelligence (AI) methods such as Advanced Neural Network and Evolutionary Algorithm

(EA) have emerged and are extensively used to reconstruct and optimize complex

surfaces. This study aims to optimize NURBS surfaces from unstructured 3D data points

with feasible control points while preserving the shape of the objects by using Differential

Evolution Algorithm (DEA). The Growing Grid Network (GGN) is implemented on a

map structure, while DEA is optimally fit on to the NURBS surfaces. In this study,

undefined or unstructured data points from several 2D and 3D datasets were used to

validate the performance of the proposed method. An error analysis was also conducted to

reconfirm the efficacy of the proposed algorithm. This is done by comparing the generated

surface with the original surface using other EAs such as: Genetic Algorithm and Particle

Swarm Optimization. Experimental results indicate that the proposed Growing Grid

Network Differential Evolution (GGNDE) has successfully generated smoother surfaces

with lesser number of control points and produced minimum feasible errors while

preserving the shape of the objects.

vi

ABSTRAK

Grafik komputer merupakan suatu bidang yang sedang berkembang pesat

disebabkan sumbangannya yang ketara dalam kemajuan teknologi moden. Rekabentuk

Berbantukan Komputer (CAD), Pembuatan Berbantukan Komputer (CAM) dan Reka

bentuk Geometri Berbantukan Komputer (CAGD) digunakan untuk membina semula

permukaan bagi mendapatkan set nilai sampel geometri yang terbatas dan tidak tersusun.

Proses pembinaan semula permukaan melibatkan dua langkah utama, iaitu

pemparameteran dan pemadanan permukaan. Banyak kaedah penyelesaian telah

digunakan dalam kajian terdahulu untuk pembinaan semula permukaan seperti Splin-b

Nisbah Tak Seragam (NURBS) dan Splin-b. Namun demikian pada masa ini kaedah

kecerdasan buatan seperti Rangkaian Saraf Lanjutan dan Algoritma Evolusi telah

diperkenalkan dan digunakan dengan meluas bagi membina semula dan mengoptimumkan

permukaan yang rumit. Kajian ini bertujuan mengoptimumkan permukaan NURBS

daripada data 3D tak berstruktur dengan bilangan titik kawalan yang tersaur dan

mengekalkan rupa bentuk objek menggunakan Algoritma Evolusi Pembezaan (DEA).

Rangkaian Tumbesaran Grid dilaksanakan pada struktur pemetaan manakala DEA

dimuatkan secara optimum pada permukaan NURBS. Dalam kajian ini beberapa set data

2D dan 3D yang tidak tertakrif atau tidak berstruktur digunakan untuk mengesahkan

prestasi kaedah cadangan. Analisis terhadap ralat juga dilakukan dengan

memperbandingkan permukaan terjana dengan permukaan asal menggunakan kaedah

Algoritma Evolusi yang lain seperti Algoritma Genetik dan Pengoptimuman Partikel

Berkelompok. Hasil kajian mendapati bahawa kaedah cadangan, iaitu Rangkaian

Tumbesaran Grid Evolusi Pembezaan telah berjaya menjana permukaan yang licin dengan

bilangan titik kawalan yang kurang dan ralat minimum tersaur di samping mengekalkan

rupa bentuk asal objek berkenaan.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xii

LIST OF ABBREVIATIONS xv

LIST OF SYMBOLS xvi

LIST OF APPENDICES xvii

1 INTRODUCTION 1 

1.1  Introduction 1 

1.2  Background of the Study 2 

1.3  Problem Statement 6 

1.4  Research Aim 7 

1.5  Objectives 7 

1.6  Research Scope 7 

1.7  Thesis Organization 8 

2 LITERATURE REVIEW 10 

2.1  Overview of Surface Reconstruction 10 

2.1.1  Surface Reconstruction Process 11 

2.2  Surface Representation 11 

2.2.1  Bezier Curve and Surface 12 

2.2.1.1  Bezier Equations 12 

viii

2.2.2  B-Spline 14 

2.2.2.1  B-Spline Equations 14 

2.2.3  Non-Uniform Rational B-Spline (NURBS) 16 

2.2.3.1  NURBS Equations 17 

2.3  Artificial Neural Network for Surface Reconstruction 18 

2.3.1  Overview of Kohonen Network 19 

2.3.1.1  Kohonen Network Architecture 19 

2.3.1.2  Kohonen Network Algorithm 22 

2.4  Evolutionary Computation for Surface Reconstruction 25 

2.4.1  Overview of Differential Evolution 25 

2.4.1.1  Properties of Differential Evolution 28 

2.4.1.2  Proper Algorithm of Differential Evolution 28 

2.4.1.3  The Basic Operation of Differential Evolution 30 

2.5  Related Studies 31 

2.6  Summary 33 

3 METHODOLOGY AND THE PROPOSED METHOD 34 

3.1  The Framework of the Study 34 

3.1.1  Data Collection 35 

3.1.2  Growing Grid Network (GGN) Generation 36 

3.1.2.1  Update the Parameter of Growing Grid Network (GGN) 44 

3.1.3  Parameterization and Knot Vector Generation of Initial Sketch 45 3.1.3.1  Parameterization 46 

3.1.3.2  Knot Vector Generation 47 

3.1.4  Differential Evolution Algorithm 48 

3.1.4.1  Population and Chromosome initialization 50 

3.1.4.2  Calculation of fitness value 50 

3.1.4.3  Mutation and Selection 51 

3.1.5  Performance Analysis and Comparison 53 

3.2  Summary 54 

4 EXPERIMENTAL RESULTS AND ANALYSIS 55 

4.1  Experiment Protocol 55 

ix

4.1.1  Data Specifications 56 

4.1.2  Parameter Setting of Growing Grid Network (GGN) 58 

4.1.3  Development of the Growing Grid Data Network 58 

4.1.4  Implementation of Parameterization and Knot Vector Generation 59 

4.1.5  Parameter Setting of Differential Evolution Algorithm (DEA) 60 

4.1.6  Parameter Setting of PSO 62 

4.1.7  Parameter Setting of GA 63 

4.2  Performance Assessments 64 

4.2.1  Results and Analysis of Curve and Surface using GGN – DEA 65 

4.2.2  Results and Analysis of Curve and Surface using PSO 68 

4.2.3  Results and Analysis of Curve and Surface using (GA) 69 

4.3  Results Comparison 71 

4.4  Experimental Results of Mathematical Function using Evolutionary Computing 73 4.4.1  Experimental Results for Mathematical

Functions 77 4.5  Summary 79 

5 CONCLUSIONS AND FUTURE WORK 80 

5.1  Research Achievements 80 

5.2  Research Contributions 81 

5.3  Future Works 82

REFERENCES 83 

APPENDICES A-C 88 

x

LIST OF TABLES

TABLE NO. TITLE PAGE

1.1 Previous Related Study on Surface Reconstruction 5 

2.1 Comparation of PSO, GA, and DE (Wdaa 2008) 26 

3.1 Sample Data for Semi-sphere 36 

4.1 Data Specifications 56 

4.2 Parameter settings of Growing Grid Network (GGN) 58 

4.3 Parameter Value of Mask Data Points 60 

4.3 Parameter Settings of DEA 61 

4.4 Parameter settings of PSO 62 

4.5 Parameter settings of GA 64 

4.6 Average error of GGN-DEA for 2D data (curve) 65 

4.7 Average error of GGN-DEA for 3D data (surface) 66 

4.8 Average error of GGN-PSO for 2D data (curve) 68 

4.9 Average error of GGN-PSO for 3D data (surface) 69 

4.10 Average error of GGN-GA for 2D data (curve) 70 

4.11 Average error of GGN-GA for 3D data (surface) 70 

4.12 Average Error between DEA, PSO, and GA for Curve Data 71 

4.13 Average Error between DEA, PSO, and GA for Surface Data 72 

4.14 Results for Mathematical Functions 77 

A.1 Sine Data 89 

A.2 Trochoid Data 90 

A.3 Spiral Data 91 

A.4 Free form curve data 92 

A.5 Sample of Mask data 95 

A.6 Sample of Ship Hull data 96 

xi

A.7 Sample of Saddle data 97 

A.8 Sample of Sphere Data 98 

A.9 Sample of Buddha statue data 99 

A.10 Sample of Free-form surface data 100 

B.1 Average error of GGN-DEA for 2D data (curve) 104 

B.2 Average error of GGN-DEA for 3D data (surface) 104 

B.3 Average error of GGN-PSO for 2D data (curve) 110 

B.4 Average error of GGN-PSO for 3D data (surface) 110 

B.5 Average error of GGN-GA for 2D data (curve) 116 

B.6 Average error of GGN-GA for 3D data (surface) 116 

B.7 Comparison of Average Error between DEA, PSO, and GA for

Curve Data 125 

B.8 Comparison of Average Error between DEA, PSO, and GA for

Surface Data 126

xii

LIST OF FIGURES

FIGURE NO TITLE PAGE

2.1 Kohonen SOM Architecture 20 

2.2 Mapping Structure of Kohonen SOM 21 

2.3 Mapping Dimension of Kohonen SOM 21 

2.4 Lattice Structure of Kohonen SOM 22 

2.5 Gaussian Graph Function 24 

2.6 Basic pseudo-code for the DE algorithm. 28 

2.7 Flowchart of DE algorithm 29 

3.1 Framework of the study 35 

3.2 Construction of initial sketch using GGN 37 

3.3 Neurons Initialization 38 

3.4 Initialization Position of Neurons 38 

3.5 Initialization Weights for each sample point 39 

3.6 Win counter value for each neuron 39 

3.7 Sample Points Ps 39 

3.8 Winning Neuron 40 

3.9 Boundary Neuron 40 

3.10 Boundary Neuron Detected 41 

3.11 Boundary Neuron Not Detected 41 

3.12 Update Position If Boundary Neuron Not Detected 41 

3.13 Update Position If Boundary Neuron Detected 42 

3.14 Maximum winning counter detected 42 

3.15 Node Addition to Column 43 

3.16 Node Addition to Row 43 

3.17 Differential Evolution Algorithm 49 

xiii

3.18 The architecture of population of DE 50 

3.19 The Mutation, Recombination, and Selection process of DE 53 

4.1 Spiral data 57 

4.2 Unstructured Mask 57 

4.3 Structured Mask 59 

4.4 Generated Spiral Curve using DEA 67 

4.5 Generated Semi Sphere Surface using DEA 67 

A.1 Sine data 93 

A.2 Trochoid data 93 

A.3 Spiral data 94 

A.4 Free form curve data 94 

A.5 Mask data 101 

A.6 Ship Hull data 101 

A.7 Saddle data 102 

A.8 Sphere data 102 

A.9 Buddha Statue data 103 

A.10 Free Form Surface data 103 

B.1 Sine curve, generated by DEA 105 

B.2 Trochoid curve, generated by DEA 105 

B.3 Spiral curve, generated by DEA 106 

B.4 Free form curve, generated by DEA 106 

B.5 Sphere in wireframe view, generated by DEA 107 

B.6 Ship in wireframe view, generated by DEA 107 

B.7 Mask in wireframe view, generated by DEA 108 

B.8 Saddle in wireframe view, generated by DEA 108 

B.9 Free Form in wireframe view, generated by DEA 109 

B.10 Buddha statue in wireframe view, generated by DEA 109 

B.11 Sine curve, generated by PSO 111 

B.12 Trochoid curve, generated by PSO 111 

B.13 Spiral curve, generated by PSO 112 

B.14 Free form curve, generated by PSO 112 

B.15 Sphere in wireframe view, generated by PSO 113 

B.16 Ship in wireframe view, generated by PSO 113 

B.17 Mask in wireframe view, generated by PSO 114 

xiv

B.18 Saddle in wireframe view, generated by PSO 114 

B.19 Free Form in wireframe view, generated by PSO 115 

B.20 Buddha statue in wireframe view, generated by PSO 115 

B.21 Sine curve, generated by GA 117 

B.22 Trochoid curve, generated by GA 117 

B.23 Spiral curve, generated by GA 118 

B.24 Free form curve, generated by GA 118 

B.25 Sphere in wireframe view, generated by GA 119 

B.26 Ship in wireframe view, generated by GA 119 

B.27 Mask in wireframe view, generated by GA 120 

B.28 Saddle in wireframe view, generated by GA 120 

B.29 Free Form in wireframe view, generated by GA 121 

B.30 Buddha Statue in wireframe view, generated by GA 121

xv

LIST OF ABBREVIATIONS

ABBREVATION DESCRIPTION

2D (x, y)-Two dimensional plane

3D (x, y, z)-Three dimensional plane

CAD Computer Aided Design

CAM Computer Aided Manufacturing

CAGD Computer Aided Geometric Design

DE Differential Evolution

GA Genetic Algorithm

GGN Growing Grid Network

NFL No Free Lunch

NURBS Non-Uniform Rational B-Spline

PSO Particle Swarm Optimization

xvi

LIST OF SYMBOLS

SYMBOLS DESCRIPTION

Bk,n(t) - basis function for Bezier curve at parameter t.

Ni,k(t) - B-Spline basis function at parameter t and kth

degree

xvii

LIST OF APPENDICES

APPENDIX TITLE PAGE

A ORIGINAL DATA 89 

B RESULTS OF THE OPTIMIZATION 104 

C PUBLICATIONS 127 

 

CHAPTER 1

INTRODUCTION

1.1 Introduction

Computer graphics is fast growing field due to its significant contributions to the

advancement of modern technology for human and nation wealth creation. One of the

major areas in computer graphics is surface reconstruction since it is widely used in

industries to design the products. Computer-Aided Design (CAD), Computer-Aided

Manufacturing (CAM) and Computer-Aided Geometric Design (CAGD) are commonly

used in reconstructing the surfaces to obtain a set of geometric sample values (in most

cases, points) which is normally limited and unorganized. Conventionally, mathematical

description represents a shape of physical surface accurately and concisely. Surface

reconstruction and surface representation has also an important part in the design

problems such as construction of scanned 3D objects, modeling of car bodies, medical

imaging and other free-form objects.

Surface representation illustrates three cases: explicit form, implicit form and

parametric form. Among these, the parametric representation is the most useful and

widely applied, since it is axis independent and applicable to complex surfaces

(Shamsuddin, Ahmed et al. 2006). Several types of parametric curve and surface

representation include Bezier, B-Spline and NURBS are the most influential methods used

by the engineers. However, the common problems encountered in surface reconstruction

from unstructured data is on how to acquire the control points of the surface. This is due to

2

the parametric surface in which to obtain suitable parameterizations of data points are

crucial.

Some scientists to overcome the problems of surface reconstruction from

unstructured data have used variety of methods. Some of these methods used multi-

quadratic functions (Franke, Hagen et al. 1994), range images (Curless 1997), discrete

fairing and variation subdivision (Kobbelt 2000), dynamic base surfaces (Azariadis

2004), approximations of the unsigned distance function (Flöry and Hofer 2010), and

Artificial Intelligence (AI).

Recent literatures have shown that the usage of AI in surface reconstruction can

produce good results (Hoppe, DeRose et al. 1992; Ivrissimtzis, Jeong et al. 2003; Junior,

Neto et al. 2004; Gálvez, Cobo et al. 2008; Tsai, Huang et al. 2009; Mendona Ernesto

Rego, Araujo et al. 2010).The common AI methods in surface reconstruction include

Artificial Neural Network (ANN), Kohonen Self Organizing Map (SOM), and Genetic

Algorithm (GA).

1.2 Background of the Study

Surface reconstruction is view from two perspectives: mathematical aspects and

Artificial Intelligence (AI). Some recent studies have shown that some AI application

shave achieved remarkable results (Ivrissimtzis, Jeong et al. 2003; Junior, Neto et al.

2004; Gálvez, Cobo et al. 2008). The input data obtained from the object can be

represented into two types: image mesh, and the 3D point from the scanned image with

have noise and scattered. These types of data, however, yield an enormous amount of

irregular and scattered digitized point that requires intensive reconstruction processing.

Surface reconstruction consists of two main steps: parameterization and surface

approximation. In 3-dimensional surfaces, parameterization represents one-to-one

3

mapping from the surface to a suitable domain. In other words, surface approximation is a

procedure of determining a control polygon that generates a curve or surface from a set of

known data points using a set of parameter obtained from the parameterization procedure.

Previous related studies used data points, which are noisy and scattered. However,

the issue arises on how to re-arrange the scattered point data to be organized, and to deal

with the algorithm’s complexity as well as processing time.

(Barhak and Fischer 2001) use Self Organizing Map (SOM) for creating a 3D

parametric grid. The main advantage of SOM relies on both the orientation of the grid and

the position of the sub-boundaries. The neural network grid converges to the sampled

shape through an adaptive learning process. However, some enhancements need to be

done to create a complete envelope of a volumetric object and accuracy.

(Jalba and Roerdink 2007) have done some studies on deploying a fast convection

algorithm to attract the evolving surface towards the data points. However, this method

has limitation, that is, surface features are smaller than the size of the smallest grid cells;

hence the reconstruction is not accurate enough.

(Gálvez, Iglesias et al. 2007)used GA to reconstruct the surface, and functional

networks for the functional constraints problems. These methods are integrated with the

least-squares approximation to yield suitable methods for surface fitting. However, this

approach still faces the difficulty to attain global optimum due to the variation of distance

error function for fitting data for different models.

(Gálvez, Cobo et al. 2008) has also used PSO to obtain a suitable parameterization

of the data points on Bezier surface reconstruction. However, the results found that there

was no correlation between the number of iterations and the quality of the results. In this

scenario, it means that exploring the space domain of the problem is not yet optimal.

Hence, adjusting PSO parameters can help solve the above problem.

4

(Kazhdan, Bolitho et al. 2006) has presented surface reconstruction problem as

Poisson problem that allows a hierarchy of locally supported basis functions. The solution

has reduced to well-conditioned sparse linear system. However, there are some limitations

on this approach due to the weaknesses in handling huge data sets.

(Shamsuddin and Ahmed 2006) proposed hybrid parameterization to solve

parameterization of data points on the surfaces. The method takes into consideration the

maximum rational B-spline basis functions as the initial values. The centripetal method

generates the parameter values of hybrid parameterization. However, some improvements

are required to refine the reconstructed object to be fairer and smoother.

(Forkan and Shamsuddin 2008) proposed a method for surface reconstruction

based on the hybridization of Kohonen Network and Particle Swarm Optimization (PSO).

In this study, PSO probes the best control points for the data with B-Spline as surface

representation. However, B-Spline surface representation is still incapable to represent

accurately most of the analytically defined shape and conic sections such as Circle,

Ellipse, and Hyperbola.

(Gálvez and Iglesias) in 2010 has presented recovery of a surface using PSO from

scattered noisy data points. The process consists of two main phases: parameterization and

surface fitting. The result is good in terms of accuracy. However, the computational time

and complexity need to be reduced.

Table 1.1 shows the summary of recent study in the field of surface reconstruction

.

5

Table 1.1: Previous Related Study on Surface Reconstruction

Author/Year Summary Limitation/Future Work

(Kazhdan,

Bolitho et al.

2006)

surface reconstruction problem can

be expressed as Poisson problem

Weak on handling huge

data sets.

(Jalba and

Roerdink 2007)

using a method that employs a fast

convection algorithm

Surface features are

smaller than the size of

the smallest grid cells.

Hence the surface are not

accurately reconstructed

(Gálvez, Iglesias

et al. 2007)

use PSO for obtaining a suitable

parameterization of the data points

on Bezier surface reconstruction

Only limited on bezier

surface parameterization

process

(Kumar et al.

2003)

Using Genetic Algorithm to

optimize the parameter

Limited to B-Spline

curve

(Shamsuddin,

Ahmed et al.

2006)

Using hybrid parameterization, Give

better accuracy

Weights value are

adjusted manually

(Barhak and

Fischer 2001)

use a neural network Self

Organizing Map (SOM) method for

creating a 3D parametric grid.

Cannot envelope the

volumetric object.

(Cheng, Wang et

al. 2004)

Using Squared Distance

Minimization (SDM)

features like edges and

corners in data sets need

to be detected first, have

some problem in

smoothness and accuracy

(Sarfraz and

Riyazuddin 2006)

Using simulated annealing to

optimalize weight and knot

parameter of NURBS

Good only on single

segment image. Limited

only on curve

(Forkan and

Shamsuddin

2008)

Using Kohonen Network for

mapping the data, B-Spline

representation and PSO Algorithm

Further exploration by

using NURBS with

weight and knot vector

6

for surface fitting optimization

(Delint Ira Setyo

2010)

Solve NURBS curve approximation

problem by using Particle Swarm

Optimization (PSO).

Limited only on 2D data

(Gálvez and

Iglesias 2010)

Using PSO for parameterization and

surface fitting

The results show the

algorithm can reconstruct

the surface accurately,

but the complexity and

computation time is still

need to be reduced

Based on the above issues, a comprehensive study needs to be conducted to

produce an algorithm that can utilize the application of AI for surface reconstruction

process. This hybridization could benefit the manufacturing and industry sectors.

1.3 Problem Statement

As mentioned previously, to solve the issues on surface reconstruction from

unstructured data points, these data need to be organized. The procedure involves by

obtaining suitable parameterization of data points and control point that can approximate

the original surfaces. Hence, the research questions for this study include:

1. What is Growing Grid Network (GGN), and how to deal with NURBS surface

reconstruction?

2. How GGN organize the unstructured data in NURBS surface reconstruction?

3. How Differential Evolution Algorithm (DEA) being implemented in

optimizing the NURBS surface?

4. How the DEA obtains an optimal control point that can approximate the data

point in NURBS surface reconstruction?

7

1.4 Research Aim

The aim of the study is to optimize NURBS surface reconstruction from

unstructured 3D data points with feasible control points while preserving the shape of the

objects using Evolutionary Algorithms.

1.5 Objectives

The main objectives of this research are as follows:

1. To develop NURBS surface reconstruction from unstructured data points using

growing grid network (GGN).

2. To propose and develop Differential Evolution Algorithm (DEA) to optimize

NURBS surface reconstruction

3. To validate and compare the proposed algorithm with the existing related

methods

1.6 Research Scope

The scope of this study includes:

1. Technique to be used:

a. The growing grid technique adopted from (Forkan and Shamsuddin

2008)are used to obtain the initial surface

b. Differential evolution optimization are used to optimize the initial

surface generated from growing grid network

8

c. Time consumption is not considered, more focus on the aspects of

accuracy. And we also we focus on parametric approach, not on the

typical graphic methods, i.e., non-parametric approach. Which is

the common issue in non-parametric, such as triangulation, is

difficulties in dealing with sharp corner (T-Junction).

2. Datasets

a. Datasets in this study are open surface which noise level is minimal

and unorganized. Examples of 2D and 3D data are sine, spiral,

semi-sphere, ships and free-form shapes.

3. Tools and environment to be used include:

a. Java Programming Language

b. Graphics API and library, OpenGL and GLUT

c. For rendering purpose, OpenGL API GLUT library will be used.

4. Experiment will be run on PC with specifications:

a. Operating System: Windows 7 Professional

b. Processor: 2GHz CPU

c. Memory: 2048MB RAM

1.7 Thesis Organization

Thesis structure is given as follows:

Chapter 1: Introduction -explains an overview of the background of the study,

development of techniques and methods used in surface reconstruction and the common

problems that are usually encountered in surface reconstruction, also the problem

statement, the aim, the objective, and the scope of this research. Finally, the thesis

organization is given below.

9

Chapter 2: Literature review - This chapter states some existing methods of surface

reconstruction. It also contains the review of the related previous works for solving

surface reconstruction and approximation methods.

Chapter 3: Methodology and The Proposed Method- This chapter describes the

system overview and the framework of proposed methods. It explains the details of every

steps of growing grid, include the process of generating initial surface using growing grid

map and differential evolution in optimizing and approximating NURBS surface

reconstruction.

Chapter 4: Experimental Results and Analysis - This chapter presents the findings

of the study, analysis of the results and the comparisons with other works.

Chapter 5: Conclusions and Future Work - This chapter provides the summary of

the research, the contribution of the work and recommendation for future study.

83

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