PLANT LEAF IDENTIFICATION USING MOMENT INVARIANTS
& GENERAL REGRESSION NEURAL NETWORK
ZALIKHA BT ZULKIFLI
UNIVERSITI TEKNOLOGI MALAYSIA
PLANT LEAF IDENTIFICATION USING MOMENT INVARIANTS
& GENERAL REGRESSION NEURAL NETWORK
ZALIKHA BT ZULKIFLI
A project report submitted in partial 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
OCTOBER 2009
iii
To my beloved Mama and Baba,
for letting me exprience the kind of love
that people freely die for.
And also to my sisters and Noor Hafizzul,
for your patience, love, friendship and humor.
iv
ACKNOWLEDGEMENTS
All praises and gratitude to Allah s.w.t, who has guided and blessed me in
finishing my project successfully.
In the first place I would like to record my gratitude to Assoc. Prof. Dr. Puteh
Saad for her supervision, advice and guidance from the very early to the final stage
of this project which enabled me to develop an understanding of this project. Her
involvement has triggered and nourished my intellectual maturity that I will benefit
from, for a long time to come.
Words fail me to express my appreciation to my parents for their dedication,
love, inseparable support and prayers. To my sisters, Shazana, Amira, Diyana and
Nur Alia, thank you for being supportive and caring siblings.
My special thanks goes to my friends, Hafizz, Umie, Ekin and Huda for their
warm support and encouragement, each from unique perspective. To others who
have assisted in any way, I express my sincere gratitude.
Finally, I would like to thank everbody who was important to the completion
of the project, as well as expressing my apology that I could not mention personally
one by one.
v
ABSTRACT
Living plant identification based on images of leaf is a very challenging task
in the field of pattern recognition and computer vision. However, leaf classification
is an important component of computerized living plant recognition. As inherent
trait, leaf definitely contains important information for plant species identification
despite its complexity. The objective of this research is to identify the effectiveness
of three moment invariant methods, namely Zernike Moment Invariant (ZMI),
Legendre Moment Invariant (LMI) and Tchebichef Moment Invariant (TMI) to
extract features from plant leaf images.
Then, the resulting set of features
representing the leaf images are classified using General Regression Neural Network
(GRNN) for recognition purposes. There are two main stages involved in plant leaf
identification. The first stage is known as feature extraction process where moment
invariant methods are applied. The output of this process is a set of a global vector
feature that represents the shape of the leaf images. It is shown that TMI can extract
vector feature with Percentage of Absolute Error (PAE) less than 10.38 percent.
Therefore, TMI vector feature will be the input to second stage. The second stage
involves classification of leaf images based on the derived feature gained in the
previous stage. It is found that GRNN classifier produces 100 percent classification
rate with average computational time of 0.47 seconds.
vi
ABSTRAK
Pengenalpastian tumbuh-tumbuhan berdasarkan imej daun adalah satu tugas
yang sangat mencabar dalam bidang pengecaman pola. Walaubagaimanapun,
pengelasan daun adalah salah satu komponen yang penting dalam pengecaman
tumbuh-tumbuhan secara berkomputer. Berdasarkan ciri-ciri yang diwarisi, daun
mengandungi maklumat penting untuk pengecaman spesis tumbuhan walaupun
terdapat ciri-cirinya yang rumit. Kajian ini bertujuan untuk mengenalpasti
keberkesanan tiga teknik momen tak varian, iaitu Zernike Moment Invariant (ZMI),
Legendre Moment Invariant (LMI) and Tchebichef Moment Invariant (TMI) dalam
pengekstrakan fitur. Kemudian, hasil daripada pengekstrakan fitur imej daun
dikelaskan menggunakan pengelas General Regression Neural Network (GRNN)
bagi tujuan pengecaman. Terdapat dua fasa utama yang terlibat dalam pengecaman
daun. Fasa pertama dikenali sebagai proses pengestrakan fitur di mana teknik
momen tak varian digunakan. Ouput yang dihasilkan daripada proses ini adalah set
vektor fitur sejagat yang mewakili bentuk imej daun. Didapati bahawa TMI dapat
mengestrak fitur dengan peratus ralat (PAE) kurang daripada 10.38 peratus. Oleh itu,
vektor fitur daripada TMI akan dijadikan sebagai input pada fasa kedua. Fasa kedua
melibatkan pengelasan imej daun. Didapati bahawa pengelas GRNN mempunyai
jumlah klasifikasi paling tepat (100 peratus) dengan purata masa menumpu 0.47 saat.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xi
LIST OF FIGURES
xiii
LIST OF ABBREVIATIONS
xv
LIST OF APPENDICES
xvi
INTRODUCTION
1
1.1
Introduction
1
1.2
Problem Background
3
1.3
Problem Statement
4
1.4
Importance of the Study
4
1.5
Objectives of the Project
5
1.6
Scopes of the Project
6
1.7
Summary
6
viii
2
LITERATURE REVIEW
7
2.1
Introduction
7
2.2
Moment Invariants
9
2.2.1
Zernike Moment Invariant
11
2.2.2
Legendre Moment Invariant
13
2.2.3
Tchebichef Moment Invariant
15
2.3
2.4
3
Implementation of Moment Invariant in
Pattern Recognition Applications
16
2.3.1
Face Recognition
17
2.3.2
Optical Character Recognition
18
2.3.3
Other Pattern Recognition Applications
19
Artificial Neural Networks
21
2.4.1
21
General Regression Neural Networks
2.5
Neural Networks Implementation
24
2.6
Summary
25
METHODOLOGY
26
3.1
Introduction
26
3.2
Research Framework
27
3.3
Software Requirement
28
3.4
Image Source
28
3.5
Image Pre-processing
29
3.6
Feature Extraction
30
3.6.1
ZMI Algorithm
31
3.6.2 LMI Algorithm
33
3.6.3 TMI Algorithm
34
3.7
Intra-class Analysis
35
3.8
Inter-class Analysis
37
3.9
Data Pre-processing
38
3.10
Classification by Neural Networks
39
3.11
Inter-class Analysis
40
3.12
Summary
42
ix
4
IMPLEMENTATION, COMPARISON OF
RESULTS AND DISCUSSION
43
4.1
Introduction
43
4.2
Leaf Images
44
4.3
Result of Intra-class Analysis
46
4.3.1 Absolute Error
48
4.3.2 Percentage Absolute Error (PAE)
50
4.3.3
Percentage Min Absolute Error 1
(PMAE1)
4.3.4
Percentage Min Absolute Error 2
(PMAE2)
4.3.5
Result of Inter-class Analysis
4.4.1
55
55
Comparison Based On Computational
Time
5
54
Comparison Based On Value of
Feature Vectors
4.4.2
53
Total Percentage Mean Absolute
Error (TPMAE)
4.4
52
58
4.5
Analysis of Feature Extraction Results
59
4.6
Classification Phase
59
4.6.1 Data Preparation
60
4.6.2
60
GRNN Spread Parameters Declaration
4.7
Classification Results of GRNN Classifier
61
4.8
Summary
64
DISCUSSION AND CONCLUSION
65
5.1
Introduction
65
5.2
Discussion of Results
65
5.3
Problems and Limitations of Research
67
5.4
Recommendation for Future Works
67
5.5
Conclusion
68
x
REFERENCES
APPENDICES A-H
70
75 - 96
xi
LIST OF TABLES
TABLE NO.
TITLE
2.1
List of face recognition applications that use
moment invariants
17
List of OCR applications that use moment
Invariants
19
List of other pattern recognition applications
that use moment invariants
20
List of applications that use neural networks as
classifier
24
3.1
The value of scaling and rotation factors
30
4.1
List of image name
44
4.2
Scaling and rotation factors of image A1
46
4.3
Value of feature vectors using ZMI
47
4.4
Value of feature vectors using LMI
47
4.5
Value of feature vectors using TMI
48
4.6
Absolute Error for ZMI
49
4.7
Absolute Error for LMI
49
4.8
Absolute Error for TMI
50
4.9
PAE for image A1 with rotate factor of 10°
51
4.10
TPMAE of different moments for image A1
54
2.2
2.3
2.4
PAGE
xii
4.11
Value of feature vectors of four images for ZMI
56
4.12
Value of feature vectors of four images for LMI
56
4.13
Value of feature vectors of four images for TMI
56
4.14
Computational time taken in seconds
58
4.15
GRNN result for each spread parameters
62
xiii
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
GRNN architecture
23
3.1
Process involve in research framework
27
3.2
Algorithm of Basic_GMI() computation
32
3.3
Algorithm of ZMI computation
32
3.4
Algorithm of Norm() computation
33
3.5
Algorithm of LMI computation
34
3.6
Algorithm of TMI computation
35
3.7
Process of training, validation and testing phase
in neural network
40
4.1
Leaf images
44
4.2
An image A1 with its various rotations and
scaling factor
45
4.3
PAE graph for image A1 with rotate factor of 10°
51
4.4
PMAE1 graph for image A1 with image variation
52
4.5
PMAE2 graph for image A1 with different dimension
53
4.6
TPMAE graph of different moments for image A1
54
4.7
Comparison value of feature vectors based on leaf
class
57
xiv
4.8
Comparison value of feature vectors based on leaf
class for original image
57
4.9
k-fold partition of the dataset
60
4.10
Graph of PCC versus spread parameter value
63
4.11
Graph of time average versus spread parameter value
63
xv
LIST OF ABBREVIATIONS
AE
–
Absolute Error
ANN
–
Artificial Neural Network
BPN
–
Back-propagation Neural Network
GMI
–
Geometric Moment Invariant
GRNN
–
General Regression Neural Network
IEC
–
Invariant Error Computation
k-NN
–
k-nearest neighbor
LMI
–
Legendre Moment Invariant
MLPN
–
Multilayer Perceptron Neural Network
MMC
–
Moving Median Centers
NCC
–
Number of Correct Classification
OCR
–
Optical Character Recognition
PAE
–
Percentage Absolute Error
PCA
–
Principal Components Analysis
PCC
–
Percentage of Correct Classification
PMAE1
–
Percentage of Mean Absolute Error 1
PNN
–
Probabilistic Neural Network
RBFNN
–
Radial Basis Function Neural Network
RBPNN
–
Radial Basis Probabilistic Neural Network
TIFF
–
Tagged Image File Format
TMI
–
Tchebichef Moment Invariant
ZMI
–
Zernike Moment Invariant
xvi
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Project 1 Gantt Chart
75
B
Project 2 Gantt Chart
77
C
Original Leaf Images
79
D
Binary Leaf Images
81
E
Image References
83
F
Value of Feature Vectors By ZMI
85
G
Value of Feature Vectors By LMI
89
H
Value of Feature Vectors By TMI
93
CHAPTER 1
INTRODUCTION
1.1
Introduction
Plant is greatly important sources for human’s living and development
whether in industry, food stuff or medicines. It is also significantly important for
environmental protection. According to World Wide Fund for Nature (WWF), there
are currently about 50, 000 - 70, 000 known species all over the world (WWF,
2007). However, many plant species are still unknown yet and with the deterioration
of environments, these unknown species might be at the margin of extinction. So it
is necessary to correctly and quickly identify the plant species in order to preserve its
genetic resources.
Plant species identification is a process in which each individual plant should
be correctly assigned to descending series of groups of related plants, as based on
common characteristics (Du et al., 2006). Currently, plant taxonomy methods still
adopt traditional classification method such as morphologic anatomy, cell biology
and molecular biological approaches. This task mainly carried out by botanists. The
2
traditional method is time consuming and less efficient. Furthermore, it can be a
troublesome task. However, due to the rapid development in computer technologies
nowadays, there are a new opportunities to improve the ability of plant species
identification such as designing a convenient and automatic recognition system of
plants.
Plants can be classified according to the shapes, colors, textures and
structures of their leaf, bark, flower, seedling and morph. Nevertheless, if the plant
classification is based on only two dimensional images, it is very difficult to study
the shapes of flowers, seedling and morph of plants because of their complex three
dimensional structures. Plant leaves are two dimensional in nature and hold
important features that can be useful for classification of various plant species.
Therefore, in this research, the identification of different plants species is based on
leaf features.
Many approaches of previous work use k-nearest neighbor (k-NN) classifier
and some adopted Artificial Neural Network (ANN) (Wu et al., 2007). There are
some disadvantages of these previous works. Some are only appropriate to certain
plant species and some methods compare the similarity between features where it
requires human to enter the query manually (Heymans et al., 1991; Ye et al., 2004).
ANN is believed to have the fastest speed and best accuracy for classification. In
previous work indicates that ANN classifiers run faster than k-NN (Du et al., 2005).
Therefore, this research adopts an ANN approach.
The main improvements of this research are on leaf feature extraction and the
classifier. The leaf features are extracted from binary leaf image by using moment
invariants technique; Zernike Moment Invariant, Legendre Moment Invariant and
Tchebichef Moment Invariant. As for classifier, General Regression Neural Network
(GRNN) is chosen. The performance of these moment invariants technique and
classifier are compared to obtain the most suitable technique for plant leaf
identification.
3
1.2
Problem Background
Plant is a living form with the largest population and has the widest
distribution on the earth. There are many kinds of plant species living on the earth,
which plays an important part in improving the environment of human life and other
lives existing on the earth. Unfortunately, more and more plant species are at the
margin of extinction. Therefore, it is important to correctly and quickly recognize the
plant species in order to understand and managing them. However, it is difficult for
layman to recognize the plant species and currently, Botanist is the expert. Due to the
limited number of experts, it is very necessary to automatically recognize the
different kind of leaves for plant classification.
Plant classification not only recognizes different plants and names of the
plant but also provides the differences of different plants and builds system for
classifying plant. There have been several approaches for plant classification based
on plant features of leaves, barks and flowers. In this study, the plant classification
will be determined by plant leaf features because the leaf carry useful information
for classification of various plants such as aspect ratio, shape and texture.
This study is to explore the effectiveness of three moment invariants
technique, namely Zernike Moment Invariant (ZMI), Legendre Moment Invariant
(LMI) and Tchebichef Moment Invariant (TMI) for extraction of binary leaf image.
This study also intends to determine the performance of GRNN as classifiers in leaf
recognition. Thus, the finding from this study can provide useful information for
developing automated plant classification tools.
4
1.3
Problem Statement
Feature Extraction and classification are two challenging phases in image
analysis applications. The following are the problem statement for this study:
•
There is no general feature extraction technique that is available for all types
of images. Thus, experimentations need to be carried out to determine
among Moment-based techniques, which one of them is most suitable for
plant leaf images.
•
An issue that in the classification phase is the matching task since the set
features extracted is numerous for a single image.
1.4
Importance of the Study
This study involves the identification of plant leaf. Many of the existing plant
species on the earth are still unknown, which might be at the margin of extinction.
There are many ways to recognize the plant species, but it usually time consuming.
This is because it involves the expert, botanists to recognize the plant species.
Therefore, it is very important to automatically recognize the plant species in order
to manage them.
It is difficult for layman to recognize plant species. By having a computeraided plant identification system, they can also identify the plant species. This will
increase an interest in studying plant taxonomy and ecology, lift biology educations
5
standards and promote the use of information technology for the management of
natural reserve parks and forest plantation.
By doing this research, the best approach for leaf features extraction and
classification have to be analyze. The performance of three moment invariants
technique and the classifiers will be compared to give best results and very high
performance in a short time. Thus, it is important to expand the knowledge of the
study so that it can give benefit for the environmental protection to preserve
endangered plant species.
1.5
i.
Objectives of the Project
To compare the effectiveness of three moment invariants techniques, namely
Zernike Moment Invariant (ZMI), Legendre Moment Invariant (LMI) and
Tchebichef Moment Invariant (TMI) to extract features from leaf images.
ii.
To classify the set of features representing the leaf images using General
Regression Neural Network (GRNN).
iii.
To evaluate the performance of features extraction techniques and the
classifier based on inter- class and intra-class invariance.
6
1.6
i.
Scopes of the Project
In this study, the binary plant leaf shape images are used and acquired by
digital camera.
ii.
Only three moment invariants techniques, namely ZMI, LMI and TMI are
used for features extraction of binary plant leaf images.
iii.
1.7
GRNN classifier is utilized for leaf classification.
Summary
This chapter discussed a general introduction of the potential and challenges
of plant identification. As stated in the chapter, the main objectives of this research is
to determine the effectiveness and performance of each methods used in this
research. Each methods used will be investigate and explore in order to achieved the
objectives and overcome the problems of this research. The next chapter will be the
literature review, where a description of the literature relevant to the research is
presented.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Automatic recognition, classification and grouping of patterns are important
problems in variety of engineering and scientific disciplines such as artificial
intelligence, computer vision, biology and medicine. Interest in the area of plant leaf
recognition has been improved recently due to emerging applications which are not
only challenging, but also computationally demanding. Classification of plants is a
very old field, which is carried out by taxonomists and botanists and it is time
consuming processes. Due to this time consuming process and the fact that the
identification is not trivial for non-experts, it could be helpful if the identification
process is by a computer-based automatic system.
In recent years, several researchers have dedicated their work to plant
identification. Till now, many works have focused on leaf feature extraction for
recognition of plant. In addition, leaf shape is one of the characteristics that play an
important role in the plant leaf classification process and need to be evaluated. Wang
8
X. F., et al. (2005) introduced a method of recognizing leaf images based on shape
features using hyper-sphere classifier. He applied image segmentation to the leaf
images. Then, he extracted eight geometric features and seven moment invariants
from preprocessed leaf images for classification. To address these shape features, he
used a moving center hyper-sphere classifier. For the experimental results, he shows
that 20 classes of plant leaves successful classified and the average recognition rate
is up to 92.2 percent. Another pervious work that focused on leaf feature extraction
for recognition of plant is done by Wu, S. G., et al. (2007). He extracted the leaf
features from digital leaf image, where 12 features are extracted and processed by
Principal Components Analysis (PCA) to form input vector. Then, he employed
Probabilistic Neural Network (PNN) as classifier to classify 32 kinds of plants and
obtained accuracy greater than 90 percent.
Previous works have some disadvantages such as it requires preprocess work
of human to enter keys manually, where this problem also happens on methods
extracting features used by botanists (Warren, 1997). Some of the previous works are
only applicable to certain species (Heymans, et al., 1991). Among all approaches
used in the previous works, Artificial Neural Network (ANN) has fastest speed and
best accuracy for classification phase. According to Du, J., et al. (2005), ANN
classifiers such as MLPN, BPN, RBFNN and RBPNN run faster than k-NN and
MMC hyper-sphere classifier. Moreover, ANN classifiers advance other classifiers
on accuracy. Therefore, this research adopts an ANN approach.
For this research, a complete plant leaf recognition system should include
two stages. The first stage is the extraction of leaf features. One part of automated
feature extraction is recognition through shape where it is based on the matching of
description of shapes. Several shape description techniques have been developed
such as moment invariants. Moment invariants technique has been often used as
features for shape recognition and classification. Moment invariants deal with the
classification of definite shapes for most applications that used the technique such as
the identification of a particular type of aircraft. Therefore, this research applies
moment invariants technique such as Zernike Moment Invariant (ZMI), Legendre
9
Moment Invariant (LMI) and Tchebichef Moment Invariant (TMI) for features
extraction of binary leaf images which yielded the best result for plant leaf
recognition in comparison between the three moment invariants technique.
The second stage involves classification of plant leaf images based on the
derived feature obtained in the previous stage. Classification is performed by
comparing descriptors of the unknown object with those of a set of standard shapes
to find the closest match. The classifier plays significant role in the plant leaf
recognition process in the second stage. The growing popularity of neural network
models to solve pattern recognition problems as a result of their low dependence on
domain specific knowledge and the availability of efficient learning algorithms. In
addition, existing feature extraction and classification algorithms can also be mapped
on neural network models for efficient implementation. Therefore, a General
Regression Neural Network (GRNN) is implemented as classifier for this research.
The performance of GRNN learning algorithm in plant leaf recognition then will be
evaluated.
2.2
Moment Invariants
Moment invariants have been used as feature extractions in variety of object
recognition applications during last 40 years. Therefore, moment invariants are one
of the most significant and frequently used as shape descriptors. It is well-known
technique of calculating and comparing the moment invariants of the shape of a
feature in image processing for recognition and classification. The characteristics of
an object are model numerically by the invariant values to uniquely represent its
shape (Keyes and Winstanley, 2000). Then, the classification of invariant shape
recognition is carry out in the multidimensional moment invariant feature space.
There have been several techniques that derive the invariant features from the
10
moments for object recognition which is distinguished by their moment definition,
the type of data exploited and the method for deriving invariants value from the
image moments (Belkasim et al., 1991).
The existing of moment invariants begin many years before the appearance
of the first computer. However, it is under the framework of the theory of algebraic
invariants which is introduced by German mathematician named David Hilbert
(Flusser, 2006). Then in 1962, Hu was the first to introduced moment invariants to
the pattern recognition area (Hu, 1962). He derived a set of invariants using
algebraic invariants and defines seven of shape descriptor values computed by
normalizing central moments that are invariant to object scale, translation and
rotation. The term invariant indicate that an image feature remains unchanged if that
image undergoes one or combination of changes of size, position and orientation
(Haddadnia et al., 2001). The main disadvantage of his moment invariants refers to
the large values of geometric moments, which lead to numerical instabilities and
noise sensitivity (Kotoulas and Andreadis, 2005).
In order to achieve high recognition performance, selection of feature
extraction method is one of the most important factors. In order to recognize
variation of plant leaf, moment invariant techniques is used for features extraction in
this research. This technique is use in features extraction of plant leaf because of the
following reasons (Puteh, 2004):
•
Global characteristics of the image shape are shown by the
computation of a set of moments from a digital image.
•
It gives a lot of information about the different types of geometrical
features inherent in the image.
•
It generates a set of features that is invariant.
11
Many works have been committed to various improvements and
generalization of Hu’s invariants. These enhancements generate new moment
invariant techniques such as Geometric Moment Invariant (GMI), Unified Moment
Invariant (UMI), Zernike Moment Invariant (ZMI), Legendre Moment Invariant
(LMI), Krawtchouk Moment Invariant (KMI) and Tchebichef Moment Invariant
(TMI). However, this research will only focused on three types of moment invariant
techniques; ZMI, LMI and TMI for features extraction of leaf image. These moment
invariant techniques will be evaluated for their performance capabilities.
ZMI technique is well known and widely used in the features extraction. ZMI
is selected because it is invariant to rotation and insensitive to noise (Puteh, 2004).
LMI technique is chosen due to its robustness to image transformations under the
noisy condition. Moreover, this technique have a near zero redundancy measure in a
feature set (Annadurai et al., 2004). TMI technique is selected since it is has low
noise sensitivity (Kotoulas and Andreadis, 2005) and prove to be effective as feature
descriptors (Mukundan, 2001).
2.2.1
Zernike Moment Invariant
The geometric moment definition has the form of the projection of f(x, y)
onto the monomials xnym. Unfortunately, the basis set {xnym} is not orthogonal.
Consequently, these moments are not optimal with regard to the information
redundancy. Moreover, the lack of orthogonal property causes the recovery of an
image from its geometric moments strongly ill-posed. Therefore, Teague (1980)
introduced the use of continuous orthogonal polynomials such as Zernike moments
to overcome the shortcomings of information redundancy present in the geometric
moments. Zernike moments are a class of orthogonal moments and because of its
12
orthogonal property, Zernike moments are the simplicity of image reconstruction
(Khotanzad and Hong, 1990).
Zernike moments are scaling and rotation invariant. Scale and translation
invariance can be applied using moment normalization. A convenient way to express
Zernike moments in terms of geometric moments in Cartesian form is given by the
equation (2.1). Then the ZMI functions are derived from the equation (2.3) which is
invariant against rotation and scaling factors. The pixel density of N x N image size
is refer as f(x, y).
n
N
M
∑
∑ ∑
k=m
x=1
,
(2.1)
y=1
where
!
!
;
!
|
!
|
(2.2)
(2.3)
From the equation (2.2) it can be seen that Zernike moments use polynomials
of the image radius instead of monomials of Cartesian coordinates and a complex
exponential factor of the angle . This makes their complex modulus invariant to
rotation. Their orthogonal property renders image reconstruction from its moments
feasible and accurate. Furthermore, numerical instabilities are rare in compare to
13
geometric moments (Kotoulas and Andreadis, 2005). There are several advantages of
Zernike moments such as:
•
The magnitudes of Zernike moments are invariant to rotation.
•
Zernike moments are robust to noise and minor variations in shape
(Khotanzad, 1998).
•
Zernike moments have minimum information redundancy since the
basis is orthogonal (Teague, 1980).
Nevertheless, the computation of Zernike moments poses some problems
such as (Mukundan et al., 2001):
•
The coordinates of the image must be mapped to unit circle.
•
Computational complexity of radial Zernike polynomial increase as
the order becomes larger.
2.2.2
Legendre Moment Invariant
The moments with Legendre polynomials as kernel function denoted as
Legendre moments were introduced by Teague (1980). Legendre Moment Invariants
(LMI) belongs to the class of orthogonal moments and they were used in several
pattern recognition applications. They can be used to attain a near zero value of
redundancy measure in a set of moments correspond to independent characteristics
of the image.
14
The Legendre moments of order (p + q) with image intensity function f (x, y)
are defined as equation (2.4). In equation (2.5) | x | ≤ 1 and (n – k) is even.
p q
∑ ∑
(2.4)
i=0 j=0
∑
∑ ∑
cos
1
!
/
!
cos
sin
! !
(2.5)
sin
,
(2.6)
where:
1
0.5 tan
(2.7)
(2.8)
15
2.2.3
Tchebichef Moment Invariant
One common problem with the continuous moments is the discrete error,
which accumulates as the order of the moments increases (Yap et al., 2003).
Mukundan et al. (2001) introduced a set of discrete orthogonal moments function
based on the discrete Tchebichef polynomials to face this problem. The discrete
orthogonal polynomials is used as basis functions for image moments to eliminates
the need for numerical approximations and satisfies the orthogonal property in the
discrete domain of image coordinate space (Kotoulas and Andreadis, 2005).
The pth order Tchebichef moment Tp of one-dimensional N point signal f (x),
is defined as (Wang G. and Wang, S., 2006):
Tp =
N-1
~
1
∑ tp(x) f (x)
~
ρ (p, N) x=0
(2.9)
where p = 0, 1, 2, …, ~tp(x) denotes the pth oder scaled Tchebichef polynomials and
ρ~ (p, N) is the squared-norm of scaled polynomials (Mukundan et al, 2001.). They
are given by:
p
~
tp (x) = p! ∑ (-1) p-k
N
(2.10)
p k=0
and
ρ~ (p, N) = N (1 - (1 / N2)) (1 – (22 / N2)) … (1 – (p2 / N2))
2p + 1
(2.11)
16
The (p + q)th order Tchebichef moment Tpq of two-dimensional image
function f (x, y) on the discrete domain of [0, N-1] x [0, M-1] is defined as (Wang G.
and Wang, S., 2006):
Tpq =
2.3
N-1 M-1 ~
~
1
∑ ∑ tp (x) tq (y) f (x, y)
~
ρ (p, N) ~ρ (p, N) x=0 y=0
(2.12)
Implementation of Moment Invariants in Pattern Recognition
Applications
Moment based features are a traditional and widely used technique in pattern
recognition due to its discrimination power (Phiasai et al., 2001). Classification of
shapes is performed using special features which are invariant under twodimensional variations such as translation, rotation and scaling. Moment invariants
method has been proven to be helpful in pattern recognition applications because of
their sensitivity to object features (Nabatchian et al., 2008). Therefore, several
pattern recognition applications such as face recognition, optical character
recognition and aircraft identification implemented moment invariants technique for
features extraction. The next sections present brief descriptions about pattern
recognition applications that implemented moment invariants technique.
17
2.3.1
Face Recognition
Lately, the face recognition has been used more frequently for human
recognition and human authentication. It has been one of the popular topics in the
area of pattern recognition because of its applications in commercial and security
applications. Nowadays, the current technologies have made this application possible
for driver’s license, passport, national ID verification or security access.
Extraction of applicable features from the human face images is a crucial part
of the recognition. Thus, when designing a face recognition system with high
recognition rate, it is important to choose the proper feature extractor. Different
moment invariants have been used to extract features from human face images for
recognition application. Table 2.1 shows the list of face recognition applications that
use moment invariants as their features extraction.
Table 2.1: List of face recognition applications that use moment invariants
Author
Annadurai et al. (2008)
Image Type
Technique
Human face
Hu Moment Invariant, Bamieh Moment
images
Invariant, Zernike Moment Invariant,
Pseudo Zernike Moment Invariant,
Teague-Zernike Moment Invariant,
Normalized Zernike Moment Invariant,
Normalized Pseudo Zernike Moment
Invariant and regular Moment Invariant
Rani, J. S. et al. (2007)
Human face
Tchebichef Moment Invariant
images
Haddadnia et al. (2001)
Human face
Zernike Moment Invariant, Pseudo
images
Zernike Moments and Legendre Moment
Invariant.
18
2.3.2
Optical Character Recognition
Optical Character Recognition (OCR) is one of the oldest areas of pattern
recognition with a lot of contribution for recognition of printed documents. OCR
provides human-machine interaction because it is used to transform human-readable
characters to machine-readable codes. OCR is widely used in many applications like
automatic processing of data, check verification or banking, business and scientific
applications. OCR provides the benefit of little human interference and higher speed
in both data entry and text processing, especially when data already exists in
machine-readable characters.
Feature extraction is one of the main processes in any OCR system. Feature
extraction is important in context of document analysis where some variations may
be caused by a number of different sources such as geometric transformation due to
low data quality and slant or stroke width variation due to font altering (Mishra et
al., 2008). In OCR, feature extraction process removes redundancy from the data and
represents the character image by a set of numerical features. It is very important
process since the classifier will only see the features. Therefore, several of previous
works have implemented moments based technique which is invariant under changes
of position, size and orientation. Table 2.2 shows the list of OCR applications that
use moment invariants as their features extraction.
19
Table 2.2: List of OCR applications that use moment invariants
Author
El affar et al. (2009)
Image Type
Arabic
Technique
Krawtchouk Moment Invariant
handwritten word
Leila and Mohammed
Arabic
(2008)
handwritten
Kunte and Samuel
Kannada printed
Geometric Moment Invariant and
(2006)
text
Zernike Moment Invariant
Sarfraz et al. (2003)
Arabic printed text
Geometric Moment Invariant
Dehghan and Faez
Farsi handwritten
Zernike Moment Invariant, Pseudo
(1997)
Tchebichef Moment Invariant
Zernike Moment Invariant and
Legendre Moment Invariant
2.3.3
Other Pattern Recognition Applications
There are other pattern recognition applications require robust feature
descriptors that are invariant to translation, rotation and scaling transformations. Its
basic idea is to describe the object by a set of features which are insensitive to
particular deformations and provide enough discrimination power to differentiate
among objects to different classes. Hence, moment invariants technique has been
choose as feature extraction in pattern recognition areas.
Other pattern recognition applications that have implemented moment
invariant technique as feature extraction is classification of aircraft. Aircraft
identification must be made from image that is usually noisy in the real world. Thus,
moment invariants are implemented in aircraft identification since the technique is
sensitive to noise (McAulay et al., 1991). In this previous work, the technique makes
20
classification of aircraft robust in the presence of noise and some measure of scale,
translation and rotation invariance (McAulay et al., 1991). The moment invariants
provide the invariance and reduce the amount of data.
Moment invariants also have been implemented for insect identification (Gao
et al., 2007). In this previous work, they focused on winged insects with their left
forewing images. The images need good quality because the blurriness and disrepair
of images will make some veins of wing pattern unusable. Moreover, most of the
existing feature extraction methods are time consuming, error-prone and lessrepeatable. Therefore, moment invariants approach has been choose for insect
identification. Table 2.3 shows the list of other pattern recognition applications that
use moment invariants as their feature extraction methods.
Table 2.3: List of other pattern recognition applications that use moment invariants
Author
Gao et al. (2007)
Image Type
Forewing images
Technique
Hybrid Moment Invariant
of dragonflies
Maaoui et al. (2005)
Color object
Zernike Moment Invariant
images
Puteh (2004)
Trademark images
Geometric Moment Invariant and
Zernike Moment Invariant
McAulay et al. (1991)
Aircraft
Geometric Moment Invariant
21
2.4
Artificial Neural Networks
An Artificial Neural Network (ANN) is usually described as a network
composed of a large number of simple neurons that are massively interconnected,
operate in parallel and learn from experience. ANN was inspired by biological
findings involving the behavior of the brain as a network of units called neurons.
ANN is much simplified and bears only resemblance on the surface even though it is
modeled after biological neurons. Some of the major attributes of ANN such as they
can learn from examples, generalize well on unseen data and able to deal with
situation where the input data are incomplete or fuzzy (Kamruzzaman et al., 2006).
There have been hundreds of different models considered as ANNs since the
first neural model been introduced by McCulloch and Pitts (1943) that includes
General Regression Neural Network, which will be briefly discuss in the next
section. The differences in those models might be the functions, the accepted value,
the topology and learning algorithm. There are also many hybrid models where each
neuron has more properties. Neural networks have the ability to recognize patterns,
even when the information consist of these patterns is noisy or incomplete. Previous
works show that neural networks are very good pattern recognizers since they have
the ability to learn and build unique structures for a particular problem (Uhrig,
1995).
2.4.1
General Regression Neural Networks
General Regression Neural Network (GRNN) is proposed by Donald F.
Specht (1991) and it is belongs to the class of supervised networks. The GRNN is
based on radial basis function and operates in a similar way to PNN but perform
22
regression tasks. The GRNN architecture as illustrated in Figure 2.1 consists of four
layers which are (Specht, 1991):
•
Input layer – There is one neuron in the input layer for each
predictor variable. In the case of categorical variables, N-1 neurons
are used where N is the number of categories. The input neurons
standardize the range of the values by subtracting the median and
dividing by the interquartile range. The input neurons then feed the
values to each of the neurons in the hidden layer.
•
Hidden layer – This layer has one neuron for each case in the
training dataset. The neuron stores the values of the predictor
variables for the case along with the target value. When presented
with the x vector of input values from the input layer, a hidden neuron
computes the Euclidean distance of the test case from the neuron’s
center point and then applies the RBF kernel function using the sigma
value. The resulting value is passed to the neurons in the summation
layer.
•
Summation layer – There are only two neurons in this layer. One
neuron is the denominator summation unit and the other is the
numerator summation unit. The denominator summation unit adds up
the weight value coming from each of the hidden neurons. The
numerator summation unit adds up the weight values multiplied by
the actual target value for each hidden unit.
•
Decision layer – This layer divides the value accumulated in the
numerator summation unit by the value in the denominator
summation unit and uses the result as the predicted target value.
23
Figure 2.1: GRNN architecture
The GRNN is a memory based neural network based on the estimation of a
probability density function. A method that expresses the functional form as a
probability density function determined empirically from the observed dataset, thus
requiring no a priori knowledge of the underlying function (Specht, 1991). The
network’s estimate Ŷ(X) can be thought of as a weighted average of all observed
values Yi, where each observed value is weighted according to its distance from X.
The equation (2.14) is for Ŷ(X), where the resulting regression which involves
summations over the observations is directly applicable to problem involving
numerical data.
∑
(2.13)
∑
where Di2 is defined as:
(2.14)
24
The GRNN provides estimates of continuous variables and converges to the
underlying regression surface. This neural network is one pass learning algorithm
with highly parallel structure. The main advantages of GRNN are fast learning and
convergence to the optimal regression surface as the number of samples becomes
very large (Erkmen et al., 2006). GRNN approximates any arbitrary function
between input and output vectors, drawing the function estimate directly from the
training data. Moreover, it is consistent as the training set size becomes large, the
estimation error approaches zero, with only mild restrictions on the function (Avci,
2002).
2.5
Neural Networks Implementation
The interest in neural networks comes from the networks’ ability to imitate
human brain as well as its ability to learn and respond. As a result, neural networks
have been used in a large number of applications and have proven to be effective in
performing complex functions in a variety of fields. These include pattern
recognition, classification, vision, control systems and prediction. Table 2.4 shows
the list of applications that implement neural networks as their classifier.
Table 2.4: List of applications that use neural networks as classifier
Author
Image Type
Classifier
Accuracy
Polat et al., (2006)
3D objects
GRNN
94.17%
Erkmen et al., (2006)
English letters
GRNN
94.78%
Wang (2008)
Human face images
RBF
90.20%
Wu et al. (2007)
Leaf images
PNN
90.31%
25
Unlike multilayer feed-forward neural network which requires a large
number of iterations in training to converge to a desired solution, GRNN need only a
single pass of learning to achieve optimal performance in classification (Bhatti et al.,
2004). The GRNN is a powerful regression tool with a dynamic network structure
whose network training speed is extremely fast (Polat et al., 2006). Due to the
simplicity of the network structure and its implementation, it has been widely
applied to a variety of fields. Therefore, the GRNN is chosen as classifier for plant
leaf identification.
2.6
Summary
This chapter explains about the domain of the research which is leaf features
extraction and classification. It gives some briefing of feature extraction which
involves the leaf extraction by features that are suitable for classification and have at
least translation, scale and rotation invariance. Several feature extraction methods
have been discussed in this chapter such as ZMI, LMI and TMI. These methods have
been implemented in various pattern recognition applications that include face
recognition, optical character recognition and aircraft identification as their feature
extraction. Therefore, these methods have been chosen to be implementing in the
plant leaf identification and the comparison of their performances will be make. This
chapter also described the neural networks as classifier for classification of plant
leaf. A brief discussion about GRNN is made and it shows that this neural networks
offer better features for recognition applications. In addition, it gives high efficient
training. Thus, for the plant leaf identification, GRNN is chosen to be implementing
as classifier for this research.
CHAPTER 3
METHODOLOGY
3.1
Introduction
At each operational step in the research process, researchers are required to
choose from multiplicity of methods, procedures and models of research
methodology which will help them to best achieve research objectives. The research
techniques, procedures and methods that form the body of research methodology are
applied to the collection of information about various aspects of a situation, issue or
problem so that the information gathered can be used in other ways. This is where
research methodology plays crucial role. Therefore, this chapter will explain the
major activities involved in this project in order to achieve the objectives and solve
the background problems. In addition, the research operational framework, software
and equipments requirements also will be discussed.
27
3.2
Research Framework
There are two main stages involve in order to accomplish the objectives of
this research. Each of these stages involves different processes to complete plant leaf
identification thoroughly and successfully. The first stage involves four processes
that include image acquisition, image pre-processing, feature extraction, intra-class
and inter-class analysis. For the second stage, three processes are involved. The
processes are data pre-processing which involve numerical image, plant leaf
classification where General Regression Neural Networks is implement as classifier
and lastly, analyzing classes of leaf. Figure 3.1 illustrate the process involve in the
research framework in whole. Each of these processes will be discussed in more
details in this chapter.
Second
First Stage
Start
Intra-class
analysis
Image
acquisition
Image preprocessing
Feature
extraction
Data preprocessing
Inter-class
analysis
Classification
Inter-class
analysis
End
Figure 3.1: Process involve in research framework
28
3.3
Software Requirement
The software requirement during feature extraction phase of this research
includes Borland C++ 5.02. A moment invariants algorithm is build using C
programming languages. Therefore, this software is used since it can handle the C
programming language. Moreover, it can run the moment invariants algorithm more
effectively. For classification phase, the software used is MATLAB 7.8. This
software provides Neural Network Toolbox which makes it easier to use neural
networks in MATLAB. The toolbox consists of a set of functions and structures that
can handle neural networks for classification task.
3.4
Image Source
The implementation of feature extraction methods and classifier will be tested
and evaluated using plant leaf images. The collection of leaf images are from variety
of plants. The procedure is that the leaf from plant is pluck and then, the digital color
image of the leaf is taken with a digital camera. When taken the leaf photo, there is
no restriction on the direction of leaves. In this research, 10 species of different
plants will be use. Each species includes 10 samples of leaf images. These leaf
images come in with different of size, shapes and class.
29
3.5
Image Pre-processing
Image pre-processing does not increase the image information content. It is
useful on a variety of situations where it helps to suppress information that is not
relevant to the specific image processing or analysis task. The aim of image preprocessing is to improve image data so that it can suppress undesired distortions and
enhances the image features that are relevant for further processing.
Pre-processing of leaf images prior to plant leaf identification is essential.
The main purpose of this process is to provide images that follow fixed scaling and
rotation factors. These images will become an input to feature extraction phase. The
images with its various rotations and scaling factor are evaluated with feature
extraction methods to examine their invariant characteristics.
Pre-processing usually contains a series of sequential operations including
prescribe the image size, conversion of gray-scale images to binary images
(monochrome) file and modify the scaling and rotation factors of the image. Brief
descriptions of the images pre-processing steps for this research are as follows:
a) The images are minimized to 210 x 314 pixel dimensions in order to
ease the computational burden.
b) In order to use pixels with values of 0 and 1, the grey-scale images are
converted to binary images. The binary images are often produced by
thresholding a grays-scale or color image, in order to separate an
object in the image from the background. Each image has its own
threshold value; therefore the value is not fixed. Then, the image is
saved using the .raw format.
30
c) Each leaf images have 12 variant images with different scaling and
rotation factors. For image scaling factor, four values are used and for
image rotation factor, four degree values are used. The remaining four
images are influenced by combination of scaling and rotation factors.
Table 3.1 illustrates the value of scaling and rotation factors. This
research used 120 binary images that represent 10 different plant
leaves.
Table 3.1: The value of scaling and rotation factors
No.
3.6
Geometric Transformations Factors
1
The image is reduced to 0.5x
2
The image is reduced to 0.75x
3
The image is enlarged to 1.2x
4
The image is enlarged to 1.4x
5
The image is rotated to 10°
6
The image is rotated to 20°
7
The image is rotated to 45°
8
The image is rotated to 90°
9
The image is reduced to 0.5x and rotated to 10°
10
The image is reduced to 0.75x and rotated to 20°
11
The image is enlarged to 1.2x and rotated to 45°
12
The image is enlarged to 1.4x and rotated to 90°
Feature Extraction
The next stage in the plant leaf identification is the feature extraction phase.
The main advantage of this phase is that it removes redundancy from the image and
31
represents the leaf image by a set of numerical features. These features are used by
the classifier to classify the data. In this research, moment invariants have been used
to build up the feature space. This technique has the desirable property of being
invariant under image scaling, rotation and translation.
The moment-based feature extraction technique is performed in order to
extract the global image shape of binary image. This technique performed well in
extracting the global image shape (Puteh and Norsalasawati, 2004). In this research,
three types of moment invariant techniques are used to generate features vector of
leaf images; namely ZMI, LMI and TMI. These features vector are analyzed to
examine their invariant characteristics.
3.6.1
ZMI Algorithm
ZMI algorithm has quite complex calculation and Figure 3.2 shows the
algorithm of ZMI computation. The GMI technique is the basic technique to others
moments technique and this technique is focused on Basic_GMI() function as shown
in Figure 3.3. Zernike’s rule can be reputedly connected with geometric moments
until third order in Step 4. The existing Zernike translation invariants can be
indirectly obtained by making use of the existing geometric central moments.
Normalized central moment,
computation is used in image scaling.
32
Basic_GMI();
Step 1: //mpq calculation
for i = 0 to p do
for j = 0 to q do
Mij 0;
for k = 1 to row+1 do
for l = 1 to column+1 do
if (xkl ≠ 0)
Mij Mij + ( ki x l j )
Step 2: // and calculation
M10/M00;
M01/M00;
Step 3: //
calculation
for i = 0 to (p+1) do
for j = 0 to (q+1) do
0;
for k = 1 to row+1 do
for l = 1 to column+1 do
if (xkl ≠ 0)
;
Step 4: //
calculation
for i = 0 do (p+1) do
for j = 0 to (q+1) do
1;
/
;
Figure 3.2: Algorithm of Basic_GMI() computation
Step 1: //Read image file
Store data as x column row
Step 2: //Set the initial variables
column;
row;
4;
3.41592654;
Step 3: Call Basic_GMI();
Step 4: //ZMI calculation
3/
3/
12/
4/
5/
5/
8
2
η
4;
η
η
;
;
4
;
3
6
3
;
6
3
4
;
Figure 3.3: Algorithm of ZMI computation
2
6
;
33
3.6.2
LMI Algorithm
A new computation function, Norm() as shown in Figure 3.4 is introduced in
LMI technique. Legendre moments computation does not have invariants
characteristics. Therefore, Norm() is used to overcome this disadvantage because the
function provide invariants characteristics to rotation, translation and scaling factors.
However, Basic_GMI() function need to be called first in order to obtain the value
of
,
and
for Norm() function in Step 1. Figure 3.5 illustrates the algorithm
of LMI computation.
Step 1: // calculation
0.5 tan 2 /
;
Step 2: // calculation
for i = 0 to p do
for j = 0 to q do
/2
1;
0;
for k = 1 to row+1 do
for l = 1 to column+1 do
if (xkl ≠ 0)
1/
cos
sin
cos
sin
;
;
Figure 3.4: Algorithm of Norm() computation
34
Step 1: //Read image file
Store data as x column row
Step 2: //Set the initial variables
column;
row;
4;
Step 3: Call Basic_GMI();
Step 4: Call Norm();
Step 5: //LMI calculation
5/4
3/2
5/4
3/2
15/4
3/2
15/4
3/2
7/4
5/2
7/4
3/2
4;
1/2
1/2
1/2
1/2
3/2
3/2
;
;
;
;
;
;
Figure 3.5: Algorithm of LMI computation
3.6.3
TMI Algorithm
The initial declaration of image size in TMI technique makes this technique
different from others moment techniques. This algorithm also used Basic_GMI()
function. Figure 3.6 shows the algorithm of TMI computation.
35
Step 1: //Read image file
Store data as x column row
Step 2: //Set the initial variables
column;
row;
4;
Step 3: Call Basic_GMI();
Step 4: Call Norm();
Step 5: //LMI calculation
M /N ;
6
3
1
6
3
1
30
30
2
/
30
30
/
2
36
18
9
1
4;
pixel size;
/
/
1
1
1
1
1
1 ;
1 ;
5
1
2 ;
5
1
2 ;
/
1 ;
Figure 3.6: Algorithm of TMI computation
3.7
Intra-class Analysis
An analysis has to be done in order to evaluate the effectiveness and the
performance of moment invariant techniques utilized in this research. This analysis is
known as intra-class analysis. Intra-class analysis is an analysis on the same object
representing the images with different scale and rotation factors.
In order to measure the invariant performance competency of the moment
techniques in this research, a series of equations introduced by Shahrul Nizam (2006)
is implemented. These equations are used to determine the similarity between
different features vectors produced from all the moment techniques in this research
36
that represent the same object of the particular image. These equations are known as
Invariant Error Computation (IEC). The features vector for original image is give by:
Ha ( γi ) = { γi , γi + 1 , γi + 2 , … , γn }
(3.1)
where a is referred as the class name with i is the feature dimension. Each class
consists of a set of images generated by transforming the original image with
different scale and orientations. Therefore, each features vector for the variations of
images is given by:
( γi ) = { γi , γi + 1 , γi + 2 , … , γn }
(3.2)
where m refers to the type of variations of class a. Therefore, there will be several
feature vectors representing the images with different scale and orientations for one
object.
To calculate the absolute error for each dimension, equations (3.3) is used. In
order to make comparison of each pattern of class a, the percentage of absolute error
of each pattern is perform using equation (3.4). Equation (3.5) is computed to obtain
the percentage of mean absolute error for the feature vector for the variations of
class. In order to examine an error distribution along the dimension of each features
vectors are illustrate in equation (3.6). Lastly, equation (3.7) and (3.8) is utilized to
calculate the total error of class a for each moment invariant method.
∆
|
|
(3.3)
37
∆
%∆
|
|
100
(3.4)
I
∑
1
(3.5)
i=1
M
∑
2
(3.6)
m=1
M
∑
m=1
1
(3.7)
n
∑
i=1
3.8
2
(3.8)
Inter-class Analysis
Inter-class analysis is conducted by making a comparison between values of
feature vectors based on the original image used in this research. Different moment
invariant methods will generate a different value of feature vectors for the same
image. Hence, each group of data produced by different technique definitely has its
own feature space. In order to analyze the different value of feature vectors for the
images, each feature vectors value of the original image is put into a table according
to the moment invariant methods used. The comparison is examined based on the
characteristic of feature vectors value, the similarity and dissimilarity of those values.
Other than that, computational time for each moment invariants methods to extract
38
feature vectors is taken. The purpose is to examine which methods have fast
computational speed so that it can be implemented later.
3.9
Data Pre-processing
Data pre-processing describes any type of processing performed on raw data
to prepare it for another processing procedure. The main purpose of the data preprocessing stage is to manipulate the raw data into a form with which a neural
network can be trained. The data pre-processing can often have a significant impact
on the performance of a neural network. Therefore, before applying any of the builtin functions for training, it is important to check that the data is reasonable.
Before this process is performed, the data obtained from the feature
extraction methods are in a numeric form or measures. Thus, in order to obtain an
effective training of neural networks, the numeric data should be scaled. This process
is known as normalization. Normalization is a scaling down transformation of the
features. There is often a large difference between the maximum and minimum
values within a feature. When normalization is performed, magnitudes of the values
is normalized and scaled to appreciably low values. This is important for many
neural network algorithms. One form of suitable data normalization can be achieved
using equation (3.9) which is known as Linear Transformation equation. The scaled
variable should be within the range of 0 to 1.
(3.9)
39
(3.10)
where:
v’ is the new feature value that been normalized.
vo is the old feature value before normalized.
vmin is minimum feature value in the data sample.
vmax is maximum feature value in the data sample.
3.10
Classification by Neural Networks
In this research, a GRNN is utilized as classifier in plant leaf identification.
This network is trained in supervised manner. Therefore, the input data is normalized
so that the feature vectors values is in the range of 0 to 1. Otherwise, normalization
process is ignored if the feature vectors values obtained is in that range.
The dataset is divided into training dataset, validation dataset and testing
dataset. Training dataset is used for learning and to fit parameters of the classifier.
To evaluate the success of the network, validation dataset is used to test the
network’s ability to generalize the data and tune the parameters of classifier. After
fixed iterations have completely been carried out, the testing dataset is used in the
trained network in order to estimate the error rate so that the network can recognize
the pattern effectively. Figure 3.2 shows the process of training, validation and
testing phase in neural network.
40
Stop
Yes
Start
Best rate error?
No
Initialize
weight
Train neural network by
fixed iterations
No
Neural network
validation phase
Best candidate
found?
Yes
Test neural network
Figure 3.7: Process of training, validation and testing phase in neural network
3.11
Inter-class Analysis
The main purpose of inter-class analysis is to find out the suitable parameter
value for GRNN classifier. This analysis is conducted during training and testing
phase in neural networks. The parameter values used during these phases are
changed in order to search for the best performance. Each data obtained from the
feature extraction process by moment invariants methods are used in the
classification process. Then, the performance of each output gained from the
classification process is evaluated in conjunction with the moment invariants
methods used.
41
The method of k-fold cross validation is used to find the generalization
performance of GRNN classifier in combination with different types of features
extraction. The k-fold cross validation is one way to improve over the holdout
method. The dataset is divided into k subsets and the holdout method is repeated k
times. Each time, one of the k subsets is used as the test dataset and the other k – 1
subsets are put together to form the training dataset. Then, the average error cross all
k trials is computed. Every data point gets to be in a test exactly once and in training
set k – 1 times.
In this research, the dataset is split into k subsets. Then, the classifier is
trained and tested k-times. The cross validation estimate is identified as the number
of correct classification divided by the number of data points. The percentage of
correct classification (PCC) is given by the equation (3.11) and the number of correct
classification (NCC) is given by (3.12). If the testing vector is true, σ (x, y)t = 1.
Otherwise, σ (x, y)t = 0.
k
100
∑
(3.11)
k=1
n
∑
,
t=1
where n refers to the number of data tested.
(3.12)
42
3.12
Summary
As a conclusion, this chapter explains briefly on this research methodology
where it involves two main stages which include feature extraction phase and
classification phase. The processes involved in this research are presented in the
research framework. Each process implemented in this research is dependent with
one another. The next chapter will describes briefly about the initial finding of this
research.
CHAPTER 4
IMPLEMENTATION, COMPARISON OF RESULTS
AND DISCUSSION
4.1
Introduction
This chapter will discussed the implementation processes of this research.
These implementation processes involve two stages. The first stage involves several
processes that include image pre-processing, feature extraction of leaf images, intraclass and inter-class analysis. Moment invariants methods such as ZMI, LMI and
TMI are applied in the feature extraction process. As for the second stage, the
processes involved are data pre-processing, classification of leaf images and interclass analysis. For classification process, GRNN is applied as the classifier. The
implementation methods and the results achieved will be describes briefly in this
chapter.
44
4.2
Leaf Images
As mentioned on the previous chapter, 10 plant leaf images are used as
dataset for this research. These images are divided into four classes based on what
plant family those leaves are classified. The list of classes is shown in Appendix D.
Figure 4.1 illustrates several plant leaf images that are used as image sample for
discussion in this chapter. Table 4.1 shows the list of leaf image name that are used.
Firstly, these images will undergo image pre-processing process as explained in
section 3.5. The images file format are changed from .JPEG to .RAW file format.
Then, the size of the images is set to 210 x 314 pixel dimensions. Each of the leaf
images will go through feature extraction process where variant shapes of each leaf
is extract. The feature vectors obtained from the feature extraction process are used
for intra-class and inter-class analysis.
(A1)
(B1)
(B2)
Figure 4.1: Leaf images
Table 4.1: List of image name
Image
Image Name
A1
Rambutan Leaf
B1
Jackfruit Leaf
B2
Cempedak Leaf
C1
Apple Mango Leaf
(C1)
45
The leaf images undergo scaling and rotation transformations to produce
variants images in order to test the robustness of the feature extraction methods
adopted. Each leaf images have 12 variant images with different scaling and rotation
factors so that moment invariants methods can extract the leaf features from variant
images. Therefore, there will be 120 images used for this research. Figure 4.2 shows
an image (A1) that go through different scaling and rotation factors and Table 4.2
illustrates the value of scaling and rotation factors for the image A1.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
Figure 4.2: An image A1 with its various rotations and scaling factor
46
Table 4.2: Scaling and rotation factors of image A1
Image
4.3
Geometric Transformations Factors
(i)
An original image
(ii)
The image is reduced to 0.5x
(iii)
The image is reduced to 0.75x
(iv)
The image is enlarged to 1.2x
(v)
The image is enlarged to 1.4x
(vi)
The image is rotated to 10°
(vii)
The image is rotated to 20°
(viii)
The image is rotated to 45°
(ix)
The image is rotated to 90°
(x)
The image is reduced to 0.5x and rotated to 10°
(xi)
The image is reduced to 0.75x and rotated to 20°
(xii)
The image is enlarged to 1.2x and rotated to 45°
(xiii)
The image is enlarged to 1.4x and rotated to 90°
Result of Intra-class Analysis
As mentioned in Chapter 3, intra-class analysis is an analysis on the same
object representing the images with different scale and rotation factors in order to
evaluate the effectiveness of three moment invariants used in this research. The ZMI,
LMI and TMI algorithm is used on the image A1 to generate value of feature
vectors. The value of feature vectors obtained is shown in Table 4.3, 4.4 and 4.5.
Based on all the tables, different moment technique will generate a different value of
feature vectors. Thus, each group of data produced by different techniques certainly
has its own feature space.
47
Table 4.3: Value of feature vectors using ZMI
Features
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
Z1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Z2
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Z3
0.635635
0.607506
0.655656
0.565031
0.562489
0.620557
0.616969
0.603451
0.602521
0.605920
0.653990
0.583170
0.638983
Z4
0.012314
0.000183
0.007104
0.012254
0.012184
0.012173
0.011781
0.010333
0.005147
0.000117
0.006534
0.010694
0.000455
Z5
0.028653
0.111753
0.081395
0.002817
0.002078
0.019799
0.018199
0.013358
0.004406
0.110258
0.080247
0.000019
0.016148
Z6
0.001663
0.007546
0.004978
0.000183
0.000120
0.001136
0.001035
0.000735
0.000278
0.007491
0.005080
0.000001
0.000981
Table 4.4: Value of feature vectors using LMI
Features
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
L1
0.538080
1.514827
0.865814
0.319003
0.308754
0.490340
0.490102
0.495826
0.538763
1.524409
0.877148
0.318934
0.515445
L2
0.594291
1.105704
0.799418
0.399411
0.388183
0.555403
0.553812
0.548580
0.551787
1.111666
0.805962
0.385297
0.504803
L3
0.139806
1.095573
0.471333
-0.042627
-0.050289
0.098727
0.100859
0.121703
0.207824
1.098006
0.481336
-0.043277
0.188827
L4
-0.351960
0.079916
-0.251064
-0.322703
-0.318163
-0.350699
-0.347579
-0.327088
-0.260746
0.082641
-0.246620
-0.301491
-0.086752
L5
0.216601
1.938061
0.673498
0.031775
0.026174
0.166669
0.166800
0.174141
0.202880
1.959498
0.692444
0.023601
0.076944
L6
-0.304855
-0.958870
-0.543087
-0.113220
-0.103035
-0.263006
-0.260992
-0.252323
-0.222183
-0.969801
-0.551981
-0.086263
-0.058644
48
Table 4.5: Value of feature vectors using TMI
Features
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
4.3.1
T1
0.793469
0.577823
0.680862
0.963832
0.972662
0.820431
0.821020
0.819524
0.726457
0.576871
0.678458
0.885322
0.490751
T2
2.134144
2.209943
2.182293
2.062425
2.029389
2.132314
2.136632
2.155781
1.724457
2.210815
2.185880
1.584306
0.406124
T3
-0.024839
0.005101
-0.001821
-0.006109
-0.003557
-0.023621
-0.020143
0.012155
0.000462
0.001488
-0.004009
-0.000679
-0.000777
T4
6.847867
7.181068
6.992699
6.747295
6.654909
6.813371
6.797773
6.740852
4.548186
7.182417
6.988950
4.793722
0.130346
T5
-0.027656
0.016385
-0.008342
-0.014567
-0.006881
-0.030528
-0.029908
-0.032643
0.000786
0.019021
0.003699
-0.000932
0.004355
T6
-14.207045
-10.369094
-12.224204
-17.224380
-17.453477
-14.698226
-14.720452
-14.681998
-13.020708
-10.349965
-12.181573
-15.871980
-8.807629
Absolute Error
By calculating the Absolute Error (AE) using equation (3.3), the invariant
characteristic of feature vectors can be examine because AE describes the difference
between an original data with its counterpart’s variations. The AE is successfully
employed when a comparison is made between variations of data produced by the
techniques only. Table 4.6, 4.7 and 4.8 illustrate the AE for the image A1. Number
one (1) to six (6) appear on the top column indicates to the feature dimension.
49
Table 4.6: Absolute Error for ZMI
ZMI
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
2
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
3
0.000000
0.028129
0.020021
0.070625
0.073146
0.015078
0.018666
0.032184
0.033114
0.029715
0.018355
0.052465
0.003348
4
0.000000
0.012131
0.005210
0.000060
0.000130
0.000141
0.000533
0.001981
0.007167
0.012197
0.005780
0.001620
0.011859
5
0.000000
0.083100
0.052742
0.025836
0.026575
0.008854
0.010454
0.015295
0.024247
0.081605
0.051594
0.028634
0.012505
6
0.000000
0.005883
0.003315
0.001480
0.001543
0.000527
0.000628
0.000928
0.001385
0.005828
0.003417
0.001662
0.000682
5
0.000000
1.721460
0.456897
0.184826
0.190427
0.049932
0.049801
0.042460
0.013721
1.742897
0.745843
0.193000
0.139657
6
0.000000
0.654015
0.238232
0.191635
0.201820
0.041849
0.043863
0.052532
0.082672
0.664946
0.247126
0.218592
0.246211
Table 4.7: Absolute Error for LMI
LMI
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
1
0.000000
0.976747
0.327734
0.219077
0.229326
0.047740
0.047978
0.042254
0.000683
0.986329
0.339068
0.219146
0.022635
2
0.000000
0.511413
0.205127
0.194880
0.206108
0.038888
0.040479
0.045711
0.042504
0.517375
0.211671
0.208994
0.089488
3
0.000000
0.955767
0.331527
0.097179
0.089517
0.041079
0.038947
0.018103
0.068018
0.958200
0.341530
0.096529
0.049021
4
0.000000
0.272044
0.100896
0.029257
0.033797
0.001261
0.004381
0.024872
0.091214
0.269319
0.105340
0.050469
0.265208
50
Table 4.8: Absolute Error for TMI
TMI
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x +90°
1
0.000000
0.215646
0.112607
0.170363
0.179193
0.026962
0.027551
0.026055
0.067012
0.216598
0.115011
0.091853
0.302718
2
0.000000
0.075799
0.048149
0.071719
0.104755
0.001830
0.002488
0.021637
0.409687
0.076671
0.051736
0.549838
1.728020
3
0.000000
0.019738
0.023018
0.081730
0.021282
0.001218
0.004696
0.012684
0.024377
0.023351
0.020830
0.024160
0.024062
4
0.000000
0.333201
0.144832
0.100572
0.192958
0.034496
0.050094
0.107015
2.299681
0.334550
0.141083
2.054145
6.717521
5
0.000000
0.011271
0.019314
0.013089
0.020775
0.002872
0.002252
0.004987
0.026870
0.008635
0.023957
0.026724
0.023301
6
0.000000
3.837951
1.982841
3.017335
3.246432
0.491181
0.513407
0.474953
1.186337
3.857080
2.025472
1.664935
5.399416
Table 4.6 shows that AE value for ZMI is consistent because the differences
value of feature vectors between the original image with the image influenced by
rotation and scale factors is in the range of 0.00 to 0.08. However, AE value for LMI
is greater than ZMI, where the range is between 0.00 to 1.74. This proves that the AE
value still showing the big differences on variation of the same image. As for TMI,
the AE value is in the range of 0.00 to 6.72 and these AE value is the higher
compared to ZMI and LMI. Apart from that, between all three moment techniques,
ZMI and LMI are better in feature extraction of leaf images because the differences
between the values of feature vectors of the image is small.
4.3.2
Percentage Absolute Error (PAE)
The absolute error is not suitable to compare different moment techniques.
This is due to it comprises only absolute number and without bringing out the
important information from the original data. Moreover, each moment technique
51
uses different equations and generated different feature spaces. Hence, PAE as refers
to equation (3.4) is used. Table 4.9 and Figure 4.3 illustrate the value of PAE for
image A1 with rotate factor of 10° and different types of moment.
Table 4.9: PAE for image A1 with rotate factor of 10°
Moment techniques
ZMI
LMI
TMI
1
0.00
8.87
3.40
2
0.00
6.54
0.09
3
2.37
29.38
4.90
4
1.15
0.36
0.50
5
30.90
23.05
10.38
6
31.69
13.73
3.46
35.00
30.00
PAE (%)
25.00
20.00
ZMI
15.00
LMI
10.00
TMI
5.00
0.00
0
1
2
3
4
5
6
7
Feature
Figure 4.3: PAE graph for image A1 with rotate factor of 10°
Based on Figure 4.3, the PAE for TMI is lower compared to ZMI and LMI.
However, from the method that has been described before, it seems that ZMI and
LMI have a good performance, where both moment techniques are better in feature
extraction of leaf images. This is because the values of feature vectors for both
moments are invariant. Therefore, more analysis need to be done in order to measure
the invariant performance competency of the moment techniques utilized in this
research.
52
4.3.3
Percentage Mean Absolute Error 1 (PMAE1)
The main purpose of PMAE1 calculation is to find out the distribution of
error arises among image variation for one object. Figure 4.4 shows the value of
PMAE1 for variation of image A1.
400
350
PMAE1
300
250
200
150
ZMI
100
LMI
50
TMI
0
Image Variation
Figure 4.4: PMAE1 graph for image A1 with image variation
From Figure 4.4, it can be seen that the scaling factor of 0.5x with rotation
factor of 10° generate the highest error compared to other factors for all moment
invariant applied. This is because when an image is under the influence of both
scaling and rotation factor, the value of PMAE1 increases. Even so, TMI generate
the lowest PMAE1 value for almost variation of image A1. On the other hand, LMI
produced the highest error compare to ZMI and TMI.
53
4.3.4
Percentage Mean Absolute Error 2 (PMAE2)
PMAE2 calculation using equation (3.6) is used to examine an error
distribution along the dimension of feature vector. Figure 4.5 illustrates the value of
PMAE2 for image A1 with different dimension of feature vector.
250
PMAE2
200
150
ZMI
100
LMI
TMI
50
0
1
2
3
4
5
6
Feature Vector Dimension
Figure 4.5: PMAE2 graph for image A1 with different dimension
Based on Figure 4.5, it is found that when the order of ZMI is increased, thus
the value of PMAE2 become higher. This is because higher order moments are more
sensitive to image noise and quantization effects (Shahrul Nizam et al., 2006). As
shown in Figure 4.5, the value of error for ZMI is rising as the dimension increases.
However, TMI shows the lowest error compare to other two of moment techniques.
Whereas, LMI generate the highest value of PMAE2 compare to other moment
techniques.
54
4.3.5 Tootal Percentage Mean Absolute Error
E
(TPM
MAE)
Laastly, the tottal error of image
i
A1 fo
or each mom
ment techniiques is calcculate
by using equation
e
(3.7) to measuure the invarriant perform
mance of thhe techniquees.
Table 4.100 and Figuree 4.6 show the value off TPMAE for
f three mooment techn
niques of
image A1..
Table 4.10:
4
TPMA
AE of differrent momennts for imagge A1
Momentt techniquess TPMAE
E value
Z
ZMI
46.89
L
LMI
100.77
T
TMI
34.997
120
TPMAE (%) Value
100
80
60
40
20
0
ZMI
L
LMI
TMI
Moment TTechniques
Figure 4.66: TPMAE graph of diifferent mom
ments for im
mage A1
p
meethod that has
h been desscribed, botth ZMI and TMI
Baased on the previous
produced good perforrmance. Hoowever, the value
v
of TP
PMAE indiccates that TM
MI
55
generate the lowest total error compare to ZMI and LMI. Therefore, between all
three moment techniques, TMI are better in feature extraction of leaf images because
it produces smaller total error.
4.4
Result of Inter-class Analysis
As mentioned in Chapter 3, inter-class analysis is conducted by making a
comparison between values of feature vectors based on the original image used in
this research and the computational time for each moment invariants methods to
extract feature vectors.
4.4.1
Comparison Based On Value of Feature Vectors
To obtain the result of inter-class analysis, a comparison between values of
feature vectors based on the original image is conducted. Therefore, four types of
leaf images are used as input data. The three moment invariants algorithm is used on
the four images as shown in Figure 4.1 in order to obtain the value of feature vectors
of each image. For this analysis, only the original leaf image is used. Table 4.11,
4.12 and 4.13 shows the value of feature vectors for each image using the ZMI, LMI
and TMI techniques.
56
Table 4.11: Value of feature vectors of four images for ZMI
ZMI
Image (A1)
Image (B1)
Image (B2)
Image (C1)
1
0.000000
0.000000
0.000000
0.000000
2
0.000000
0.000000
0.000000
0.000000
3
0.635635
0.618260
0.685912
0.669314
4
0.012314
0.012979
0.018290
0.007497
5
0.028653
0.019577
0.043595
0.145911
6
0.001663
0.001217
0.002760
0.008900
Table 4.12: Value of feature vectors of four images for LMI
LMI
Image (A1)
Image (B1)
Image (B2)
Image (C1)
1
0.538080
0.452960
0.511409
0.936573
2
0.594291
0.527403
0.589186
0.841079
3
0.139806
0.061426
0.085868
0.533608
4
-0.351960
-0.355419
-0.394231
-0.226067
5
0.216601
0.130443
0.179605
0.792434
6
-0.304855
-0.235014
-0.300754
-0.599251
Table 4.13: Value of feature vectors of four images for TMI
TMI
Image (A1)
Image (B1)
Image (B2)
Image (C1)
1
0.793469
0.843832
0.799002
0.665351
2
2.134144
2.121805
2.045261
2.192707
3
-0.024839
-0.035599
-0.037482
-0.011157
4
6.847867
6.851434
6.940548
7.155110
5
-0.027656
-0.019132
-0.002248
-0.014760
6
-14.207045
-15.085694
-14.278451
-11.913172
As shown from all the tables, leaf images that belong to the same class (B1
and B2) produce a dissimilar value of feature vectors. Nevertheless, the difference
between the value of feature vectors for both B1 and B2 images is small. This is
because B1 and B2 leaf are belonging to the same family plant.
57
Figure 4.7: Comparison value of feature vectors based on leaf class
Figure 4.8: Comparison value of feature vectors based on leaf class for original
image
58
The class or family of the leaf can be differentiated through the value of
feature vectors. However, Figure 4.7 and Figure 4.8 show that it is not so accurate
because there are leaf images that produce value of feature vectors that is almost near
to the value of feature vectors from other class or family. From the graph, the value
of feature vectors produce by Apple Mango (C1) leaf, Malgoa Mango (C2) leaf and
Kuini (C3) leaf are near to each other since these leaf belong to the same family,
Anacardiaceae. On the other hand, Rambutan (A1) leaf, Pulasan (A2) leaf and Mata
Kucing (A3) leaf are belong to the same family, Sapindaceae but the feature vectors
value from one of this leaf is near to the value of other leaf belong to Moraceae
family.
4.4.2
Comparison Based On Computational Time
Plant leaf identification is time consuming processes. Thus, the
computational time for ZMI, LMI and TMI algorithms to extract feature vectors of
each image is taken in order to find the fastest moments techniques. This could help
the experts to identify the plant leaf much faster. Table 4.14 lists down the
computational time taken in seconds for the three moment invariants techniques.
Based on table, the computational time taken for ZMI and TMI is the fastest
compared to LMI. The average time adopted for both moments is 0.24 seconds. This
is because ZMI and TMI algorithm used the same Basic_GMI() function.
Table 4.14: Computational time taken in seconds
Image
(A1)
(B1)
(B2)
(C1)
Average
ZMI
0.24
0.25
0.24
0.22
0.24
LMI
0.86
0.85
0.88
0.91
0.86
TMI
0.24
0.25
0.24
0.22
0.24
59
4.5
Analysis of Feature Extraction Results
In the intra-class analysis, a set of equations known as AE, PAE, PMAE1,
PMAE2 and TPMAE is used to examine the invariant performance properties of
moment invariant techniques. It is found that TMI generated the lowest error value
for PAE, PMAE1, PMAE2 and TPMAE when compared to ZMI and LMI. For
instance, PAE for TMI is the lowest with less than 10.38 percent of error while ZMI
less than 31.69 percent and LMI less than 29.38 percent. In addition, it is found that
TMI technique generated feature vectors that are more invariant capability compared
to ZMI and LMI. For the inter-class analysis, it is found that three moment invariant
techniques have different value of feature vectors depending on the image used. The
difference between the values is small for certain leaf image that belonging to the
same family plant. However, there are certain leaf images that produce value of
feature vectors which is near to other family plant. As for the computational time
taken for each moment techniques, it is found that ZMI and TMI have a same
average time, which is 0.24 seconds. Therefore, the value of feature vectors from
TMI will be used as input data for classification phase.
4.6
Classification Phase
This section is focused on the classification approaches adopted to be studied
in this research as described in Chapter 3. This section begins with a description of
data preparation and followed by an explanation of the implementation of GRNN
classification method. This section is then continues with the presentation of the
results obtained. The results produced are then analyzed and discussed.
60
4.6.1
Data Preparation
Classification process was performed by using MATLAB 7.8. The input data
for the classifier are numerical values extracted from leaf images using TMI
technique. The dataset consists of 130 total samples. The entire dataset was split into
four subsets where two subsets composed of 32 data (fold 1 and 3) and another two
subsets consists of 33 data (fold 2 and 4).
The network training and testing are repeated four times (k times). Each time
of training and testing, the k-fold cross validation is used to determine the
performance of learning algorithm. Therefore, three subsets are put together to form
the training dataset and one subset is used as the test dataset. The advantage of these
data representation is that all the samples in the dataset are eventually used for both
training and testing. Figure 4.9 reflects the k-fold partition of the dataset.
Total number of samples
Fold 1
Fold 2
Fold 3
Fold 4
Experiment 1
Experiment 2
Experiment 3
Experiment 4
Train data
Figure 4.9: k-fold partition of the dataset
Test data
61
4.6.2 GRNN Spread Parameters Declaration
The performance of the classifier is affected by the choice of spread
parameters. Spread value should be large enough that neurons respond strongly to
overlapping regions of the input space. The spread values are chosen 0.001, 0.005,
0.01, 0.05, and 0.1 for GRNN to achieve maximum classification rate. Each spread
value will be compared and discussed.
4.7
Classification Results of GRNN Classifier
GRNN have been used for classification scheme. The classification results of
GRNN classifier for each spread parameters are given in Table 4.15 respectively.
From Figure 4.10, it can be concluded that the percentage of correct classification
(PCC) is 100 percent for each spread parameters. However, these results shown that
each spread value do not give impact to leaf image recognition because all produced
100 percent classification rate.
62
Table 4.15: GRNN result for each spread parameters
Spread
0.001
0.005
0.01
0.05
0.1
k
Data
Unit
NCC
32
33
32
33
Time
Taken
(Second)
4.49
0.60
0.44
0.42
1
2
3
4
1
2
3
4
32
33
32
33
0.46
0.51
0.51
0.41
32
33
32
33
1
2
3
4
32
33
32
33
2.42
0.50
0.47
0.48
32
33
32
33
1
2
3
4
32
33
32
33
0.67
0.50
0.44
0.39
32
33
32
33
1
2
3
4
32
33
32
33
1.12
1.24
0.50
0.62
32
33
32
33
k = Group of sample data
Data Unit = Number of data in sample group
32
33
32
33
PCC
Time Average
(Second)
100
1.49
100
0.47
100
0.98
100
0.50
100
0.87
63
PCC vs. Spread parameter 120
PCC (%)
100
80
60
40
20
0
0.001
0.005
0.01
0.05
0.1
Spread Parameter
Figure 4.10: Graph of PCC versus spread parameter value
Since all the PCC values are same, the most suitable spread parameter is
chosen based on the time average adopted by each spread values. Figure 4.11 shows
that the spread value of 0.005 acquired the quickest time average compared to others
value.
Time Average vs. Spread Parameter 1.6
Time Average (s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.001
0.005
0.01
0.05
0.1
Spread Parameter
Figure 4.11: Graph of time average versus spread parameter value
64
4.8
Summary
This chapter discussed a comparison of effectiveness between ZMI, LMI and
TMI in representing the description of an image. From the intra-class and inter-class
analysis that have been conducted, it shows that a TMI technique is suitable for
feature extraction of leaf images. In order to continue the second stage of this
research that involved classification, the value of feature vectors from TMI will be
used as input data. This chapter then continues to discuss the performance of the
GRNN classification approach. The results of recognition and classification of leaf
images is 100 percent accuracy rate. However, the spread parameters of the GRNN
are unaffected with the leaf image recognition.
CHAPTER 5
DISCUSSION AND CONCLUSION
5.1
Introduction
The suitable moment invariant techniques for feature extraction prior to
classification task are investigated in this research. This chapter draws general
conclusion of the results obtained from this research. In addition, disadvantages of
this research will be discussed in this chapter and several potentials areas for future
work will be suggested.
5.2
Discussion of Results
With plant leaf imaging playing an increasingly prominent role in the plant
identification, the problem of extracting useful information has become important.
66
For example, leaf imaging helps to define the shape of the plant leaf and aiding in
plant recognition. To facilitate an effective and efficient review of this plant leaf
information, techniques for performing feature extraction become the key, which
lead to the main focus of this research that is to extract features from plant leaf
images using moment-based methods.
The first objective of this research is to compare the effectiveness of three
moment invariants techniques, namely ZMI, LMI and TMI to extract features from
leaf images. The effectiveness and strength of these moment invariants technique are
further studied and tested in order to see which moment invariants suitably
represents an image and would give a promising recognition rate. Then, the extracted
features from moment technique are used in training and classification using GRNN
classifier. Results of classification using these features are compared in terms of
accuracy with different spread parameters and computational time. With the
implementation of image pre-processing, feature extraction using moment invariants
technique and GRNN classification on the set of features representing the leaf
images as described in Chapter 4, the objectives of this research has been fulfilled.
In general, experiments conducted revealed the performance of the
techniques used in this research and help achieved the objectives of the research.
Moreover, it provided sufficient information and knowledge for further
investigations and future works. Experimentations from Chapter 4 have shown that
TMI are better than ZMI and LMI in representing feature extraction of leaf images.
By using Invariant Error Computation (IEC) in order to measure the invariant
performance competency of the moment techniques in this research, it shown that
TMI is suitable technique to be use for feature extraction with percentage of absolute
error is less than 10.38 percent. However, the result reveal that there are certain leaf
images produce value of feature vectors which is near to other different family of
leaf, even though the difference between the for certain leaf image that belonging to
the same family plant is small. As for classification phase, the experimental results
demonstrated that GRNN classifier produced 100 percent classification rate with
67
average computational time is 0.47 seconds. Nevertheless, the findings reveal that
the spread values do not influenced the classification rate.
5.3
Problems and Limitations of Research
There were some problems and limitations present while conducting this
research. Below are the problems and limitations:
•
One of the disadvantages in this research is the use of limited sample of leaf
images. In this research, only 10 species of different plants were used and
each species includes 10 samples of leaf images. Thus, it does not generalize
the classification method.
•
The occurrences of different leaf classes in the world are not taken into
account to improve the classification accuracy.
5.4
Recommendation for Future Works
Below are some suggestions that could lead to the improvement of the
classification of leaf images and some possible points that could lead to future
research.
68
•
Utilization of other moment techniques that could be used in feature
extraction of an image. For instance, weighted central moments and cross
weighted moments can be applied in image analysis application.
•
Utilization of other neural network model for comparison such as recurrent
neural network or Radial Basis Function (RBF). These models can be applied
in network training and classification to compare with GRNN classification
implemented in this research.
•
The methodologies used in feature extraction with moment techniques and
GRNN classification can be extended to other approaches for plant
classification based on plant features of colors, textures and structures of their
bark, flower, seedling and morph.
5.5
Conclusion
The work described in this research has been concerned with the two
challenging phases in image analysis applications which are feature extraction and
classification phase. Since there is no general feature extraction method that is
available for all type of images, an experiment needs to be conducted in order to
determine the suitable methods for plant leaf images. Therefore, an investigation of
the moment invariants techniques was presented which were used to be implemented
in the feature extraction of plant leaf images. This research presents a comparison of
effectiveness between ZMI, LMI and TMI in representing the description of an
image. In this research, feature extraction methodologies using ZMI, LMI and TMI
are applied. Then, the comparison between these moment invariants technique is
presented by using IEC in terms of image representation efficiency. Based on the
experimentations performed, it can be concluded that TMI is better in representing
an image description compare with ZMI and LMI. Thus, the first objective of this
project has been achieved. Network training and classification is conducted using
69
GRNN with different spread parameters. However, these spread parameters do not
influenced the classification rate since each spread values generated 100 percent
classification rate. Future study is suggested in order to improve the recognition
accuracy of leaf images and to seek for any improvement in feature extraction where
applicable.
70
REFERENCES
Annadurai, S. and Saradha, A. (2004). Face Recognition Using Legendre Moments.
Fourth Indian Conference on Computer Vision, Graphics & Image
Processing ICVGIP 2004. 16-18 December 2004. Kolkata, India.
Belkasim, S., et al. (2004). Radial Zernike Moment Invariants. Computer and
Information Technology, 2004. CIT '04. The Fourth International
Conference. Sept 2004. Vol 1: 790- 795.
Chergui, L. and Benmohammed, M. (2008) Fuzzy Art Network for Arabic
Handwritten Recognition System. The International Arab Conference on
Information Technology (ACIT).
Dehghan, M. and Faez, K. (1997). Farsi handwritten character recognition with
moment invariants. Digital Signal Processing Proceedings, 1997. 1997 13th
International Conference. July 1997. Patras, Greece. Vol 3: 507-510.
Du, J. X. et al. (2005). Shape recognition based on radial basis probabilistic neural
network and application to plant species identification. Proceedings of 2005
International Symposium of Neural Networks. 30 May – 1 June 2005.
Chongqing, China. Vol 3498: 281-285.
Du, J. X. et al. (2006). Computer-aided plant species identification (CAPSI) based
on leaf shape matching technique. Transaction of the Institue of
Measurement and Control. Vol 28: 275-284.
Du, J. X. et al.(2007). Leaf shape based plant species recognition. Applied
Mathematics and Computation. 15 Feb 2007. Vol 185(2): 883-893.
El affar, A. et al. (2009). Krawtchouk Moment Feature Extraction for Neural Arabic
Handwritten Words Recogntion. IJCSNS International Journal of Computer
Science and Network Security. January 2009. Vol 9 (1): 417-423.
71
Flusser, J. (2006). Moment Invariants in Image Analysis. Proceedings of World
Academy of Science, Engineering and Technology. 11 February 2006. Vol
11: 196-201.
Gao, Y. et al.(2007). Identification Algorithm of Winged Insects Based On Hybrid
Moment Invariants. The 1st International Conference on Bioinformatics and
Biomedical Engineering, 2007. 6-8 July 2007. Wuhan, China. 531-534.
George, M. (2007). Radial Basis Function Neural Networks and Principal
Component Analysis for Pattern Classification. 2007 International
Conference on Computational Intelligence and Multimedia Applications. 1315 December 2007. Sivakasi, Tamil Nadu. 200-206.
Haddadnia, J. et al. (2001). Neural Network Based Face Recognition With Moment
Invariants. Proceedings.2001 International Conference on Image Processing.
5 – 9 November 2001. Singapore. Centre for Remote Imaging, Sensing and
Processing (CRISP), University of Singapore. Vol 1: 1018 – 1021.
Haddadnia, J. et al. (2002). A Hybrid Learning RBF Neural Network For Human
Face Recognitionwith Pseudo Zernike Moment Invariant. IEEE International
Joint Conference On Neural Network (IJCNN'02). May 2002. Honolulu,
Hawaii, USA. IEEE. 11-16.
Heymans, B. C. et al. (1991). A neural network for opuntia leaf-form recognition.
Proceedings of IEEE International Joint Conference on Neural Networks,
1991. 18-21 November 1991. Singapore. Vol 3: 2116-2121.
Hu, M. K., (1962). Visual Pattern Recognition by Moment Invariants. IRE Trans.
Inform. Theory. IT-E. 179-187.
Keyes, L. and Winstanley, A. (). Moment Invariants as a Classification Tool for
Cartographic Shapes on Large Scale Maps. 3rd AGILE Conference on
Geographic Information Science. 25-27 May 2000. Finland. 1-13.
Khotanzad, A. and Hong, Y. H. (1990) Invariant image recognition by Zernike
moments. IEEE Transactions on Pattern Analysis and Machine Intelligence.
May 1990. Vol 12 (5): 489-497.
Kotoulas, L. and Andreadis, I. (2005). Real-time computation of Zernike Moments.
IEEE Transactions on Circuits and Systems for Video Technology. June
2005. Vol 15 (6): 801-809.
72
Kunte, R. S. and Samuel R. D. S. (2006). A simple and efficient optical character
recognition system for basic symbols inprinted Kannada text. Sadhana
Bangalore. October 2007. India. Vol 32: 521-533.
Li, B. (2008). An Algorithm for License Plate Recognition Using Radial Basis
Function Neural Network. International Symposium on Computer Science
and Computional Technology, 2008. 20-22 December 2008. Shanghai,
China. Vol 1: 569-572.
Maaoui, C. et al. (2005). 2D Color Shape Recognition Using Zernike Moments.
IEEE International Confererence on Image Processing, 2005. 11-14
September 2005. Vol 3: 976-980.
McAulay, A. et al. (1991). Effect of Noise in Moment Invariant Neural Network
Aircraft Classification. Proceedings of the IEEE 1991 National Aerospace
and Electronic Conference, NAECON. 20-24 May 1991. Dayton, OH, USA:
IEEE, Vol 2: 743-749.
Mishra, C. K. et al. (2008). An Approach for Feature Extraction Using Spline
Approximation for Indian Characters (SAIC) in Recognition Engines.
TENCON 2008 IEEE Region 10 Conference. 19-21 November 2008.
Hyderabad. 1-4.
Mukundan, R. Et al. (2001) .Image Analysis by Tchebichef Moments. IEEE
Transactions on Image Processing. September 2001. Vol 20 (9): 1357-1364.
Mukundan, R. (2005). Radial Tchebichef Invariants for Pattern Recognition.
TENCON 2005 IEEE Region 10. 21-24 November 2005. Melbourne,
Australia. 1-6.
Mukundan, R., and Ramakrishnan, K. R. (1998). Moment Function in Image
Analysis. Farrer Road, Singapore: World Scientific Publishing.
Nabatchian, A. et al.(2008). Human Face Recognition Using Different Moment
Invariants: A Comparative Study. Congress on Image and Signal Processing,
2008. 27-30 May 2008. Sanya, China. Vol 3: 661-666.
Pan, F. and Keane, M. (1994). A New Set Of Moment Invariants For Handwritten
Numeral Recognition. 1994 Proceedings IEEE International Conference on
Image Processing, ICIP-94. 13 – 16 November 1994. Austin, Texas, USA :
IEEE. Vol 1: 245-249.
73
Phiasai T. et al. (2001). Face recognition system with PCA and moment invariant
method. The 2001 IEEE International Symposium on Circiuts and Systems,
ISCAS 2001, 6-9 May 2001. Sydney, NSW. Vol 2: 165-168.
Puteh Saad. (2004). Feature Extraction of Trademark Images Using Geometric
Invariant Moment and Zernike Moment-A Comparison. Chiang Mai Journal
of Science. Vol 31: 217-222.
Rani, J. S. et al. (2007). A Novel Feature Extraction Technique for Face
Recognition. International Conference on Computational Intelligence and
Multimedia Applications, 2007. 13-15 December 2007. Sivakasi, Tamil
Nadu. Vol 2: 428-435.
Sarfraz, M. et al. (2003). Offline Arabic text recognition system. The Proceedings of
IEEE International Conference on Geoemetric Modeling and GraphicsGMAG’2003. 16 – 18 July 2003. London, England, UK: IEEE. 30 -35.
Shahrul Nizam Yaakob and Puteh Saad. (2007). Generalization Performance
Analysis between Fuzzy ARTMAP and Gaussian ARTMAP Neural
Network. Malaysian Journal of Computer Science. Vol 2: 13-22.
Teague, M. R (1980). Image Analysis Via The General Theory Of Moments. Journal
of the Optical Society of America. Vol. 70: 920-930.
Uhrig, R. E. (1995). Introduction to Artificial Neural Networks. Proceedings of the
1995 IEEE IECON 21st International Conference on Industrial Electronics,
Control and Intrumentation, 1995. 6-10 November 1995. Orlando FL. Vol 1:
33-37.
Wang, D. et al.(2002). Protein sequences classification using radial basis function
(RBF) neural networks. Proceedings of the 9th International Conference on
Neural Information Processing, 2002. 18-22 November 2002. Vol 2: 764768.
Wang, G. and Wang, S. (2006). Recursive computation of Tchebichef moment and
its inverse transform. The Journal of the Pattern Recognition Society. Vol 39
(1): 47-56.
Wang, W. (2008). Face Recognition Based On Radial Basis Function Neural
Networks. International Seminar on Future Information Technology and
Management Engineering, 2008. 20 November 2008. Leicestershire, UK. 4144.
74
Wang, X.F.et al. (2005). Recognition of Leaf Images Based on Shape Features
Using a Hypersphere Classifier. International Conference on Intelligent
Computing, 2005. 23-26 August 2005. Hefei, China. Vol 3644: 87–96.
Warren, D. (1997). Automated leaf shape description for variety testing in
chrysanthemums. Proceedings of IEEE 6th International Conference on
Image Processing and Its Applications, 1997. 14-17 July 1997. Dublin,
Ireland. Vol 2: 497-501.
Warwick, K. And Craddock, R. (1996). An introduction to Radial Basis Functions
for system identification: A comparison with other neural network methods.
Proceedings of the 35th IEEE on Decision and Control, 1996, 11-13
December 1996. Kobe, Japan. Vol 1: 464-496.
Wu, S. G. et al. (2007). A Leaf Recognition Algorithm for Plant Classification Using
Probabilistic Neural Network. 2007 IEEE International Symposium on Signal
Processing and Information Technology. 15-18 December 2007. Giza. 11-16.
Yap, P. T. et al.(2003). Image Analysis by Krawtchouk Moments. IEEE
Transactions on Image Processing. November 2003. Vol 12 (11): 13671377.
Yu, P. F. and Xu D. (2008). Palmprint Recognition Based On Modified DCT
Features and RBF Neural Network. 2008 International Conference on
Machine Learning and Cybernetics. 12-15 July 2008. Kunming, China.
2982-2986.
APPENDIX A
PROJECT 1 GANTT CHART
76
APPENDIX B
PROJECT 2 GANTT CHART
78
APPENDIX C
ORIGINAL LEAF IMAGES
80
Rambutan Leaf
Pulasan Leaf
Jackfruit Leaf
Cempedak Leaf
Apple Mango Leaf
Malgoa Mango Leaf
Water Apple Leaf
Guava Leaf
Mata Kucing Leaf
Kuini Leaf
APPENDIX D
BINARY LEAF IMAGES
82
A1
A2
B1
B2
C1
C2
D1
D2
A3
C3
APPENDIX E
IMAGE REFERENCES
84
Class
A
B
C
D
Family
Sapindaceae
Moraceae
Anacardiaceae
Myrtaceae
Image Reference
Image Name
A1
Rambutan Leaf
A2
Pulasan Leaf
A3
Mata Kucing Leaf
B1
Jackfruit Leaf
B2
Cempedak Leaf
C1
Apple Mango Leaf
C2
Malgoa Mango Leaf
C3
Kuini Leaf
D1
Water Apple Leaf
D2
Guava Leaf
APPENDIX F
VALUE OF FEATURE VECTORS BY ZMI
86
Image
A1
Variations
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
Z1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Z2
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Z3
0.635635
0.607506
0.655656
0.565031
0.562489
0.620557
0.616969
0.603451
0.602521
0.605920
0.653990
0.583170
0.638983
Z4
0.012314
0.000183
0.007104
0.012254
0.012184
0.012173
0.011781
0.010333
0.005147
0.000117
0.006534
0.010694
0.000455
Z5
0.028653
0.111753
0.081395
0.002817
0.002078
0.019799
0.018199
0.013358
0.004406
0.110258
0.080247
0.000019
0.016148
Z6
0.001663
0.007546
0.004978
0.000183
0.000120
0.001136
0.001035
0.000735
0.000278
0.007491
0.005080
0.000001
0.000981
A2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.647436
0.590291
0.649283
0.572045
0.516953
0.636369
0.633819
0.620990
0.826289
0.594622
0.651233
0.594460
0.615908
0.010966
0.000017
0.005250
0.010113
0.014088
0.011413
0.010983
0.009074
0.000732
0.000015
0.005474
0.007752
0.027219
0.054974
0.101750
0.099790
0.013415
0.010361
0.042978
0.040791
0.030428
0.000325
0.104842
0.094883
0.003343
0.002744
0.003283
0.006743
0.006028
0.000904
0.000648
0.002470
0.002419
0.001915
0.000025
0.007122
0.006144
0.000204
0.007236
A3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.625028
0.573056
0.628142
0.580341
0.521963
0.615167
0.612193
0.598542
0.751236
0.576910
0.629263
0.557239
0.647095
0.007624
0.000194
0.003012
0.009788
0.013570
0.007773
0.007149
0.005945
0.001266
0.000146
0.002979
0.006119
0.036839
0.057360
0.085407
0.093564
0.021852
0.016480
0.046233
0.043328
0.029321
0.006937
0.087829
0.087343
0.006019
0.002510
0.003390
0.005805
0.005875
0.001363
0.001024
0.002837
0.002843
0.001868
0.000430
0.006137
0.006015
0.000375
0.008642
B1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.618260
0.619330
0.652744
0.533810
0.494805
0.012837
0.599736
0.585839
0.765629
0.622332
0.012979
0.000658
0.008816
0.012285
0.016355
0.012837
0.012480
0.011005
0.000146
0.000767
0.019577
0.115626
0.067368
0.002914
0.002166
0.013243
0.011917
0.007136
0.014500
0.115880
0.001217
0.007603
0.004015
0.000185
0.000135
0.000769
0.000665
0.000382
0.000903
0.007766
87
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.650031
0.556970
0.510344
0.008871
0.005008
0.013897
0.059688
0.000904
0.000703
0.003687
0.000057
0.001934
B2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.685912
0.623475
0.688150
0.583518
0.500996
0.670811
0.657723
0.655865
0.936719
0.624143
0.683670
0.703748
0.559116
0.018290
0.000529
0.010350
0.017505
0.012614
0.017959
0.017087
0.012661
0.001239
0.000489
0.009746
0.002775
0.016586
0.043595
0.134621
0.107836
0.011338
0.001116
0.033813
0.026769
0.017675
0.037231
0.135200
0.101401
0.013738
0.004464
0.002760
0.008872
0.006564
0.000712
0.000070
0.002058
0.001532
0.001107
0.002341
0.009094
0.006339
0.000844
0.003720
C1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.669314
0.554841
0.633123
0.588043
0.533404
0.666942
0.666632
0.685743
1.510689
0.572809
0.643593
1.052552
0.694531
0.007497
0.000935
0.001800
0.013125
0.018212
0.008569
0.007727
0.003876
0.023263
0.000350
0.002181
0.004485
0.033740
0.145911
0.079309
0.141431
0.091401
0.059018
0.132049
0.126142
0.083279
0.461573
0.094866
0.139222
0.046963
0.077335
0.008900
0.005273
0.008692
0.005715
0.003730
0.008091
0.008620
0.005215
0.034424
0.006597
0.009810
0.002950
0.027662
C2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.655626
0.572520
0.641911
0.570523
0.529225
0.650464
0.647310
0.782935
1.265528
0.581017
0.642852
0.909395
0.647947
0.009054
0.000280
0.003344
0.014038
0.019317
0.009811
0.008858
0.004514
0.028049
0.000127
0.002720
0.000091
0.031518
0.097518
0.091758
0.124971
0.057531
0.039070
0.084150
0.078169
0.050164
0.259619
0.098107
0.120651
0.005575
0.032128
0.005797
0.006204
0.007734
0.003578
0.002424
0.005201
0.005390
0.003140
0.017580
0.006888
0.008612
0.000371
0.017228
C3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.711298
0.573631
0.668573
0.620065
0.571247
0.701716
0.702671
0.013738
0.000448
0.004741
0.017540
0.023564
0.014753
0.014321
0.146580
0.100282
0.159268
0.073590
0.057664
0.117078
0.113331
0.009227
0.006748
0.009695
0.004604
0.003578
0.007043
0.007146
88
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.690453
1.195519
0.590746
0.678429
0.891044
0.654210
0.007819
0.000131
0.000153
0.005589
0.003707
0.030513
0.066603
0.075281
0.114679
0.155321
0.000017
0.028750
0.004173
0.004930
0.007990
0.010540
0.000002
0.018265
D1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.597423
0.607555
0.639680
0.508845
0.466781
0.584299
0.580846
0.568454
0.756281
0.609393
0.638012
0.538109
0.528630
0.010917
0.000305
0.007552
0.008529
0.012113
0.010785
0.010347
0.009020
0.000033
0.000321
0.007004
0.003593
0.015728
0.016519
0.105456
0.061046
0.001851
0.001184
0.011663
0.010365
0.006111
0.011719
0.106475
0.058695
0.000781
0.001012
0.000933
0.007034
0.003625
0.000152
0.000075
0.000634
0.000588
0.000331
0.000750
0.007260
0.003738
0.000050
0.002342
D2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.618918
0.590722
0.636616
0.557030
0.502838
0.613991
0.601413
0.587112
0.679978
0.586863
0.633525
0.524397
0.581507
0.009511
0.000013
0.004948
0.010029
0.010602
0.009253
0.008843
0.007438
0.002054
0.000024
0.004651
0.007554
0.023563
0.034097
0.096478
0.079355
0.007239
0.002301
0.031428
0.024101
0.016726
0.003488
0.094753
0.073311
0.002006
0.001316
0.001937
0.006498
0.004858
0.000484
0.000144
0.001829
0.001450
0.000985
0.000219
0.006540
0.004801
0.000099
0.003582
APPENDIX G
VALUE OF FEATURE VECTORS BY LMI
90
Image
A1
Variations
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
L1
0.538080
1.514827
0.865814
0.319003
0.308754
0.490340
0.490102
0.495826
0.538763
1.524409
0.877148
0.318934
0.515445
L2
0.594291
1.105704
0.799418
0.399411
0.388183
0.555403
0.553812
0.548580
0.551787
1.111666
0.805962
0.385297
0.504803
L3
0.139806
1.095573
0.471333
-0.042627
-0.050289
0.098727
0.100859
0.121703
0.207824
1.098006
0.481336
-0.043277
0.188827
L4
-0.351960
0.079916
-0.251064
-0.322703
-0.318163
-0.350699
-0.347579
-0.327088
-0.260746
0.082641
-0.246620
-0.301491
-0.086752
L5
0.216601
1.938061
0.673498
0.031775
0.026174
0.166669
0.166800
0.174141
0.202880
1.959498
0.692444
0.023601
0.076944
L6
-0.304855
-0.958870
-0.543087
-0.113220
-0.103035
-0.263006
-0.260992
-0.252323
-0.222183
-0.969801
-0.551981
-0.086263
-0.058644
A2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.641242
1.621959
0.982281
0.391887
0.323230
0.582353
0.580809
0.587310
0.997316
1.595338
0.940090
0.399761
0.343985
0.663444
1.150270
0.861957
0.456608
0.448915
0.618990
0.616649
0.615573
0.846259
1.140290
0.838069
0.443949
0.427602
0.245424
1.185174
0.584371
0.043186
-0.070132
0.194324
0.194793
0.209960
0.845121
1.158995
0.545210
0.065700
0.375679
-0.324644
0.141612
-0.200499
-0.305964
-0.391591
-0.328875
-0.326461
-0.313460
-0.012744
0.124868
-0.216024
-0.270077
0.523508
0.343481
2.180189
0.872057
0.083993
0.054307
0.270389
0.268917
0.278254
0.649083
2.119591
0.799272
0.073476
-0.009594
-0.381594
-1.027026
-0.624273
-0.163025
-0.209600
-0.330129
-0.327222
-0.324348
-0.311881
-1.013019
-0.592765
-0.125056
0.009600
A3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.710385
1.696934
1.070523
0.437983
0.355450
0.652452
0.653911
0.657226
1.002445
1.671702
1.026783
0.435665
0.460183
0.693649
1.180278
0.900923
0.493676
0.480540
0.653449
0.653850
0.656654
0.824401
1.170844
0.877155
0.478773
0.519959
0.333124
1.240778
0.672156
0.087286
-0.045886
0.281171
0.283023
0.284358
0.825405
1.216461
0.631356
0.104089
0.480278
-0.275749
0.186629
-0.148214
-0.303191
-0.395839
-0.284659
-0.282811
-0.282382
-0.047144
0.170850
-0.165253
-0.275640
0.613552
0.443452
2.355384
1.035120
0.124118
0.079417
0.365078
0.367593
0.372808
0.750378
2.296790
0.956092
0.116858
0.023953
-0.413594
-1.075538
-0.675992
-0.198224
-0.245400
-0.366782
-0.367489
-0.371830
-0.382663
-1.062063
-0.644412
-0.171993
0.070848
B1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.452960
1.402378
0.757462
0.281017
0.244940
0.415108
0.414830
0.418391
0.736521
1.375908
0.527403
1.059440
0.738687
0.367341
0.373929
0.494459
0.491796
0.485957
0.685270
1.047910
0.061426
0.988446
0.358124
-0.068306
-0.136676
0.030102
0.033249
0.050036
0.454090
0.963447
-0.355419
0.014213
-0.294064
-0.317039
-0.386011
-0.350864
-0.346189
-0.328028
-0.086461
-0.000362
0.130443
1.693646
0.505492
0.018088
0.012478
0.097871
0.097724
0.101525
0.273712
1.637588
-0.235014
-0.893288
-0.470651
-0.095205
-0.140658
-0.202114
-0.198697
-0.190326
-0.165317
-0.876976
91
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.725999
0.299276
0.211900
0.715717
0.337971
0.258073
0.333166
-0.020784
0.196729
-0.297529
-0.221221
0.271152
0.459805
0.009798
-0.031042
-0.441089
-0.035239
-0.064955
B2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.511409
1.486282
0.835196
0.335085
0.248542
0.479601
0.466808
0.499334
0.912355
1.486694
0.832680
0.426649
0.210707
0.589186
1.097286
0.794631
0.443222
0.345937
0.561462
0.546931
0.552036
0.870470
1.098201
0.789613
0.446168
0.325621
0.085868
1.071795
0.425712
-0.064307
-0.099740
0.059902
0.054812
0.116189
0.510074
1.069958
0.428680
-0.015119
0.190891
-0.394231
0.057637
-0.288385
-0.384797
-0.326578
-0.389758
-0.380710
-0.336225
-0.135355
0.056798
-0.282611
-0.208795
0.344740
0.179605
1.872622
0.618756
0.047764
0.009249
0.148656
0.137399
0.162649
0.359077
1.873578
0.615609
0.022181
-0.011732
-0.300754
-0.945718
-0.539646
-0.177136
-0.088886
-0.270882
-0.254589
-0.241111
-0.237592
-0.947839
-0.531808
-0.053438
-0.071131
C1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.936573
1.873779
1.290898
0.606622
0.498579
0.849270
0.847819
1.000277
3.550078
1.763706
1.209821
1.380908
0.324841
0.841079
1.248181
1.012691
0.692347
0.703543
0.788984
0.786311
0.839586
3.095672
1.206235
0.973809
1.112799
0.795302
0.533608
1.398479
0.875217
0.134396
-0.059848
0.452392
0.453707
0.690067
4.445548
1.305510
0.801543
1.400588
0.653622
-0.226067
0.294442
-0.049996
-0.402445
-0.549489
-0.253828
-0.250967
-0.135147
1.226387
0.226733
-0.089838
0.147387
1.108176
0.792434
2.777145
1.461507
0.351857
0.256142
0.649316
0.647445
0.856884
3.863485
2.511576
1.298798
1.188087
-0.036508
-0.599251
-1.177828
-0.830371
-0.505404
-0.638221
-0.532236
-0.528147
-0.529951
-0.986985
-1.112816
-0.775307
-0.468722
0.283184
C2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.774654
1.745294
1.125224
0.509095
0.427174
0.701735
0.700480
0.981153
2.486116
1.685112
1.120460
0.867053
0.288652
0.742292
1.199008
0.931779
0.619776
0.633813
0.694009
0.690633
0.823057
2.227663
1.176273
0.927996
0.815427
0.568322
0.384206
1.288755
0.723881
0.057188
-0.097394
0.317519
0.320087
0.760502
2.767775
1.232305
0.719941
0.530764
0.519491
-0.274813
0.215732
-0.129478
-0.416527
-0.542907
-0.290717
-0.285923
-0.097272
1.014159
0.178421
-0.129085
-0.007002
0.808615
0.533642
2.468099
1.135077
0.230375
0.167882
0.428406
0.427392
0.737626
2.022915
2.327123
1.126875
0.306329
-0.046925
-0.473224
-1.102047
-0.716920
-0.413384
-0.525175
-0.414631
-0.409874
-0.413811
-0.324192
-1.068917
-0.711801
-0.151412
0.060713
C3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
0.793129
1.815422
1.137358
0.510357
0.426197
0.705515
0.705646
0.778963
1.223073
0.948176
0.612226
0.621426
0.718289
0.711596
0.369529
1.371503
0.729548
0.065434
-0.091969
0.291611
0.303632
-0.321013
0.261876
-0.142885
-0.416633
-0.541428
-0.336927
-0.325013
0.551079
2.633198
1.152563
0.216075
0.156279
0.423857
0.423866
-0.523954
-1.131643
-0.739900
-0.382264
-0.487894
-0.447076
-0.434689
92
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.801483
1.814527
1.701168
1.065092
0.789385
0.303336
0.744207
1.547634
1.179138
0.906119
0.743450
0.582507
0.451125
1.963902
1.270779
0.673900
0.465017
0.456552
-0.242910
0.411315
0.192082
-0.167732
-0.122244
0.770150
0.542442
1.535573
2.361925
1.016711
0.306124
-0.022855
-0.434871
-0.480946
-1.065096
-0.677757
-0.199946
0.082605
D1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.462384
1.454409
0.773719
0.277108
0.237829
0.426375
0.425809
0.431118
0.756653
1.444752
0.774921
0.309177
0.247730
0.524039
1.078568
0.739196
0.349027
0.349728
0.492065
0.491350
0.497308
0.692914
1.074448
0.739125
0.342581
0.287295
0.089226
1.039155
0.388167
-0.042433
-0.110139
0.060908
0.061146
0.064943
0.502644
1.030208
0.390073
0.010409
0.231321
-0.329853
0.048444
-0.270858
-0.277674
-0.340914
-0.324074
-0.323122
-0.324612
-0.065113
0.043034
-0.268709
-0.204500
0.301949
0.142420
1.806840
0.532120
0.015619
0.010024
0.110122
0.109900
0.115398
0.303784
1.785936
0.534449
0.014675
-0.027415
-0.229253
-0.919893
-0.467686
-0.072552
-0.107716
-0.197112
-0.196508
-0.203440
-0.170717
-0.913977
-0.467648
-0.037697
-0.047046
D2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.593471
1.577574
0.935178
0.357230
0.260517
0.583051
0.546508
0.550972
0.763964
1.618075
0.913433
0.358940
0.359179
0.618877
1.131084
0.830497
0.428001
0.344719
0.609232
0.581052
0.582520
0.679817
1.148286
0.816426
0.427008
0.389837
0.217383
1.143843
0.545832
0.010372
-0.069830
0.211022
0.179347
0.186299
0.509804
1.177610
0.526837
0.020556
0.333449
-0.309887
0.117555
-0.208733
-0.306420
-0.298906
-0.306784
-0.307262
-0.301797
-0.145048
0.140229
-0.213275
-0.296910
0.402461
0.286380
2.080326
0.792903
0.058308
0.011955
0.274381
0.232772
0.239336
0.432754
2.172334
0.757125
0.061445
-0.003010
-0.327988
-0.999084
-0.581945
-0.137667
-0.078884
-0.316781
-0.286531
-0.288474
-0.281994
-1.025435
-0.563685
-0.137226
0.003798
APPENDIX H
VALUE OF FEATURE VECTORS BY TMI
94
Image
A1
Variations
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
T1
0.793469
0.577823
0.680862
0.963832
0.972662
0.820431
0.821020
0.819524
0.726457
0.576871
0.678458
0.885322
0.490751
T2
2.134144
2.209943
2.182293
2.062425
2.029389
2.132314
2.136632
2.155781
1.724457
2.210815
2.185880
1.584306
0.406124
T3
-0.024839
0.005101
-0.001821
-0.006109
-0.003557
-0.023621
-0.020143
0.012155
0.000462
0.001488
-0.004009
-0.000679
-0.000777
T4
6.847867
7.181068
6.992699
6.747295
6.654909
6.813371
6.797773
6.740852
4.548186
7.182417
6.988950
4.793722
0.130346
T5
-0.027656
0.016385
-0.008342
-0.014567
-0.006881
-0.030528
-0.029908
-0.032643
0.000786
0.019021
0.003699
-0.000932
0.004355
T6
-14.207045
-10.369094
-12.224204
-17.224380
-17.453477
-14.698226
-14.720452
-14.681998
-13.020708
-10.349965
-12.181573
-15.871980
-8.807629
A2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.749864
0.567120
0.656077
0.898776
0.000648
0.775057
0.776145
0.774807
0.431321
0.569819
0.665147
0.795220
0.408835
2.167696
2.214620
2.197390
2.194924
3.642581
2.166691
2.170114
2.185398
0.737205
2.214718
2.197781
1.598761
-0.033540
-0.016849
0.001390
-0.004034
0.010314
0.002172
-0.009863
-0.007598
0.004487
0.000753
0.000063
-0.002807
0.002944
0.177336
6.959532
7.208514
7.057294
6.903477
14.336623
6.940084
6.927001
6.853955
1.200169
7.203467
7.040257
4.345430
-0.652676
-0.025186
0.011024
-0.011858
0.001782
-0.000075
-0.032964
-0.023428
0.001331
-0.000856
0.017341
0.008001
-0.003756
0.219091
-13.411918
-10.177834
-11.775160
-16.007047
-21.362187
-13.896431
-13.948427
-13.883328
-7.735926
-10.229504
-11.961397
-14.253060
-7.454383
A3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.730454
0.560703
0.641769
0.865057
1.152630
0.752177
0.752200
0.751746
0.489870
0.563152
0.650431
0.831533
0.377131
2.221299
2.222182
2.222302
2.226552
3.679278
2.228241
2.233496
2.240824
1.080957
2.222741
2.226202
2.011977
0.019833
7.003506
7.223472
7.087973
6.979962
14.429203
6.978279
6.960121
6.877725
2.243302
7.218339
7.066167
5.679544
-0.655689
-0.023667
0.015147
-0.002760
-0.014568
-0.002075
-0.017117
0.001802
0.004178
-0.000629
0.021809
0.025144
-0.000340
0.200162
-13.095347
-10.064492
-11.529398
-15.416269
-20.652120
-13.515781
-13.534255
-13.463160
-8.782565
-10.109498
-11.697170
-14.901885
-6.841299
B1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.843832
0.590748
0.710340
1.068707
1.394240
0.871882
0.872222
0.870748
0.449753
0.594059
2.121805
2.208691
2.177419
2.370866
3.941221
2.118097
2.119310
2.133680
0.554760
2.208822
6.851434
7.158125
6.967336
8.183490
16.742971
6.829875
6.807263
6.704992
0.643179
7.151002
-0.019132
0.012301
-0.015661
0.002097
-0.000556
-0.033489
-0.031848
-0.038182
0.001410
0.017661
-15.085694
-10.603928
-12.743150
-19.156328
-24.983674
-15.596182
-15.636693
-15.594915
-8.072425
-10.664826
0.000907
-0.003658
-0.004163
0.005823
0.000319
0.003393
0.001873
-0.005054
-0.001261
-0.004839
-0.004372
-0.001774
0.148518
-0.035599
0.002546
-0.014127
0.001252
-0.000722
-0.039107
-0.027183
0.012795
0.002562
0.002141
95
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.720227
0.802760
0.501641
2.176515
1.075105
-0.079296
-0.005841
-0.000909
0.156120
6.942665
2.569161
-0.714813
-0.004151
0.001038
0.183917
-12.944421
-14.392584
-9.146039
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.799002
0.579796
0.684512
1.046032
1.206281
0.818481
0.827528
0.774177
0.383848
0.579773
0.685737
0.557531
0.455345
2.045261
2.199050
2.144899
2.672342
2.823280
2.046086
2.052653
1.855220
0.392276
2.199265
2.149829
0.394589
-0.223602
-0.037482
0.006559
-0.011429
0.002382
-0.003908
-0.038335
-0.028385
-0.000363
-0.002249
0.005240
0.001887
0.205715
6.940548
7.184007
7.023082
10.035240
10.410801
6.929253
6.894524
5.729676
0.487362
7.183397
7.010150
0.725190
-0.576310
-0.002248
0.011944
-0.007488
0.002022
0.000490
-0.016287
-0.031576
0.001204
-0.002235
0.017402
0.000650
0.001268
0.207436
-14.278451
-10.404269
-12.272005
-18.735330
-21.618214
-14.625148
-14.832108
-13.875641
-6.884242
-10.405364
-12.318851
-10.001852
-8.319920
C1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.665351
0.544853
0.605465
0.928934
1.267279
0.686780
0.687483
0.594493
0.176744
0.554127
0.617234
0.362833
0.340578
2.192707
2.218342
2.210326
3.424438
5.825161
2.197433
2.200332
1.768180
0.153160
2.217660
2.210335
0.620480
-0.131537
-0.011157
-0.001245
-0.004804
-0.000496
0.019236
-0.010405
-0.007650
-0.001217
-0.010445
-0.000558
-0.003698
-0.000221
0.308490
7.155110
7.264560
7.185085
13.487342
28.822324
7.156944
7.133893
5.044380
-0.048556
7.246283
7.156539
1.148706
-0.472099
-0.014760
0.008976
-0.006148
0.000557
0.006830
-0.012354
0.024054
0.001429
-0.016169
0.017948
0.031881
-0.000078
0.252814
-11.913172
-9.778989
-10.870059
-16.643144
-22.705326
-12.347338
-12.411635
-10.655211
-3.172981
-9.951942
-11.117378
-6.509607
-6.445462
C2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.707256
0.555850
0.630748
0.998435
1.321995
0.730794
0.731882
0.514358
0.207467
0.561519
0.632041
0.374146
0.377153
2.196118
2.218388
2.209803
3.535786
5.709870
2.197937
2.203784
1.204551
0.159280
2.219475
2.214080
0.327310
-0.105380
-0.005972
-0.000896
-0.002267
-0.000878
-0.016961
-0.002271
0.001896
-0.000234
0.006873
-0.002841
-0.002320
0.003072
0.274054
7.088638
7.239572
7.135169
13.976765
27.828795
7.080160
7.053351
2.885169
-0.116029
7.229597
7.119797
0.277814
-0.552529
-0.022300
0.013048
-0.005982
-0.002157
-0.011147
-0.013926
0.018742
0.001036
-0.002136
0.021954
0.036042
-0.000346
0.239750
-12.677429
-9.980157
-11.332166
-17.889673
-23.702125
-13.149561
-13.200801
-9.222314
-3.726902
-10.086685
-11.379373
-6.711868
-7.081538
C3
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
0.694059
0.548776
0.625510
0.938571
1.258163
0.721293
0.721111
2.126780
2.209738
2.181496
3.053502
5.088730
2.124420
2.123195
-0.027649
0.008600
-0.003683
-0.000456
0.003949
-0.019927
0.001328
7.107538
7.253217
7.142840
11.779724
24.435040
7.096893
7.078833
-0.005755
0.009741
-0.005905
0.000299
-0.008239
-0.011887
0.006611
-12.402844
-9.847257
-11.224167
-16.816051
-22.542348
-12.938956
-12.994534
B2
-0.005793
96
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.648819
0.271554
0.558889
0.637642
0.432236
0.372008
1.829286
0.315010
2.209211
2.177112
0.519355
-0.144831
-0.001321
0.002564
0.008861
0.009565
-0.001410
0.265457
5.414568
0.274447
7.234128
7.119269
0.858648
-0.550914
-0.000427
-0.005262
0.017212
0.025312
-0.002643
0.253757
-11.631053
-4.871015
-10.036747
-11.482221
-7.750722
-6.909521
D1
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.841859
0.585193
0.707438
1.031587
1.351383
0.867506
0.868345
0.865805
0.447490
0.586327
0.707506
0.796875
0.482647
2.166389
2.214715
2.195263
2.179202
3.641897
2.164362
2.168466
2.183545
0.574775
2.214737
2.199029
1.148934
-0.032737
-0.007769
0.004287
0.001802
-0.009723
-0.001355
-0.000377
-0.001622
-0.017652
0.000566
0.004218
0.001597
0.002114
0.149357
6.827104
7.167180
6.961667
6.789709
14.210130
6.802024
6.777564
6.687670
0.649595
7.164436
6.950152
2.522131
-0.710183
-0.038828
0.013137
-0.019354
0.011889
0.000056
-0.043100
-0.036416
-0.034847
-0.000115
0.018424
-0.000827
-0.000253
0.184461
-15.047584
-10.503972
-12.700236
-18.376423
-24.214423
-15.538571
-15.577672
-15.491890
-8.027032
-10.525480
-12.713082
-14.290583
-8.781217
D2
Original
0.5x
0.75x
1.2x
1.4x
10°
20°
45°
90°
0.5x + 10°
0.75x + 20°
1.2x + 45°
1.4x + 90°
0.772744
0.572063
0.667596
0.929796
1.123764
0.778073
0.796780
0.796304
0.565694
0.567959
0.673175
0.932608
0.427456
2.192073
2.218489
2.209752
2.175027
2.525901
2.197409
2.206170
2.222749
1.237013
2.218720
2.215445
2.224621
0.027077
6.895987
7.195347
7.024362
6.824214
8.635690
6.887288
6.856858
6.792501
2.689540
7.203446
7.009061
6.645534
-0.685293
-0.033461
0.014160
-0.010542
-0.009096
0.002741
-0.029759
-0.022382
-0.018685
0.000357
0.019762
0.007960
-0.044255
0.176789
-13.842373
-10.267235
-11.989031
-16.561351
-20.137434
-13.963968
-14.308606
-14.267214
-10.144975
-10.193018
-12.095483
-16.693879
-7.747035
0.004430
0.000479
0.000639
0.001824
0.001096
0.009058
0.008891
0.005704
0.001452
-0.000706
0.001546
-0.008552
0.131219