C O P U L A B A S E D R IG ID -B O D Y IM A G E R E G IS T R A T IO N

by

F aram arz K ashanchi

M.Sc. Computer Science
University of Mysore, Mysore, India.

THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MATHEMATICAL, COMPUTER, AND PHYSICAL SCIENCES
(MATHEMATICS)

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
2012

(c) Faramarz Kashanchi, 2012

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A bstract
The literature presents a wide number of algorithms in the field of image registration.
However, analysis of the literature revealed that much emphasis has not been placed
on copula based image registration. Thus this thesis seeks to explain the image regis­
tration problem and how it may be solved using copula based measures. Here we are
aiming to combine the MATLAB fminsearch optimization method with copula based
alignment measure, in order to monitor the performance of copula, based alignment
measure in image registration. Performance of four copula functions namely, Clayton,
Frank, Gaussian and Marshal-Olkin are tested in image registration algorithm. A
comparison is then posited of the performance of the four copula functions in image
registration. These four copula measures are then compared with the well known
method of image registration alignment measure, th a t is the joint histogram based
mutual information. The accuracy and speed of the image registration algorithm
was monitored on aerial and medical (MRI) images. Since we are using rigid-body
transformations, the image registration algorithm is categorized as rigid body image
registration.

C on ten ts

A b stra ct

i

L ist o f F igures

iv

L ist o f T ables

v ii

A ck n ow led gem en ts

v iii

1

2

In tro d u ctio n

1

1.1 Image Registration Overview.........................................................................

2

1.2

Copulas in Image P ro c e s s in g .......................................................................

5

1.3

Literature Review of Image Registration

................................................

6

1.4

Map of T h e s i s ................................................................................................

7

S ta tem en t o f T h e R esearch an d M a th em a tic a l C oncep ts

8

2.1 Statement of R e s e a rc h ...................................................................................

8

2.1.1

Image Registration A lg o rith m .........................................................

9

2.2 Rigid-body Transform ation.................................................................................10
2.3 Calculating the Alignment M e a su re .................................................................12
2.3.1

Joint Histogram Based M utual Information

ii

....................................13

2.3.2 Copula Based Alignment M easu re........................................................16
2.3.3 Examples of Copula Based Alignment M e a s u r e ...............................30
2.4

3

O p tim iz a tio n .....................................................................................................35

E xp erim en ts

3.1

39

ExperimentalIm a g e s ......................................................................................... 39
3.1.1 Magnetic Resonance Imaging (MRI)

................................................. 40

3.1.2 Which Copula ? ....................................................................................... 43
3.2

Results of ImageRegistration Experiments

...................................................43

3.2.1 Joint Histogram Mutual In fo rm a tio n ................................................. 46
3.2.2 Clayton C o p u la ....................................................................................... 48
3.2.3 Frank C o p u la ...........................................................................................50
3.2.4 Marshal-Olkin C o p u l a ...........................................................................51
3.2.5 Gaussian C o p u l a .................................................................................... 52
3.3

4

Summary of Experim ents..................................................................................54

Sum m ary and C o n clu d in g R em a rk s

60

B ib liograp h y

63

A p p en d ix A Softw are L icen ce and V ersion s

67

A .l

M A T L A B ........................................................................................................... 67

A.2

M a p l e ................................................................................................................. 67

A p p en d ix B D escrip tio n o f Im a g es in E x p erim en ts

68

B .l

M R I .....................................................................................................................68

B.2

A e r i a l ..................................................................................................................68

iii

List o f F igures
1.1 Flowchart of image registration stag e s..........................................................

4

2.1 Flowchart of image registration stages in this t h e s i s ................................

9

2.2 Right: matrix of x coordinate (xcoords) Left: m atrix of y coordinate
(ycoords)

.............................................................................................................10

2.3 (Left: Image A Right: Image B ) Two gray scale images of size 3x3 and
their corresponding intensity matrices below t h e m .......................................14
2.4 Clayton copula cumulative distribution fu n c tio n ...........................................20
2.5 Clayton copula Probability density curves for 8 = 0.1 (firstrow left),
9 = 0.4 (first row right) and 9 = 0.9 (second row) ....................................... 21
2.6 Frank copula cumulative distribution f u n c tio n .............................................. 22
2.7 Frank copula Probability density curves for 6 = 0.1 (first row left),
9 = 0.4 (first row right) and 9 = 0.9 (second row)

.......................................24

2.8 Marshal-Olkin copula cumulative distribution f u n c tio n .............................. 25
2.9 Marshal-Olkin copula Probability density curves for 9 = 0.1 (first row
left), 8 = 0.4 (first row right) and 8 = 0.9 (second r o w ) ............................. 27
2.10 Gaussian copula cumulative distribution f u n c t i o n ....................................... 28
2.11 Gaussian copula Probability density curves for p = 0.1 (first row left),
p = 0.4 (first row right) and p = 0.9 (second r o w ) .......................................30

iv

2.12 Divergence measures and mutual information calculated using four cop­
ulas

......................................................................................................................32

2.13 Left: Column matrix of Image A Right Column matrix of Image B

. . 34

2.14 Left: Reference Image Right: Test Image bottom: Registered test im­
age using the fminsearch and joint histogram based mutual information

37

2.15 Plot of the convergence of the fminsearch in 25 ite ra tio n s .......................... 38
3.1 P I to P8 are the reference images and T P 1 to T p8 are the testing images 41
3.2 A l to A8 are the reference images and TA l to TA8 are the testing images 42
3.3

First aerial image which mis-registered using the joint histogram mu­
tual information method. (First row left: the reference image, first row
right: the test image, second row left: the registered test image, second
row right: the overlap of registered test image on the reference image
which is mis-registered)...................................................................................... 48

3.4

First aerial image which registered using the Gaussian copula method.
(first row left: the reference image, first row right: the test image,
second row left: the registered test image, second row right: the overlap
of registered test image on the reference image)

.......................................... 53

3.5 PSNR values for the 8 MRI im a g e s ......................................................

54

3.6 PSNR values for the 8 Aerial i m a g e s ..............................................................55
3.7

The optimization result of the Gaussian Copula for the 4th aerial image
after 87 iterations

3.8

.............................................................................................56

The optimization result of the Gaussian Copula for the 4th MRI image
after 73 iterations

.............................................................................................56

3.9 Registered image of the 4th aerial image.(first row left: the reference
image, first row right: the test image, second row left: the registered
test image, second row right: the overlap of registered test image on
the reference image)

..........................................................................................57

3.10 Registered image of the 4th MRI image, (first row left: the reference
image with ROI, first row right: the test image with ROI, second row
left: the registered test image with ROI, second row right: the overlap
of registered test image on the reference image)

vi

..........................................58

List o f Tables
2.1 Joint Histogram of Image A and Image B

.................................................... 15

2.2 Values of the divergence measures and mutual information in Figure 2.4 33
3.1 Image registration results for joint histogram mutual information . . . .
3.2 Image registration results for Clayton Copula

46

.............................................49

3.3 Image registration results for Frank C o p u la................................................... 50
3.4 Image registration results for Marshal-Olkin C o p u la ................................... 51
3.5 Image registration results Gaussian C o p u la .................................................... 52

vii

A ck n ow ledgem ents
I am thankful to all my friends who encouraged and supported me whenever and
wherever. This thesis taught me just like life, research also has ups and downs and
my friends were with me in both places.
I would like to thank Dr.Edward Dobrowolski who mentored me about the m ath­
ematical ideas and Maple software operation.
Love and thanks to my family for their unconditional support. My family’s kind­
ness and love is the reason of my being.
Most importantly I thank my supervisor Dr.Pranesh Kumar who showed me the
way of life in the first place. His patience, motivation and great ideas, enlightened
the way of research for me.
Last but not the least I am thankful to the professors of University of Northern
British Columbia and my supervisory committee , Dr.Liang Chen and Dr.Iliya Bluskov
for their guidance and support.

C hapter 1

In trod u ction

In this research we attem pt to examine the information theoretic and stochastic ap­
proach in the context of image alignment. Image alignment, or in other words image
registration, is an ongoing research area and there are various publications on it.
Excellent reviews can be found in [10] and [2]. This thesis can be considered as a
continuation of research [14] on finding a simple mathematical calculation method
for alignment measure in image registration. Image registration is the problem of
finding the best possible geometrical transformation in order to align the moving or
test image based on the fixed or reference image. In this thesis we apply a very pow­
erful statistical function called copula function to find the best possible geometrical
transformation for the tested images. Medical (MRI) and aerial images are used as
the main test cases, or in other words, input to the copula based image registration
approach. In this chapter we are going to talk about various types and applications of
image registration, types of images used in image registration and copula functions.

1

1.1

Im age R e g istr a tio n O verview

Image registration is known as one of the preprocessing and image restoration stages
of digital image processing [11]. Inputs to the image registration algorithms are test
image and reference image which are from a same scene. The test image may be
obtained at different time, view, sensor or a computer generated image. In image
registration we intend to align test image based on the reference image. The differences
between the images of same scene such as geometrical or intensity differences may
prevent the best possible alignment of test image based on reference image. The goal of
image registration is to achieve the perfect alignment between images of the same scene
in spite of the geometrical or intensity differences. In the following paragraphs we will
demonstrate various types of image registration methodologies and the particular
method used in this thesis.
In image registration the differences or variations between the images of the same
scene are of two types. First, the differences that we want to preserve; for example, a
tumor in a MRI image. Second, the differences th a t we want to correct, which may
be due to multi-modal, multi-temporal, multi-view and model-image issues of image
acquisition. These image acquisition issues are explained in the following [2]:
M u lti-m o d al: Modality of an image, depends on the type of sensor which acquires
it. In image registration we may deal with images from the same scene taken with
different types of sensors which is referred to as multi-modal images. Hence due to
intensity differences, the registration of multi-modal images is more difficult compared
to mono-modal images which are acquired using the same sensor.
M u lti-te m p o ra l: Two images of the same scene may be taken at different times and
hence their alignment is a challenging problem in image registration field.
M ulti-view : Images of the same scene obtained from different view points may
2

change the geometrical angle and add noise to the images so image registration algo­
rithms attem pt to align these images.
M odel-im age: Images of a scene and its computer generated models may need to
be aligned in order to obtain their similarities and differences.
The above classification of image distortions are used in various applications such
as remote sensing [3], computer vision[4] and medical imaging[5]. It is very im portant
to decide which one of the above four differences may be corrected using the developed
image registration algorithm. In this thesis we are going to simulate the multi-view
test images by intentional translation and rotation of the reference images.
Image registration methods are divided into two major categories namely featurebased and area-based which are explained in the following [2]:
F eatu re-b ased : Also called landmark-based approach. Here the alignment of the
reference image and test image is obtained by aligning the chosen landmarks. These
landmarks could be surface, points or lines. Feature-based image registration algo­
rithms are faster than the area-base image registration algorithms. Also the mentioned
drawback of feature-based methods is that their accuracy depends on the accuracy of
feature identification and selection.
A rea-based: Also called content-base approach. Here the alignment is obtained
based on the intensity values of the images. Here we take the whole image as the
feature and hence it is not dependent on the feature identification and selection. The
drawback of this method is that it is slower than the feature-based method.
There are three steps to solve any image registration problem. These three steps
are alignment measure, transformation and optimization. In Figure 1.1, the diagram
shows steps of image registration algorithms.

3

Reference Image

Alignment Measure

Test Image

Perform Rigid-Body
Transformation

Optimizer (generating
the transforma­
tion parameters)

no

Converged?

yes

Test Image is Aligned

Figure 1.1: Flowchart of image registration stages

A lignm ent M easure: While matching the feature (in our thesis gray scale inten­
sity of pixels) the alignment measure may be calculated to obtain the best alignment
between the reference and test images. There are various alignment measures in the
literature. Some of the popular alignment measures are the joint histogram based
mutual information, divergence measure, sum of squared differences and local corre­
lation [10].
T ran sfo rm atio n : There are two types of geometrical transformations namely para­
metric and non-parametric transformations [2]. The instances of parametric trans­
formation are rigid-body, affine and spline transformations. In non-parametric trans­
4

formation all the pixels can individually change their position. Implementation of
rigid-body transformation based image registration is the target of this thesis. Rigidbody transformation consists of rotation, translation and scaling from which we make
use of rotation and translation in this thesis.
O p tim izatio n : Image registration is also an optimization problem. Optimization is
part of the image registration algorithm and it may find the most optimum trans­
formation of the test image to align to the reference image. Some optimization al­
gorithms are Powell’s method, Downhill simplex method (fminsearch in MATLAB),
gradient-descent and Levenberg-Marquardt method[23].
In summary in this thesis we are going to use the multi-view image, area-base al­
gorithm, mutual information and divergence measure based alignment measure, rigidbody transformation and Downhill simplex optimization.

1.2

C opulas in Im age P r o c e ssin g

As of our knowledge in image processing, copula functions are used in image change
detection algorithms [12]. Copula function is a strong statistical tool used to represent
the joint distribution in terms of the marginal distributions and dependence parame­
ters. As obtaining the joint distribution may be a critical task, copula functions come
to the rescue by generating the joint distribution by the knowledge of the marginal
distributions. Unlike other joint distribution estimation methods, copula based joint
distribution estimation, does not use Gaussian or Gamma distribution parameters.
This is an advantage in image processing since we may not have Gaussian or Gamma
distribution. Mercier uses the copula based random number generation from the two
images in order to obtain a closer distribution for the best possible image change
5

detection between the test and reference images [13].
There are few publications on the use of copula functions in image registration.
The most important publication in the field of copula based image registration is [14].
Durrani and Zeng use the Clayton copula density functions to calculate the divergence
measure as the alignment measure for image registration. There are various copula
families and their performance as an alignment measure in image registration can be
investigated. In this thesis we are going to compare the performance of four copula
functions namely, Clayton, Frank, Gaussian and Marshal- Olkin in image registration.

1.3

L iteratu re R e v ie w o f Im age R eg istr a tio n

There are many publications about image registration, here we give a brief summary.
In 1996 Fredrik Mess investigated the joint histogram based mutual information to
estimate the rigid-body transformation in image registration. Here the performance of
various interpolation and degradation and optimization functions have been analysed
in mutual information based image registration [15].
In 1997 William M.Wells found that mutual information is a good measure for
multi-modal image registration and it performs better than correlation and sum of
squared difference base methods. He found an estimate of the transformation th at
registered the reference volume u and test volume v by maximizing their mutual
information. Here a Parzen window based mutual information method was proposed
[16].
The main inspiration for the idea in this thesis comes from TS Durrani 2001 pa­

6

per [14] where they applied various copula based divergence measures as alignment
measure for image registration. They found th at the copula based divergence mea­
sures works well in area based image registration. They used Clayton copula which
is an Archimedean copula family and they did not investigate other copula families.
In other paper [17], the same author, further studied the various copula based di­
vergence measures for random numbers of normal shape and found th a t the copula
based divergence measures behave differently on different distributions. Hence Dur­
rani stated that for various distributions we observe different behaviour of the copula
based divergence measures. In the area based image registration problem we normally
have images with varied distributions. Therefore we need to find, which copula based
divergence measure will give us the best result in image registration.
In this thesis we are going to investigate the performance of four copula functions
namely, Clayton, Frank, Gaussian and Marshal Olkin, in calculation of copula based
divergence and mutual information measure. This measure is used as the alignment
measure in the area based image registration. A description of the copula concepts
and their families is given in [1].

1.4

M ap o f T h esis

In chapter 2, we are going to define the statem ent of the research and also we will
present the mathematical concepts and algorithm used in this thesis. In chapter 3, we
will present the experiments performed on the medical (MRI) and aerial images and
we will report the pros and cons of our algorithm. In chapter 4, we will summarize
the results of this thesis and provide the future work potential ahead to this research.

7

C hapter 2

S ta tem en t o f T h e R esearch and
M ath em atical C on cep ts

In this chapter we are going to present the statement of research and mathematical
concepts used in this thesis. After we state the research problem, we first demonstrate
the overall algorithm used for image registration and then we describe rigid body
transformation, alignment measures and optimization in detail.

2.1

S ta tem en t o f R esearch

In this thesis we are going to combine the MATLAB fminsearch optimization with
copula based divergence and mutual information measure and joint histogram based
mutual information in order to develop an image registration algorithm. This image

registration algorithm is area based and consist of rigid body transformation. In the
following we will see how the parts of the image registration algorithm are assembled.

2.1.1

Im age R egistration A lgorith m

The steps used in the image registration algorithm in this thesis are as presented in
Figure 2.1.

Reference Image

Alignment Measure

Perform rigid-body
translation and
rotation on reference
image to simulate
the test image

Perform Rigid-Body
Transformation

Optimizer(generating
the transforma­
tion parameters)

no

Converged?

yes

Test Image is Aligned

Figure 2.1: Flowchart of image registration stages in this thesis

In the following sections we will see how the rigid body transformation simulates
the testing images. Further the various alignment measures which are based on diver­
gence measure and mutual information will be explained. In the end we are going to
9

look at the fminsearch optimization algorithm which is based on the Downhill simplex
optimization method.

2.2

R ig id -b o d y T ran sform ation

In this thesis we used two dimensional images because of the simplicity and the avail­
ability. We used the rigid-body transformation which refers to the transformations
without shape distortion effects on images to be registered. Here the rigid-body trans­
formation consists of rotation and horizontal and vertical translation. Any transfor­
mation consists of two stages. First is the transformation coordinate formation using
the transformation matrix. Second is the interpolating the pixels to the new coordi­
nates.
The algorithm which is used in this thesis for rigid-body transformation is as
follows:
Stepl: First we make x and y axis coordinates matrices using the meshgrid func­
tion in MATLAB. For instance for an image with three rows and three columns the
x and y coordinate matrices are as follows:

'1 2 3'
1 2 3
1 2 3

'1 2 3'
1 2 3
1 2 3
Figure 2.2:
(ycoords)

Right: matrix of x coordinate (xcoords) Left: matrix of y coordinate

10

Step 2: In this step we multiply the transformation matrix with the x and y
coordinate matrices to form new coordinate matrices. For instance if there are 0.5
radius rotation and 2 cm translating on x and y axis, then the new x and y coordinates
are calculated as follows:

XCOOvds-new ~ m l 1{xCOOrds

Xcenter)

ml2(t/COOT'ds

2/c e n te r ) "t* XQff sv.i :

ycoordsnew —77?21(xCOOrds

Xcenter)

777-22(yC()OT(isVcenter')

Voffset-

where
%center ^

xsize
5
't'shift

ysize
V center ~

5

V s h ift

%o f f s e t ==

center

3C sh ift 1

V o f / s e t — V center

V s h ift 1

m i l = cos (angle),
m l2 = —sin {angle),
m 21 — —m l2,
m 22 = m il,
Here xsize — 3and ysize = 3 and xshift =

2 and yshift — 2 may make

coordinates whichare rotated by 0.5 radius and translated on x and y axis

11

a new

by 2 cm.

Step 3: MATLAB function called ”interp2” performs the interpolation. interp2
function receives the image matrix and the xcoordsnew and ycoordsnew and the type
of the interpolation as input. The types of interpolation could be Nearest neighbor
interpolation, Linear interpolation and Cubic spline interpolation. In this thesis we
use the linear interpolation because we know there are no shape changes on the images.

2.3

C alcu latin g th e A lig n m e n t M easu re

Among the alignment measures which are used for the area based rigid-body image
registration, the joint histogram mutual information is one of the reference measures.
The aim of this thesis is to develop a copula based divergence and mutual information
measure based image registration algorithm and compare it with the joint histogram
mutual information image registration.
There are various copula families in the literature [1]. Among available copula
functions we chose four, out of which only Clayton copula from Archimedean family
is used in image registration [14]. Four copula functions are Clayton, Frank, Gaussian,
Marshal-Olkin [1]. The main reasons to chose these copulas are the simple form of
alignment measure calculations and their interval of Kendall’s tau.
In the following we will describe the joint histogram based mutual information and
copula based divergence and mutual information measure concepts. These concepts
are demonstrated by a simple example.

12

2.3.1

Joint H istogram B ased M u tual Inform ation

The mutual information registration alignment metric is known to be a powerful
method in image registration [19]. In the year 1995 Colligon [18] and MIT group [16],
started the use of mutual information in image registration. Considering Image A as
the base image and image B as the image to be registered, the following equation will
define the mutual information between images A and B:

I ( A ( x ), B{x)) = H(A(x)) + H(B(x)) + H{A(x),B{x)),

(2.1)

where A{x) and B(x) are the intensity values matrices of images and H (A (x )) and
H(B(x)) are the marginal entropies of images A and B. H ( A {X ), B {x )) is the joint
entropy of images A and B.
In Image registration the test image (Image B ) should be geometrically transferred
until the maximum mutual information is obtained. Hence we can write the Equation
(2.1) as follows:

I(A(x), T( B(x )) ) = H{A{x)) + H(T( B(x )) ) + H ( A ( x ) , T ( B ( x ))),

(2.2)

where T(B(x)) is the transformed version of image B. Therefore the value obtained
from the Equation (2.2) is going to be used as the image registration alignment mea­
sure metric.
While working with images the joint histogram obtains the entropies for mutual
information calculation. Joint histogram will find the frequencies of couple of inten­
sities one from image A and another from image B. By dividing these frequencies by

13

the total number of pixels in any one of the images (since the reference and test im­
ages are of same size in this thesis), we will obtain the joint and marginal probability
distributions. P(a,i, bj) is the joint probabilities for images A and B. p(a,i) and p(bj)
are the marginal probabilities for the images A and B respectively. The following
equation calculates the mutual information between images A and B using the joint
and marginal probabilities of images A and B:

I { A , B ) = Y J Y ^ P { a i,b0)log

P(ai,bj )

(2.3)

.p(ai)P(bi).

To illustrate the steps used to calculate the joint histogram mutual information
between two matrices, we use two 3 pixels by 3 pixels gray scale images and their
corresponding intensity matrices which consist of three rows and three columns. The
gray scale images have the intensity values between 0 and 255. Figure 2.3 shows two
images and the corresponding matrices.

'220 220 10'
'220 20 250
10 250 5

'1 0 250 220'
220 20 220
5 220 10

Figure 2.3: (Left: Image A Right: Image B ) Two gray scale images of size 3x3 and
their corresponding intensity matrices below them

In the next step of calculating the mutual information we calculate the joint histogram
for the two matrices in Figure 2.3 as follows:
14

Table 2.1: Joint Histogram of Image A and Image B
B

5

10

20

220

250

P(<n)
1
9
2
9
1
9
3
9
2
9
9
9

A
5

0

|

0

0

0

10

|

0

0

i

0

20

0

0

I

0

0

220

u0

-9

u0

-9

-9

250

0

0

0

I

0

P(bj)

1 2
9
9

1
9

9

1
9

4

In the Table 2.1 the values on the margins are the marginal probabilities and
the values in the central cells are the joint probabilities. In the following, mutual
information for image A and image B is calculated using Equation (2.1):

I { A B ) = h o g -f- + hog-%- + hog-%- + \ l o g \ a
81 y 81 y
81 y
81
1
1
1
2
+
+ q lo9T2 + q log-j- + q1o91[
81
81
81
y
81
= 0.9068,

' The mutualinformation between Image A and image B is 0.9068. This value will
be used as the alignment measure between image A and image B.

15

2.3.2

C opula B ased A lign m en t M easure

Copulas are functions which join uniform marginal distributions of random variables
to form their multivariate distribution functions. Copulas separate joint distributions
for two contributions: (i) marginal distributions of each variable and (ii) copula as
a measure of dependence. Sklar’s theorem [20] states th at any multivariate distribu­
tion can be expressed as the fc-copula function C(ui,

...,uk) evaluated at each

of the marginal distributions. Copula is not unique unless the marginal distribu­
tions are continuous. Using probability integral transform, each continuous marginal
Ui = Fi(xi) has a uniform distribution on Ie[0, 1] where F i fa ) is the cumulative in­
tegral of fi(xi) for the random variable Xie(—oo,oo). The /c-dimensional probability
distribution function F has a unique copula representation

F ( x i , x 2, - , x k) = C(F i (x i ),F 2(x 2), ...,Fk(xk)) = C(ui,u2, ...,uk).

(2.4)

The joint probability density function is

f ( x i , x 2,...,xk) = Ui=1fi(xi)c(Fi{xi), F2(x2) , F k(xk)),

(2.5)

where fi{xi) is each marginal density and coupling is provided by copula density

,
c(uu u2,...,uk)

{dkC ( u i, u 2,...,uk))
(3UJ0U2...0U*) ’

. .
(2-6)

if it exists. In case of independent random variables, copula density c(u\,u2, ...,uk)
is identically equal to one. The importance of the above equation f ( x i , x 2,

x k) is

th at the independent portion expressed as the product of the marginals can be sepa­

16

rated from the function c(ui, u2,

, u k) describing the dependence structure or shape.

The dependence structure summarized by a copula is invariant under increasing and
continuous transformations of the marginals.
The simplest copula is independent copula

II := C{ui,u2, ..., it*) = u xu2...uk,

(2.7)

with uniform density functions for independent random variables.
An empirical copula may be estimated from the N pairs of data (xi;t, x 2;t)0<t<N by

C(n/N, m / N ) = £ t l r M < n , r t .2 < m ,
where r t|i and rt2 are

(2 .8 )

the rank statistics of [ou;t]t and [x2;t]t respectively. Frecht [21]

Hoeffding [22] lower and upper bounds for copula are respectively

W ( u x,u 2, ...,uk) :=max( 1 ~n-\-EiUi,0) < C (u x, u2, ..., u k),

C (u i, u 2, ...,uk) <

= M ( u u u2, ...,uk).

(2.9)

(2.10)

Relationships between copula and concordance measures Kendall’s r , Spearm an’s
p, Gini’s index 7 are as follows:

17

r = 4J J

C(ui, U2)dC(ui,U2) — 1,

p = 12 f J , UiU2dC(ui,U2) —3,

7 = 2

(|«! + U 2 — 1| — |ltl — U2\ )dC(Ux, U2).

( 2 .11 )

( 2 . 12 )

(2.13)

2

In probability theory and statistics, a copula can be used to describe the de­
pendence between random variables. They express joint structure among random
variables with any marginal distributions.
The dependence between two sets of random variables can be calculated in terms
of divergence measure. In this thesis for Clayton and Frank copulas, we used Kol­
mogorov copula based divergence to calculate the copula based divergence measure.
Kolmogorov divergence is given in terms of copula density function as follows:

(2.14)
[o,iJ2
where c(x, y) is the copula density function. The copula density functions are obtained
by performing differentiation on the copula distribution functions.
It is very important to choose the right copula function. One of the constraints
to choose the appropriate copula is the interval of the copula parameters. As we will
see in chapter 3, for the used data (MRI and Aerial images), four copula functions
satisfied the copula parameter (6) interval. The four copula functions and their family
18

names which are used in this thesis:
i. Clayton from the Archimedean family.
ii. Frank copula from Archimedean family.
iii. Marshal-Olkin copula from non-Archimedean copula family with simultaneous
presence of an absolutely continuous and a singular component.
iv. Gaussian copula from elliptical family.

Here we present copula density function of each copula family and the calculation
of the copula parameter in terms of Kendall’s Tau.

C layton C opula
The Clayton copula cumulative distribution function (cdf) is provided in Equation
(2.15) and it is plotted in Figure 2.4.

C(x,y) = {x-* + y - ° - \ ) - l ,

19

(2.15)

Figure 2.4: Clayton copula cumulative distribution function

where x and y are random variables between 0 and 1 and 9 is (0 < 9 < oo). After
performing the Equation (2.6), we will obtain the Clayton copula probability density
function (pdf) as:

c(x,y) = (1 + 9 ) x - l~ey ~1- e( - 1 + x~e + ir® )-2_(' \

(2-16)

where x and y are random variables between 0 and 1 and 6 is (0 < 6 < oo). c(x, y)
is the Clayton copula density function. The relationship between Clayton copula
parameter 6 and Kendalls rank correlation r is:

0 = IT T -

where r is (0 < r < 1) and 9 is (0 < 9 < oo).

20

<2-17)

In Figure 2.5 we can see the density curves of the Clayton copula for 9 = 0.1 (first
row left), 9 = 0.4 (first row right) and 9 = 0.9 (second row). Smaller 9 values represent
weaker association and as the 9 value increases the association gets stronger. Figure
2.5 shows how the shape of Clayton copula density curve changes as the association
gets stronger.

Figure 2.5: Clayton copula Probability density curves for 9 = 0.1 (first row left),
9 = 0.4 (first row right) and 9 — 0.9 (second row)

21

F ran k C o p u la
The Frank copula cumulative distribution function (cdf) is provided in Equation (2.18)
and it is plotted in Figure 2.6.

Figure 2.6: Frank copula cumulative distribution function

where x and y are random variables between 0 and 1 and 6 is (-oo < 6 < oo). After
performing the Equation (2.6), we will obtain the Frank copula probability density
function (pdf) as:

.

.

c(z, V ) -

e - ex6e-6y{e-e - 1)
( e - e X( e - e y _ x) + e-0 _ e - e y y >

(2‘19)

where x and y are random variables between 0 and 1 and 0 is (—oo < 9 < oo).
c(x, y) is the Frank copula density function. The relationship between Frank copula
22

parameter 0 and Kendalls rank correlation t is:

where r is (—1 < r < 1) and 0 is (—oo < 0 < oo).
In Figure 2.7 we can see the density curves of the Frank copula for 0 = 0.1 (first
row left), 0 = 0.4 (first row right) and 0 = 0.9 (second row). Smaller 0 values represent
weaker association and as the 0 value increases the association gets stronger. Figure
2.7 shows how the shape of Frank copula density curve changes as the association gets
stronger.

23

Figure 2.7: Frank copula Probability density curves for 9 = 0.1 (first row left), 9 = 0.4
(first row right) and 9 = 0.9 (second row)

24

M a rsh a l-O lk in C o p u la

The Marshal-Olkin copula cumulative distribution function (cdf) is provided in Equa­
tion (2.21) and it is plotted in Figure 2.8.

C{x,y) = m in { x l ey , x y 1 0),

«

«

( 2 . 21 )

I

m m *

Figure 2.8: Marshal-Olkin copula cumulative distribution function

where x and y are random variables between 0 and 1 and 9 is (0 < 9 < 1). After
performing the Equation (2.6), we will obtain the Marshal-Olkin copula probability
density function (pdf) as:

c(x,y)

(1 —9)x 6

if y < x

(1 - 9)y-°

if x < y,

25

( 2 . 22 )

where x and y are random variables between 0 and 1 and 9 is (0 < 9 < 1). c(x,y) is
the Marshal-Olkin copula density function. The relationship between Marshal-Olkin
copula parameter 9 and Kendalls rank correlation r is:

9=

1+ T

(2.23)
V '

where r is (0 < t < 1) and 9 is (0 < 9 < 1).
In Figure 2.9 we can see the density curves of the Marshal-Olkin copula for 9 — 0.1
(first row left), 9 = 0.4 (first row right) and 9 = 0.9 (second row). Smaller 9 values
represent weaker association and as the 9 value increases the association gets stronger.
Figure 2.9 shows how the shape of Marshal-Olkin copula density curve changes as the
association gets stronger.

26

1.4 -

12 "

0.4

1.4"
1.0 -

06 -

0.40 .2-

O.S
0.4
0.2

Figure 2.9: Marshal-Olkin copula Probability density curves for 9 = 0.1 (first row
left), 9 ~ 0.4 (first row right) and 9 — 0.9 (second row)

27

G a u ssia n C o p u la

The Gaussian copula cumulative distribution function (cdf) is provided in Equation
(2.24) and it is plotted in Figure 2.10.

- C

"

C

i W - ’l T

# 5) ->■ ■*»

Figure 2.10: Gaussian copula cumulative distribution function

where x and y are random variables between 0 and 1 and p is (—1 < p < 1) and
Xin = <3>_1(:r) and yin = $ -1(y). After performing the Equation (2.6), we will obtain
the Gaussian copula probability density function (pdf) as:

c(x,y) = (1 - p2) h x p ^ - ^ ( 1 - P2) l (xfn + Vin - 2PXinVin)^ exp Q ( 4 * + Vin)^ >
(2.25)
28

where x and y are random variables between 0 and 1 and p is (-1 < p < 1). x in —
$ _1(x) and yin — 4>-1 (r/) and c (x, y) is the Gaussian copula density function. Here
<j> G [—1,1] is the parameter of Gaussian copula and $ -1 is the inverse standard
Gaussian cumulative distribution function. The relationship between Gaussian copula
parameter p and Kendalls rank correlation r is:

1* ),
\
p = « ■n (r—

(2.26)

where r is (—1 < r < 1) and p is (—1 < p < 1).
In Figure 2.11 we can see the density curves of the Gaussian copula for p — 0.1 (first
row left), p = 0.4 (first row right) and p = 0.9 (second row). Smaller p values represent
weaker association and as the p value increases the association gets stronger. Figure
2.11 shows how the shape of Gaussian copula density curve changes as the association
gets stronger.

29

Figure 2.11: Gaussian copula Probability density curves for p = 0.1 (first row left),
p = 0.4 (first row right) and p = 0.9 (second row)

2.3.3

Exam ples o f C opula B ased A lign m en t M easure

Clayton and Frank copula density functions are used in the Equation (2.14) in or­
der to calculate the copula based divergence as alignment measure. Equation (2.14)

30

is the Kolmogorov copula based divergence measure which finds the divergence be­
tween two random variables. There are various copula based divergence functions
such as Kullback-Leibler, Tsallis, Renyi and Kolmogorov. In this thesis we use the
Kolmogorov copula based divergence measure due to its simplicity and faster calcu­
lation.
For Marshal Olkin and Gaussian copulas the mutual information is provided in Equa­
tions (2.27) and (2.28) respectively. Marshal Olkin and Gaussian based m utual infor­
mation metrics are used as alignment measures. Here we use the m utual information
measure for Marshal Olkin and Gaussian copulas since the simple forms are provided.

I( x ,y ) = : | ± | % ( 1 - 9) +

(2.27)

where 6 is Marshal Olkin copula parameter from Equation (2.23).

y) =

- p2),

(2.28)

where p is the Pearson correlation between the two random variables x and y.
In Figure 2.12 we can see the calculated divergence and mutual information values
between two random numbers using the Clayton, Frank, Marshal and Gaussian cop­
ulas. Here we observed th at divergence measure does not exist for Clayton copula for
r > 0.8. Also mutual information decreases for Marshal Olkin for r > 0.8. In Figure
2.12 and Table 2.2, we use the Pearson correlation for the presentation purpose.

31

Clayton
Frank
1.4

Divergence

1.2

0.6

0.4
0.2

Pearson product-moment correlation

Mutual Information

• Marshal
■Gaussian

0.6

0.4
0.2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Pearson product-moment correlation

0.9

1

Figure 2.12: Divergence measures and mutual information calculated using four cop­
ulas

32

The values of divergence measure and mutual information calculated using Clayton,
Frank, Marshal-Olkin and Gaussian copulas are presented in Table 2.2.
Table 2.2: Values of the divergence measures and mutual information in Figure 2.4
Pearson Correlation

Marshal

Clayton

Frank

Gaussian

0.010

0.030

0.011

0.011

0.000

0.110

0.307

0.122

0.125

0.006

0.210

0.547

0.243

0.244

0.023

0.310

0.752

0.376

0.370

0.051

0.410

0.924

0.522

0.507

0.092

0.510

1.061

0.684

0.660

0.151

0.610

1.163

0.865

0.832

0.233

0.710

1.225

1.065

1.034

0.351

0.810

1.238

NaN

1.276

0.534

0.910

1.185

NaN

1.583

0.880

Now we shall demonstrate an example which shows the calculation of copula based
divergence measure and mutual information using the Equation (2.14). Let us use the
sample images in Figure 2.3. The following are the steps to calculate the divergence
measure using the Frank copula density in Kolmogorov divergence function:
Stepl: Here we arrange pixels for both the images in Figure 2.3 in column matrices.
In order to make the column matrix we have to append each column of the m atrix to
the end of the first column (Figure 2.13).
Step 2: In this step we calculate the Kendall’s Tau rank correlation between the
two column matrices.
r = 0.4336,
33

'1 0 '
220
5
250
20
220
220
220
10

'220'
220
10
220
20
250
10
250
5

Figure 2.13: Left: Column m atrix of Image A Right Column matrix of Image B

Step 3:Using Equation (2.20) to calculate the Frank copula parameter 6 using the
Kendall’s Tau.
0 = 4.6433,

Step 4: Using the Equation (2.14), in this step we calculate the double integration
of the Frank density function with the calculated copula parameter:

ff

-

e~ex0e~ey(e~6 —11

JJ l ( - (e-*.(e- % - l ) V e - » - eV

[0,1]2

-

* * * = °-54185'

The divergence measure of images A and B in Figure 2.3 by using the Frank copula
based Kolmogorove divergence is 0.54185.
The Clayton copula based divergence measure also calculated using the same four
steps as Frank copula (in steps 3 and 4 we use the Clayton copula parameter and den­
sity function) . For Marshal-Olkin and Gaussian copulas the calculations are simpler.
34

In Marshal-Olkin copula after the step 3 and calculating the copula parameter using
the Equation (2.23). we perform Equation (2.27) and find the mutual information
between images A and B.

9

i

<yr\

s\

I ( x ,y) = ~2_Q lo9 il ~ e) + 2 ~Zq = °-9596>
where 9 = ^

= 0.6049.

In Gaussian copula we use the Pearson correlation between the two column matrices
and calculate the mutual information using Equation (2.28) as follows:

!{ X,V) =

- iP2)) = 0.1993,

where p is the Pearson correlation between the two matrices and its value in this case
is 0.5733.

2.4

O p tim iza tio n

For the images which are similar and need few numbers of transformations to be
registered, optimization is not required and this method is called direct image regis­
tration. Direct image registration uses the registration metric (alignment measure)
and after few transformations, by reaching to the maximum or minimum metric the
images get registered. On the other hand when the images are not similar and there
are many numbers of transformations, the optimization is used in image registration
algorithms. There are various optimization techniques of which we use the MATLAB fminsearch optimization method [23]. The fminsearch uses the Nelder-Mead

35

simplex method. The fminsearch is also called the direct search in which we search
for the smallest scalar of a function iteratively. In this algorithm we are searching for
the smallest scalar of alignment metric function. In this thesis the alignment metric
function provides the negated value of divergence measure and mutual information.
Higher values of divergence measure and mutual information are closer to the reg­
istration point, therefore fminsearch finds the maximum absolute value of them. In
the Nelder-Mead search method we should provide the metric function, initial values
of transformation and the predefined parameters. These parameters are the starting
point of the transformations for image registration. After the initial transformation
of the test image the alignment measure will be calculated between the transformed
test image and reference image. The optimization algorithm will decide weather the
alignment measure is optimum or not. The transformation will be continued with
the transformation parameters from the optimization algorithm until the optimum
alignment measure is found.
The fminsearch algorithm and the predefined parameters are obtained from the
[23] paper and presented here. The predefined parameters are:

p = l , X = 2 ,7 = ^ , a = ^,

(2.29)

By using the above parameters and based on the Nelder-Mead search algorithm
constraints on the scalar value (negated divergence measure or mutual information) ,
the passing transformation parameters to the metric function gets updated.
In the following we will see an example of the image registration of the Figure
2.3 by using the fminsearch and the joint histogram based mutual information as the
alignment metric function. Here the left image in Figure 2.3 is the reference image
36

and the right image is the test image.

Figure 2.14: Left: Reference Image Right: Test Image bottom: Registered test image
using the fminsearch and joint histogram based mutual information

By looking at Figure 2.14 we can observe th at if we rotate the test image by 90 degree
counter-clockwise, we can obtain the best possible registration. The fminsearch opti­
mization algorithm is implemented in MATLAB and we obtained the transformation
metric (90 degree) in 25 iterations. In these 25 iterations fminsearch reffers to the
alignment metric function to calculate the negated mutual information and finds the
best transformation metric. Figure 2.15 shows the iterations and the values of the
negated mutual information in each iteration.

37

-

1.1

0

5

10
15
Iterations

20

25

Figure 2.15: Plot of the convergence of the fminsearch in 25 iterations

Here we can see after 25 iterations the value of -1.21489 is chosen to be the min­
imum negated mutual information value and the x which is the transformation pa­
rameter is 90 degree. This means the counter clockwise rotation of 90 degree should
be performed on the image B to be registered or aligned to image A.

38

C hapter 3

E xp erim en ts

In this chapter we will present the results of the experiments of copula based image
registrations and compare them with the standard image registration method. These
experiments are performed on the MRI and Aerial images. This chapter starts with
the description about the images used in this thesis and continues with the results
of the image registration for four copula functions namely, Clayton, Frank, Gaussian
and Marshal- Olkin copulas. In the end we shall summarize the copula based image
registration and the standard image registration (joint histogram) m ethod’s results.

3.1

E xp erim en ta l Im a g es

Two main applications of image registration are its use in medical and aerial image
processing. Most of the medical images are in three dimensions. In the literature
in order to test the performance of a new image registration metric they use a slice
39

of these three dimensional images [14]. These slices are of two dimension and are
simpler and less time consuming, in comparison with the three dimension. In this
thesis our aim is to monitor the performance of the copula based image registration
and we are using the gray-scale two dimensional slices of the Magnetic Resonance
Imaging (MRI) and aerial images.

These gray-scale images contain the intensity

values in the range of 0 to 255 in which 0 is white shade and as we move to 255 the
shades get darker. The images are collected as the reference images and geometrical
transformation algorithm has been applied on these reference images to generate the
test images. The transformation algorithm to make the test images, consisted of 2
centimetres translation on x and y axis, and 0.5 radius counter-clockwise rotation.
Simulation of test images, tells us the exact transformation factors and hence we can
compare the results of the registration with the transformation factors. For instance
now we know that if the registration algorithms obtain the geometrical transformation
values as translation of -2 centimetres in x and y axis and rotation of -0.5 radius
(clockwise) may provide the best possible registration.

3.1.1

M agnetic R eson an ce Im aging (M R I)

We use one slice gray scale of Magnetic Resonance Image (MRI) for 8 patients which
are obtained from the Vanderbilt University’s medical image research website [25].
These images are with Joint Photographic Experts Group (JPEG) format. In Figure
3.1 we can see the reference images P I (patient 1) to P8 (patient 8) and the test images
(T P 1 to TP8) which are images from the same patients as P i to P8, translated 2
centimetres in x and y axis and rotated counter-clockwise 0.5 radius. These images
are two dimensional with the resolution of 256 X 256. In the experiments of the MRI
images we obtain a region of interest (ROI) of rectangular shape with coordinates:
40

[xi=30, 2/1=30, £2=225, 2/2=225] and the calculations are performed on the region
of interest. Region of interest may discard the unimportant portions of the image
such as the background. We use the region of interest for the MRI images and not
for the aerial images because all parts of the aerial images are im portant for image
registration.

pi

y
ps

Tp1

■
TpS ;

Figure 3.1: P I to P8 are the reference images and TP1 to Tp8 are the testing images

Aerial Im ages
Aerial images in this thesis are acquired from Quick Bird sensor (QUICKBIRD SATEL­
LITE IMAGES) [24] with Joint Photographic Experts Group (JPEG) format. Eight
2-D gray scale aerial images axe presented in the first two rows of Figure 3.2 and
41

labelled as A1 to A8. The description and the geographic location of these images are
provided in Appendix B. These images originally were color images of resolution 512
X 512 X 3 . By using the MATLAB function called rgb2gray() we converted these im­
ages to gray scale images th at are with resolution 512 X 512. With another MATLAB
function called imresizeQ we down-sampled these images to 188 X 188 resolution for
all the images except the seventh image (A7) which is of resolution 177 X 177. As one
can see in the Figure 3.2, the images were padded on the margins which is called zero
padding. In Figure 3.2 the images with labels A1 to A8 are the reference images and
images with TA1 to TA8 are 2 centimetres translated on x and y axis and rotated 0.5
radius counted-clockwise version of the reference images. TA l to TA8 are called the
test images (moving images) in this thesis.

Figure 3.2: A l to A8 are the reference images and TA l to TA8 are the testing images

42

3.1.2

W hich C opula ?

Before starting the experiments we had to check which copula function is appropriate
for the images used in this thesis. In order to know which copula function is ap­
propriate we shall obtain the Kendall’s Tau intervals of the paired images (reference
and test images) to see if atleast the initial intervals are with in the used copula’s
constraints. The four copula functions which are used in this thesis consist of the
following constraints:
Gaussian copula: r is( —1 < r < 1) and p is (—1 < p < 1).
Clayton copula: r is (0 < r < 1) and 9 is (0 < 9 < oo).
Marshal Olkin copula: r is (0 < r < 1) and 9 is (0 < 9 < 1).
Prank copula: r is (—1 < r < 1) and 9 is (—oo < 9 < oo).
Apart from the above conditions, as we saw in the previous chapter for r > 0.8,
Marshal Olkin and Clayton copulas do not provide appropriate measures. Hence the
appropriateness of copula functions must be investigated in the image registration
experiments.

3.2

R esu lts o f Im age R e g istr a tio n E x p erim en ts

As mentioned in the previous section we have sixteen cases of image registration from
which eight are MRI images and eight are aerial images. At this section we are going
to report the performance of the image registration of these sixteen images using the
five distinct alignment measure methods. These five methods are joint histogram,
Gaussian copula, Marshal-Olkin copula, Clayton copula and Frank copula. We shall
measure the performance of these image registration methods by using the peak signal43

to-noise ratio (PSNR) method, in which the closer the value of PSNR to zero is, the
better the performance of the image registration algorithm. The calculation of PSNR
for two intensity matrices of two images is as follows:

P S N R = lOlogio

(3.1)

M SE

5^[/i(x,y) - l2{x,y)?
where MSE (Mean Square Error) is M S E = —---------------------

>and

x and y

are the numbers of pixels in each of the respective axis. h { x , y) and h { x , y) are the
intensity values of image 1 and image 2.
As we discussed in chapter 2, the fminsearch optimization algorithm searches for
the global minimum, hence we use the negated values of divergence measure and mu­
tual information. Here the maximum value of negated values of divergence measure
and mutual information will be the global minimum value for the fminsearch opti­
mization algorithm. In this point the fminsearch algorithm converges and announces
the obtained geometrical transformation values. When we transform the test image
using the resulted value of geometrical transformation from the optimization algo­
rithm, we are supposed to obtain an image in which the PSNR value is close to zero.
If the PSNR value is close to zero, it means the tested image is nearest possible to the
reference image. The header columns of the tables in this section from left to right
report the following: (Tables 3.1, 3.2, 3.3, 3.4 and 3.5)

Column 1. Name of the images which are registered.
Column 2. Initial AM (Alignment Measure) is the negated divergence measure or
mutual information between the reference and the test images before starting the reg­

44

istration algorithm.
Column 3. Final AM (Alignment Measure) is the final negated divergence measure
or mutual information value which is the global minimum value that fminsearch con­
verged for it and declared the corresponding transformations optimum.
Column 4. Final T is the final transformation th a t the image registration algorithm
converges to it. The transformations are shown as three values in which from left to
right, the first value is the translation on the x axis, second value is the translation
on the y axis and the third value is the rotation angle in terms of radius.
Column 5. Iteration is the number of iterations in which the fminsearch algorithm
converges.
Column 6. PSNR is peak signal-to-noise ratio between the registered test image and
the reference image.

Also for all the registration algorithms we need to provide the initial values for
the geometrical transformation in which we provided as 5 centimetres translation on
x and y axis and 0.2 radius counter-clockwise rotation. Note that these initial values
were provided in the literature [26] and we did not change them because of comparison
of the new copula based image registration algorithm with the reported results in the
literature.
In the following we will see the tabulated results of the proposed image registration
methods.

45

3.2.1

Joint H istogram M u tual Inform ation

The joint histogram mutual information based image registration is one of the popular
methods in area based image registration. Hence here in fminsearch optimization
method, the negated joint histogram mutual information is used as the objective or
metric function (Alignment Measure) which guides the fminsearch optimization to
converge and find the best possible geometrical transformation. The results of the
joint histogram based image registration method are reported in Table 3.1.
Table 3.1: Image registration results for joint histogram mutual information

Iterations

PSNR

(-2.0081 -2.0083 -0.5000)

59

-12.76

-2.08

(-1.9988 -2.0105 -0.5000)

85

-12.85

-0.83

-2.08

(-2.0025 -2.0169 -0.5002)

71

-12.71

P4

-0.93

-2.08

(-1.9699 -2.0028 -0.5000)

78

-12.18

P5

-0.90

-2.13

(-2.0071 -2.0053 -0.5000)

61

-11.05

P6

-0.86

-2.00

(-2.0120 -2.0079 -0.5001)

84

-11.42

P7

-0.93

-2.14

(-1.9972 -1.9938 -0.5001)

82

-10.52

P8

-0.96

-2.12

(-2.0102 -2.0050 -0.5001)

69

-11.37

A1

-0.46

-0.99

(1.5018 3.2268 -0.4990)

84

-32.03

A2

-0.45

-1.74

(-2.0010 -2.0136 -0.5000)

72

-22.83

A3

-0.32

-1.51

(-1.9959 -1.9857 -0.5000)

79

-22.79

A4

-0.37

-1.68

(-2.0089 -1.9842 -0.4999)

75

-22.30

A5

-0.48

-1.75

(-2.0101 -2.0140 -0.5002)

74

-24.33

A6

-0.53

-1.95

(-1.9946 -1.9998 -0.5000)

74

-21.41

A7

-0.53

-2.18

(-2.0030 -1.9891 -0.5001)

72

-16.72

A8

-0.45

-1.73

(-2.0117 -1.9958 -0.4999)

76

-24.99

InitialAM

Final AM

FinalT

PI

-0.86

-2.07

P2

-0.89

P3

Images

46

In Table 3.1 we observe the first aerial image (Al) is a miss-registered case, in
which the PSNR value between the registered image and the reference image is very
small (far from zero) and hence it is not registered properly. Also we can observe
that the PSNR values of the MRI images are larger than the aerial images. This is
because the corners of the aerial test images disappear while simulating them (effect
of rotation on rectangular images in Figure 3.3). While registering, the corners of
the aerial test images are missing which leads to smaller PSNR value in comparison
with the MRI images. For MRI test images simulation there are no lost parts, since
their background are black and does not have the effect of rotation on rectangular
images. In Figure 3.3 we can see the first aerial miss-registered image. In this figure
the red channel, belongs to the reference image and blue channel, belongs to the test
image. Here the joint histogram based algorithm obtained the transformation which
is (1.5018 3.2268 -0.4990). Here the test image is translated 1.5018 centimetres on
x axis and 3.2268 centimetres on y axis and rotated -0.4990 radius which is very far
from the reference transformation values th at should be close to (-2, -2, -0.5).

47

Figure 3.3: First aerial image which mis-registered using the joint histogram mutual
information method. (First row left: the reference image, first row right: the test
image, second row left: the registered test image, second row right: the overlap of
registered test image on the reference image which is mis-registered)

3.2.2

C layton C opula

Clayton copula is from Archimedean copula family. It is one of the registration align­
ment metric functions, which may lead the test images to get aligned to the reference
48

images. Here the alignment measure is the negated Kolmogorove divergence measure.
As we can see in the Table 3.2 the PSNR values of this method are lower than the
other four methods (far from zero). Also we can see in Final Am column, the values
are -1.26 which indicates that r > 0.8, (section 2.3.3) where the divergence measure
does not exist for the Clayton copula and the optimization algorithm stops at this
point. Hence the combination of Clayton copula and fminsearch optimization was not
successful in registering the images in this thesis.
Table 3.2: Image registration results for Clayton Copula

InitialAM

Final AM

FinalT

Iterations

PSNR

PI

-0.64

-1.26

(-1.5478 -2.2116 -0.5068)

72

-17.65

P2

-0.69

-1.26

(-2.6049 -1.6625 -0.5025)

78

-19.17

P3

-0.67

-1.26

(-2.3662 -1.8764 -0.4910)

69

-17.57

P4

-0.67

-1.26

(-2.0666 -2.7114 -0.4999)

61

-18.41

P5

-0.68

-1.26

(-1.9466 -1.2580 -0.4969)

53

-17.78

P6

-0.71

-1.26

(-1.4237 -2.5233 -0.4993)

50

-18.46

P7

-0.72

-1.26

(-2.5197 -2.3777 -0.5088)

35

-16.53

P8

-0.69

-1.26

(-1.6440 -1.4109 -0.5004)

36

-16.56

Al

-0.49

-1.26

(-1.5507 -0.3933 -0.4961)

70

-28.12

A2

-0.47

-1.26

(-1.7790 -0.8507 -0.4936)

70

-28.61

A3

-0.43

-1.26

(-1.4453 -2.1648 -0.4929)

87

-24.91

A4

-0.43

-1.26

(-2.4282 -0.2177 -0.5096)

99

-26.61

A5

-0.36

-1.26

(-1.5795 -2.0429 -0.5140)

77

-28.98

A6

-0.60

-1.26

(1.6422 1.3176 -0.4720)

56

-29.85

A7

-0.65

-1.26

(0.7547 1.5196 -0.4670)

71

-27.52

A8

-0.37

-1.26

(-2.8610 -2.0608 -0.5052)

88

-29.54

Images

49

3 .2 .3

F ran k C o p u la

Frank copula is from Archimedean copula family, which is used as the registration
metric function in fminsearch optimization. Here the alignment measure is the negated
Kolmogorove divergence measure. Table 3.3 shows the results of the 16 cases of image
registration. The performance of Frank copula was satisfactory because the PSNR
values are closer to zero.
Table 3.3: Image registration results for Frank Copula

InitialAM

FinalAM

FinalT

Iterations

PSNR

PI

-0 .5 4

-1.58

(-1.9952 -1.9982 -0.5000)

67

-12.76

P2

-0.58

-1.58

(-2.0020 -1.9985 -0.5000)

72

-12.84

P3

-0.57

-1.58

(-1.9951 -1.9984 -0.5000)

71

-12.71

P4

-0 .5 7

-1.56

(-1.9982 -2.0026 -0.4999)

92

-12.14

P5

-0.57

-1.60

(-2.0005 -1.9982 -0.5000)

94

-11.05

P6

-0.59

-1.56

(-2.0003 -2.0026 -0.4999 )

80

-11.41

P7

-0.60

-1.61

(-1.9994 -1.9997 -0.5000)

67

-10.52

P8

-0 .5 7

-1.60

(-2.0023 -2.0008 -0.5000)

75

-11.37

A1

-0 .4 4

-1.68

(-1.9997 -1.9996 -0.5000)

92

-20.67

A2

-0.43

-1.59

(-2.0001 -2.0007 -0.5000)

78

-22.83

A3

-0.39

-1.51

(-2.0004 -2.0000 -0.5000)

76

-22.79

A4

-0.39

-1.59

(-2.0099 -2.0099 -0.5000)

100

-22.30

A5

-0.33

-1.60

(-2.0029 -2.0029 -0.5000)

74

-24.32

A6

-0.52

-1.66

(-2.0038 -2.0142 -0.5001)

77

-21.41

A7

-0.55

-1.78

(-1.9991 -2.0098 -0.4999)

76

-16.71

A8

-0.34

-1.61

(-2.0014 -2.0054 -0.5001)

84

-24.99

Images

50

3 .2 .4

M a rsh a l-O lk in C o p u la

Marshal-OIkin copula is from non-Archimedean copula family. It is used in image
change detection algorithms (13] and here we are applying it to the area based image
registration. Here the alignment measure is the negated mutual information. As can
be seen in Table 3.4, the registration is not accurate as PSNR values are very small
(far from zero). Also we can see in Final AM column, the values are -1.24 which
indicates that r > 0.8, (section 2.3.3) where the mutual information decreases for
Marshal Olkin copula and the optimization algorithm stops at this point. Hence the
Marshal-Olkin copula was not successful in order to register the tested images.
Table 3.4: mage registration results for Marshal-O kin Copula
Iterations

PSNR

(-1.8183 -2.5884 -0.5063)

75

-17.79

-1.24

(-2.4841 -2.5740 -0.4954)

79

-18.01

-1.05

-1.24

(-2.7930 -1.8269 -0.4972)

72

-20.48

P4

-1.05

-1.24

(-2.0579 -2.7539 -0.5016)

75

-18.82

P5

-1.06

-1.24

(-1.7279 -1.2338 -0.5031)

80

-18.08

P6

-1.06

-1.24

(-1.9167 -2.5406 -0.5008)

71

-15.56

P7

-1.08

-1.24

(-1.2947 -1.5164 -0.5056)

68

-17.66

P8

-1.06

-1.24

(-1.3496 -2.1672 -0.5052 )

81

-17.68

A1

-0.89

-1.24

(-1.4370 -0.2514 -0.4963)

65

-27.74

A2

-0.87

-1.24

(-1.1359 -1.4990 -0.4890 )

77

-25.90

A3

-0.82

-1.24

(-1.7766 -3.0360 -0.5053)

78

-21.50

A4

-0.89

-1.24

(-1.4370 -0.2514 -0.4963)

65

-25.62

A5

-0.73

-1.2 4

(-1.4682 -1.8742 -0.5149)

60

-25.13

A6

-0.99

-1.24

(2.7180 -1.4284 -0.4501)

54

-29.47

A7

-1.03

-1.24

(1.2575 1.6143 -0.4559)

73

-27.57

A8

-0.75

-1.24

(-2.9131 -2.2967 -0.5020)

73

-28.08

Images

InitialAM

Final AM

FinalT

PI

-1.03

-1.24

P2

-1.07

P3

51

3 .2 .5

G a u ssia n C o p u la

Gaussian copula as defined in the previous chapter is one of the elliptical copula
families and used to calculate the mutual information. Here the alignment measure is
the negated mutual information. Table 3.5 presents the results of the Gaussian copula
based image registrations for 16 images.
Table 3.5: Image registration results Gaussian Copula

Iterations

PSNR

(-1.9979 -2.0003 -0.5001)

81

-12.76

-2.55

( -1.9992 -1.9983 -0.5000)

71

-12.84

-0.47

-2.33

(-2.0015 -2.0021 -0.5000)

72

-12.70

P4

-0.48

-2.55

(-1.9989 -1.9996 -0.4999)

73

-12.14

P5

-0.51

-2.59

(-2.0012 -1.9996 -0.5000)

74

-11.05

P6

-0.72

-2.45

(-1.9973 -2.0037 -0.4999)

82

-11.41

P7

-0.52

-2.68

( -2.0020 -2.0012 -0.4999)

90

-10.52

P8

-0.54

-2.70

(-2.0001 -1.9987 -0.5000)

82

-11.37

Al

-0.15

-1.73

(-2.0058 -2.0015 -0.5000)

97

-20.67

A2

-0.08

-1.40

( -2.0043 -2.0046 -0.5000)

82

-22.83

A3

-0.17

-1.34

(-2.0110 -2.0044 -0.4999)

80

-22.79

A4

-0.12

-1.40

(-2.0078 -2.0156 -0.4999)

87

-22.30

A5

-0.07

-1.40

(-2.0065 -2.0069 -0.4999)

74

-24.32

A6

-0.10

-1.48

(-2.0148 -2.0092 -0.4999)

79

-21.41

A7

-0.21

-2.01

(-1.9816 -1.9816 -0.4997)

76

-16.70

A8

-0.09

-1.41

(-2.0025 -2.0066 -0.4999)

84

-24.99

InitialAM

Final AM

FinalT

PI

-0.38

-2.37

P2

-0.51

P3

Images

52

In this method we obtained better result than the joint histogram method as the PSNR
values are closer to zero. In Figure 3.4 the registration of the first aerial image (A l),
using the Gaussian copula is presented. Here the Gaussian copula image registration
base algorithm obtains the transformation as (-2.0058 -2.0015 -0.5000) in which the
test image is translated -2.0058 centimetres on the x axis and -2.0015 centimetres on
the y axis and rotated -0.5000 radius. The obtained transformation is very close to
the reference transformation values that is (-2, -2, -0.5).

Figure 3.4: First aerial image which registered using the Gaussian copula method,
(first row left: the reference image, first row right: the test image, second row left:
the registered test image, second row right: the overlap of registered test image on
the reference image)

53

3.3

Sum m ary o f E x p erim en ts

We may summarize the image registration results obtained in the previous section
using Figures 3.5 and 3.6. In Figure 3.5, we can see the PSNR values of the eight
registered MRI images. According to this figure the joint histogram, Gaussian and
Frank copulas obtained similar results and Clayton copula and Marshal-Olkin were
not successful copulas for the tested images. Figure 3.6, presents the PSNR results
for the 8 aerial images. In this figure also we see the similar pattern as Figure 3.5.
Figure 3.6, shows that the joint histogram method miss-registered the first aerial
image. Also on these figures we can see th a t the Marshal-Olkin copula and Clayton
copula resulting in miss-registered images in comparison with the other copula based
methods. In Figure 3.6, we can see the Marshal-Olkin registration, have the best
performance for third aerial image and worst performance for fifth aerial image.

-10

Joint Hist
Gaussian
Clayton
— Marshal-Olkin
♦
Frank

-12

Pi

-1 4

z

-m

-1 8
-2 0

1

2

3

4
5 6
MRI Images

7

8

Figure 3.5: PSNR values for the 8 MRI images

54

-1 5

• Joint Hist
Gaussian
Clayton
-Marshal-Olkin
Frank

-2 0

-3 5
-4 0
Aerial Images
Figure 3.6: PSNR values for the 8 Aerial images

Another summary of the results in this thesis can be seen from the following figures.
In Figure 3.7 and 3.8, we can see the path for optimization of the rotation radius,
which starts from 0.2 radius rotation angle and reaches to the optimum rotation angle
that is -0.5 radius. Figures 3.7 and 3.8, belong to the Gaussian copula registration
of the fourth MRI and aerial images respectively. In these figures the rotation angle
starts from the initial angle point th at we provided to the fminsearch algorithm that
is 0.2 radius and finally converges to a point close to -0.5 radius. The -0.5 radius
rotation angle is the best possible rotation angle th a t can align the test image to the
reference image. The fminsearch MATLAB optimization as described in [23], performs
the initial simplex, contract outside, contract inside, expand and reflect stages in the
number of iterations, until it reaches to the optimum point.

55

oc
£
a
HH

0.5

«8

3

-1

1.5
-0 .8

-0 .6

-0 .4 -0 .2
0
Rotation radius

0.2

Figure 3.7: The optimization result of the Gaussian Copula for the 4th aerial image
after 87 iterations

§ - 0 .5
-1
3 - 1 .5

bO
® -2 .5
0.6

0.4
0.2
0
Rotation radius

0.2

Figure 3.8: The optimization result of the Gaussian Copula for the 4th MRI image
after 73 iterations

Figures 3.9 and 3.10, are the demonstration of the Gaussian copula based image
registration of the fourth aerial and MRI images. We demonstrated the convergence
of the rotation radius of these images in Figures 3.7 and 3.8. Here the Gaussian copula
method obtain the transformation values as ( -2.0078 -2.0156 -0.4999) and (-1.9989
-1.9996 -0.4999) for A4 and P4 images respectively. Note th a t in Figure 3.10 we can
56

see the reference and test MRI images along with the rectangular region of interest
(ROI) of coordinates: [xx=30, $fi=30, X2=225, y2=225].

Figure 3.9: Registered image of the 4th aerial image.(first row left: the reference
image, first row right: the test image, second row left: the registered test image,
second row right: the overlap of registered test image on the reference image)

57

Figure 3.10: Registered image of the 4th MRI image.(first row left: the reference
image with ROI, first row right: the test image with ROI, second row left: the
registered test image with ROI, second row right: the overlap of registered test image
on the reference image)

In image registration algorithms one im portant concern is the speed of the al­
gorithms. The computer used in this research is an Intel Core i3 with 6 GB RAM
on Windows 7 operating system. In this thesis the average speed of the performed
algorithms are as follows:
Joint histogram: 29.494 seconds
Clayton:7663.788 seconds
58

Frank: 6025.555 seconds
Marshal:5995.018 seconds
Gaussian: 7.900 seconds
The Gaussian copula performs faster than the other methods. The reason is th at
the major time consuming process of copula based algorithms are the Kendall’s Tau
calculation and Gaussian copula uses Pearson product correlation which is faster th at
Kendall’s Tau. The reason for Kendall’s ta u ’s slow performance is the large size of
the data. The data size in this thesis are 256 x 256 =65536 for each MRI image and
188 x 188 = 35344 and 177 x 177 = 31329 for each aerial image. Also aerial images
take longer to process as they have more intensity value variations. Hence one of the
future research topics can be the use of methods, in order to improve the speed of the
Kendall’s tau calculations.

59

C hapter 4

Sum m ary and C on clu d in g R em arks

Image registration is an ongoing area of research.

There exists many crucial ap­

plications related to image registration and improvement of the image registration
algorithm is an achievement.
In this thesis we tried to chose a particular algorithm of image registration. This
particular algorithm consists of multi-view image, area-base algorithm, mutual infor­
mation and divergence measure based alignment measure, rigid-body transformation
and Downhill simplex optimization. We aim to monitor the performance of align­
ment measure. Here the performance of five alignment measures namely Clayton
copula, Frank copula, Marshal-Olkin copula, Gaussian copula and joint histogram are
monitored for the area based rigid body image registration. We only monitored the
performance of the alignment metric function, where all other parameters were equal
in experiments. For instance in the beginning of the research, to simulate the test
images from the reference images we added random noise to the test images. The
random noise may affect the equality of the experiments in order to compare the per­
60

formance of the alignment metric functions in image registration. Hence we did not
add random noise and only performed geometrical translation and rotation on the
reference images, in order to simulate the multi-view test images. The following are
the main conclusions we could observe in this thesis:

1. We know that each copula family follows some constraints about the copula
parameters used in the copula density functions, but for image registration we need
to perform the experiments as we did in this thesis and see which copula function
may suit the image registration algorithm. For the tested images and the particular
image registration algorithm, the Gaussian and Frank copulas performed better than
other tested copula functions.

2. The usage of copulas in image registration indeed is a new topic and this re­
search may be continued in order to obtain faster and more accurate methods.

3. Slow algorithm was the main drawback of this research and in the core of it the
Kendall’s tau calculation, was time consuming due to the large size of the images. The
slow operation of the Kendall’s tau is mainly due to the large size of data. The size of
MRI images used were 256 X 256 = 65536 for each image and the size of aerial images
used were 188 x 188 = 35344 and 177 x 177 = 31329 for each image. The Gaussian
copula performed faster than other copulas, due to its usage of Pearson correlation
instead of Kendall’s Tau rank correlation. Hence applying methods, in order to make
faster copula based image registration algorithm, might be a topic of future research.

61

4.

As this is the beginning of the copula based image registration, these methods

must be performed on the real life application such as 3 dimensional medical images
and large aerial images.

Here a version of copula based divergence and mutual information based image
registration algorithm was assembled in which four copula functions namely, Clayton,
Frank, Marshal-Olkin and Gaussian copulas were used as alignment measures. These
algorithms were tested on 16, two-dimensional gray-scale MRI and aerial images. In
this combination the Frank and Gaussian copulas performed better than other copula
functions. There are other copula functions th a t may be used in image registration
algorithms. Also as we mentioned in chapter one, there are various image registration
algorithms. Hence this research may be continued with other combinations of image
registrations and other copula families in order to improve the speed and accuracy of
the image registration outcome and they may be compared with the present thesis.

62

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[22] W. Hoeffding, Masstabinvariance korrelationsmasse, Schriften des Mathematischen Instituts fu r Angewandte Mathematik der Universitat Berlin, 5, 3, 179-233,
1998.
[23] J.C. Lagrias, J.A. Reeds, M.H. Wright, P.E. Wrighht, Convergence Properties
of the Nelder Mead Simplex in Low Dimensions, SIAM J. OPTIM Vol 9, No 1 ,
112-147, 1998.
[24] http://www.satimagingcorp.com/gallery-quickbird.html.
[25] http://www.vuse.vanderbilt.edu/ image/registration
[26] William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J,

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Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare
(http://ocw .m it.edu), Massachusetts Institute of Technology.

66

A p p en d ix A

Software L icence and V ersions

A .l

M ATLAB

MATLAB Version 7.7.0471 (R2008b) September 17, 2008
Licence Number: 257533.

A .2

M ap le

Maple 13.01 Wednesday, July 8, 2009
Product Build ID 413217
Single User Profile
Licensed to: University of Northern British Columbia
Serial Number: YVKYTN55WTAZNVUF
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A p p en d ix B

D escrip tion o f Im ages in
E xp erim en ts

B .l

MRI

14th slice of patients 1 to 8 MRI images with gray scale intensity values and JPEG
format and 256 x 256 resolution.

B .2

A erial

Al: Location: Ukraine, Acquisition date: 9-Oct-2002, Sensor: Quick Bird
A2: Location: Greece, Acquisition date: 12-May-2004, Sensor: Quick Bird

68

A3: Location: Peru, Acquisition date: 9-Jan-2005, Sensor: Quick Bird
A4: Location: Guam, Acquisition date: 9-July-2002, Sensor: Quick Bird
A5: Location: Iran, Acquisition date: 26-June-2009, Sensor: Quick Bird
A6: Location: Russia, Acquisition date: 9-June-2004, Sensor: Quick Bird
A7: Location: Bahamas, Acquisition date: 27-Dec-2003, Sensor: Quick Bird
A8: Location: Saudi Arabia, Acquisition date: 30-Dec-2005, Sensor: Quick Bird
These aerial images are with gray scale intensity values and JPEG format. A l ,A2,A3,A4,A5,A6
and A8 are with 188 x 188 resolution and A6 is with 177 x 177 resolution.

69