magnetic resonance image segmentation engineering project report for b.sc. degree

MAGNETIC RESONANCE IMAGE

SEGMENTATION

by

Mehmet Serdar TEKE

Engineering Project Report

Yeditepe University

Faculty of Engineering and Architecture

Department of Electrical and Electronics Engineering

Istanbul, 2012

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MAGNETIC RESONANCE IMAGE

SEGMENTATION

APPROVED BY:

Assoc. Prof. Dr. Cem ÜNSALAN ……………………

(Supervisor)

Prof. Dr. Canbolat UÇAK ……………………

Assoc. Prof. Dr. Cahit CANBAY …………………….

Assist. Prof. Dr. Esin ÖZTÜRK IŞIK …………………….

DATE OF APPROVAL: 09.01.2012

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS ……………………………………………………………. vi

ABSTRACT ……………………………………………………………………............. vii

ÖZET …………………………………………………………………………….............viii

LIST OF FIGURES …………………………………………………………………….. ix

LIST OF SYMBOLS/ABBREVIATIONS …………………………………………….. xii

1. INTRODUCTION …………………………………………………………………….. 1

2. MAGNETIC RESONANCE IMAGING BASICS ……………………………... 2

2.1. Description of the Magnetic Resonance Imaging System ……………………... 2

2.2. The Working Principles of a Magnetic Resonance Imaging System ……………... 5

2.2.1. Main Magnetic Field …………………………………………………………….. 5

2.2.2. Radio-frequency Field …………………………………………………………….. 6

2.2.2.1. Longitudinal Relaxation …………………………………………………….. 6

2.2.2.2. Transverse Relaxation …………………………………………………….. 8

2.2.3. Linear Gradient Fields …………………………………………………………….. 10

2.3. Techniques Used in MR Imaging Systems …………………………………….. 11

2.3.1. Slice Selection …………………………………………………………………….. 11

2.3.2. Frequency Encoding …………………………………………………………….. 12

2.3.3. Phase Encoding …………………………………………………………….. 13

2.3.4. k-Space Formation …………………………………………………………….. 14

2.3.5. Data Acquisition …………………………………………………………….. 15

3. PREFILTERING PHASE BEFORE SEGMENTATION ……………………………... 18

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3.1. Bilateral Filtering …………………………………………………………….. 18

3.2. Median Filtering …………………………………………………………………….. 22

4. IMAGE SEGMENTATION …………………………………………………….. 26

4.1. Thresholding …………………………………………………………………….. 26

4.2. Threshold Detection Methods …………………………………………………….. 27

4.3. Multi-thresholding …………………………………………………………….. 30

5. COMBINATION OF PREFILTERING AND SEGMENTATION FOR FINAL

DESIGN ……………………………………………………………………………. 33

5.1. Graphical User Interface …………………………………………………………….. 33

5.2. Tests Applied to Different MR Images using Prepared GUI ……………………... 34

5.2.1. Tests on T1 Contrasted MR Image of Brain …………………………………….. 34

5.2.1.1. Bilateral Filtering …………………………………………………………….. 35

5.2.1.2. Median Filtering …………………………………………………………….. 39

5.2.2. Tests on MRA Image …………………………………………………………….. 40

5.2.2.1. Bilateral Filtering …………………………………………………………….. 40

5.2.2.2. Median Filtering …………………………………………………………….. 41

6. CONCLUSION …………………………………………………………………….. 43

REFERENCES …………………………………………………………………….. 44

APPENDIX A: THE SOURCE CODE FOR THE EFFECT OF WIDTH CHANGE

ON GAUSSIAN DISTRIBUTION IN MATLAB …………………………………….. 46

APPENDIX B: THE SOURCE CODE FOR BILATERAL FILTERING IN

MATLAB ……………………………………………………………………………. 47

APPENDIX C: THE SOURCE CODE FOR COMPARING HISTOGRAM SHAPES

OF ORIGINAL IMAGE AND BILATERAL-FILTERED IMAGE IN MATLAB……... 49

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APPENDIX D: THE SOURCE CODE FOR MEDIAN FILTERING AND

HISTOGRAM COMPARISON BETWEEN ORIGINAL IMAGES AND

MEDIAN-FILTERED IMAGES IN MATLAB …………………………………….. 50

APPENDIX E: THE SOURCE CODE FOR SIMPLE THRESHOLDING IN

MATLAB ……………………………………………………………………………. 51

APPENDIX F: THE SOURCE CODE FOR IMAGE SEGMENTATION WITH

DIFFERENT THRESHOLD VALUES DEPENDING ON ITS HISTOGRAM IN

MATLAB ……………………………………………………………………………. 52

APPENDIX G: THE SOURCE CODE FOR OPTIMUM THRESHOLDING USING

THE IMAGE HISTOGRAM …………………………………………………………….. 54

APPENDIX H: THE SOURCE CODE FOR MULTI-THRESHOLDING IN

MATLAB ……………………………………………………………………………. 55

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ACKNOWLEDGEMENTS

Associate Professor Cem ÜNSALAN, who is my supervisor for this project,

showed me the ways to follow for reaching the purpose of this project. Without his

tremendous efforts to help me, this project would not have been materialized. Also,

Assistant Professor Esin ÖZTÜRK IŞIK from Biomedical Engineering helped me

understand important concepts of magnetic resonance imaging needed for the background

of this project.

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ABSTRACT

In this project, segmentation of various regions from an MR image showing the

anatomy of human body is executed. The problem to segment these regions is that intensity

values of different tissues in an MR image get through each other at region boundaries.

Therefore, two types of image filtering techniques which are bilateral and median filtering

are applied before segmentation procedure. Multi-thresholding is used as a segmentation

method after filtering procedure. Image eroding is executed to get rid of irrelevant regions

usually appearing as little dots after segmentation takes place. As a result, multi-

thresholding process after bilateral filtering is applied makes a slightly better job to

differentiate regions in an MR image than after median filtering is applied. However,

median filtering should also be useful before segmentation process as it is computationally

faster than bilateral filtering.

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ÖZET

Bu projede, MR imgelerindeki çeşitli bölgelerin bölütleme işlemi kullanılarak

birbirlerinden ayrılması hedeflenmiştir. MR imgelerindeki farklı dokular farklı seviyedeki

gri tonlarına sahip olduklarından bu işlem uygulanabilmektedir. Ancak, bu dokuların gri

tonları kesiştikleri bölümlerde birbirlerine çok yaklaştıklarından bölütleme işleminden

önce MR imgesinin filtrelenmesi gerekmektedir. Medyan ve çift yönlü imge filtreleme

yöntemleri bu amaç için kullanılmıştır. Bu filtrelerden herhangi biri uygulandıktan sonra

bölütleme işlemi filtrelenmiş imgenin üstünde denenmiştir. Bölütleme yöntemi olarak

çoklu eşik uygulaması kullanılmıştır. Bu işlemin sonunda elde edilen imgeye, istenen

dokuyla alakası olmayan ve genellikle küçük noktalar şeklinde beliren noktaları ortadan

kaldırmak için imge aşındırma işlemi uygulanmıştır. Sonuç olarak, çoklu eşik

uygulamasından önce kullanılan çift yönlü filtrenin medyan filtreden biraz daha iyi bir

sonuç verdiği ortaya çıkmıştır. Ancak, medyan filtre de bu uygulama için kullanışlı olabilir

çünkü bilgisayar ortamında çift yönlü filtreden daha hızlı bir sonuç vermektedir.

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LIST OF FIGURES

Figure 1. The magnetic field created in a solenoid ……………………………... 3

Figure 2. Gradient coils in the magnetic resonance imaging system ……………... 3

Figure 3. The whole magnetic resonance imaging system ……………………... 4

Figure 4. Longitudinal relaxation …………………………………………….. 7

Figure 5. Longitudinal relaxation plots for fat and blood ……………………... 8

Figure 6. Transverse relaxation …………………………………………………….. 9

Figure 7. Transverse relaxation plots for fat and blood ……………………... 9

Figure 8. Selective excitation technique …………………………………….. 12

Figure 9. Phase encoding and frequency encoding gradient waveforms ……... 13

Figure 10. Representation for the 2D Fourier transformation of k-space ……... 14

Figure 11. The relationship among the gradient field, the spatial frequency, and the

duration of the gradient …………………………………………….. 15

Figure 12. Gradient waveform, the corresponding change in spatial frequency, and

the echo obtained at the end of the process ……………………………... 16

Figure 13. Total magnet field composed of the static field B0 and the gradient

fraction …………………………………………………………….. 17

Figure 14. Effect of width on Gaussian distribution ……………………………... 19

Figure 15. Original MR Image and its output image after applying bilateral filter….. 20

Figure 16. Histogram shape analysis for a T2 weighted brain MR image for bilateral

filtering …………………………………………………………….. 21


filtering …………………………………………………………….. 22

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Figure 18. Median filtering edge preserving property ……………………………... 22

Figure 19. Original MR Image and its output image after applying median filter……23

Figure 20. Histogram shape analysis for a T2 weighted brain MR image for median

filtering …………………………………………………………….. 24


filtering …………………………………………………………….. 24

Figure 22. Simple thresholding where (a) is the original T2 contrasted MR image,

(b) is the gray-matter and cerebrospinal fluid segmentation of the

original MR image, (c) is the white-matter segmentation of the original

MR image …………………………………………………………….. 27

Figure 23. Bimodal histogram example …………………………………………….. 28

Figure 24. Image segmentation using different threshold values ……………... 29

Figure 25. Zoomed histogram in Figure 15 between the intensity values 60 and 189

showing the lowest-frequency intensity with a line at intensity 150……... 30

Figure 26. Optimum thresholding using the image histogram ……………………... 30

Figure 27. A T1 contrasted brain MR image and its histogram distribution ……... 31

Figure 28. T1 contrasted MR image of brain and its histogram with thresholding

intensity levels …………………………………………………….. 32

Figure 29. Multi-thresholding of T1 contrasted MR image of brain ……………... 32

Figure 30. GUI prepared for final design …………………………………….. 33

Figure 31. Selection of test image …………………………………………….. 34

Figure 32. Selected image shown in GUI with its histogram ……………………... 35

Figure 33. Bilateral filtered image and its histogram in GUI ……………………... 35

Figure 34. CSF segmentation from T1 contrasted MR image of brain before and

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after filtering …………………………………………………………….. 36

Figure 35. Gray matter segmentation from T1 contrasted MR image of brain before

and after filtering …………………………………………………….. 36

Figure 36 White matter segmentation from T1 contrasted MR image of brain before

and after filtering …………………………………………………….. 37

Figure 37. Final CSF extraction after applying erosion ……………………………... 37

Figure 38. Final gray matter extraction after applying erosion ……………………... 38

Figure 39. Final white matter extraction after applying erosion ……………………... 38

Figure 40. CSF extraction from T1 contrasted MR image of brain before and after

median filtering …………………………………………………….. 39

Figure 41. Gray matter extraction from T1 contrasted MR image of brain before

and after median filtering …………………………………………….. 39

Figure 42. White matter extraction from T1 contrasted MR image of brain before

and after median filtering …………………………………………….. 40

Figure 43. Vessel extraction from MRA image before and after bilateral filtering….. 41

Figure 44. Vessel extraction from MRA image before and after median filtering…... 42

xii

LIST OF SYMBOLS/ABBREVIATIONS

B Magnetic Field Strength

dl Differential Length

µ Permeability

I Electrical Current

ω Angular Resonance Frequency

f Resonance Frequency

γ Gyromagnetic Ratio

Mz Magnetic Moment along Direction z

M0 Initial Magnetic Moment

t Time

T1 Longitudinal Relaxation Time

T2 Transverse Relaxation Time

Gx Gradient Field Strength along Direction x

B0 Main Magnetic Field Strength

Bt Total Magnetic Field Strength

GRO Readout Gradient Field

GPE Phase Encoding Gradient Field

kRO Spatial Frequency in Frequency Encoding Direction

kPE Spatial Frequency in Phase Encoding Direction

xiii

MR Magnetic Resonance

RF Radio Frequency

NMR Nuclear Magnetic Resonance

MATLAB Software Development Package of the Mathworks Inc. Standing for Matrix

Laboratory

CSF Cerebrospinal Fluid

GUI Graphical User Interface

MRA Magnetic Resonance Angiography

1

1. INTRODUCTION

In this project, it is aimed to extract certain areas from various MR images. For

instance, MR image of a brain consists of three major parts which are white matter, gray

matter, and cerebrospinal fluid. Each of these regions has a different range of intensities.

However, their intensity ranges get through each other at the boundaries of the image.

Therefore, some filtering is needed to differentiate these regions.

Two types of filters are used to considerably differentiate regions included in an

MR image. These filters are bilateral filter and median filter which are to be explained in

the upcoming parts of this report. The reason why these two filters are used before

segmenting regions from an MR image is that they smooth image while preserving edges.

Smoothing makes intensity values of the same regions in an image closer to each other and

preserving edges makes intensity values at the region boundaries in an image

differentiable. After applying one of these filters, segmentation process takes place. For the

segmentation process, histogram dependent multi-thresholding technique is used. This

method is to be explained in more details in the upcoming parts of this report. Erosion

process takes place after segmentation to get rid of some irrelevant regions appearing as

little dots. Final image showing extracted region is obtained after erosion process.

As MR images showing human anatomy are input images for this project, some

background information is to be given at the beginning of this report for the understanding

of why different anatomical regions have different intensity values in an MR image. Also,

techniques used to fill an MR image up are to be explained in this background.

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2. MAGNETIC RESONANCE IMAGING BASICS

2.1. Description of the Magnetic Resonance Imaging System

Magnetic resonance imaging is a powerful noninvasive medical imaging modality

that uses a large magnetic field, gradient coils and radio-frequency coils [2].

The large magnetic field is created by a large electric current depending on the

Ampere’s law which states that for any closed loop path, the sum of length elements times

the magnetic field in the direction of length element is equal to the permeability times the

electric current enclosed in the loop [1].

(1)

As it is seen in Equation 1, the magnitude of the magnetic field, that is B, is directly

proportional to the amount of the current passing through the wire that is I. In this

equation, µ is the permeability of the medium which is constant and dl is the differential

length of the wire [1].

The large magnet that surrounds the whole MR device might be considered as a

large solenoid which creates a constant magnetic field inside itself. In Figure 1, shape of a

solenoid is illustrated.

3

Figure 1. The magnetic field created in a solenoid [1]

Gradient coils in the magnetic resonance imaging system are used to localize the

signal coming from body. Since main magnet produces a magnetic field in only one

direction inside itself as seen in Figure 1, gradient coils are necessary to construct an image

of body. There are three gradient coils in the magnetic resonance imaging system as the

world is a three-dimensional environment. Gradient coils must work in harmony to obtain

the image. For instance, one of three gradient coils is used to select the slice of body and

the other two gradient coils are used to fill the image up by using detected signals [2].

Figure 2. Gradient coils in the magnetic resonance imaging system [11]

4

When we take a look at Figure 2, it is seen that configuration of the gradient coils is

such that each gradient coils can create a magnetic field along one direction. Moreover,

they are named depending on the direction along which they produce the magnetic field.

For example, Z coil in this figure creates a magnetic field along z-direction when it is

active [2].

Radio-frequency coils are used to excite the protons which are hydrogen atoms in

body. Since the body is mostly composed by water which has the chemical formula of

H2O, hydrogen is the most abundant element in body. That is why magnetic resonance

imaging can be applied in clinical applications. The excitation of protons is to be explained

in details in the upcoming section of the report [2].

Figure 3. The whole magnetic resonance imaging system [12]

Figure 3 shows all components in the magnetic resonance imaging system. The

most important three parts of this system; which are magnet, gradient coils, and RF coil;

5

have already been explained. The other parts are used to process the signal coming from

body.

2.2. The Working Principles of a Magnetic Resonance Imaging System

Magnetic resonance imaging is based on the principles of nuclear magnetic

resonance that is a spectroscopic technique used by scientists to obtain microscopic

chemical and physical information about molecules. In the magnetic resonance imaging

system, the water molecule or simply the hydrogen element that consists of only one

proton is utilized according to nuclear magnetic resonance [3].

Atoms that have an odd number of protons possess a nuclear spin angular

momentum, and therefore exhibit MR phenomenon according to NMR physics. These

protons can be visualized as spinning charged spheres that give rise to a small magnetic

moment because of the fact that moving point charges, such as protons and electrons,

produce complicated but well known magnetic fields that depend on the charge, velocity,

and acceleration of particles. Since hydrogen, with a single proton, is the most abundant in

biological specimens, hydrogen based NMR is mostly used as the magnetic resonance

imaging technique [2].

2.2.1. Main Magnetic Field

The main magnetic field is created by a large solenoid discussed in the section 2.1.

In the absence of an external magnetic field, the spins are oriented randomly and the net

macroscopic magnetic moment is zero. However, in the presence of an external magnetic

field created by the large solenoid, magnetic moment vectors of spins tend to align in the

direction of magnetic field creating a net magnetic moment and the nuclear spins exhibit

resonance at a frequency which is called as the Larmor frequency [2].

http://en.wikipedia.org/wiki/Point_particle

http://en.wikipedia.org/wiki/Electron

6

(2)

(3)

In Equation 2, ω is the Larmor frequency in radians, γ is the gyromagnetic ratio,

that is a known constant unique for each type of atom, and B is the applied magnetic field

in the large solenoid. In Equation 3, f is the Larmor frequency in Hertz. According to both

equations, Larmor frequency is directly proportional to applied magnetic field. The Larmor

frequency is an important concept to excite the protons. The RF signal sent from the RF

coil must be at the Larmor frequency to be able to image protons and therefore body [2].

2.2.2. Radio-frequency Field

RF pulse in the magnetic resonance imaging system is to excite protons in the

biologic specimen. It is turned on for a short time and then turned off. After it is turned off,

relaxation process starts to begin.

2.2.2.1. Longitudinal Relaxation

Before applying the radio-frequency field, protons have a precession movement and

align in the z-direction due to main magnetic field. When a 900 of RF pulse is applied to

those protons, they start to lose magnitude in z-direction and gets closer to x-y plane as

time passes. After enough time passes, they reach x-y plane and rotate on x-y plane with

zero magnitude along z-direction since the net magnetic moment is rotated by 900 using RF

pulse. After RF pulse is removed, net magnetic moment of protons starts to rise on the z-

direction. This procedure is described as longitudinal relaxation or T1 relaxation. The

reason why it is called longitudinal relaxation is that the longitudinal axis is used for z-axis

[2].

7

(4)

The Equation 4 is obtained by using the Bloch equation, which is used to create

mathematical models for the signal acquired through the magnetic resonance imaging

processes, for a 900 of RF pulse. The Bloch equation is also given in Equation 5 [2].

(5)

Figure 4. Longitudinal relaxation [13]

In Figure 4, movement of net magnetic moment is illustrated on the left hand side

of figure after turning off RF pulse. In other words, process of longitudinal relaxation is

schematized on the left side of this figure. On the right hand side of Figure 4, change in net

magnetic moment vector on z-axis is plotted versus time according to Equation 4.

Each tissue of human body has a different T1 which enables us to image body. In

order to explain this concept, an example might be useful. T1 of fat is 180ms while T1 of

blood is 800ms. According to these values, fat has a greater magnetic moment on z-axis

than blood which means that, at a specific time for both tissues, fat has a greater intensity

value on z-axis than blood. Thus, in T1 contrasted MR image, fat is brighter than blood. To

be able to increase the intensity difference between fat and blood for T1 relaxation, signal

8

should be acquired when their difference is at its highest value. Figure 5 shows the T1

relaxation plots for both tissues and their difference [5].

Figure 5. Longitudinal relaxation plots for fat and blood [5]

2.2.2.2. Transverse Relaxation

Before applying radio-frequency field, protons have a precession movement and

align in z-direction due to main magnetic field. When a 900 of RF pulse is applied to those

protons, they start to lose magnitude in z-direction and gets closer to x-y plane as time

passes. After enough time passes, they reach x-y plane and rotate on x-y plane with zero

magnitude along z-direction since net magnetic moment is rotated by 900 using RF pulse.

After RF pulse is removed, the net magnetic moment of protons starts to rise on the z-

direction. Thus, net magnetic moment of protons gets smaller on the x-y plane. This

procedure is described as transverse relaxation or T2 relaxation. The reason why it is called

transverse relaxation is that x-y plane is known as transverse plane [2].

(6)

The Equation 6 is also obtained by using the Bloch equation, which is given in the

Equation 5.

9

Figure 6. Transverse relaxation [13]

In Figure 6, movement of net magnetic moment is illustrated on the left hand side

of figure after turning off RF pulse. In other words, process of transverse relaxation is

schematized on the left side of this figure. On the right hand side of Figure 6, change in net

magnetic moment vector on x-y plane is plotted versus time according to Equation 6.

Each tissue of human body has a different T2 which enables us to image body. In

order to explain this concept, an example might be useful. T2 of fat is 90ms while T2 of

blood is 180ms. According to these values, fat has a smaller magnetic moment on x-y

plane than blood which means that, at a specific time for both tissues, fat has a lower

intensity value on x-y plane than blood. Thus, in T2 contrasted MR image, fat is darker

than blood. To be able to increase the intensity difference between fat and blood for T2

relaxation, signal should be acquired when their difference is at its highest value. Figure 7

shows the T2 relaxation plots for both tissues and their difference [5].

Figure 7. Transverse relaxation plots for fat and blood [5]

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2.2.3. Linear Gradient Fields

If applied magnetic field is only main static field in z-direction, then all spins

possess the same resonance frequency that is the Larmor frequency shown in Equation 2.

When excited by RF coils, they behave like oscillators inducing signals at that frequency.

As the excitation RF coil encompasses entire region of interest, it is not possible to excite a

selected portion of volume nor is possible to distinguish the signals generated from

different spatial locations if only main magnetic field exists [2].

In magnetic resonance imaging, spatial localization is achieved by applying linear

gradient fields in addition to main magnetic field that is B0. For instance, if a gradient field

is applied in x-direction that is Gx, then applied field becomes as in Equation 7 where Bt is

total magnetic field applied in the system [2].

(7)

It should be noted that this gradient field leaves combined magnetic field still

pointing in the z-direction but now the field strength varies with x-location. Thus, Gx can

be defined as in Equation 8 where Bz is magnetic field applied along z-direction [2].

(8)

Thus, frequency of spins becomes a function of x-location as given in Equation 9

where ω0 is the Larmor frequency equal to γB0.

(9)

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Since oscillation frequency is a function of x-location, spins in one x-location are

excited when RF pulse is applied as it is needed to match the Larmor frequency to excite

spins. Therefore, localization of signal acquired is succeeded in the process [2].

The idea of creating an inhomogeneous field with a gradient on purpose plays a

vital role in MR imaging. By making magnetic field inhomogeneous, resonant frequency

of spins changes instantaneously when magnetic field strength changes enabling us to

localize incoming signal [2].

2.3. Techniques Used in MR Imaging Systems

2.3.1. Slice Selection

According to Larmor equation given in Equation 2, excitation frequency is directly

proportional to applied magnetic field. This can be utilized to selectively excite distinct

parts of body. A gradient field in z-direction, Gz, causes a linearly varying magnetic field

along z-axis and a defined slice can be excited using an RF pulse of certain frequency

width shown in Figure 8 [2].

By either increasing gradient strength or decreasing RF bandwidth, slice thickness

can be reduced. As gradient strength is increased, slope of changing magnetic field is

increased causing the reduction in slice thickness. Similarly, decreasing RF bandwidth

causes slice thickness to become smaller as seen in Figure 8 [4].

In magnetic resonance imaging, slice selection allows for a two dimensional

imaging by making slice as small as possible. As thickness of slice becomes smaller, error

in two-dimensional image gets smaller. To make error zero, slice must be at a unique point

on z-axis, not in a width on it, which is possible with an infinite gradient field which is not

possible.

12

Figure 8. Selective excitation technique [4]

2.3.2. Frequency Encoding

After applying slice selection, encoding of spatial information is to be applied in

two dimensions. This is succeeded by magnetic field gradients in respective directions.

These are differentiated by time of gradient switching before or during data acquisition. In

frequency encoding, a readout gradient, GRO, is turned on during data acquisition. Thus,

gradient direction is called the readout direction [4].

GRO produces an additional and linearly varying magnetic field. Because of the

proportionality between magnetic field and frequency depending on the Larmor equation,

the latter also changes linearly. Therefore, spins at different positions emit radiation with

different frequencies which can be distinguished using Fourier transform. Each frequency

is related to a specific position on readout direction. Also, the intensity of radiation with

this frequency is proportional to the number of spins at corresponding position [4].

13

2.3.3. Phase Encoding

Phase encoding gradient, GPE, is turned on and off before data acquisition in order

to set spatial-dependent phase of spins and kPE, that is the spatial frequency in phase

encoding direction. For N points in PE direction, it is needed to have exactly N values of

kPE obtained by stepwise modification of GPE that is shown on the left hand side of Figure

9. As a result, N acquisition cycles are needed with GPE modified by a constant ΔGPE per

time step as shown in Figure 9. By using a constant gradient duration τ, kPE changes by a

value ΔkPE in each data acquisition cycle. GRO amplitude during data acquisition is the

same for entire MR imaging process. However, a continuous variation of kRO is realized by

acquiring the data points at different time points of i*Δt, where i changes from 1 to N.

Figure 9 also explains this concept. Then, all values of kRO are measured within a single

acquisition cycle. Therefore, the total imaging time is mainly defined by the number of

phase encoding steps. Considering k-space, full row filling during each cycle is achieved

which implies that the row number is defined by the number of phase encoding steps [4].

Slice selection, frequency encoding, and phase encoding are used together to build

an MR image. Each one is applied using one of gradient coils as there are three gradients

due to three-dimensional environment of the world. In summary, slice selection procedure

is applied at the beginning. Then, phase encoding and frequency encoding processes are

applied together to form k-space which is used to obtain the image by inverse Fourier

transform.

Figure 9. Phase encoding and frequency encoding gradient waveforms [4]

14

2.3.4. k-Space Formation

k-space is a two-dimensional space for the raw data matrix in MR imaging. These

data can be converted into an image using Fourier transform illustrated in Figure 10.

Imaging is associated with two reciprocal parameters which are position and spatial

frequency. An MR image is a position-dependent intensity distribution obtained using slice

selection, frequency encoding, and phase encoding. Therefore, it corresponds to the

position domain. Accordingly, the variable k describes the other domain of the Fourier pair

and it is the spatial frequency [4].

Figure 10. Representation for the 2D Fourier transformation of k-space [4]

The value kx is the number of phase cycles per meter distance from origin through

where a magnetization vector passes because of application of gradient, Gx. Thus, kx has

the unit of cycles per meter similar to the unit of frequency that is cycles per second. That

is why k is called the spatial frequency [4].

Phase changes of a magnetization vector depend on gradient magnetic field strength

and its duration. Therefore, kx is directly proportional to gradient strength and its duration

as illustrated in Figure 11 [4].

15

Figure 11. The relationship among the gradient field, the spatial frequency, and the

duration of the gradient [4]

2.3.5. Data Acquisition

The acquisition procedure is applied during positive readout gradient switching as

seen at the bottom of Figure 12. The raw data signal which is called an echo is shown in a

box at the bottom Figure 12. On the top of Figure 12, the variable kRO is drawn as a

function of time. As it is seen in this figure, kRO changes from minimum to maximum

within the acquisition interval between the time points A and B due to Equations 10 and 11

which imply that the area under gradient waveform is directly proportional to the spatial

frequency [4].

(10)

(11)

The negative start value at A is produced by turning on a negative gradient before

the acquisition. The change in magnetization vector produced by readout gradient is

different for different positions on the readout axis. Thus, it produces a net de-phasing of

the initial magnetization vector. Subsequently, turning on a positive gradient reverses these

16

phase manipulations until the initial phases are recovered by making areas of negative and

positive gradients equal to each other. At this point, kRO is equal to zero and the echo is at

its maximum. By continuing to apply the readout gradient at positive direction, it again

produces a net de-phasing and a continuously decreasing signal as an echo. The whole

process is illustrated in figure 12 [4].

Figure 12. Gradient waveform, the corresponding change in spatial frequency, and the

echo obtained at the end of the process [4]

The meaning of positive and negative gradients is illustrated in Figure 13 which

shows total magnetic field composed of the static field B0 and the gradient field fraction.

When gradient field is turned on, an additional magnetic field is applied in B0 direction

with its amplitude linearly changing along the gradient direction. The origin in this figure

is generally set as the center of object imaged using MR.

Examining Figure 13, increasing line shows magnetic field strength of positive

gradient addition to main magnetic field and decreasing line shows magnetic field strength

of negative gradient addition to main magnetic field. For positive gradient application, a

17

magnetic field anti-parallel to B0 is overlaid when x<0 and a magnet field parallel to B0 is

overlaid when x>0. For negative gradient application, a magnetic field parallel to B0 is

overlaid when x<0 and a magnetic field anti-parallel to B0 is overlaid when x>0. Thus, the

magnetization vector is static at x=0 while neighboring spins on both sides are moving

symmetrically but in different directions on x-y plane [4].

Figure 13. Total magnet field composed of the static field B0 and the gradient fraction [4]

18

3. PREFILTERING PHASE BEFORE SEGMENTATION

Filtering is the most fundamental operation of image processing and computer

vision. Basically, filtering means that the value of filtered image at a given location is a

function of the values of the input image in a small neighborhood of the same location. For

example, Gaussian low-pass filtering computes a weighted average of pixel values in the

neighborhood, in which the weights decrease with distance from the neighborhood center,

smoothing the input image [6].

3.1. Bilateral Filtering

Bilateral filtering applies smoothing on images while it preserves edges in input

image. To make both smoothing and edge preserving, a bilateral filter needs to be

nonlinear. It combines gray levels or colors depending on both their geometric closeness

and their photometric similarity and also prefers near value to distant values in both

domain and range. This process of bilateral filtering makes itself nonlinear [6].

The bilateral filter is a normalized convolution where weighting for each pixel y is

determined by the spatial distance from the center x, as wells as its relative intensity. The

spatial and intensity weighting functions f and g are typically Gaussian distributions. These

functions are multiplied together to produce the weighting for each pixel. For input image

I, output image O, and neighborhood window N, the output of bilateral filtering is defined

in Equation 12 [7].

(12)

19

By applying Gaussian distribution function for both f and g functions in Equation

12, Equation 13 is obtained, where C is the normalization constant, σd and σr are

parameters controlling the fall-off of weights in spatial and intensity domains, respectively

[8].

(13)

To understand the effect of fall-off weights, Gaussian function should be examined.

In mathematics, a Gaussian function has the form shown in Equation 14 in which x is the

variable, µ is the center, and σ is the controlling parameter for the width of the bell.

(14)

Effect of σ on Gaussian function is illustrated on Figure 14 which is created on

MATLAB and whose source code is given in appendix A. As seen in Figure 14, the width

is increased by σ.

Figure 14. Effect of width on Gaussian distribution

20

By examining back the bilateral filtering function given in Equation 13, σr controls

edge detection process of the bilateral filter. This is because the difference between

intensities of x and y gets very large on edges. Therefore, σr executes edge preserving

function of bilateral filter. Since bilateral filter computes a weighted sum of neighboring

pixels, it still executes smoothing for small intensity differences. This is because, for

similar intensity values, second part of Equation 13 becomes ineffective since I(x)-I(y) gets

equal to almost zero and σr has no effect on the filter.

Due to the fact that bilateral filters apply smoothing on images while preserving

edges, it is commonly used before segmentation procedures take place.

Figure 15 shows a brain MR image and its output obtained by applying bilateral

filter. The source code for this procedure in MATLAB is given at the end of the report in

appendix B with its corresponding function.

Figure 15. Original MR Image and its output image after applying bilateral filter

As it is seen in Figure 15, neighbor pixels having similar intensity values are

smoothed and neighbor pixels having large intensity value differences keep their

differences preserving edges.

21

It is needed to see the difference between the histograms of original image and

bilateral-filtered image. To be able to see this difference, a MATLAB code is written.

Figure 16 and Figure 17 shows this for two images and their bilateral-filtered versions.

Also, the source code for this operation in MATLAB is added to the report in appendix C.

As it seen in Figures 16 and 17, peaks are amplified after applying bilateral filters

to both images, that is the result of edge preserving for high intensity differences and

smoothing for small intensity differences due to the application of bilateral filter.

For histogram based segmentation, the change in histogram shape after applying

bilateral filter is significantly important since it makes our job easier for determining

threshold values. Also, the rate of segmentation error is reduced after applying bilateral

filter to original image because of the same reason. In the upcoming of this report,

differences between segmented original image and segmented bilateral-filtered image are

to be examined after trying another image filtering process which is the median filter.


filtering

22


filtering

3.2. Median Filtering

Median filtering is also a type of filter which smoothes the image while preserving

edges. As it calculates the median of neighboring pixels’ intensities, not the mean, it

preserves edges unlike mean filters which calculates the mean of neighboring pixels’

intensities not considering large intensity differences between the central pixel and its

neighboring pixels. However, median filter takes large intensity differences between

central pixel and neighboring pixels into consideration to preserve edges. To examine this

property of median filter, an example is useful.

Figure 18. Median filtering edge preserving property

23

Considering Figure 18, median filter’s edge preserving property is to be explained.

As seen in this figure, large intensity differences between pixels on the right of edge and

on the left of edge. While calculating the median, all numbers are sorted from the smallest

to the largest. For this example, intensities of pixels are sorted as 24-43-47-120-123-124-

125-125-127. The middle number in this sorting, which is 123, is to be new intensity value

for central pixel after applying median filter. If mean filter is applied to this window, new

intensity value for central pixel becomes 95 which is the average of all values. Therefore,

mean filter still makes smoothing at edges while median filter preserves edges and still

does the smoothing for small intensity differences.

In MATLAB, median filtering is applied to the image on which bilateral filtering is

also applied in section 3.1. The result is shown in Figure 19. Also, histogram differences

between original images and median-filtered images are shown in Figures 20 and 21. The

source code for these operations is added to the report in appendix D.

Figure 19. Original MR Image and its output image after applying median filter

24


filtering


filtering

25

Examining Figure 19, edges are preserved and smoothing is also applied for small

intensity differences. However, it is seen in that bilateral filter is more effective in this

operation than median filter comparing Figures 15 and 19. In histogram shape analysis,

this is better seen while comparing Figures 17 and 21. In this comparison, it is clearly seen

that, peaks are much more amplified and edge-neighboring pixels are much more

suppressed after applying bilateral filter in the comparison of Figures 17 and 21.

26

4. IMAGE SEGMENTATION

Image segmentation is basically an image processing algorithm which subdivides

an image into its constituent regions or objects. The level of detail to which the subdivision

is carried depends on the problem being solved. Thus, segmentation should stop when the

objects or regions of interest in an application have been detected [10].

4.1. Thresholding

Gray-level thresholding is the simplest segmentation process. Many objects or

image regions are characterized by constant reflectivity or light absorption of their

surfaces. Therefore, a brightness constant or threshold can be determined to segment

objects and background. Thresholding is computationally inexpensive and fast. It is the

oldest image segmentation method and is still widely used in simple applications [9].

Complete segmentation of an image might result from thresholding in simple

scenes. Thresholding is the transformation of an input image f to an output, or segmented,

binary image g as seen in Equation 15 where T is the threshold and i,j together represent

one pixel. According to Equation 15, all pixels in the input image are covered for

segmentation [9].

(15)

Examining Equation 15, g(i,j) = 1 for image elements of objects and g(i,j) = 0 for

image elements of background, or vice versa. As the resulting image is a binary image, it

has only black, for binary 0, and white, for binary 1, regions helping us strictly divide the

image into its constituents [9].

27

Figure 22 shows a simple thresholding of a brain MR image implemented using

MATLAB, software for engineering design. In Figure 22(a), original MR image is

visualized. In Figure 22(b), gray-matter and cerebrospinal fluid segmentation of original

image is executed and, in Figure 22(c), white-matter segmentation of original image is

succeeded. Note that the same threshold values are used in both segmentations. The source

code of this program in MATLAB is added to the report in appendix E.

Figure 22. Simple thresholding where (a) is the original T2 contrasted MR image, (b) is the

gray-matter and cerebrospinal fluid segmentation of the original MR image, (c) is the

white-matter segmentation of the original MR image

4.2. Threshold Detection Methods

Useful methods of threshold detection are based on histogram shape analysis. If an

image consists of objects of approximately the same gray-level that differs from the gray

level of background, then resulting histogram is bi-modal. Pixels of objects form one of its

peaks as pixels of background form the second peak. In Figure 23, a typical example is

shown, which is obtained using MATLAB. The chosen threshold must meet minimum

segmentation error requirements. It makes intuitive sense to determine the threshold as the

gray-level that has minimum histogram value between the two mentioned maxima [9].

28

Figure 23. Bimodal histogram example

To the image whose histogram is obtained using MATLAB in Figure 23 where

there are two maxima at intensities of around 40 and 255, different threshold values are

applied for segmentation. The result obtained is shown in Figure 24. As seen in Figure 24,

results obtained with threshold values of 50 and 200 are not good due to segmentation

errors. When the histogram of this image is analyzed using Figure 23, it is realized that

maximum points are at the intensities of about 40 and 250. Thus, these threshold values

lead to high segmentation errors. A threshold value between 100 and 175 should be ideal to

minimize the segmentation error depending on image histogram as illustrated in Figure 24

with the intensities of 100 and 150. The source code for obtaining Figures 23 and 24 is

added to the report in appendix F.

29

Figure 24. Image segmentation using different threshold values

To find the optimum threshold, which results in minimum segmentation error for

the corresponding finger-print image, intensity value that has minimum-frequency value

between two peaks is to be chosen as the threshold for the segmentation. Figure 25 is used

for this purpose. The histogram shown in Figure 23 is taken and zooming procedure on it is

performed between the intensity values of 66 and 178. Using the zoomed histogram, the

lowest-frequency intensity is determined as 150 as shown in Figure 25. Applying threshold

of 150 on MRA image for segmentation, the result in Figure 26 is obtained in which the

original image is also shown. The source code for obtaining this figure in MATLAB is

added to the report in appendix G.

30

Figure 25. Zoomed histogram in Figure 15 between the intensity values 60 and 189

showing the lowest-frequency intensity with a line at intensity 150

Figure 26. Optimum thresholding using the image histogram

4.3. Multi-thresholding

In the histogram of an image, there might be more than two peaks corresponding to

different regions. A T1 contrasted MR image might be an example for this situation.

31

Actually, three regions are included in a brain MR image. They are CSF standing for

cerebrospinal fluid, white matter, and gray matter. The histogram of a T1 contrasted MR

image is illustrated in Figure 27 where corresponding intensity levels for each tissue are

shown.

Figure 27. A T1 contrasted brain MR image and its histogram distribution

32

Two thresholds are needed to divide this kind of image into its constituent regions.

The smallest intensity levels between two peaks are chosen for minimum segmentation

error. This job is executed in MATLAB. Figure 28 shows T1 contrasted MR image and its

histogram. In Figure 29, segmented images with multi-thresholding are shown. The source

code for obtaining Figures 28 and 29 in MATLAB is added to the report in appendix H.

Figure 28. T1 contrasted MR image of brain and its histogram with thresholding intensity

levels

Figure 29. Multi-thresholding of T1 contrasted MR image of brain

33

5. COMBINATION OF PREFILTERING AND SEGMENTATION FOR

FINAL DESIGN

To be able to see the benefit of filtering before segmentation, segmentation is

needed to be executed after filtering. For this purpose, a GUI, graphical user interface, is

made using MATLAB.

5.1. Graphical User Interface

In the graphical user interface prepared, there are four major parts. The first one is

to load a sample image. Load image button is used for this purpose. This button shows the

sample image and its histogram after selecting sample image. The second part is for the

pre-filtering procedure There are two filtering processes which are bilateral filtering and

median filtering in this part. Parameters of these filtering processes are to be entered by the

user. The third part is for segmentation. There are four parameters which are smaller

threshold values and larger threshold values for original image and filtered image. These

four threshold values are to be used for multi-thresholding. The fourth and the last part is

for erosion process to get rid of the unwanted regions after segmentation applied. Complete

GUI is shown in Figure 30.

Figure 30. GUI prepared for final design

34

5.2. Tests Applied to Different MR Images using Prepared GUI

5.2.1. Tests on T1 Contrasted MR Image of Brain

T1 contrasted MR image of brain used in previous parts of the project is also used in

GUI prepared for final design.

Figure 31. Selection of test image

After pushing load image button in GUI, window in Figure 31 appears to select a

sample image. In this window, T1 contrasted MR image of brain is selected to apply the

experiments.

After selecting T1 contrasted MR image of brain from this menu, selected image is

shown in GUI with its histogram as seen in Figure 32. After this procedure, one of the

filters from pre-filtering menu is to be chosen. Therefore, two parts of this experiment

exist.

35

Figure 32. Selected image shown in GUI with its histogram

5.2.1.1. Bilateral Filtering

Before clicking on bilateral filtering button, bilateral filtering parameters should be

chosen. These parameters are explained in Section 3.1. After entering these parameters,

bilateral filtering button is to be clicked on for bilateral filtering procedure. After clicking

on this button, filtered image and its histogram are shown in GUI as seen in Figure 33.

Figure 33. Bilateral filtered image and its histogram in GUI

36

After bilateral filtering is applied to test image for pre-filtering, segmentation

procedure takes place. Two threshold values for original and filtered images are entered for

multi-thresholding depending on image histograms. After they are entered, multi-

thresholding button is pushed for segmentation. For different threshold values, results are

shown in Figures 34, 35, and 36.

Figure 34. CSF segmentation from T1 contrasted MR image of brain before and after

filtering

Figure 35. Gray matter segmentation from T1 contrasted MR image of brain before and

after filtering

37

Figure 36. White matter segmentation from T1 contrasted MR image of brain before and

after filtering

After segmentation is applied, erosion procedure takes place to remove irrelevant

parts, usually occurring as dots, from the segmented image. After erosion process is

applied, final results obtained are shown in Figures 37, 38, and 39.

Figure 37. Final CSF extraction after applying erosion

38

Figure 38. Final gray matter extraction after applying erosion

Figure 39. Final white matter extraction after applying erosion

39

5.2.1.2. Median Filtering

After entering median filtering parameters and median filtering is applied as the

pre-filtering process, multi-thresholding is applied for different threshold values depending

on image histograms. Then, erosion procedure is applied. Results obtained are shown in

Figures 40, 41, and 42.

Figure 40. CSF extraction from T1 contrasted MR image of brain before and after median

filtering

Figure 41. Gray matter extraction from T1 contrasted MR image of brain before and after

median filtering

40

Figure 42. White matter extraction from T1 contrasted MR image of brain before and after

median filtering

5.2.2. Tests on MRA Image

MRA image used in previous parts of the project is also used in GUI prepared for

final design.

5.2.2.1. Bilateral Filtering

After bilateral filtering is applied to test image for pre-filtering, segmentation

procedure takes place. Two threshold values for original and filtered image are entered for

multi-thresholding depending on image histogram. After they are entered, multi-

thresholding button is pushed for segmentation. Then, erosion procedure is applied. Results

obtained are shown in Figure 43.

41

Figure 43. Vessel extraction from MRA image before and after bilateral filtering

5.2.2.2. Median Filtering

After median filtering is applied to test image for pre-filtering, segmentation

procedure takes place. Two threshold values for original and filtered image are entered for

multi-thresholding depending on image histogram. After they are entered, multi-

thresholding button is pushed for segmentation. Then, erosion procedure is applied. Results

obtained are shown in Figure 44.

42

Figure 44. Vessel extraction from MRA image before and after median filtering

43

6. CONCLUSION

Before beginning this project, it was targeted to segment various regions from an

MR image with an acceptable rate of error. To reduce error of segmentation, bilateral and

median filters are applied as pre-filtering procedures before segmenting relevant regions

from MR image. After applying both filters, segmentation and erosion procedures take

place to obtain final segmented image.

Filtering affects segmentation procedure in a good manner. When filtered and non-

filtered images are segmented, resultant images show that there are unwanted segmented

parts for non-filtered images. These parts are usually seen as dots in irrelevant regions.

These dots are considerably reduced after applying both filters.

Both filters change the histogram of MR images making regions included in it more

differentiable for multi-thresholding, which is used as a segmentation technique in this

project, than unfiltered image. Therefore, threshold values of multi-thresholding are

determined in such a way that segmentation error especially at boundaries of regions is

reduced.

By comparing results obtained using bilateral filter to results obtained using median

filter, it is observed that bilateral filtered and segmented MR image has less irrelevant

regions than median filtered and segmented image meaning that bilateral filtering is

preferable before segmentation. However, median filtering makes a good job before

segmentation, too. As it is computationally faster than bilateral filtering, it might be useful

in applications where fast image processing is needed.

Median filtering causes some relevant regions not to be included in segmented

images while bilateral filtering makes a better job of preserving relevant parts in

segmented image than median filtering. However, difference between the results obtained

after both filters is not too much as both filters are good enough for segmentation process.

44

REFERENCES

[1] Nave, C. R., Electricity and Magnetism, Ampere’s Law, http://hyperphysics.phy-

astr.gsu.edu/hbase/hframe.html.

[2] Nishimura, D. G., Principles of Magnetic Resonance Imaging, Stanford University,

2010.

[3] Flögel, U. and Jacoby, C., Nuclear Magnetic Resonance, http://www.nmr.uni-

duesseldorf.de/sets/theory.html.

[4] Flögel, U. and Jacoby, C., MR Imaging, http://www.nmr.uni-

duesseldorf.de/sets/theory2.html.

[5] Flögel, U. and Jacoby, C., Relaxation, http://www.nmr.uni-

duesseldorf.de/sets/theory3.html.

[6] Tomasi, C. and Manduchi R., Bilateral Filtering for Gray and Color Images,

Proceedings of the IEEE International Conference on Computer Vision, pp. 839-

846, http://www.cs.duke.edu/~tomasi/papers/tomasi/tomasiIccv98.pdf, January

1998.

[7] Weiss, B., Fast Median and Bilateral Filtering, Proc. SIGGRAPH,

http://www.shellandslate.com/download/fastmedian_5506.pdf, 2006.

[8] Zhang, M., Bilateral Filter in Image Processing Master Thesis,

etd.lsu.edu/docs/available/etd-05132009-120913/unrestricted/Zhang_thesis.pdf,

August 2009.

[9] Sonka, M., Hlavac, V., and Boyle, R., Image Processing, Analysis, and Machine

Vision 3rd

Edition, Cengage-Engineering, 2007.

[10] Gonzalez, R.C. and Woods, R.E., Digital Image Processing 3rd

Edition, Pearson

Prentice Hall, New Jersey, 2008.

[11] Internet source whose link is http://www.ovaltech.ca/philyexp.html

[12] Internet source whose link is http://www.rell.com/pages/Product-Group-

Category.aspx?applicationId=197

45

[13] Internet source whose link is http://www.med.univ-

rennes1.fr/cerf/edicerf/BASES/BA004_cv_rb_9.html

46

APPENDIX A: THE SOURCE CODE FOR THE EFFECT OF WIDTH

CHANGE ON GAUSSIAN DISTRIBUTION IN MATLAB

%% THE SOURCE CODE FOR THE EFFECT OF WIDTH CHANGE ON GAUSSIAN

DISTRIBUTION

%% IN MATLAB

x=-5:0.01:5; %set variable range

sigma=[0.2,0.4,0.8,1.6,3.2]; %set width of the bell

center=0; %set center point

y1=10.*exp(-((x-center).^2)/(2.*((sigma(1)).^2)));





figure;

plot(x,y1,'r');

hold on;

plot(x,y2,'b');

hold on;

plot(x,y3,'g');

hold on;

plot(x,y4,'m');

hold on;

plot(x,y5,'k');

title('Effect of Width Change on Gaussian Distribution');

xlabel('x');

ylabel('f(x)');

sigma_legend=legend('sigma=0.2','sigma=0.4','sigma=0.8','sigma=1.6','sigma=3.2');

set(sigma_legend,'Location','NorthWest');

47

APPENDIX B: THE SOURCE CODE FOR BILATERAL FILTERING

IN MATLAB

B.1. Function Written on MATLAB for Bilateral Filtering

% bilateral_filter2 applies two dimensional bilateral filtering using

% the method outlined in:

% B = bilateral_filter2(A,W,SIGMA) performs 2-D bilateral filtering

% to the image 'A'. 'A' should be a double precision matrix of size

% NxMx1 (size NxMx1 for grayscale image) with normalized values in

% the closed interval [0,1]. The half-size of the Gaussian bilateral

% filter window is defined by W. The standard deviations of the

% bilateral filter are given by SIGMA, where the spatial-domain

% standard deviation is given by SIGMA(1) and the intensity-domain

% standard deviation is given by SIGMA(2).

%% Pre-process input and select appropriate filter.

function B = bilateral_filter2(A,w,sigma)

%% Apply bilateral filtering.

%% Pre-compute Gaussian distance weights.

[X,Y] = meshgrid(-w:w,-w:w);

G = exp(-(X.^2+Y.^2)/(2*sigma(1)^2));

% Create waitbar.

h = waitbar(0,'Applying bilateral filter...');

set(h,'Name','Bilateral Filter Progress');

%% Apply bilateral filter.

dim = size(A);

B = zeros(dim);

for i = 1:dim(1)

for j = 1:dim(2)

% Extract local region.

iMin = max(i-w,1);

iMax = min(i+w,dim(1));

jMin = max(j-w,1);

jMax = min(j+w,dim(2));

I = A(iMin:iMax,jMin:jMax);

% Compute Gaussian intensity weights.

H = exp(-(I-A(i,j)).^2/(2*sigma(2)^2));

% Calculate bilateral filter response.

F = H.*G((iMin:iMax)-i+w+1,(jMin:jMax)-j+w+1);

B(i,j) = sum(F(:).*I(:))/sum(F(:));

48

end

waitbar(i/dim(1));

end

% Close waitbar.

close(h);

B.2. Main Function Written on MATLAB for Bilateral Filtering

% Main_Program makes the use of 'bilateral_filter2' function.

% Bilateral filtering is applied by the function 'bilateral_filter2',

% that is written for 2-dimensional bilateral filtering in the second

% part of this source code

%% Reading test images

% Note that they must be double precision in the interval [0,1].

img1 = double(imread('proje3_test.jpg'))/255;

img1 = rgb2gray(img1);

%% BILATERAL FILTERING

%% Introduce AWGN into test images.

% Note that this will show the benefit of bilateral filtering.

img1 = img1+0.03*randn(size(img1));

img1(img1<0) = 0; img1(img1>1) = 1;

%% Set bilateral filter parameters.

w = 5; % bilateral filter half-width

sigma = [3 0.1]; % bilateral filter standard deviations

%% Apply bilateral filter to the image.

bflt_img1 = bilateral_filter2(img1,w,sigma);

%% Display grayscale input image and filtered output.

figure; clf;

set(gcf,'Name','Grayscale Bilateral Filtering Results');

subplot(1,2,1);

imagesc(img1);

axis image; colormap gray;

title('Input Image');

subplot(1,2,2);

imagesc(bflt_img1);

axis image; colormap gray;

title('Result of Bilateral Filtering');

drawnow;

49

APPENDIX C: THE SOURCE CODE FOR COMPARING

HISTOGRAM SHAPES OF ORIGINAL IMAGE AND BILATERAL-

FILTERED IMAGE IN MATLAB

%% THE SOURCE CODE FOR COMPARING HISTOGRAM SHAPES OF ORIGINAL

IMAGE AND

%% BILATERAL-FILTERED IMAGE IN MATLAB

%% IMAGE READING

img1=imread('proje1_test.jpg');


img3=imread('proje1_test_result.jpg');

img4=imread('proje3_test_result.jpg');

img1=rgb2gray(img1);




%% PLOTTING THE HISTOGRAM

figure;

subplot(2,2,1);imshow(img1,'DisplayRange',[]);title('ORIGINAL IMAGE',...

'FontWeight','bold');

subplot(2,2,2);imhist(img1,256);title('HISTOGRAM OF ORIGINAL IMAGE',...


subplot(2,2,3);imshow(img3,'DisplayRange',[]);title('BILATERAL-FILTERED

IMAGE',...


subplot(2,2,4);imhist(img3,256);title('HISTOGRAM OF BILATERAL-FILTERED

IMAGE',...


figure;



subplot(2,2,2);imhist(img2,256);title('HISTOGRAM OF ORIGINAL IMAGE',...


subplot(2,2,3);imshow(img4,'DisplayRange',[]);title('BILATERAL-FILTERED

IMAGE',...


subplot(2,2,4);imhist(img4,256);title('HISTOGRAM OF BILATERAL-FILTERED

IMAGE',...


50

APPENDIX D: THE SOURCE CODE FOR MEDIAN FILTERING

AND HISTOGRAM COMPARISON BETWEEN ORIGINAL IMAGES

AND MEDIAN-FILTERED IMAGES IN MATLAB

%% THE SOURCE CODE FOR MEDIAN FILTERING AND HISTOGRAM

COMPARISON

%% BETWEEN ORIGINAL IMAGES AND MEDIAN-FILTERED IMAGES IN

MATLAB

%% IMAGE READING





%% MEDIAN FILTERING

img1_median=medfilt2(img1);

img2_median=medfilt2(img2);

%% SHOWING ORIGINAL AND MEDIAN-FILTERED IMAGES

figure;



subplot(1,2,2);imshow(img2_median,'DisplayRange',[]);title('MEDIAN-FILTERED

IMAGE',...


%% HISTOGRAM COMPARISON

figure;



subplot(2,2,2);imhist(img1);title('HISTOGRAM OF ORIGINAL IMAGE',...



IMAGE',...


subplot(2,2,4);imhist(img1_median);title('HISTOGRAM OF MEDIAN-FILTERED

IMAGE',...


figure;



subplot(2,2,2);imhist(img2);title('HISTOGRAM OF ORIGINAL IMAGE',...



IMAGE',...


subplot(2,2,4);imhist(img2_median);title('HISTOGRAM OF MEDIAN-FILTERED

IMAGE',...


51

APPENDIX E: THE SOURCE CODE FOR SIMPLE THRESHOLDING

IN MATLAB

%% THE SOURCE CODE FOR SIMPLE THRESHOLDING IN MATLAB

%% IMAGE READING

img=imread('proje1_test.jpg ');

img=rgb2gray(img);

%% WHITE-MATTER SEGMENTATION BY SIMPLE THRESHOLDING

img1=img;

[m n]=size(img1);

for i=1:m;

for j=1:n;

if img1(i,j)<=80;

img1(i,j)=0;

else

img1(i,j)=255;

end

end

end

%% GRAY-MATTER SEGMENTATION BY SIMPLE THRESHOLDING

img2=img;

[m n]=size(img2);

for i=1:m;

for j=1:n;

if img2(i,j)>80;

img2(i,j)=0;

else

img2(i,j)=255;

end

end

end

%% PLOTTING THE ORIGINAL IMAGE AND SEGMENTED IMAGES

figure;

subplot(1,3,1);imshow(img,'DisplayRange',[]);

title('(a)',...


subplot(1,3,2);imshow(img1,'DisplayRange',[]);

title('(b)',...



title('(c)',...


52

APPENDIX F: THE SOURCE CODE FOR IMAGE SEGMENTATION

WITH DIFFERENT THRESHOLD VALUES DEPENDING ON ITS

HISTOGRAM IN MATLAB

%% THE SOURCE CODE FOR IMAGE SEGMENTATION

%% WITH DIFFERENT THRESHOLD VALUES

%% DEPENDING ON ITS HISTOGRAM IN MATLAB

%% IMAGE READING

img=imread('proje2_test.jpg');

img=rgb2gray(img);


figure;


title('Original Image',...


subplot(1,2,2);imhist(img,256);

title('Histogram of Original Image',...


ylabel('Frequency');

%% THRESHOLDING USING HISTOGRAM

img1=img;

[m n]=size(img1);

for i=1:m;

for j=1:n;

if img1(i,j)<=50;

img1(i,j)=0;

else

img1(i,j)=255;

end

end

end

img2=img;

[m n]=size(img2);

for i=1:m;

for j=1:n;

if img2(i,j)<=100;

img2(i,j)=0;

else

img2(i,j)=255;

end

end

end

img3=img;

[m n]=size(img3);

for i=1:m;

for j=1:n;

if img3(i,j)<=150;

img3(i,j)=0;

53

else

img3(i,j)=255;

end

end

end

img4=img;

[m n]=size(img4);

for i=1:m;

for j=1:n;

if img4(i,j)<=200;

img4(i,j)=0;

else

img4(i,j)=255;

end

end

end

%% PLOTTING THE SEGMENTED IMAGES

figure;


title('Thresholded with 50',...











54

APPENDIX G: THE SOURCE CODE FOR OPTIMUM

THRESHOLDING USING THE IMAGE HISTOGRAM

%% THE SOURCE CODE FOR OPTIMUM THRESHOLDING USING THE IMAGE

HISTOGRAM

%% IMAGE READING

img=imread('proje2_test.jpg');

img=rgb2gray(img);

%% SEGMENTATION WITH OPTIMUM THESHOLD

img1=img;

[m n]=size(img1);

for i=1:m;

for j=1:n;

if img1(i,j)<=150;

img1(i,j)=0;

else

img1(i,j)=255;

end

end

end

%% PLOTTING THE ORIGINAL IMAGE AND SEGMENTED IMAGE

figure;


title('ORIGINAL IMAGE',...



title('SEGMENTED IMAGE WITH OPTIMUM THRESHOLD',...


55

APPENDIX H: THE SOURCE CODE FOR MULTI-THRESHOLDING

IN MATLAB

%% THE SOURCE CODE FOR MULTI-THRESHOLDING IN MATLAB

%% IMAGE READING

img=imread('proje4_test.jpg ');

img=rgb2gray(img);


figure;


title('Original Image',...


subplot(1,2,2);

imhist(img,256);

title('Histogram of Original Image',...


ylabel('Frequency');

%% MULTITHRESHOLDING

img1=img;

[m n]=size(img1);

for i=1:m;

for j=1:n;

if img1(i,j)<=120;

if img1(i,j)>41;

img1(i,j)=255;

end

else

img1(i,j)=0;

end

end

end

img2=img;

[m n]=size(img2);

for i=1:m;

for j=1:n;

if img2(i,j)<=41;

if img2(i,j)>0;

img2(i,j)=255;

end

else

img2(i,j)=0;

end

end

end

img3=img;

[m n]=size(img3);

for i=1:m;

for j=1:n;

56

if img3(i,j)>120;

img3(i,j)=255;

else

img3(i,j)=0;

end

end

end

%% PLOTTING THE ORIGINAL IMAGE AND SEGMENTED IMAGES

figure;


title('ORIGINAL IMAGE',...



title('GRAY MATTER SEGMENTATION',...



title('CSF SEGMENTATION',...



title('WHITE MATTER SEGMENTATION',...


magnetic resonance image segmentation engineering project report for b.sc. degree

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