List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Niazi, M. K. K., Ibrahim M. T., Guan L., Nyström I., (2011) An Itera-tive Method for Intensity Inhomogeneity Correction based on the Gray-weighted Distance Transform of the Magnitude Spectrum. Submittedfor journal publication.

II Niazi, M. K. K., Ibrahim M. T., Guan L., Nyström I., (2011) Bias FieldCorrection Using Grey-weighted Distance Transform Applied on MRVolumes. In Proceedings of the 8th IEEE International Symposium onBiomedical Imaging: From Nano to Macro (ISBI), pp. 357-360.

III Niazi, M. K. K., Ibrahim M. T., Nilsson M. F., Sköld A.,Guan L., Nys-tröm I., (2011) Robust Signal Generation and Analysis of Rat Embry-onic Heart Rate In Vitro using Laplacian Eigenmaps and EmpiricalMode Decomposition. In Proceedings of 14th International Conferenceon Computer Analysis of Images and Patterns (CAIP), Springer LectureNotes in Computer Science, pp. 2302-2305.

IV Niazi, M. K. K., Nilsson M. F., Danielsson B. R., Bengtsson E., (2009)Fully Automatic Heart Beat Rate Determination in Digital VideoRecordings of Rat Embryos. Transactions on Mass-Data Analysis ofImages and Signals, Volume 1, Issue 2, pp. 132-146.

V Nilsson M. F., Danielsson C., Sköld A., Johansson A., Blomgren B.,Wilson J. , Niazi K. M. K., Bengtsson E., Kultima K., Webster W.S.,Danielsson B.R., (2010) Improved Methodology for Identifying the Ter-atogenic Potential in Early Drug Development of hERG Channel Block-ing Drugs. Reproductive Toxicology, Volume 29, Issue 2, pp. 156-163.

VI Niazi, M. K. K., Nyström I., (2010) A Modified Particle Swarm Opti-mization Applied in Image Registration. In Proceedings of 20th Interna-tional Conference on Pattern Recognition (ICPR), pp. 2302-2305.

For each paper, the authors are ordered according to their individual contributions. InPaper I, and Paper II, the author has had the major responsibility of method devel-opment, its implementation, and write-up. While in Paper III, all other authors weremainly involved in the data acquisition process and write-up. In Paper V, the contri-bution was mainly on the implementation.Reprints were made with permission from the publishers.

Related Work

In the process of performing the research leading to this thesis, the author hasalso contributed to the following publications.

1. Ibrahim M. T., Niazi M. K. K., Guan L., (2011) Horizontal Features basedIllumination Normalization Method for Face Recognition. In IEEE Inter-national Symposium on Image and Signal Processing and Analysis (ISPA).

2. Ibrahim M. T., Khan M. A., Alimgeer K. S., Niazi M. K. K., Taj I. A., ,Guan L., (2010) Velocity and pressure-based partitions of horizontal andvertical trajectories for on-line signature verification. Pattern Recognition,Volume 43, Issue 8, pp. 2817-2832.

3. Abela D., Ritchie H., Ababneh D., Gavin C., Nilsson M. F., Niazi M. K. K.,Carlsson K., Webster W. S., (2010) The effect of drugs with ion channel-blocking activity on the early embryonic rat heart. Birth Defects ResearchPart B: Developmental and Reproductive Toxicology, Volume 89, Issue 5,pp. 429-440.

4. Niazi M. K. K., Hedrich J., Nyström I., (2010) Image Registration Us-ing Particle Swarm Optimization Approach. Swedish Symposium on Im-age Analysis (SSBA), pp. 31-34.

5. Selig B., Niazi, M. K. K., Nyström I., (2010) Using a Ring-Shaped RegionAround the Optic Disk in Retinal Image Registration of Glaucoma Patients.Swedish Symposium on Image Analysis (SSBA), pp. 35-38.

6. Niazi M. K. K., Nilsson M. F., Danielsson B. R., Bengtsson E., (2008) FullyAutomatic Heart Beat Rate Determination in Digital Video Recordings ofRat Embryos. In Proceedings of 3rd International Conference on Advancesin Mass Data Analysis of Images and Signals in Medicine, Biotechnology,Chemistry and Food Industry (MDA), Springer Lecture Notes in ArtificialIntelligence 5108, pp. 27-37.

7. Niazi M. K. K., Bengtsson E., (2008) Measuring Heart Rate from Rat Em-bryo Videos. Swedish Symposium on Image Analysis (SSBA), pp. 35-38.

8. Khan M. A. U., Niazi, M. K. K., Khan M. A., Ibrahim M. T., (2007) En-dothelial Cell Image Enhancement using Non-Subsampled Image Pyramid.Information Technology Journal, Volume 6, Issue 7, pp. 1057-1062.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Basic image analysis concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Digital image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Image filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 2D sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Computational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Directional features and DFT . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Frequency domain filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Topics closely related to thesis contributions . . . . . . . . . . . . . . . . 25

3.1 Decimation-free directional filter bank . . . . . . . . . . . . . . . . . . . 253.2 Grey-weighted distance transform . . . . . . . . . . . . . . . . . . . . . . 303.3 Empirical mode decomposition . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Particle swarm optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Neighborhood topologies . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.2 Basic particle swarm optimization algorithm . . . . . . . . . . 37

4 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Intensity inhomogeneity correction in 2D medical images . . . . 39

4.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Intensity inhomogeneity correction in MR volumes . . . . . . . . . 40

4.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Heart beat analysis of rat embryos . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Particle swarm optimization for image registration . . . . . . . . . . 44

4.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Acknowledgements

First and foremost, all praises are due to Allah, who has given me thestrength, knowledge and patience to accomplish my goals. Without Hisblessings, mercy and grace, this dissertation would not have been possible.I am blessed to have so many people around me who have helped methroughout these years. Without their guidance and support, this thesis wouldnot have been possible. Every individual played a key role and it gives megreat pleasure to thank them all.

First of all, I would like to express my sincere gratitude to my supervisorProf. Ewert Bengsston, who kept me motivated and led me throughout thewhole process both through good and bad days. His excellent guidance,encouragement and moral support always kept me going for more.I owe my deepest gratitude to my co-supervisor, Prof. Ingela Nyström, whohas stood by me since day one and who has always been willing to discussand share her ideas with me. Her friendly and active participation has beena major boost to achieve my goals. Without any doubt, I can easily statethat she is a master in the art of supervising.Working with my co-authors, M. T. Ibrahim, M. F. Nilsson, W. S. Websterand L. Guan has been a pleasure for me. Frequent discussions with themhas always provided me with numerous valuable ideas.I am grateful to L. Wadelius who has always provided us with a home-likeenvironment and always been there when in need of any administrativehelp.Heartiest thanks to O. Eriksson, whose job to keep the computer networkup to date and running, made my life so much easier.Many thanks to my colleagues past and current, who have made this jour-ney so much more interesting and knowledgeable for me.I would also like to thank COMSATS Institute of Information Techonology,Islamabad for their partial support during my PhD studies.My indebtedness towards V. Curic and M. Gavrilovic for their friendshipand late night discussions on social and scientific subjects.My friends here and back in Pakistan, for always being in contact with me.My special thanks to Mazhar, Sohaib, Ramzan, and Murad for providingme with a learning environment that helped me stabilize my life.Saeed Shah, Hayat Ullah Khan, Aman Ullah Khan, Mansoor Niazi,Ghufran Niazi for making me feel that I am still in Mianwali, mywonderful hometown.

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Special thanks to R. Khan, M. T. Ibrahim, J. Azar, M. Gavrilovic, H. Sarveand V. Curic for helping me with the language and formulations during thethesis write-up.

None of this would have been possible if it were not for my family. Their moralsupport and patience with my tight schedule provided me with the strength andthe will to run this marathon and complete it successfully.

Uppsala, October 2011M. Khalid Khan Niazi

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1. Introduction

The field of digital image analysis enables computers to extract, modify, en-hance, and help humans understand digital images. For this reason, it has aprofound impact on many different fields, ranging from astronomy to nan-otechnology. The development of new imaging devices has revolutionized thefield of medical sciences. However, data (images/videos/volumes) acquiredwith these devices is often assessed visually, a practice which is tedious anderror prone. Image analysis provides methods and tools which help biolo-gists/clinicians make accurate assessment, diagnosis, delineation, and drawconclusions about different medical conditions. It helps in eliminating userbias and reinforces standardization of analysis among research labs.

Digital image filtering refers to modifying the pixels in an image basedon some function of a local neighborhood of the pixels [1]. Filtering is of-ten considered as a prerequisite to many image analysis tasks. It can eitherbe performed in the spatial domain or in a transformed domain. Selectionof an appropriate filter along with its parameter is a crucial task. This oftenrequires human intervention which tends to make the results highly subjec-tive and highly dependent upon user experience. The selection of appropriatefilter parameters often turn out be an optimization problem in itself. Intrinsi-cally, these filter parameters should be automatically selected depending onthe content of an image.

The main objective of this thesis is to develop image dependent filteringmethods for biomedical image analysis with an aim to extract relevant in-formation from medical images/videos/volumes with as little user interactionas possible. In this regard, two different image analysis application areas areconsidered. The methods are specifically developed for intensity inhomogene-ity correction in medical images/volumes, and for robust signal detection andanalysis from video recordings of rat embryos.

The thesis also presents an automated method to facilitate the assessmentof Glaucoma. In Glaucoma, an eye suffers from disorders associated with in-creased intraocular pressure which may lead to loss of visual function. De-tection of Glaucoma often requires ophthalmologists to assess the develop-ment/movement of blood vessels in the optic disc. To visually perform thistask is a highly subjective and difficult process. In order to facilitate this te-dious process, the thesis presents a robust method for retinal image registra-tion.

13

1.1 Thesis outlineA brief description of basic concepts in image analysis are presented in Chap-ter 2. Key concepts closely related to thesis contributions are covered in Chap-ter 3. Chapter 4 presents the contribution of the papers included in the thesis.Summary of the contributions and future directions are described in Chapter 5.

14

2. Basic image analysis concepts

To make the thesis self contained, this chapter presents some basic conceptsin image analysis.

2.1 Digital image analysisAn n-dimensional (nD) continuous image is often represented as an nD func-tion/signal f (n), where n represents the vector of independent variables [2].An nD digital image can be considered as a sampled and quantized versionof a continuous image. Digital images are stored in arrays, where each ele-ment in the array is known as a spatial element (spel). Spels are also know aspicture elements (pixels) in two-dimensional (2D) images and volume pictureelements (voxels) in three-dimensional (3D) images. As the thesis deals withdigital images, for simplicity we will use the term image and digital imageinterchangeably. In the remainder of this chapter, we will limit our discussionto 2D images for the sake of easy visualization and notational convenience.However, all these concepts can easily be generalized to nD images.

The field of digital image analysis enables computers to enhance, extract,modify, and helps humans understand digital images. This analysis can ei-ther be performed in the spatial domain or in the transformed domain. In thespatial domain, pixels are directly manipulated to perform the analysis. In thetransformed domain, an image is processed by modifying the transformed rep-resentation of an image. The analysis is performed in the domain which bestrepresents the essential characteristics of an image.

2.2 Image filteringDigital image filtering often refers to modifying the pixels in an image basedon some function of a local neighborhood of the pixels [1, 2]. It amplifiesor attenuates certain frequency components in an image. If the filter attenu-ates certain high frequencies of an image, it is commonly known as a low-pass filter. Such filters bring smoothness to an image by suppressing the highfrequency components. On contrary, high-pass filters suppress low-frequencycomponents of an image. The simplest form of filtering in the spatial domain

15

is accomplished through convolution. Convolution is defined as:

g(x,y) =a

∑s=−a

b

∑t=−b

h(s, t) f (x− s,y− t)), (2.1)

where f represents the 2D image, and h is a 2D filter having (2a+1) ·(2b+1)coefficients. In most circumstances, the coefficients of a low-pass filter are ei-ther designed to sum to unity or the convolution is followed by a normaliza-tion operation which divides the result by the sum of the coefficients. For thisreason, a low-pass filter is often realized as a weighted average, and depend-ing on the coefficients, it can be efficiently implemented in a variety of ways.However, such low-pass filters have the disadvantage that they treat each pixelin the same way irrespective of their neighborhood content. High-pass filtercoefficients usually sum to zero, making the filter sensitive to sharp changes.In case of a large number of filter coefficients, the equivalence of convolutioncan be achieved by point-wise multiplication of the filter with an image in thefrequency domain. To better understand the frequency domain and filtering inthe frequency domain, a brief discussion about 2D sinusoids and the Fouriertransform is presented in the following sections.

2.3 2D sinusoidsA 2D sinusoid has two frequencies which represent the frequency of oscilla-tion along the horizontal and vertical spatial dimensions of an image. A 2Dsinusoid can be defined as:

S(x,y) = A sin(2π(Ux+V y)+ϕ), (2.2)

where ϕ represents the phase shift which accounts for the translation of a

Figure 2.1: Cross-sectional view of a 2D sinusoid. The vertical frequency of this si-nusoid is zero.

16

peak of a sinusoid from the origin. Here the amplitude of the sinusoid S(x,y)is represented by A. U and V represent the horizontal and vertical frequen-cies of a 2D sinusoid which are measured in cycles per pixel. To make thefrequency units visually intuitive, scaled frequency units are often preferred,where frequency is defined in terms of cycles per image. An M×N 2D sinu-soid with scaled frequencies u and v can be written as:

S(x,y) = A sin(2π(uM

x+vN

y)+ϕ). (2.3)

Fig. 2.1 shows the cross-sectional view of a 2D sinusoid having only horizon-tal frequency. In Fig. 2.1, period represents the inverse of horizontal frequency.

Fig. 2.2 shows multiple sinusoids, with different frequencies, phase,and orientation (θ ). Here θ represents the direction of fastest oscillation.Fig. 2.2(a) shows a sinusoid having horizontal frequency, zero phase (as thepeak of the sinusoid coincides with the origin), and θ = 0. Fig. 2.2(b) showsanother sinusoid with θ = π

8 and ϕ = 0. Fig. 2.2(c) shows the same sinusoidas shown in Fig. 2.2(b) but with arbitrary ϕ .

(a) (b) (c)

Figure 2.2: a) A 2D sinusoid with horizontal frequency, θ and ϕ equal to zero. b) Asinusoid where θ = π/8, and ϕ = 0. c) Another example where the 2D sinusoid haveθ = π/8 along with some arbitrary ϕ .

2.4 Fourier transformFourier transform is a widely used image analysis tool that decomposes animage into superposition of sinusoids. The output of the Fourier transformis the frequency representation of the image, where each point represents aparticular frequency contained in the spatial domain image. For this reason,it is also known as the frequency domain representation of an image. Thediscussion to follow will briefly present the discrete version of the 2D Fouriertransform, i.e., the 2D discrete Fourier transform (DFT).

17

According to the 2D DFT, an image can be resolved into sinusoids of vary-ing frequencies [3, 4]:

F(u,v) =1

MN

M−1

∑x=0

N−1

∑y=0

f (x,y)e−2π j( uxM + vy

N ), (2.4)

where F(u,v) is a complex-valued function of the sinusoid passing through thewhole image f (x,y) of size M×N. Here, frequency (u,v) is defined in terms ofcycles per image for the same reason as mentioned in Section 2.3. Generally,a 2D DFT is characterised by the frequency (ω), magnitude (|F(u,v)|), phase(ϕ ), and the direction of the fastest oscillation (θ ). Mathematically, the fourcharacteristics of a 2D DFT can be expressed as:

ω =√

u2 + v2,

θ = tan−1(

vu

),

| F(u,v) |=√

ℜ(F(u,v))2 +ℑ(F(u,v))2,

ϕ = tan−1(

ℑ(F(u,v))ℜ(F(u,v))

).

(2.5)

Fig. 2.3 shows the graphical version of the first two characteristics mentioned

Figure 2.3: Fourier plane of an image.

in Eq. 2.5. The plane shows a single point whose Euclidean distance from theorigin represents the frequency ω , of a 2D sinusoid. This sinusoid will con-tribute to each pixel in the spatial domain. For this reason, it is often said that

18

the DFT provides excellent localization in the frequency domain but at thecost of poor localization in the spatial domain. Fig. 2.3 also shows the direc-tion of the fastest oscillation θ . The third characteristic in Eq. 2.5 is shownin Fig. 2.4. Fig. 2.4(a) shows a retinal image f (x,y), where Fig. 2.4(b) showsthe magnitude spectrum (| F(u,v) |) of f (x,y). The centre of the magnitudespectrum is often known as the DC component or the zeroth frequency compo-nent and it represents the average brightness in an image. The DC componentand its neighbors are known as low frequency components while componentsnear the borders represent high frequency components. Low frequency com-ponents are usually by far the largest components of an image. This makes thedynamic range of the Fourier coefficients too large to be properly displayed ona computer screen. To properly display the magnitude spectrum of an imageon a computer screen, logarithmic transformation is often followed by linearstretching [2].

(a) (b)

Figure 2.4: a) Retinal image. b) Magnitude Fourier transform of the retinal imageshown in (a).

The inverse DFT of F(u,v) can be written as:

f (x,y) =M−1

∑u=0

N−1

∑v=0

F(u,v)e2π j( uxM + vy

N ). (2.6)

In the polar form, F(u,v) can be written as:

F(u,v) = |F(u,v)|e jϕ (2.7)

where |F(u,v)| is the magnitude and ϕ represents the phase.DFT is one of the best analysis tools when the objective is to determine

the frequency content constituting an image. However, it provides no spatial

19

information about these frequencies as all these sinusoids exist throughout theimage plane.

2.4.1 Computational issuesThe separability property of the 2D DFT leads to an efficient implementationas it can be computed by taking the one-dimensional (1D) row-wise DFTfollowed by a 1D column-wise DFT. Even with these computational savings,the computational complexity of an N-point DFT is O(N2). However, thiscomputational complexity can be reduced to O(N log2N) by using the FastFourier transform.

2.4.2 Directional features and DFTDFT is one of the most common transforms in image analysis to carry outdirectional feature analysis for around three decades [5]. It is commonly be-lieved that features which appear in some particular direction ψ in an imageare localized in the DFT on a line at ψ + π

2 . Lets consider an example to un-derstand this simple concept. Suppose we have an image having two impulsesplaced horizontally opposite to each other and equidistant from the centre ofan image.

Fig. 2.5(a) shows one such image (257×257) with two impulses placed at(128,124), (128,132). Here, the image co-ordinates start from (0,0) and thecentre of the image lies in (128,128). Now if we treat this image as a DFT,the centre will represent the DC component (which has zero magnitude). Asthe impulses are horizontally placed 4 pixels apart from the DC component,they should represent a sinusoid with only horizontal frequency. This sinusoidshould have a frequency equal to 4, as the distance between these impulsesfrom the centre of an image is 4.

(a) (b)

Figure 2.5: a) A synthetic DFT image having two impulses. b) Inverse DFT of (a).

20

Fig. 2.5(b) shows the inverse DFT of the image shown in Fig. 2.5(a). Asexpected, it’s a sinusoid with only horizontal frequency. The visual inspectionreveals that the sinusoid completes 4 cycles in the image, making its normal-ized frequency equal to 4.

Now if we forget about the sinusoidal interpretation of Fig. 2.5(b) for awhile, we can interpret it as an image having vertical features. The later inter-pretation is often preferred as it is more aligned with human intuition and per-ception. So, according to this interpretation, the vertical features in Fig. 2.5(b)are localized horizontally in Fig. 2.5(a). This interpretation leads to the con-cept that directional features having direction ψ in an image plane are local-ized on a line in the DFT at ψ + π

2 .

(a) (b)

Figure 2.6: Linear features. a) Synthetic image having vertical lines. b) DFT of (a).

(a) (b)

Figure 2.7: Linear features. a) Synthetic image having both horizontal and verticallines. b) DFT of (a).

21

Fig. 2.6(a) shows an image having multiple vertical features of variousthickness. It is worth mentioning that the distance among these vertical fea-tures is not constant, which means that the DFT contains more than one sinu-soid. It is trivial that vertical features can only be created by adding differentsinusoids of horizontal frequencies (vertical features). Fig. 2.6(b) shows theDFT of Fig. 2.6(a). The DFT only consist of multiple horizontal frequencies.This also shows that the directional features at ψ in an image plane are local-ized on a line in the DFT at ψ + π

2 .Fig. 2.7(a) shows a bit complicated image having both horizontal and verti-

cal features of different thickness. Its DFT is shown in Fig. 2.7(b) which con-sists of two dominant lines. The vertical line (vertical frequency sinusoids)in the DFT represents the horizontal features in the image plane while thehorizontal line (horizontal frequency sinusoids) in the DFT represents all thevertical lines in the image plane.

By now, we know the location of low frequency components, high fre-quency components, the location of directional features in the DFT, etc., butthe question is how to extract this information? Frequency domain filteringprovides one such mechanism to extract such information.

2.5 Frequency domain filteringIn the frequency domain, filtering can be performed by point-wise multiplica-tion of the DFT of the filter h(x,y) with the DFT of an image f (x,y). It can bewritten as:

G(u,v) = F(u,v)H(u,v), (2.8)

where F(u,v) and H(u,v) are the DFT of f (x,y) and h(x,y), respectively. Thisis followed by the inverse DFT of G(u,v) to complete the filtering.

Not only filters with a large number of coefficients can be implementedquite efficiently, it also becomes trivial to predict the filter behaviour from itsDFT. For instance, Fig. 2.8(a) shows the DFT of a low-pass filter. It is trivialfrom the DFT that its point-wise multiplication with the DFT of an image willstop the high frequencies in an image. As only low frequency components ofan image will pass through this filter, for this reason it is known as a low-pass filter. A high-pass filter can be visualized as a filter which allows highfrequency components of an image to pass through.

Fig. 2.8(b) shows an orientation selective filter whose application will ex-tract features oriented around −π

4 at a certain scale. This behaviour can easilybe judged from the two oriented Gaussian like structures at an angle of +π

4 .Directional filtering can be accomplished in a number of ways. The chapter tofollow will present one such directional filtering method.

Homomorphic filtering is a common frequency domain filter to decoupleintensity inhomogeneity from an image. Intensity inhomogeneity is often re-

22

alised as a multiplicative field which induces slow spatial variations acrossthe whole image. A homomorphic filter converts the multiplicative nature ofintensity inhomogeneity into an additive one by logarithmic transformation.Once it becomes additive, the slow variational nature of intensity inhomo-geneity can be decoupled with a filter similar to a high-pass filter. Fig. 2.9shows the radial cross-section of a homomorphic filter. Here, the magnitudeof low frequencies is suppressed by γL (< 1) while the magnitude of high fre-quencies is amplified by γH (> 1).

In most circumstances these filter parameters (γL,γH) along with the cut-offfrequency are manually selected which makes the results quite subjective.

(a)

(b)

Figure 2.8: a) Magnitude frequency response of a low-pass digital filter. b) Magnitudefrequency response of an orientation selective filter.

23

H

L

Figure 2.9: Radial cross-section of a homomorphic filter.

24

3. Topics closely related to thesiscontributions

This chapter presents a detailed description of the key concepts necessary forbetter a understanding of the thesis contributions.

3.1 Decimation-free directional filter bankDirectional analysis plays a vital role in many computer vision and image pro-cessing tasks like, image compression, texture analysis, motion analysis, edgedetection, and feature extraction [6, 7, 8, 9]. Directional information fromimages can be extracted with a variety of analysis tools. The directional fil-ter bank [7] is one such tool which decomposes the Fourier transform of animage into wedge shaped maximally decimated subbands. Each subband cor-responds to global features in some particular direction in the spatial domain.However, these subbands are visually distorted due to the presence of fre-quency scrambling [10]. These subbands were later improved in [10] by shift-ing all the sampling matrices to the end of the analysis section of directionalfilter bank.

Directional filter bank was originally designed to provide image compres-sion [7]. It was later adopted for image enhancement in [11]. These subbandsoften require an interpolation prior to image enhancement as two pixels lo-cated at the same spatial location in two different subbands will not necessar-ily correspond to the same spatial region in the original image. Moreover, forbetter insight, it is important that the spatial location of a feature be preservedin its subbands.

To preserve the spatial correspondence among the pixels, Khan et al., [12]presented a decimation-free directional filter bank which decomposes theFourier transform of an image into wedge shaped directional images of thesame size as the input image.

Fig. 3.1 shows the structure of a decimation-free directional filter bank anda directional filter bank. Q is a Quincunx matrix and can be written as:

Q =

[1 1−1 1

].

(3.1)

25

Figure 3.1: Tree structure directional decomposition. a) Directional filter bank. b)Decimation-free directional filter bank.

26

1

2

3

4

56 7

8

1

2

3

4

567

8

Ho(ω1,ω2)

H1(ω1,ω2)

1

2

3

4 1

2

3

4

56 7

8

567

8

(a)

Ho(QT(ω1,ω2))

1

2

3

4 1

2

3

4

3

4

3

4

1

2

1

2

H1(QT(ω1,ω2))

(b)

Ho(RiQTQT(ω1,ω2))

1

2

1

2

H1(RiQTQT(ω1,ω2))

1

1

2

2

(c)

Figure 3.2: a) First stage of a decimation-free directional filter bank. b) Second stageof a decimation-free directional filter bank. The output of the first stage becomes aninput to the second stage. c) Third stage of a decimation-free directional filter bank.Here, the output of the second stage becomes an input to the third stage.

27

It simultaneously produces diagonal downsampling as well as rotation aroundthe centre of the image. R represents the resampling matrices in Fig. 3.1. Intotal, there are 4 resampling matrices in the whole structure of both filter bank,which are:

R1 =

[1 10 1

]R2 =

[1 −10 1

]

R3 =

[1 01 1

]R4 =

[1 0−1 1

].

They map the parallelogram shaped passband to a diamond shaped pass-band. The matrix B in the directional filter bank is the visualization matrixwhich corrects the frequency scrambling associated with each subband [10].All the resulting filters (H) are shown in Fig. 3.2. For easy visualization, thesefilters are depicted as ideal filters, however, all these filters are non-ideal in na-ture. Fig. 3.2(a) shows how the Fourier transform of an image can be decom-posed into two wedge shaped directional images with the help of filter H0 andH1. This decomposition is often termed as the first stage in decimation-freedirectional filter bank. Fig. 3.2(b) and Fig. 3.2(c) partially show the secondand third stage of a decimation-free directional filter bank, respectively.

To make the discussion self-contained, we have enlisted the steps which arenecessary to construct the prototype filters, H0 and H1.1. Construct a 1D linear phase FIR low-pass filter u, with a cut-off frequency

of π2 .

2. Construct a 2D filter by taking the tensor product of the 1D filter u, withitself. For instance, a 1D filter u = (u1,u2, . . . ,um) having m coefficients,after the tensor product can be expressed as:

x = u⊗u =

u1u1 u1u2 . . . u1um

u2u1 u2u2 . . . u2um...

.... . .

...umu1 umu2 . . . umum

where the ⊗ symbol defines the tensor product.Fig. 3.3 shows the Fourier transform of the 2D filter x constructed by usingthe tensor product.

3. This is followed by Quincunx downsampling of the 2D filter x, resultingin a filter xd . Quincunx downsampling maps the square shaped passband inFig. 3.3 to a diamond shaped passband as shown in Fig. 3.4.

28

Figure 3.3: Frequency response of a 2D low-pass filter, x.

Figure 3.4: Frequency response of a 2D filter after applying Quincunx downsamplingmatrix.

4. As a last step to construct the prototype filter pair, we modulate the 2D filterxd with π in each frequency dimension. Mathematically, it can be stated as:

h0(x,y) = xd(x,y)e j2π( πxr )

h1(x,y) = xd(x,y)e j2π( πyc )

(3.2)

r and c represents the number of rows and columns of the filter xd . The con-volution of h0(x,y) with an image will capture the vertical components ofan image, while h1(x,y) will capture the horizontal components of an im-age. It is trivial from Eq. 3.2 that this transformation will map the diamondshaped passband to an hour glass shaped passband. Fig. 3.5(a), and 3.5(b)show the Fourier transform after this transformation. It is also worth men-tioning that an application of these filters will decompose the input imageinto its directional components irrespective of their scale.

29

(a) (b)

Figure 3.5: a) Frequency response of h0. h0 will capture the vertical features in animage. b) Frequency response of h1. It will capture the horizontal features in an image.

3.2 Grey-weighted distance transformDistance transforms are one of the widely used tools in computer vision, im-age analysis and pattern recognition. A distance transform computes the short-est distance from feature elements (seeds) to all other elements in a binaryimage [13]. It results in a grey-level image where each grey-level shows thespatial distance to the nearest feature element.

Grey-weighted distance transforms typically compute minimal cost paths,where the grey-value of a pixel represents the cost of traversing the pixel [14].They provide a framework to combine spatial information with grey-level in-formation. In the discussion to follow, the definition of some key concepts willbe followed by different definitions of the grey-weighted distance transform.

An nD digital space usually refers to an nD grid space that only containsinteger points in nD Euclidean space, i.e., Zn ⊂Rn. An nD digital image f, canbe defined as a function on Zn. Let p, q ∈ Zn and Ppq represent the set of allpossible paths between pixels p and q. Let π ∈ Ppq represents a path of lengthn as an n-tuple such that π = (p = p0, p1, . . . , pn−1 = q), where pi and pi+1are adjacent pixels. Then, the length L of the path π is defined as:

L(π) =n−1

∑i=0| f (pi)− f (pi+1) | · ∥ pi− pi+1 ∥ . (3.3)

Here, ∥ · ∥ denotes the Euclidean norm. Now the grey-weighted distance be-tween p and q can be defined as:

d(p,q) =

minπ∈Ppq

L(π) if p = q

0 if p = q.(3.4)

30

This definition of distance function was utilized in [15] to extract texture in-formation from a 3D grey-level distribution.

The length L of the path π can be defined in many different ways. Theidea of grey-weighted distance was first coined by Rutovitz in [16], where thelength of the path was defined as the sum of grey-level values along the path.In [17], the length of the path between pixel p and q was defined as:

L(π) =n−1

∑i=0

f (pi)+ f (pi+1)

2· ∥ pi− pi+1 ∥ . (3.5)

This definition clearly favours the paths with low grey-level values. In [18],instead of favouring the low grey-level values, the length of the path was foundby calculating the grey-level difference between adjacent pixels along the min-imal path. It can be represented as:

L(π) =n−1

∑i=0| f (pi)− f (pi+1) |+ ∥ pi− pi+1 ∥ . (3.6)

A distance transform computed using this distance definition is termed as adistance transform on curved spaces. In [19], the distance transform on curvedspaces was combined with a nearest neighbor to produce a roughness mapshowing the average roughness of image regions. Toivanen [18] also proposeda similar distance to Eq.3.6, a weighted distance on curved spaces, which canbe written as:

L(π) =n−1

∑i=0

√| f (pi)− f (pi+1) |2 + ∥ pi− pi+1 ∥2. (3.7)

Saha et al., [20] presented a framework to compute a fuzzy distance transformfor fuzzy objects. Fuzzy distance was defined as:

L(π) =n−1

∑i=0

µ(pi)+µ(pi+1)

2· ∥ pi− pi+1 ∥, (3.8)

where µ represents the membership function of a fuzzy set. In [21], a fuzzydistance transform was utilized to accurately measure the thickness of trabec-ular bone. Svensson [22] decomposed the fuzzy objects into simpler parts withthe help of a fuzzy distance transform.

All these distance definitions are application dependent. For instance, if theobjective is to find the edges of an object, then a weighted distance on curvedspaces is an appropriate choice as it will show a sharp change in distanceimage over edge regions.

31

3.3 Empirical mode decompositionMost of the medical images and signals are often non-linear andnon-stationary in nature. For instance, the heart activity of a rat embryo whenexposed to different drugs often turns out to be non-linear and non-stationary.For better insight, this type of data is often analysed with the help of differentanalysis techniques.

Fourier transform is one of the most commonly used signal analysis tech-niques that has sinusoidal basis functions which are global in nature. Here,global explains the fact that each of these sinusoidal basis functions existthroughout the signal/image at each time/spatial location. The global nature ofsinusoidal basis functions makes the Fourier transform a weak tool to performnon-stationary signal analysis. Due to this global nature, the Fourier trans-form provides excellent frequency resolution but at the cost of time/spatialresolution, i.e., it provides information about which frequency componentsconstitute a signal/image irrespective of where these components appear inthe signal/image.

The short time Fourier transform (STFT) is often preferred over the Fouriertransform to perform non-stationary signal and image analysis. In STFT, anon-stationary signal is often divided into small chunks. It is assumed that thesignal is almost stationary in these small chunks. Each of these chunks areFourier transformed after multiplication with a window function (having thesame size as the chunk). A small window in the STFT determines excellenttime resolution but at the cost of frequency resolution and vice versa.

Wavelets are one of the most widely used mathematical tools for the anal-ysis of non-linear and non-stationary signals and images. In wavelets, themother wavelet is used as a prototype to define analysis functions (scaled ver-sions of the mother wavelet). During wavelet analysis, the mother waveletremains the same irrespective of the data. Ideally, the wavelet analysis func-tions should be adapt to the data, i.e., depending on the data, a differentmother wavelet should be for each analysis function. Selection of an appro-priate mother wavelet according to the data is a challenging problem in itself.

To overcome all these issues with the Fourier transform, STFT, andwavelets, Empirical mode decomposition (EMD) [23] automaticallygenerates data dependent decomposition without the need for defining:• basis functions,• windowing functions,• mother wavelets and scales.EMD decomposes a non-linear and non-stationary signal into a superpositionof components having a well defined instantaneous frequency. These compo-nents are commonly referred to as intrinsic mode functions (IMF). Here, in-trinsic is used to emphasise the fact that each mode represents the oscillationmode embedded in the data.

32

The data dependent decomposition ability of EMD often makes it supe-rior to the Fourier transform, STFT, and wavelets. For this reason, EMD hasfound applications in areas like oceanography, financial analysis, meteorolog-ical data analysis, and biomedical signal and image analysis, just to name afew. To have a better insight into EMD, IMF seems to be a good starting pointas they are often considered as the basic building block of EMD.

An IMF is a zero mean function whose number of local extrema and num-ber of zero crossings at most differ by one. The conventional EMD [23] usesAlgorithm 1 to extract IMFs from the data.

Algorithm 1 Empirical Mode Decomposition

1: i←−−1.2: j←− 0.3: repeat4: Find all the extrema of a given signal z(t).5: Create an upper and lower envelope, zup(t) and zlow(t), by fitting a cu-

bic spline through maxima and minima, respectively.6: Generate a mean envelope m(t) by taking the average of zup(t) and

zlow(t).7: if (z(t)−m(t)) is an im f then8: i← i+1,9: hi(t)← d(t),

10: z(t)← z(t)−hi(t),11: j←− 0.12: else13: p j(t)← z(t),14: z(t)← z(t)−m(t),15: j← j+1.16: end if17: until (z(t) is monotonic or one extremum is left)

At the end of this process, the original signal z(t) can then be reconstructed,using the following equation:

z(t) =n

∑i=1

hi(t)+ r(t), (3.9)

where n is the number of IMFs, r(t) denotes the final residue which is inter-preted as the trend term of the signal and hi(t) is an IMF. h0(t) will capturethe fastest oscillating component and later IMFs will capture the less oscillat-ing components of z(t). It is important to mention that it generally takes morethan one iteration to generate a single IMF, as the change of coordinate system

33

from Cartesian to curvilinear often produces new extrema. So, these iterationsare often controlled by some stopping criterion. In conventional EMD [23], aCauchy-type convergence criterion was proposed where iterations were con-tinued until the difference between the successive p(t) was smaller than apre-set limit. It can be defined as:

T

∑t=0

[ | p j−1(t)− p j(t) |2

p2j−1(t)

]≤ SD. (3.10)

Often the value of SD is chosen between 0.2 and 0.3. As the definition ofSD was unrelated to the definition of IMF, it was later improved by Huanget al., [24] by stopping the iterations when the number of zero crossings andextrema remained the same for S successive p(t). It was also suggested thatthe range of S should be chosen between 4 and 8.

A lot of current research is focused on improving the IMF extraction pro-cess with an aim to reduce the mode-mixing problem and to improve the per-formance of EMD. It is normally believed that the mean envelope computedusing splines makes it impossible to decompose intrinsically different modeslying within an octave [25].

3.4 Particle swarm optimizationParticle swarm optimization (PSO) belongs to the class of population basedalgorithms, where each sample represents a potential solution. Inspired fromthe social behaviour of bird flocking or fish schooling, Kennedy et al., firstintroduced the idea of PSO in [26]. In PSO, samples are known as particlesand each particle is a candidate solution in the search space. These particlesmove in the multi-dimensional search space under certain constraints. Thissearch process can be considered as a filtering process which filters out thelocal minima in the search space.

Each particle has its own memory which is used to keep track of the bestposition visited by the particle. Moreover, the movement of each particle inthe search space is affected by the neighboring particles.

Each particle updates its location in the search space based on its velocityand the current location. It is often expressed as:

xt+1i = xk

i + vk+1i , (3.11)

where xki is the current location of the particle xk

i at kth step in nD space, whilexk+1

i is the new location of the same particle. vk+1i represents the velocity of

the xk+1i particle computed using:

vk+1i j = β [vk

i + c1r1{yki − xk

i }+ c2r2{yki − xk

i }], (3.12)

34

yki is the best position found by particle xk

i since its initialization. For thisreason it is often called personal best. yk

i is the best position found by theneighborhood of particle xk

i since initialization and is referred to as neighbor-hood best. r1 and r2 are random vectors in nD space drawn from a standarduniform distribution on the interval [0,1]. These vectors are responsible forthe stochastic nature of the algorithm. vk

i represents the velocity of the particlexk

i , and prevents the particle from drastically changing its direction by provid-ing a bias towards the previous direction of the particle. β is a constrictioncoefficient to control the velocity of particles and to ensure convergence [27].It is represented as:

β =2κ

|2−ψ−√

ψ(ψ−4))|, (3.13)

where κ ∈ [0, 1] and ψ = c1 + c2. A smaller value of κ results in fast conver-gence with local exploitation, while larger values results in slow convergencebut with higher degree of exploration. κ is often set to 1 which is success-ful for most applications [27]. The remaining two parameters are often setto c1 = c2 = 2.05 [28], providing the particle with equal opportunity to beattracted towards personal best and neighborhood best.

The term c1r1{yki − xk

i } in Eq. 3.12 is often called the cognitive componentas it is responsible for attracting the particle back to its own best position.The term c2r2{yk

i − xki } in Eq. 3.12 is known as the social component as it

quantifies the performance of particle xki relative to its neighbors. The section

to follow will describe the commonly used neighborhood topologies in PSOliterature.

3.4.1 Neighborhood topologiesThe social component is mainly responsible for flow of information throughthe swarm of particles. The performance of this social interaction is governedby the neighborhood topology. In [29, 30], it was shown that neighborhoodtopology is the key factor directly influencing the performance of informationflow in the swarm. This has lead to the development of different neighbor-hood topologies over the past decade. It is important to mention that all thesestrategies are problem dependent, i.e., some work better for some problemswhile others perform better on some other types of optimization problems.In general, these can be categorized into static and dynamic neighborhoodtopologies.

3.4.1.1 Static neighborhoodThe basic PSO algorithm [26] uses a star topology in which each particle in thesearch space is considered as a neighbor to every other particle in the swarm.So, each particle adjusts its position depending on its own best position and

35

the best position found by the whole swarm. The best position found by thewhole swarm since initialization is often known as the global best (gbest).Fig. 3.6(a) presents a graphical representation of this topology.

In many situations, it results in quick convergence, but at the cost of be-coming susceptible to getting trapped in local minima. The poor performanceassociated with star topology is often related to the fact that if the gbest islocated in a different area of the search space than the global optimum, thengbest will probably make it difficult for the swarm to explore other areas ofthe search space.

(a) (b)

Figure 3.6: Star and ring topology. Particles are represented by circles, while neigh-bors are defined by arcs. a) Star topology. Each particle is connected to all particlesin the search space. b) Ring topology. Each particle is connected to its immediateneighbors.

Ring topology is often preferred over star topology. In ring topology, eachparticle communicates with its l neighbors which enables the particles to flowaround local optima, as each particle’s neighborhood explores different re-gions in the search space [31]. Each particle updates its position dependingon its own best position it has found since initialization and the best positionfound by its l neighbors since initialization. For this reason, the best positionfound by the l neighbors of any particle since initialization is often knownas local best (lbest). Fig. 3.6(b) shows the graphical representation of ringtopology.

It is evident from Fig. 3.6(b) that ring topology enforces an overlappingneighborhood. This overlapping neighborhood ensures that the information isexchanged throughout the whole swarm [32]. In ring topology, a neighbor-hood is often defined at swarm initialization. However, there is also a possi-bility to define a neighborhood dynamically but at the expense of additionalcomputational cost.

36

3.4.1.2 Dynamic neighborhoodDynamic neighborhood topology is defined as the topology in which the par-ticles change their neighbors as the algorithm progresses to explore the searchspace [33]. The dynamic neighborhood was first proposed in [34], where thealgorithm gradually increases the number of neighbors from no neighbors tothe whole swarm. In [35], the performance of lbest was compared with somedynamic strategies where particles were clustered according to their locationin the search space. The study revealed that PSO is more effective if parti-cles are attracted toward the centre of their own cluster. Alizahen et al., [36]presented a fuzzy version of PSO where neighborhood was decided basedon fuzzy c-means clustering [37, 38]. Once again, clustering was performedbased on their location in the search space. In [39], each particle was ini-tialized with a small number of neighbors to improve the exploration abilityof PSO. During the search process, the number of neighbors was graduallyincreased to bring about the exploitation ability to the algorithm. The neigh-bors were selected randomly and these neighbors were defined using directedgraphs.

3.4.2 Basic particle swarm optimization algorithmPSO is often regarded as one of the best global optimization algorithms whenit comes to both, simplicity and efficiency. It has fewer adjustable parameterscompared to the other global optimization algorithms. Algorithm 2 shows thebasic version of the PSO algorithm for global optimization.

37

Algorithm 2 Modified PSO as a Minimization Problem

1: uniformally distribute n particles in the search space2: repeat3: for each particle i = 1, ...,n do4: % set the personal best position5: if f(xi) < f(yi) then6: yi = xi;7: end if8: % set the neighborhood best position9: if f(yi) < f(yi) then

10: yi = yi;11: end if12: end for13: for each particle i = 1, ...,n do14: pi

vec = yi - xi;15: ni

vec = yi - xi;16: update the velocity using Eq. 3.12 with U(0,1)17: update the position using Eq. 3.1118: end for19: until stopping condition is true

38

4. Thesis contributions

This chapter summarises the methods and results described in the six ap-pended papers.

4.1 Intensity inhomogeneity correction in 2D medicalimagesPaper I presents an iterative method for intensity inhomogeneity correction in2D medical images. The method iteratively suppresses the magnitude of thefrequency components which are affected by intensity inhomogeneity. It usesthe grey-weighted distance transform in the frequency domain to localize thefrequency components which are affected by intensity inhomogeneity.

The method is based on the observation that if intensity inhomogeneityis visible in an image, the frequency component responsible for such inho-mogeneity should have a higher magnitude than its neighboring components.These components are localized using the grey-weighted distance transform.The DC component of the Fourier transform serves as the only seed for thegrey-weighted distance transform. Here, distance D between adjacent pixelspi, and pi+1 is defined as:

D(pi, pi+1) =| f (pi)− f (pi+1) | · ∥ pi− pi+1 ∥ . (4.1)

We choose this distance definition for the reason that it favours frequencycomponents having similar magnitude values to the DC component. So, itsuits our objective as we are interested in finding all the frequency compo-nents having similar magnitude to the DC component. This objective can alsobe fulfilled by the distance transform on curved spaces and the weighted dis-tance transform on curved spaces. It is also important to mention that the grey-level difference in Eq. 4.1 contributes significantly to the distance function asthe magnitude near the DC component is usually far higher than other com-ponents in the Fourier transform [40].

The method results in an image-dependent high-pass filter which can beapplied to reduce the intensity inhomogeneity in an image. The method iter-atively uses high-pass filter to reduce the intensity inhomogeneity until thedifference in the correlation between images from consecutive iterations withthe original image becomes less than a predefined threshold.

39

(a) (b)

Figure 4.1: Retinal image from VICAVR database [41]. a) Original image sufferingfrom intensity inhomogeneity. b) Image after intensity inhomogeneity correction withthe proposed method.

4.1.1 ResultsThe method has been tested on images acquired from different imagingmodalities. We have used a publicly available database for retinalimages [41], a publicly available magnetic resonance images database [42],images of transmission electron microscope, binary retinal images from thedigital retinal images for vessel extraction (DRIVE) database [43], and 1200synthetic images. Both quantitative as well as qualitative evaluation has beencarried out to demonstrate the effectiveness of the method. The method hasbeen compared with a few recent methods and a state of the art method [44].Fig. 4.1(a) shows a retinal image suffering from intensity inhomogeneity. Theresult of applying the proposed method at 5th iteration is shown in Fig. 4.1(b).It is obvious that the proposed method removes the strong inhomogeneitypresent in the lower part of the input image, still maintaining the natural lookof the image.

4.2 Intensity inhomogeneity correction in MR volumesCorrection of intensity inhomogeneity induced by the radio-frequency coilin magnetic resonance volumes is a challenging problem in image analy-sis [45, 46]. Application of 2D intensity inhomogeneity correction methods(slice by slice) to magnetic resonance volumes often result in inter-slice vari-ations which may prove problematic for subsequent image analysis. Paper IIpresents the extension of the idea presented in Paper I to magnetic resonancevolumes. It is basically a non-iterative method which constructs a high-passfilter whose parameters are automatically determined from the magnetic reso-

40

50

100

150

(a) (b)

50

100

150

(c)

50

100

150

(d)

Figure 4.2: a) Image from the brainweb database [42]. b) Result after applying themethod proposed in [47]. c) Result after applying the method proposed in [48]. d)Result after applying our 3D method.

nance volume. The method starts by computing the 3D grey-weighted distancetransform of the 3D Fourier transform of the magnetic resonance volume. Thefrequencies which are affected by inhomogeneity are localized by defining aball whose radius changes according to the grey-weighted distance transform.

Grey-weighted distance transformation is computationally quite expensive.However, the proposed 3D method does not require the computation of thegrey-weighted distance transform for the full volume. As intensity inhomo-geneity is often represented by slow spatial variations, i.e., it is represented bylow frequency components in the Fourier transform, one can limit the compu-tation of the grey-weighted distance transform to a small neighborhood aroundthe DC component of the Fourier transform.

41

It is worth mentioning that all the filtering is performed using the normal-ized convolution [49]. The black background of the magnetic resonance vol-ume serves as the uncertain region while its complement serves as the certainregion for normalized convolution. Normalized convolution is preferred oversimple convolution as it helps to avoid the halo artifacts [50] which often arevisible at sharp image borders.

4.2.1 ResultsTo demonstrate the effectiveness of the proposed 3D method, it has beenquantitatively evaluated on close to 100 realistic magnetic resonance vol-umes. These volumes were produced by a magnetic resonance imaging simu-lator [42]. The method was compared with methods developed in [47, 48].

Fig. 4.2(a) shows a slice of a simulated magnetic resonance volume [42]suffering from intensity inhomogeneity. The result of applying the methodproposed in [47] is shown in Fig. 4.2(b). The method in [47] is quite simi-lar to the proposed method in a sense that both methods try to suppress themagnitude of the frequency components with the help of the Fourier trans-form. Moreover, it is demonstrated in [47] that the method works better thanstate of the art methods like N3 [44]. Fig. 4.2(c) shows the result of applyingthe method proposed in [48] and Fig. 4.2(d) shows the result of applying theproposed 3D method on Fig. 4.2(a).

4.3 Heart beat analysis of rat embryosThere is a large number of pharmaceutical drugs that block one or more ionchannels of embryonic heart either as a primary effect or as a side effect [51],with the potential of having adverse affects on embryonic development. Priorto dosing in human subjects, these pharmaceutical drugs are often tested onrat embryos. The blocking effect is often assessed by visually monitoring therat embryonic heart rate under a light microscope, a technique which is sub-jective, tedious, and error prone. Paper IV presents an automated method todetermine the heart rate of rat embryos in vitro from digital video recordings.

For this purpose, an embryo is placed in a liquid culture medium duringthe video recoding. These videos often suffer from motion artifacts caused bythe:• Non-rigid nature of the shape of the embryo.• Spatial movement of the embryo due to placement in a liquid culture

medium.It becomes a challenging problem to separate heart motion from these two mo-tion artifacts. These artifacts can be separated on the basis of spatial location,i.e., artifacts are produced by the embryonic boundary, while heart motion isproduced by the heart.

42

Paper IV presents a methodology to suppress these motion artifacts with thehelp of a decimation-free directional filter bank [12]. Here, a decimation-freedirectional filter bank is used in conjunction with directional energy estimatesto generate a boundary mask for the rat embryo. The topographic structure(non-flat) of the embryo gives rise to both, sharp and blurry boundaries. Thedecimation-free directional filter bank provides a mechanism to extract allthese boundaries irrespective of the scale. Once the boundary motion is sup-pressed, the remaining dominant motion is utilized to localize the heart regionon the basis of spectral energy distribution.

The heart spatial location is localized with the help of motion informationinstead of color information (blood color) because there are instances whenthe blood color is not easily visible due to poor imaging conditions. Fig. 4.3shows some typical frames from video recordings of rat embryos, where it isdifficult to localize the heart location based on color information. There arealso situations when one can easily see some embryonic regions having thesame color as blood. Finally, the heart motion is represented as a 1D heartsignal by taking the cross-correlation between a template frame and the restof the frames within the heart region.

The template must represent either of the extreme states in the heart cycle,i.e., either end-systole or end-diastole. In Paper IV, it was chosen by selectingthe two frames which produce the minimum cross-correlation in the first twoseconds of the video. It becomes a challenging problem to select an appropri-ate template once the motion of the heart becomes abnormal.

It is normally observed that the spatial movement of the embryo often pro-duces a trend term in the 1D heart signal. There are a variety of ways to solvethis problem. Perhaps a straight forward way would be to perform image regis-tration between consecutive frames. But image registration is computationallytoo expensive. Another alternative is to estimate the trend term from the 1Dheart signal. This essentially means that we need to design a high-pass filterwhich can filter out the trend term. However, the selection of an appropriatetype of high-pass filter with its parameter, is a complex problem.

In Paper III, these above mentioned issues are tackled with the help ofLaplacian eigenmaps (LEM) and empirical mode decomposition (EMD).Here, LEM together with k-medoids [52] is used to find the template frame.Whereas, EMD is modified to perform data dependent automatic trend andnoise filtering. Here EMD provides better insight into the data as it can beused to compute instantaneous frequency of the heart. EMD also provides asimple mechanism to find local extrema in the 1D heart signal.

4.3.1 ResultsThe method was evaluated on 151 different videos of rat embryos. Each em-bryo was recorded with the help of a video camera (uEye UI- 2210-M/C)mounted on a light microscope (Olympus SZ-40). Each embryo was filmed

43

(a) (b)

(c)

Figure 4.3: Some images of embryos. a) A typical example where color informationcannot be used to detect the heart location. b) An example where lack of color in-formation makes it difficult to detect heart location. c) Embryo with hardly visibleboundaries. Color information may lead to false heart location detection.

for 30-seconds. The method was compared with state of the art methods forheart rate determination, and was shown to provide far better results than thesemethods.

Paper V utilizes the methodology presented in Paper IV in conjunction withthe template selection method presented in Paper III to identifying the terato-genic potential in early drug development.

4.4 Particle swarm optimization for image registrationRetinal imaging is one of the main sources in ophthalmology for studying theoptical nerve head and the retina. Retinal images are often used for analyzing,diagnosing, and treating a number of diseases of the human retina. Imageregistration plays an important role in determining the progression of retinalillness. Image registration is the process of systematically placing separateimages in a common frame of reference so that the information they contain

44

(a) (b)

Figure 4.4: Retinal images. The image in a) was was taken 5 months prior to b). Here,image registration is necessary before making a comparison between the images.

can be optimally integrated or compared [53]. It transforms the floating imageto secure a nearly exact correspondence to the reference image.

To assess the progression of Glaucoma, images of the humanretina are acquired at different times with an aim of determining themovement/development of blood vessels. Such an assessment is oftenproceeded by image registration. Fig. 4.4 shows a couple of retinal imagesof the same eye which requires image registration in order to assess theprogression of Glaucoma.

Image registration using particle swarm optimization was first introducedby [54]. Since its introduction, it has been widely used in image registrationdue to its robustness and few parameters. Paper VI presents a modified par-ticle swarm optimization algorithm for retinal image registration. The algo-rithm presents a stochastic scheme to bring a balance between explorationand exploitation abilities of the swarm during the search process. It uses both,uniform and Gaussian distributions to update the velocity equation dependingon the direction of the social and cognitive component.

It is worth mentioning that the particles were initialized using a uniformdistribution. The proposed algorithm was initialized with 30 particles in thesearch space using ring topology with l-neighbors. Here l was set to 2 duringall the experiments.

4.4.1 ResultsThe algorithm was tested on 100 pairs of color (RGB) retinal images obtainedwith a fundus camera by the Department of Neuroscience, Ophthalmology atUppsala University Hospital over a period of two years. Two such images areshown in Fig. 4.4. For each pair of images, translational (horizontal and verti-cal), rotational (along x-, y- and z-axis), and scaling misalignment were con-sidered. Here, the rotations are caused by the positioning of the camera, tilting

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Figure 4.5: Registration of a retinal image in Fig. 4.4(a) with Fig. 4.4(b). Here imagesare overlaid to show the accuracy of the registration process.

of the head, and through ocular torsion. Only the green channel was utilizedin the registration process as it has been reported to give the highest contrastbetween the vessels and background [55, 56]. Fig. 4.5 shows the registrationof images in Fig. 4.4 acquired with the help of the proposed algorithm.

In the proposed algorithm, the scaling parameters in the velocity equationcan be set to any value ≥ 2.05. We have rigorously tested the algorithm forthese scaling parameters in the range [2.05 30.05]. Both scaling parameterswere varied from 2.05 to 30.05 in steps of 1. The algorithm was run 30 timesfor each value of these parameters on different pairs of images. The proposedalgorithm produced 98% correct results with these scaling parameters, whilethe basic particle swarm optimization algorithm with the same parameter set-tings produced only 26% correct results. Fig. 4.6 shows the registration resultsfor different values of these scaling parameters. Computationally the modi-fied algorithm takes a few more iterations to converge than the basic particleswarm optimization with uniform distribution. Fig. 4.7 shows the convergencetime for the proposed algorithm. The particle swarm optimization with uni-form distribution takes fewer less iterations than the modified particle swarmoptimization. However, it is often trapped in local minima.

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1 2 30

50

100

150

200

250

300

350

400

450

500

Comparison of Algorithms

Successful RegistrationFailed Registration

PSO with Gaussian DistributionMPSO

PSO with Uniform Distrubution

Figure 4.6: Registration results. Comparison of modified particle swarm optimiza-tion (MPSO) with particle swarm optimization (PSO) with uniform distribution andparticle swarm optimization with Gaussian distribution.

0 20 40 60 80 100 120 140 160 180 2000.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of Iterations

Cro

ss C

orre

latio

n

Proposed MethodGaussian DistributionUniform Distribution

Figure 4.7: Convergence rate. The x-axis represents the number of iterations, whiley-axis represents the correlation values.

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5. Conclusions

This chapter summarises the contributions and presents some directions forfuture work.

5.1 Summary of contributionsThe thesis presents several image analysis methods for biomedical applica-tions. The overall objective has been to develop data dependent filtering meth-ods for intensity inhomogeneity correction, trend filtering, noise removal, andretinal image registration. The main contribution of the thesis is the devel-opment of methods for intensity inhomogeneity correction for 2D medicalimages and its extension to 3D magnetic resonance volumes, the developmentof an automated method for the detection of heart location and the represen-tation of the heart signal as a function of time, the development of a robustmethod for heart signal generation and analysis with the help of empiricalmode decomposition, and development of a novel method for retinal imageregistration.

A new frequency domain iterative method for intensity inhomogeneity cor-rection in 2D medical images is presented. It is based on the observation that ifintensity inhomogeneity is visible in an image, its corresponding frequenciesin the frequency domain should have a higher magnitude than its surroundingfrequencies. Those frequencies are localized in the frequency domain withthe help of the grey-weighted distance transform. Since the method deter-mines all the filter parameters from the input image, it falls in the class ofdata-dependent filtering. The method was quantitatively as well as qualita-tively evaluated on images acquired from different imaging modalities. Themethod was also quantitatively evaluated on 1200 synthetic images. Finally,the method was extended to 3D magnetic resonance images, where it wasquantitatively evaluated on close to 100 realistic magnetic resonance volumes.

A method for heart location and signal detection from video recordingsof rat embryo is presented. The method uses a decimation-free directionalfilter bank to mask the motion produced by the embryonic boundary. It thenutilizes motion information to localize the heart region. The correspondingheart signal is generated by taking the cross-correlation between the templateframe and the rest of the frames within the heart region. Another method forrobust template frame detection is also presented. It uses Laplacian eigenmaps

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to detect the key frame during normal and abnormal heart activity. The methodalso provides a mechanism to remove the trend term and noise from the heartsignal with the help of empirical mode decomposition. The method was testedon over 150 videos of both normal and abnormal embryonic hearts.

A modified particle swarm optimization algorithm for retinal image regis-tration is also presented. It uses the directional information of cognitive andsocial components to update the velocity equation. Depending on the direc-tion of these components, the velocity equation of particle swarm optimizationis either updated by utilizing a uniform distribution or Gaussian distribution.The method was tested on 100 pairs of retinal images acquired over a periodof two years.

The thesis demonstrates that there is a clear need of designing data depen-dent filtering methods for biomedical applications. Data dependent filteringwill help in eliminating user bias and will help standardize analysis amongresearch labs.

5.2 Future workThe method presented in this thesis can be further improved and adopted fordifferent image processing applications. For instance,• The methods presented for intensity inhomogeneity correction can be fur-

ther improved by employing a better method to convert an ideal filter intoa non-ideal filter.

• The methods can be further extended to color images and facial images. Itwould be interesting to see how the method would work in the scale spaceframework in presence of shadow artifacts for facial images.

• The method produces halo artifacts at sharp discontinuities within imageregions (not at image boundaries) which can be further improved with thehelp of a suitable weighting function.

• We have some preliminary results on how the proposed method for inten-sity inhomogeneity correction improves different descriptors like the scaleinvariant feature transform (SIFT) [57] etc. As the grey-weighted distancetransform needs to be computed in a small region, the method can be em-bedded in feature descriptors at a cost of few extra computations.

• The grey-weighted distance transform in the frequency domain can bemodified to be used as a texture measure.

• The grey-weighted distance transform in the frequency domain can be usedas a method to detect dominant frequency regions in images (images withrich directional information).

• The methods presented for embryonic heart signal analysis can be furtherextended to include instantaneous frequency, which will provide better in-sight into the heart activity over time.

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• The modified particle swarm optimization method can be embedded intorecent particle swarm optimization strategies.

• The particle swarm optimization method can be further extended by defin-ing a new neighborhood topology, where neighbors are decided on the basisof a linear combination of the nearest neighbors in the search space and thenearest neighbors in the function space.

• The fusion of a decimation-free directional filter bank with 2D empiri-cal mode decomposition can produce a comparable decomposition to con-tourlet transform. However, the challenge would be to find the local minimain the directional images. One possible solution could be to use 1D em-pirical mode decomposition to decompose the output of decimation-freedirectional filter bank.

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Summary in Swedish

Digital bildanalys erbjuder metoder för att bearbeta digitala bilder med hjälpav en dator. Genom digital bildfiltrering kan bilder förändras så att önskadekomponenter framhävs, oftast utgående från frekvenser i bilden. Den häravhandlingen presenterar digitala bildfiltreringsmetoder för biomedicinskatillämpningar.

Först presenteras en filtreringsmetod för korrigering av ojämnheter i inten-sitetsvärden i medicinska 2D-bilder. Senare utökas samma idé till 3D-bilderfrån magnetkamera. Avhandlingen presenterar också en robust filtreringsme-tod för att lokalisera hjärtat och detektera hjärtsignaler från videosekvenserav råttembryon. Slutligen presenteras en optimeringsmetod baserad på en såkallad partikelsvärm för att utföra registrering av retinabilder, d.v.s. en ge-ometrisk filtrering.

Korrigering av ojämnheter i bilderOjämnheter i bildens intensitet är ett centralt problem i medicinska bilder. Denkarakteriseras av långsam rumslig variation som uppstår p.g.a. ofullkomligutrustning för datainsamling, ljuskällans riktning och objektets topologi, m.m.Denna variation kan kraftigt påverka den visuella utvärderingen såväl som denautomatiserade segmenteringen baserad på pixelvärden.

Det finns ett flertal metoder för korrektion av intensitetsinhomogenitet somkräver omfattande finjusteringar av olika parametrar. Detta gör att resultatenblir högst subjektiva och beroende av användarens erfarenhet. Ett exempelär anpassning av en intensitetsyta till bilden som ofta kräver att användarenbestämmer graden av ett polynom och en lämplig viktfunktion för att dämpabildregioner med skarpa diskontinuiteter. För att komma ifrån dessa prob-lem med existerande metoder, föreslås en automatisk metod för korrektionav intensitetshomogenitet av medicinska 2D-bilder som inte behöver någraparametrar.

Eftersom ojämnheterna i intensitet i regel innebär att intensiteten varierarlångsamt över bilden karaktäriseras den av låga frekvenskomponenter ifrekvensdomänen. Den föreslagna metoden konstruerar ett högpassfilter ifrekvensdomänen med filterparametrar som automatiskt räknas fram från denaktuella bilden. Metoden baseras på antagandet att lågfrekvenskomponenterpåverkade av intensitetsinhomogenitet kommer att ha en högre magnitud

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än de kringliggande frekvenskomponenterna. Dessa frekvenskomponenterlokaliseras i frekvensdomänen med hjälp av en metod som kallas gråviktadavståndstransform.

Med utgångspunkt från centrum i magnitudspektrat av den aktuella bilden,DC-komponenten, beräknas avstånd viktade så att avståndet till frekvenskom-ponenter med likartad styrka blir korta. Metoden dämpar därigenom iterativtmagnituden av de frekvenskomponenter som behöver dämpas för att inten-sitetsvariationerna skall minska.

Metoden har testats på en offentlig databas som innehåller retinabilder somhar alstrats med konfokal laserskanner ett s.k. ophthalmoscope, en offentligttillgänglig databas som innehåller magnetresonansvolymer och en offentligttillgänglig databas av binära retinabilder. Vidare har metoden testats på 1200syntetiska bilder med varierande inhomogenitetsmönster. De experimentellaresultaten visar den föreslagna metodens överlägsenhet i jämförelse med ettantal existerande metoder.

När intensitetsinhomogeniteter korrigeras snitt för snitt i magnetreso-nansvolymer resulterar det ofta i variation mellan snitten vilket kan orsakaproblem för den efterkommande bildanalysen. För att överbrygga dettaproblem, har den föreslagna 2D-korrektionmetoden utökats till att direktkorrigera 3D-magnetresonansvolymer. Den föreslagna metoden konstruerarett 3D-högpassfilter med hjälp av en 3D gråviktad avståndstransform.Metoden har testats på 92 magnetresonansvolymer av hjärnan. Jämförelsenmellan de experimentella resultaten och några aktuella metoder visar på denföreslagna metodens mångsidighet och robusthet.

Analys av videosekvenserVid utveckling av nya läkemedel är det viktigt att säkerställa att dessainte har farliga biverkningar. En sådan kontroll är huruvida de fungerarsom jonkanalsblockerare på embryohjärtan och därmed stör hjärtrytmen.Potentiella läkemedelssubstanser testas därför ofta på råttembryon.Kontrollen uppnås genom att räkna hjärtslagen under en viss tid för ettråttembryo som har placerats i ett ljusmikroskop efter att det utsatts försubstansen. Att räkna hjärtslagen manuellt är tidskrävande och är dessutomen felkälla. I avhandlingen presenterar en filtreringsmetod som förenklarkontrollprocessen genom att en automatisk metod för lokalisering av hjärtati videosekvenser tagna genom mikroskopet samt automatisk räkning avhjärtfrekvensen genom analys av videosignalen. Parametrarna för analysenerhålles autmatiskt ur de analyserade bilderna.

Den föreslagna metoden har testats på 151 videosekvenser. Deexperimentella resultaten visar att den föreslagna metoden är robust samtidigtsom den är arbetsbesparande.

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Registrering av ögonbilderGlaukom eller starr är en ögonsjukdom som är associerad med ökat tryck iögat vilket kan leda till förlust av synen. För att analysera utvecklingen avstörningen, krävs ofta att man kan jämföra bilder tagna vid olika tidpunkter såatt man kan bestämma förflyttningen och utvecklingen av blodkärl i ögat. Detär omöjligt att ta dessa bilder i exakt samma vinkel så jämförelsen kräver attbildernas geometrier anpassas till varandra genom så kallad bildregistrering.

I avhandlingen presenteras en metod för registrering av retinabilder som an-vänder en optimeringsalgoritm baserad på en modifierad partikelsvärm. Meto-den använder riktningen av partikelsvärmens sociala och kognitiva kompo-nenter för att uppdatera hastigheten av en partikel. Den använder likformigfördelning för att ge svärmen en utforskande karaktär medan den gaussiskafördelningen bidrar med exploateringsförmåga till svärmen.

Den föreslagna algoritmen har testats på 100 par av retinabilder. Robus-theten av den föreslagna algoritmen har testats genom att variera de viktigaskalningsparametrarna i hastighetsekvationen i ett stort intervall, mellan 2.05och 30.05. De uppnådda resultaten visar på metodens robusthet samt dess för-måga att hitta ett global minimum.

Sammanfattningsvis har avhandlingen bidragit till bildanalysensverktygslåda genom nya filtreringsmetoder för att förbättra och analyserabilder. Ett genomgående tema har varit att skapa metoder som själva tarfram sina styrparametrar ur de analyserade bilderna och därigenom minskaranvändarens påverkan på resultatet. Förhoppningsvis kan detta minskavariationerna i resultat mellan olika forskningsgrupper.

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