Automatic 3D segmentation of the breast in MRI · Automatic 3D segmentation of the breast in MRI Cristina Gallego Ortiz Master of Science Graduate Department of Medical Biophysics

Automatic 3D segmentation of the breast in MRI

by

Cristina Gallego Ortiz

A thesis submitted in conformity with the requirementsfor the degree of Master of Science

Graduate Department of Medical BiophysicsUniversity of Toronto

Copyright © 2011 by Cristina Gallego Ortiz

Abstract

Automatic 3D segmentation of the breast in MRI

Cristina Gallego Ortiz

Master of Science

Graduate Department of Medical Biophysics

University of Toronto

2011

Breast cancer is currently the most common diagnosed cancer among women and

a significant cause of death. Breast density is considered a significant risk factor and

an important biomarker influencing the later risk of breast cancer. Therefore, ongoing

epidemiological studies using MRI are evaluating quantitatively breast density in young

women. One of the challenges is segmenting the breast in order to calculate total breast

volume and exclude non-breast surrounding tissues. This thesis describes an automatic

3D breast volume segmentation based on 3D local edge detection using phase congru-

ency and Poisson surface reconstruction to extract the total breast volume in 3D. The

boundary localization framework is integrated on a subsequent atlas-based segmentation

using a Laplacian framework. The 3D segmentation achieves breast-air and breast-chest

wall boundary localization errors with a median of 1.36 mm and 2.68 mm respectively

when tested on 409 MRI datasets.

ii

To my parents,

Cecilia Ortiz and

Conrado Gallego in memoriam

iii

Acknowledgements

I want to express my gratitude to my supervisor, Dr. Anne Martel who has offered in-

valuable guidance, advice, encouragement and mentorship throughout.

I have accumulated many debts to my supervisory committee members, Dr. Martin Yafee

and Dr. John Sled. I would like to thank them for their time and patience in helping me

plan and revise my work at every stage of my studies.

I owe special thanks to all Martel lab members, their friendship, constructive criticism

and professional collaboration meant a great deal to me.

To Dr. Norman Boyd and his research group, especially Anoma Gunasekara and Sofia

Chavez, for providing the high quality Breast MRI datasets and for their valuable col-

laboration.

Last but not least, I wish to express my gratitude to my beloved family. Specially to my

mother, for her unconditional love, understanding, and for supporting me on whatever I

have decided to pursue in life.

iv

Contents

1 Introduction 1

1.1 Breast anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Breast development and cancer susceptibility . . . . . . . . . . . . . . . . 3

1.3 Mammographic density as a risk factor . . . . . . . . . . . . . . . . . . . 5

1.3.1 Assessment of breast density . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Measuring breast density with ultrasound . . . . . . . . . . . . . 9

1.3.3 Measuring breast density with MRI . . . . . . . . . . . . . . . . . 11

1.4 Segmentation of the breast in MRI . . . . . . . . . . . . . . . . . . . . . 14

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Breast boundary localization and 3D extraction 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Breast MRI datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Detecting the breast boundary . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Features of edge strength . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Phase congruency edge detector . . . . . . . . . . . . . . . . . . . 22

2.3.3 Extracting local frequency via local energy . . . . . . . . . . . . . 23

2.4 Breast surface via Poisson reconstruction . . . . . . . . . . . . . . . . . . 26

2.4.1 Poisson surface reconstruction . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Maximal phase congruency and edge point orientation . . . . . . 28

v

2.4.3 Implementation via Octree structures . . . . . . . . . . . . . . . . 29

2.4.4 Extracting the breast boundary as an isosurface . . . . . . . . . . 32

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Atlas-based segmentation of the breast in 3D 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Extracting a shape atlas of the breast in 3D . . . . . . . . . . . . . . . . 35

3.2.1 Groupwise registration and average volume extraction . . . . . . . 35

3.2.2 Population shape average and 3D shape representation . . . . . . 37

3.3 Mapping 3D atlas landmarks via a Laplacian framework . . . . . . . . . 38

3.4 Validation: Segmentation of population cases . . . . . . . . . . . . . . . . 40

3.4.1 Distance errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.2 Volume errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 Evaluation of standard overlap metrics . . . . . . . . . . . . . . . 45

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Conclusions and future directions 50

4.1 Contribution of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.1 Segmentation for breast density assessment . . . . . . . . . . . . . 50

4.1.2 Segmentation in the context of CAD systems . . . . . . . . . . . . 51

4.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A Statistical shape model of the breast in 3D 56

Bibliography 57

vi

List of Tables

1.1 Factors that increase the relative risk for breast cancer in women. . . . . 7

3.1 Number of landmarks and triangles generated after surface mesh decimation 38

3.2 Distances from manually annotated surfaces discriminated by boundary

region after Poisson surface reconstruction. . . . . . . . . . . . . . . . . . 42

3.3 Distances from manually annotated surfaces discriminated by boundary

region after Laplacian correspondences. . . . . . . . . . . . . . . . . . . . 42

3.4 Agreement and error overlap results. . . . . . . . . . . . . . . . . . . . . 46

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List of Figures

1.1 Anatomy of the breast . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Pike’s model of breast tissue exposure and age-incidence curve of breast

cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Computer-assisted measurement of percent mammographic density. . . . 8

1.4 Breast density measurement by Ultrasound. . . . . . . . . . . . . . . . . 10

1.5 Breast density measurement using T1-weighted imaging. . . . . . . . . . 11

1.6 Breast density measurement using chemical shift imaging. . . . . . . . . 13

1.7 Breast segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 Overall segmentation pipeline and thesis organization. . . . . . . . . . . . 16

2.1 Fourier series of a square waveform. . . . . . . . . . . . . . . . . . . . . . 20

2.2 Interpolation of a step feature to a line feature. . . . . . . . . . . . . . . 21

2.3 Polar plot of local energy model for phase congruency. . . . . . . . . . . 22

2.4 1D Log-Gabor transfer function and Quadrature pairs. . . . . . . . . . . 23

2.5 Phase congruency detection examples . . . . . . . . . . . . . . . . . . . . 25

2.6 Schematic of Poisson surface reconstruction in 2D. . . . . . . . . . . . . . 27

2.7 Extracting points of maximal phase congruency and estimating their ori-

entation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8 Schematic drawing of an octree hierarchical subdivision. . . . . . . . . . 30

2.9 Initial breast boundary surface via Poisson surface reconstruction . . . . 32

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3.1 Population-based construction of the average volume . . . . . . . . . . . 36

3.2 Average shape representation with varying decimation factors . . . . . . 37

3.3 Initializing model points into optimal isosurface. . . . . . . . . . . . . . . 39

3.4 Experimental design to assess performance of segmentation. . . . . . . . 41

3.5 Box-and-Whisker plots: Comparison of target distance errors for the two

stages of segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Representative results of 2 breast boundary reconstructions. . . . . . . . 45

3.7 Bland-Altman plot showing the difference vs average of results . . . . . . 46

3.8 Overlap metric assessment for segmentation results . . . . . . . . . . . . 47

3.9 Distribution of overlap metrics, breast volumes and surface areas. . . . . 48

3.10 Percentage and Absolute percentage Volume and Surface errors. . . . . . 49

4.1 Model fitting compared to boundary localization distance error results . . 54

ix

List of Abbreviations

2D Two-dimensions. 14

3D Three-dimensions. 1

ABUS Automated whole breast ultrasound. 10

BIRADS Breast imaging reporting and data system. 8

BRCA1 Breast cancer gene 1. 5

BRCA2 Breast cancer gene 2. 5

CAD Computer-aided diagnosis. 17

CNNs Cellular neural networks. 50

DCE-MRI Dynamic contrast enhanced magnetic resonance imaging. 12

ETL Echo Train Length. 19

FCM Fuzzy C-Means algorithm. 11

FFD Free form deformation. 35

FOV Field-of-View. 15

FSE Fast spin-echo. 19

x

GUI graphical user interface. 51

GVF Gradient vector flow. 15

IQR interquartile range. 42

MLO Mediolateral oblique view. 13

MRI Magnetic Resonance Imaging. 1

PC Phase congruency. 22

SSM statistical shape model. 52

TDLU Terminal duct lobular unit. 3

TE Time to Echo. 19

TR Time of Repetition. 19

USPD Volumetric ultrasound percent density. 10

xi

Nomenclature

An(x) n-th Fourier term amplitude. 22

D octree depth. 30

E(x) local signal Energy. 22

FN false negative volume overlap. 45

FP false positive volume overlap. 45

Fo function space of sampled points. 30

Mo mean volume overlap or Dice coefficient. 45

NgbrD(p) eight depth-D nodes. 31

Siso isosurface binary representation. 39

Sm atlas binary representation. 39

T noise threshold. 25

To total volume overlap. 44

αo,p trilinear interpolation weights for point p. 31

φ(x) mean local phase of Fourier terms. 22

δ scaling factor on bank of filters. 25

xii

γ scalar parameter for surface reconstruction. 32

λmin smallest wavelength of the Log-Gabor filter. 25

C(s) parameterization of field line. 40

N normalized negative gradient of Laplace solution. 40

xi i-th atlas point. 42

xtargeti i-th closest point on target surface. 42

µ mean of the local Energy. 25

∇χM indicator function gradient. 26

∂M reconstructed isosurface. 32

φn(x) n-th Fourier term phase. 22

ψ Laplace solution. 39

σ standard deviation of the local Energy. 25

χM indicator function. 26

~V vector field of point normals. 27

b shape parameter of a SSM. 52

k/wo filter’s bandwidth. 24

wo filter’s center frequency. 24

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Chapter 1

Introduction

While the incidence and mortality rates of breast cancer vary internationally, currently

it is the most commonly diagnosed cancer among women in most parts of the world

[1]. In Canada, breast cancer accounts for the second cause of cancer deaths despite the

significant improvement in survival rates since the mid-80s [2]. Currently the 5-year sur-

vival rate is 87% likely as a result of advances in treatment and breast cancer screening.

Breast cancer develops through multiple stages and the reason why some tumors even-

tually become invasive and metastatic and some of them remain non-invasive precancers

is still under investigation. Current research has looked at breast composition in an at-

tempt to determine significant risk factors that could be associated with the development

of cancer. Breast density, a representation of the amount of breast dense parenchyma

present in the breast, has been identified as a significant risk factor and an important

biomarker influencing the later risk of breast cancer. Ongoing epidemiological studies

conducted by Boyd et al. [3] are looking at quantitative assessment of breast density in

young women using MRI. However, it is known that quantitative evaluation of breast

density using MRI suffers from several limitations including inconsistent breast boundary

segmentation methods. This thesis presents a method to automatically segment the 3D

anatomy of the breast in MRI. Key contributions of this work are a proposed automatic

1

Chapter 1. Introduction 2

framework for extracting the boundaries of the breast region and for estimating the total

breast volume. The objective of the first chapter is to give an overview of the clinical

motivation and impact of this work. The following sections of this chapter review the

anatomy of the breast, introduce breast cancer risks and highlight breast density as an

important risk factor.

1.1 Breast anatomy

Figure 1.1: Anatomy of the breast: a) Gross anatomy of the breast. b) Terminal duct lobular

unit TDLU schematic. Adapted from: Schunke, et al. Thieme atlas of anatomy: general

anatomy and musculoskeletal system. Thieme 2006.

The breast is a modified skin gland that lies on top of the musculature that encases the

chest wall. The breast is not completely separated from these muscles, in fact only a layer

of adipose tissue and connective fascia separate the breast from the pectoral muscle. The

breast is composed of three major tissue types: glandular tissue (parenchyma), fibrous

stroma and fatty tissue. The stroma is composed of connective tissue, ligaments, blood

vessels, lymphatics, lymph nodes, and nerves [4] and its main function is to provide

support and to nurture the breast. The glandular tissue is organized in a ductal system


with a distribution that is essentially bilateral between the right and left breast. Current

literature agrees that the parenchyma of the breast consists of about 10 to 20 lobes,

each of which has a lactiferous major duct that opens on the nipple through a little

antechamber called the lactiferous sinus (see figure 1.1-a). Starting at the nipple, the

ductal system splits up in branches that reach the back of the breast. At the end of

each branch are the lobules that produce milk. Each lobule is composed of acini that

empty into the terminal ducts. The acini and terminal ducts form a complex also known

as the terminal duct lobular unit (TDLU) (see figure 1.1-b). The TDLU is the basic

secretory unit of the female breast and is of key importance in histopathology since there

is evidence to suggest that is the site where most malignant cancers originate [5].

1.2 Breast development and cancer susceptibility

Breast tissue unlike other tissues in the body is very sensitive to develop cancer. The

basis of this difference is the fact that cells in the breast mostly divide and differentiate

after birth and therefore are at a higher risk of acquiring mutations during development.

Cells in the breast of a newborn are very immature and remain this way until the onset of

puberty, when cells start to develop in response to sexual hormones. At this point, stem

cells in the breast begin to differentiate into duct cells or lobular cells. Research has found

that some stem cells remain undifferentiated perhaps to replenish injured cells. These

findings have motivated a new hypothesis focusing on these cells as origins of cancer [6].

On a different line of thought, some researchers have studied the factors that con-

tribute to breast cancer susceptibility. Pike [7] showed that breast tissue exposure to

carcinogens has the greatest susceptibility to cause abnormal mutations around the time

of puberty but decreases with the first pregnancy and continues to reduce further in pre-

menopausal and menopausal periods (see figure 1.2-a). Pike’s work also explained how

the area under the exposure-age curve could be used to describe the increasing incidence


of breast cancer with age (see figure 1.2-b). The incidence of breast cancer with age

has a distinctive curve. Below the age of 50 years, the incidence is about the same in

widely different geographical locations. However, around the age of menopause and after

menopause (> 50 years) the incidence is lower for non-western countries than for western

countries. The high incidence rates in white women in north America and in several

western European countries can be explained with the high prevalence of well known

reproductive factors associated with an increased risk, such as early menarche, late age

at first full-term pregnancy, fewer pregnancies and use of postmenopausal hormone ther-

apy (see table 1.1), but also non-reproductive factors such as increased detection with

mammography screening [1].

Rate(per 100,000)1000

100

10

1

0.1

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 80+ Age

Figure 1.2: a) Pike’s model of breast tissue exposure. fo, f1, f2 are variables of the model,

FFTP=first full-term pregnancy, b is a variable to calculate age at menarche. Adapted from:

[8] and b) age-incidence of breast cancer in selected cancer registries. Adapted from: The

changing global patterns of female breast cancer incidence and mortality [9].

The pattern of premenopausal cancer have attracted the attention of several re-

searchers. While postmenopausal cancer is more likely to be explained by reproductive


risk factors, premenopausal risk factors, prognosis of the disease and tumor biology have

been found to be somehow different in women below 40 years [10], suggesting that cancer

in this age subgroup is a different entity. The causes of some of these cancers seem re-

lated to hereditary mutations but their prevalence varies among different ethnic groups.

In Ashkenazi Jewish the prevalence of BRCA1 and BRCA2 mutations was 20% and 8%

respectively in women with breast cancer diagnosed before 42 years [11, 12]. In Iceland,

BRCA2 mutations were found in 24% of cancers diagnosed before 40 years [13]. However,

another study in the UK [14] reported a prevalence of only 3.1% and 3.0% on BRCA1

and BRCA2 mutation carriers respectively among patients from an out-bred population

with breast cancer who were younger than 50 years. This high prevalence of mutation

carriers among specific ethnic groups has been explained by genetic founder effects but

the fact that this prevalence is much lower on a more out-bred population reflects that

there are other underlying causes still to be understood.

1.3 Mammographic density as a risk factor

Mammographic density is defined as the proportion of breast that comprises epithelial

and stromal tissue on a mammogram. Dense tissue appears bright on a mammogram

while fat tissue, which is less dense, appears darker due to different x-ray attenuation

characteristics. The reason why some breasts are radiologically denser than others re-

lates to breast tissue composition. A systematic meta-analysis of 42 studies looked at the

strength of the association between breast density and breast cancer risk in the general

population. The study found an increasing relative risk of breast cancer with an in-

creasing percent density as measured on pre-diagnostic mammograms [15]. In addition,

the relative risk was found to be independent of age, ethnicity or menopausal status at

mammography. In women with > 75% mammographic density, the risk of breast cancer

was found to be 4.64 (95% CI 3.64 - 5.91) times the risk in women with little or no breast


density (< 5% mammographic density), a relative risk even stronger than the relative

risk variables considered on Gail’s model [16, 17], a model that has been validated for use

in women 35 years or older to provide 5-year and lifetime risk estimates for developing

breast cancer. Table 1.1 summarizes other known relative risks factors for comparison.

Epidemiological data have shown that there are parallel similarities between breast

density and breast tissue susceptibility with age [8]. Both breast tissue exposure based

on Pike’s model and average percent mammographic density decreases with age. Breast

density is also altered by pregnancy and menopause and decreases with age of stromal

and epithelial tissue. A study looked at the relationship between mammographic density

and histological features on randomly selected breast tissue samples taken from forensic

autopsies (Li et al.[19]). Using quantitative microscopy, the biopsy portion occupied

by cells (estimated by nuclear area), glandular structures, and collagen was correlated

with percent mammographic density. They found that percent mammographic density

was associated with greater total nuclear areas (p<0.001), greater proportion of collagen

(p<0.001), greater nuclear areas of both epithelial and non-epithelial cells as well as

greater area of glandular structures. The authors concluded that the age-related decline

of mammographic density could be explained by the reduction in epithelium and stroma,

a physiological process known as involution.

Some researches have thought of breast density as an intermediate phenotype for can-

cer [8], motivating new research on elucidating the genes that determine breast density.

A current hypothesis states that breast cancer originates on epithelial cells, and therefore

a greater content of fibroglandular tissue in the breast exposes a woman to a higher risk

due to carcinogenic influences.

1.3.1 Assessment of breast density

In 1976 the first classification system of the appearances of mammograms with their

association to breast cancer risk was proposed by Wolfe [20]. Wolfe’s classified the


Relative Risk Factor

> 4.0 • Age (65+ vs. <65 years, although risk increases across all ages until

age 80)

• BRCA1 and/or BRCA2

• ≥ 2 first-degree relatives with breast cancer diagnosed at an early age

• Personal history or breast cancer

2.1 - 4.0 • One first degree relative with breast cancer

• Biopsy confirmed atypical hyperplasia

• High dose radiation to chest

• High bone density (postmenopausal)

1.1 - 2.0 Reproductive factors

• Late age at first full-term pregnancy (>30 y)

• Early menarche (<12 y)

• Late menopause (>55 y)

• No full-term pregnancies

• Never breast fed a child

Factors that affect circulating hormones

• Recent oral contraceptive use

• Recent and long term use of hormone-replacement therapy

• Obesity (postmenopausal)

Table 1.1: Factors that increase the relative risk for breast cancer in women. Adapted from:

Dr. Susan Love’s Breast book. Fifth edition. [18]


breast parenchyma into four categories based solely on its appearance. The categories

were N1(essentially normal breast), P1(prominent ductal pattern to a minimal degree),

P2(prominent ductal pattern of a moderate to severe degree), and DY(extremely dense

parenchyma). Using this system on a sample screening population Wolfe determined that

by screening P2 and DY women, 76% of the total cancers were found, and by examining

P1 in addition to P2 and DY, 93.3% of the cancers could be found. Despite being a

preliminary study at the time it highlighted the potential benefits of routine repeated

screening examination for the P2 and DY group of patients. The breast imaging reporting

and data system (BIRADS) [21] have proposed a qualitative system to classify mammo-

graphic breast density. The BIRADS system includes 4 different categories: 1 (predom-

inately fat <25% glandular), 2 (scattered fibroglandular densities 25-50% glandular), 3

(heterogeneously dense 51-75% glandular), and 4 (extremely dense <75% glandular).

Figure 1.3: Computer-assisted measurement of percent mammographic density. a) Thresholded

digital mammogram: red contour defines edge of breast; green contour shows edge of dense

region [8]. b) Histogram with corresponding user-selected grey-level thresholds: breast edge

(iEdge) and density edge (iDY ) Adapted from: [22].

Computer-assisted measurement of percent mammographic density has also been pro-


posed. One approach uses interactive thresholding to separate tissue types based on their

radiographic appearances. With an interactive thresholding scheme, an observer selects

2 thresholds: first, a threshold at the boundary of the breast to exclude the background

and second, a threshold at the edge of a dense area to segment areas of high density in the

mammogram (see figure 1.3). The proportion of radiographic density is calculated as the

percentage of the total area of the breast that corresponds to dense tissue. Quantitative

approaches over qualitative methods for determining breast density have provided more

consistent results and larger gradients in risk. McCormack et al. [15] observed a linear

increasing trend in both incidence and prevalence in general population studies. The

area of the dense tissue has also been found to be a strong predictor of risk but debate

exists on whether or not percentage density is a stronger predictor.

Despite the current understanding of breast density as a strong predictor of breast

cancer risk, the incorporation of breast density assessment into a risk assessment model

have not made a big difference yet [23]. X-ray mammography for breast density assess-

ment has several limitations. It has been suggested that the measured mammographic

densities may vary with different projections, level and angle of compression, and scanner

calibration. X-ray mammography is a 2D imaging modality that offers only a projec-

tion image rather than a volumetric equivalent of the three-dimensional breast. More

importantly is perhaps the radiation dose limiting factor for measuring density in young

women. As a consequence, even though the majority of epidemiological evidence on

breast density as a risk factor is based on mammographic breast screening data, some

researchers have recognized the benefits of studying breast density with different imaging

modalities such as ultrasound and MRI.

1.3.2 Measuring breast density with ultrasound

Ultrasound imaging is relatively cheap, highly available and does not use ionizing radi-

ation. Ultrasound can measure breast density knowing that the speed at which sound


travels through a medium is related to tissue density and elasticity. One method for

measuring percent density by ultrasound tomography consists on segmenting areas of

fast sound speed from each speed tomogram, integrating these areas over the entire vol-

ume, and dividing by whole-breast volume to derive the volumetric ultrasound percent

density (USPD) (see figure 1.4-b) [24]. The authors reported a Spearman correlation

coefficient of 0.75 (P < .001) between USPD and percent mammographic density among

90 subjects (figure 1.4-c). Other methods for measuring density use automated whole

Figure 1.4: Breast density measurement by ultrasound. a) Automated whole breast ultrasound

(ABUS) images [25]. Structures labeled in letters are: (a) Skin, (b) nipple, (c) subcutaneous

fat, (d) glandular tissues, (e) retromammary fat, (f) pectoral muscles and (g) rib. b) Images

acquired using ultrasound tomography of the breast. c) Scatter plot of volumetric ultrasound

percent density (USPD) vs. percent mammographic density [24].

breast ultrasound (ABUS) to acquire the volume of the whole breast and then, by man-

ually excluding non-breast regions it is possible to derive the total fibroglandular area

(see figure 1.4-a). The percent density is calculated based on the ratio of the fibroglan-

dular area over the whole breast area for all of the acquired 2D slices [25]. However,

the authors did not report correlation with percent mammography density as measured

by mammography but found a high correlation with 3D breast MRI density measure-

ments using T1-weighted images and the fuzzy C-means (FCM) classifier to differentiate

fibroglandular tissue from fatty tissue [26] [27].


1.3.3 Measuring breast density with MRI

MRI is a very versatile imaging modality that provides a 3D view of the breast for

assessment of volumetric breast density without exposure to ionizing radiation. Based

solely on image analysis one can measure breast density using MRI. T1-weighted imaging

can be used to distinguish between fat tissue and dense tissue. The contrast mechanism in

MRI is dictated by tissue relaxation. Particularly, T1 relaxation describes the recovery of

the longitudinal magnetization due to the thermal interactions between hydrogen nuclei

(the spins) and large macromolecules within the tissue microenvironment (the lattice).

Due to differences in tissue composition, different tissues have different T1 values. In

the breast, T1 values are shorter for fat (about 250 ms at 1.5T) than for fibroglandular

tissues (about 700 ms at 1.5T) [28]. As a consequence, T1 effects tend to cause fat

tissue to have higher signal (appear brighter) than fibroglandular tissue. A number of

Figure 1.5: Breast density measurement using T1-weighted imaging. a) T1-weighted v-cut

segmented breast region, b) FCM clustering for 4 different tissue clusters. c) Cluster #1 (air

and lung tissue) is removed, c) Segmented breast after exclusion of chest wall muscle and skin,

d) Fibroglandular tissue segmentation via FCM-clustering, e) Breast outline and fibroglandular

region masks. Adapted from [27].


previous studies have used T1-weighted imaging to assess breast density. Lee et al.

[29] segmented fatty and fibroglandular volumes on 3D T1-weighted MRI estimating

pure-tissue averaged signal intensities via a two-compartmental model. Other authors

have proposed to measure breast density directly on standard clinical dynamic contrast

enhanced (DCE-MRI) data. Nie et al. [27] suggested the fuzzy C-means (FCM) algorithm

to determine the whole breast volume and fibroglandular volume on pre-contrast T1-

weighted images without fat saturation. Tissue cluster #1 on figure 1.5-d, corresponds

to fibroglandular tissue whereas clusters #2 and #3 both represent fatty tissues. Khazen

et al. in the UK [30] measured the proportion of fibroglandular tissue by interactive

thresholding of the signal intensities in pre-contrast T1-weighted images.

An alternative way in which MRI can evaluate breast density is by measuring breast

water as a surrogate for fibroglandular tissue and stroma. Research motivated on assess-

ing ex-vivo breast tissue properties using MRI revealed that the estimated water content

using MR spectroscopy was strongly correlated with the volumetric water content mea-

sured by enzymatic extraction. Water and fat are two spectrally different components

and therefore can be imaged separately using chemical shift imaging. Most fat in the

human body has a chemical shift with respect to water equal to 3.5 ppm (3.5 x 10−6),

which is equivalent to a frequency shift from the resonant frequency of water of 220 Hz at

1.5 T. The goal of fat/water imaging is to determine the relative signal contribution from

water and fat respectively in each voxel. Most fat and water separation MRI sequences

are based on the Dixon sequence published originally in 1984. Dixon’s sequence acquires

two separate images: the first is a conventional spin echo with water and fat signals in-

phase, and the second is acquired so that water and fat signals are 180°out-of-phase [31].

Dixon’s technique uses summation and subtraction of both complex signals to generate

fat-only and water-only images allowing direct fat and water quantification. Bright areas

on a water image correspond to water-containing tissues and represent a quantitative

surrogate for fibroglandular tissue (see figure 1.6-a, b). Further modifications have been


made to the original Dixon technique mainly to reduce the sensitivity of the method to

magnetic field inhomogeneities [32]. [33].

Figure 1.6: Breast density measurement using chemical shift imaging. Sagittal MRI slice of a)

breast fat and b) breast water, green contour shows the expert annotated outline of the breast.

c) Scatter plot of percent water by MRI vs percent density by X-ray mammography. [3].

Regardless of the quantification method of breast density using MRI, numerous stud-

ies have reported strong correlations between percent mammographic density as mea-

sured by X-ray mammography and MRI. Khazen et al. [30] found a correlation r=0.78

between MRI percent density and percent mammographic density estimated with Cumu-

lus software [34] for 138 high-risk women part of the UK MARIBS breast MRI screening

program. Wei et al. [35] found a correlation of r=0.89 between the volumetric fibrog-

landular tissue on MRI and the percent dense area on mediolateral oblique view (MLO)

mammograms for 65 patients. In a cross-sectional study by Boyd et al. [3], breast den-

sity was assessed in 100 woman using percent mammographic density as measured by

X-ray mammography and percent water as measured by MRI. The measurements had a

Spearman correlation coefficient r=0.85, p<0.0001 (see figure 1.6-c).


1.4 Segmentation of the breast in MRI

It is known that quantitative evaluation of breast density using MRI suffers from sev-

eral limitations including inconsistent breast boundary segmentation, different pulse se-

quences, and lack of standardized computerized algorithms for accurate quantification.

During breast density assessment using MRI it is necessary to segment the breast in or-

der to calculate total breast volume and exclude non-breast surrounding tissues. Image

segmentation is commonly used to subdivide the image into its constituent regions. In

our case, we are interested on segmenting the gross anatomy of the breast represented on

2D as the breast-air boundary, breast-chest wall boundary and upper and lower limits of

breast for the entire extent of the breast, and on 3D as a convex surface delimiting the

entire volume of the breast (see figure 1.7).

Figure 1.7: Left: 2D segmentation consisting on the breast-air boundary (green contour) and

breast-chest wall boundary (purple contour). Right: 3D segmentation consisting on a convex

surface.

Breast segmentation can be performed using manual contouring which is assured by

expert annotation and even though this process is highly time consuming, tedious, and

inconsistent among different operators, it is commonly used as a gold standard. Semi-


automatic approaches have been proposed in order to facilitate the segmentation task

and improve reproducibility but they still require subjective operator initialization or

correction. Often times, simple grey-level intensity separation methods are used. In the

UK study [30], total breast volume segmentation was limited to a volume defined by

a thresholding operation excluding background air and a straight line anterior to the

pectoral muscles on each axial slice, an arbitrary constraint that clearly underestimates

the total breast volume. For the two-compartmental model approach by Lee et al. [29] the

total volume of the breast is determined on a slice-by-slice fashion using edge detection for

the breast-air interface and manual outlining of the breast-chest wall interface. Nie et al.

[27] used a V-shape cut to exclude the thoracic region based on anatomical landmarks on

each axial slice followed by tissue clustering and dynamic searching to exclude the chest

wall muscles and the skin. Such an approach however, lacks generalization to different

acquisition orientations and FOV.

Robust and reliable automatic segmentation is ideal although it is challenging. In

breast MRI, the image contrast is dependent on imaging protocols and acquisition param-

eters so that segmentation based on separation of grey-level intensities, such as threshold-

ing or region growing suffer from generalization ability. In addition, the contrast between

adjacent structures such as between breast tissue and the pectoral muscles is not distinc-

tively defined. In the computer vision literature, other researchers have proposed the use

of active contours also known as Snakes to perform low-level tasks such as edge detection

combined with high-level models for the purpose of segmentation [36]. Snakes are energy

minimizing splines that move within images to find object boundaries. In the original

formulation, the snake had an internal term which aims to impose smoothness on the

shape, and an external term which encourages movement towards the edges of the image.

Snakes however, have the disadvantage of distorting the anatomy of the structure to be

segmented since they lack global shape constraints. A more robust algorithm [37] uses

the Gradient vector flow (GVF) to model a physical object that has a tendency to both


stretch and bend to adapt better to regions of high curvature, has a higher capture range

which often solves the problem of stopping at local minima but since no force (other

than smoothness) is imposed, the model can still fold and deform to non-realistic shapes.

A model-based segmentation has the flexibility of the snakes to deform but the allowed

deformation is limited within a range of variability introduced using prior knowledge

about the structure of the object. This captured biological variability is then used to

add flexibility to the deformation so that the model can be fitted appropriately to the

object to be segmented by adjusting the model’s parameters in such a way that the fit

of the model to the image is maximized.

1.5 Thesis Outline

In the remaining of this thesis, the main steps proposed for the automatic segmentation

of the breast volume in MRI are described. The complete segmentation pipeline can be

briefly summarized as follows (see Figure 1.8). For each case to segment, image features

Localization of the

edges of the breast

1Extraction of the

breast boundary

2

For each case to segment:

3Refinement

via shape atlas

Chapter 2

Build population

3D atlas

For all cases on population:

Chapter 3

4Segmented

breast volume

Figure 1.8: Overall segmentation pipeline and thesis organization.


that highlight the edges of the breast are identified in order to extract the boundary of

the breast region, which in 3D corresponds to a surface. Then, the initial estimate of

the boundary of the breast is further refined by incorporating shape-prior information

in the form of a shape atlas representing the breast anatomy in 3D. A shape atlas can

be derived from all the cases in the population by building a consensus of the shape of

the breast in 3D. The final result of this pipeline is an automatically segmented breast

volume.

Chapter 2 describes the 500 breast MRI datasets from a breast density study in

young women [3], the localization of the edges of the breast region based on 3D local

edge detection, and the extraction of the boundary of the breast in 3D via a Poisson

surface reconstruction method.

Chapter 3 presents the construction of a breast atlas using 3D groupwise registration

to automatically generate subject correspondences in a population. Then follows the

description of the refinement of the breast surface by incorporating 3D shape information

from an atlas using a Laplacian framework.

Chapter 4 summarizes the contribution of this thesis and illustrates the relevance of

the proposed segmentation pipeline in the context of computer-aided diagnosis (CAD)

systems. In addition, this chapter elaborates on how the breast boundary localization

framework can be integrated as the starting point for a subsequent model-based segmen-

tation and presents preliminary results contained in Appendix A. Finally, this chapter

draws conclusions and presents ideas for future work.

Chapter 2

Breast boundary localization and 3D

extraction

2.1 Introduction

One of the challenges for volumetric breast segmentation is to provide optimal features

of edge strength so that the boundary of the whole breast is highlighted and accurately

localized to serve as a template during segmentation. This chapter focuses on the method-

ology applied to localize significant edges of the breast region and to extract the breast

boundary as a surface. The population of previously segmented breast cases as part of a

young women breast density study [3] is presented in section 2.2. Section 2.3 describes

phase congruency edge detectors, their advantages over gradient-based edge detectors

and addresses the implementation of such filters in this work. Section 2.4 presents a

surface reconstruction algorithm based on a Poisson system built from the extracted fea-

tures of edge strength. The bulk of this work has been submitted and accepted to the

Breast Image Analysis workshop, part of the MICAAI 2011 conference.

18

Chapter 2. Breast boundary localization and 3D extraction 19

2.2 Breast MRI datasets

A population of breast MRI datasets (n = 500) was used from a previous study by Boyd

et al. [3]. The population consisted on 400 young women aged 15 - 30 years and a random

sample (n=100) of their mothers, aged 40 - 60 years. Briefly, breast images were acquired

in the sagittal plane with a slightly modified version of the GE FSE Dixon sequence. A 28

cm field of view was used with a 256 x 128 acquisition matrix. The slice thickness was 7

mm interleaved with TE 14x8 ms, ETL 8 and TR 2500. The total imaging time for both

breasts was about 13 min to obtain 45 slices covering the entire volume of both breasts.

Each scan consists of three images of water and fat signals with phase shifts of 0, π and

2π, according to the three-point Dixon method [38]. The image corresponding to the zero

degree phase shift corresponds to an image were both the fat and water signals are in

phase. This in-phase image of the right breast was used for further segmentations. Each

breast was semi-automatically segmented by 3 observers (inter-reader and intra-reader

agreement was more than 0.94) using an active contour approach with manual correction

[3]. In this work, the 2D delineated contours for each breast were stacked together in

adjacent cross-sections and resampled to an isotropic voxel size of 2.56 mm. Surface

meshes representing the 3D volume of the breast were finally obtained and these surfaces

were used as the gold-standard to measure the accuracy of the automatic selection of

landmarks and the overall segmentation accuracy (see chapter 3).

2.3 Detecting the breast boundary

2.3.1 Features of edge strength

The edge detection literature provides several definitions of an edge feature. One defi-

nition refers to an edge as a place in the image where there is a jump in intensity or in

brightness also known as a step discontinuity. Gradient magnitude and Laplacian mag-


nitude edge detectors such as Canny filters [39] attempt to detect edges based on local

extrema of the directional derivatives of the grey-level intensity function taken across

the edge [40] and therefore detect edges as points of maximal intensity gradient. Points

of maximal intensity gradient only occur at edges on cases of high image contrast and

homogeneous illumination, i.e when the structure of interest stands out clearly from the

background. Another alternative are phase congruency filters. Extensive work in this

area by Morrone and Owens [41] [42] and Kovesi [43] explained the relationship between

the phase and the perception of structures and showed that ”edges” are image features

perceived by the human brain as a wide variety of features besides points of maximal

gradient intensity. Perona and Malik [44] identified that most edge features in images are

composed of combinations of steps, roofs, ramps and mach bands feature types. Phase

congruency filters detect image features where the local Fourier components are max-

imally in phase. Phase congruency can be demonstrated in the 1D case by a square

waveform where the Fourier components are all in phase at the step of the square wave

(see figure 2.1).

Figure 2.1: Fourier series of a square waveform and sum of the first four terms. For a square

waveform p = 1 and φ = 0. Adapted from Kovesi [43].

The square waveform represents a pure step feature. The effect of different feature

orientations and illuminations on the perception of an ”edge” is exemplified on figure

2.2. The gradual interpolating from a step feature (top of image on figures 2.2-a,b) to a

line feature (bottom of image on figures 2.2-a,b) is generated by varying the phase offset


φ from 0 to π/2 on the square waveform Fourier series with amplitude decay p = 1 (see

equation on figure 2.1). Despite the variation in the sharpness of the edge due to the

differences of illumination and orientations, our visual perception of the feature type as

an edge remains.

Figure 2.2: Interpolation of a step feature to a line feature. a) 2D grating pattern by varying

φ = [0, π/2] from top to bottom and using p = 1.0, b) Canny edge detector of (a), c) Phase

congruency edge detector of (a). Adapted from Kovesi [43].

If we attempt to detect the edges of the grating in image 2.2-a using both a Canny

edge detector [39] (shown in figure 2.2-b) and phase congruency (shown in figure 2.2-c)

for comparison, it is clear that phase congruency captures more accurately the location

of the edge. The Canny filter erroneously detects the location of the edge where the

feature corresponds to a line feature. At the bottom of the image, the response of the

Canny filter is doubled as it captures points of maximal change in gradient intensity at

each side of the line edge feature. This example clearly exhibits some of the limitations of

gradient based edge detectors. This work focuses on phase congruency filters for detecting

edge features on breast MRI since these filters provide a more uniform response and are

less affected by low contrast. Low contrast between tissue types on the chest wall and

potential variations in illumination that occur due to field inhomogeneities clearly affect

edge detection in breast MRI by maximal gradient intensity methods.


2.3.2 Phase congruency edge detector

Points of phase congruency can be detected as points of maximal local energy. Morrone

and Owens [42] proposed a local energy model based on physiological evidence to ex-

plain the relationship between phase congruency, energy and Fourier components. The

schematic on figure 2.3 represents the Fourier components of a signal at a given location,

where the n-th Fourier term has an amplitude given by An(x) and a phase by φn(x). The

local energy of the signal E(x) is represented as the sum of the components added head

to tail, and the mean local phase of all Fourier terms is given by φ(x).

Figure 2.3: Polar plot of local energy model for phase congruency. The Fourier components of

the signal at a location x are plotted head to tail. Adapted from Kovesi [43].

The phase congruency function (PC) intends to measure the total phase deviation of

each Fourier component from the phase of the local energy of the signal E(x) [45] :

PC(x) = maxφ(x)∈[0,2π]

∑nAncos(φn(x)− φ(x))∑

nAn, n ≥ 0 (2.1)

A point of maximum phase congruency is defined as a point x where PC(x) reaches

a maximum. In this context, points of local phase congruency occur for all n where

φn(x)− φ(x) = 0 and thus PC(x) = 1. Furthermore, Venkatesh et al. [45] demonstrated


that these points also correspond to maxima of E(x). Since E(x) is a sum of infinite

number of components, if they follow a distribution with a small standard deviation

about the phase of E(x), then the magnitude of E(x) will attain a local maximum. It is

shown that the energy of the signal is equal to the phase congruency scaled by the sum

of the Fourier component amplitudes:

E(x) = PC(x)∑n

An (2.2)

2.3.3 Extracting local frequency via local energy

The Gabor function can be used as band-pass filter to obtain frequency information at

a particular region of the image spectrum.

Figure 2.4: 1D Log-Gabor transfer functions in a) linear frequency scale and b) logarithmic

frequency scale. Quadrature pairs of log-Gabor wavelets in the spatial domain tuned to the

same center frequency wo = 0.05 and bandwidths of c) 1 octave, d) 2 octaves and e) 3 octaves.

The logarithmic Gabor function has additional advantages; It can be designed with

arbitrarily large bandwidths while maintaining zero DC component. In addition, Field


[46] have shown that the log-Gabor function is more consistent with measurements on

mammalian visual systems indicating that cells in the visual cortex have responses that

are symmetric on the logarithmic frequency scale (see figure 2.4-a). The 1D log-Gabor

transfer function is defined as:

G(w) = exp

[− log (w/wo)

2

2 log (k/wo)2

](2.3)

where wo is the center frequency and k/wo is the bandwidth of the filter. k/wo should be

kept constant to achieve filters with equal bandwidths at different scales. Figure 2.4-a,b

shows examples of log-Gabor transfer functions with bandwidth of approximately 1, 2

and 3 octaves. Figure 2.4-c,d,e shows three log-Gabor filters in the spatial domain with

the same center frequency but different bandwidths. As expected, if the filter’s width in

the frequency domain increases the wavelet’s bandwidth in the spatial domain becomes

narrower. Filters should be designed to achieve a fairly even spectral coverage, avoiding

overlap between filters and minimizing aliasing artifacts.

Kovesi [47] described how phase congruency can be calculated in 2D images using

wavelets to obtain local frequency information via banks of filters. His approach consists

of building log-Gabor quadrature pairs, which are basically two linear operators with the

same amplitude response but phase responses shifted by 90°, to then apply them over

several scales and orientations via a bank of filters. Phase congruency information is

finally combined from each scale and orientation. A bank of filters is created by rescaling

a minimum wavelength to different scales so that an optimal overlap is obtained and the

sum of all transfer functions forms a relatively uniform coverage of the spectrum.

Computing phase congruency in 3D using a bank of filter imposes the complexity of

defining the number of appropriate orientations and the angular spread of the filters in

order to evenly cover the image spectrum. As an alternative, points of maximal phase

congruency can be detected as points of maximal local energy [42]. In this work, the

practical implementation of a phase congruency detector in 3D is simplified with the

computation of the local energy of the signal via a multidimensional generalization of


quadrature filters [48]. In contrast to the bank of oriented filters approach, there is

no need for an additional summation along different orientations. The filter’s center

frequency at a given scale is determined by the following equation:

ws =1

λmin(δ)s−1 , s = 1, 2, ..., n (2.4)

here λmin is the smallest wavelength of the Log-Gabor filter and δ is a scaling factor

between successive scales. λmin is scaled up to the total number of scales n. A noise

threshold (T ) is applied to the computation of phase congruency in order to suppress the

noise response. T is calculated at the smallest scale assuming a Rayleigh distribution as

T = µ + kσ, where µ is the mean and σ is the standard deviation of the local energy

distribution, k is a positive constant set to 2 in this work. To investigate the effect

of different parameters on edge detection results one can scan the filter response over

different parameters. Values of λmin = 3, k/wo = 0.65, and δ = 2.1 over a total of 6

scales gave good edge localization results (see fig 2.5-b).

Figure 2.5: Phase congruency detection examples with different filter parameters. a) λmin = 1,

b) λmin = 3, c) λmin = 6.

The effect of decreasing λmin can be appreciated in fig 2.5-a, where the filter enhances

local features at relatively higher frequencies. In contrast, increasing λmin (fig 2.5-c)

appears to blur out some features and local structure is detected at lower frequencies.


2.4 Breast surface via Poisson reconstruction

2.4.1 Poisson surface reconstruction

In this section, the Poisson surface reconstruction method is tested to reconstruct the

breast surface from oriented points obtained from phase congruency filtering in order to

extract an initial estimation of the breast region boundary.

Seminal work on surface reconstruction methods [49, 50, 51, 52] have been motivated

on reconstructing surfaces from 3D range scanning data. Surface reconstruction from

point samples is a vast studied problem in computer graphics and has been previously

treated as an inverse problem. Hoppe et al. [49] proposed to infer automatically the

geometry and topology of a surface (or two dimensional manifold) S given as input par-

tial information about sampled points on or near the surface and information about the

sampling process. Furthermore, Kazhdan et al. [51] showed that the surface reconstruc-

tion problem from a set of oriented surface points could be treated as a spatial Poisson

problem. Solving a surface reconstruction problem as a Poisson system offers resilience

to noise and the fitting process is global to all data. The key insight to the Poisson for-

mulation is that there is a very close relationship between oriented set of points sampled

on a surface and the gradient of the surface’s indicator function χM , which is a labeling

function whose value is zero at points outside of the region enclosed by the surface and

is one at points inside, i.e:

χM(p) =

1 p ∈M

0 p /∈M(2.5)

This relationship is strengthen by the properties of the gradient function. First, the

gradient of the surface’s indicator function ∇χM is a vector field whose components are

the partial derivatives of χM . Second, the gradient function is zero almost everywhere

(since the indicator function is mostly constant) except at points near the surface, where

is equal to the inward surface normal. Thus, based on this geometric property of the


gradient, a set of normal surface vectors can be viewed as samples of the gradient of the

indicator function.

Figure 2.6: Schematic of Poisson surface reconstruction in 2D. Adapted from [51].

In this context, Poisson surface reconstruction is an implicit function fitting method,

in which an implicit function in 3D is fitted to sampled points and then the surface is

extracted as an iso-surface of the implicit function (see figure 2.6). The idea behind this

approach is to infer the topology of an unknown surface by solving the 3D indicator

function χM ,

Khazhdan et al. [51] proposed to estimate the gradient of the indicator function χM ,

as the function that best approximates the vector field defined by point normals ~V , which

is equivalent to:

minχM‖∇χM − ~V ‖ (2.6)

This variational problem can be transformed into a Poisson problem: finding the best

solution involves computing a least-squared approximate solution of the scalar function

χM whose Laplacian (divergence of the gradient) equals the divergence of the vector field

~V :

∆χM ≡ ∇.∇χM = ∇.~V (2.7)


The Poisson system on equation 2.7 is solved based on the fundamental theorem of

calculus which formalizes the relationship between a solid model and samples of oriented

points on its surface. The Divergence Theorem states that the integral of the divergence

of a vector field over a solid is equal to the surface integral of the vector field over the

boundary. So that, for example, integrating the divergence of ~V is equal to the value

obtained by walking over the boundary of the region M (see figure 2.6) and summing the

values of the dot-product between the vector field and the surface normals ~V (s).~n(s).

2.4.2 Maximal phase congruency and edge point orientation

Points of maximal phase congruency coincide with features of high edge strength and

therefore can be thought as points of maximal edge potential. To estimate the orientation

of points at the breast boundary that correspond to maximal phase congruency, points are

sampled with an orientation corresponding to the gradient of the image. The gradient of

an image is a vector field that encodes the directional changes of the grey-level intensities

in the image. Broadly speaking, at the air-breast boundary interface, the gradient vectors

are normal vectors in the direction of the maximum change in intensity (i.e point inward

the breast-air boundary). The change in intensity at the breast-chest wall for Dixon

sequences is in the expected direction as well. In the lower and upper portions of the

breast no real edge information is present and therefore points are not sampled on these

regions. Hole-filling is one of the appealing properties of implicit surface reconstruction

methods [50]. Even in the presence of holes or missing point samples, the reconstruction

algorithm is guaranteed to return the surface of a solid model under appropriate sample

density. Then, the reconstructed mesh will be a closed surface and all the holes will have

been filled in. One of the limitations of this approach is the necessity to invert the sign

of the gradient vectors if for a different MRI breast sequence, the breast region is hypo-

intense with respect to the chest wall tissue. Figure 2.7 shows maximal phase congruency

points and their corresponding orientations based on the gradient of the image at the


Figure 2.7: Extracting points of maximal phase congruency and estimating their orientation.

a) mask of maximal phase congruency points, b) Image gradient for sampling orientation of

points.

same locations. In figure 2.7-a, a thresholded phase congruency image is shown where

bright areas correspond to regions of maximal phase congruency. Figure 2.7-b shows the

directional vectors of the corresponding locations. The color represents the magnitude

of the gradient at a given point. Also the inward direction of the vectors pointing to the

inside of the breast can be appreciated.

2.4.3 Implementation via Octree structures

The starting point for the Poisson problem is to obtain the vector field ~V from boundary

points. The Poisson inverse gradient method solves the inverse problem for approximating

a scalar field ( χM) when ~V is a non-conservative force field and therefore cannot be

expressed as the gradient of any scalar function. However, χM can be approximated over

a region such that the gradient of χM is the closest to ~V . Discretization of the problem

can be done in a natural manner using the pixel grid over which the image is defined, but


such a uniform 3D regular grid becomes impractical for high resolution reconstructions

[50]. A more efficient discretization is possible using an adaptive octree. An adaptive

octree is a 3D hierarchical data structure, where the subdivision unit is a cube that can

recursively be subdivided into eight smaller cubes or octants, hence the name octree.

Octrees use a node structure to store volumetric elements. Figure 2.8 shows an octree

with 2 levels of subdivision or octree depths (D = 2). This sequential subdivision is

represented on a spanning tree, where each node has eight children nodes. In an octree,

the resolution in each dimension increases by two at each subdivision level. Therefore,

to reach a resolution grid of 256x256x256, an octree with depth equal to 8 is required

(28 = 256). Kazhdan et al. [51] approach uses the positions of the sample points to

Internal node

Leaf nodeRoot

Level 1

Level 2

Figure 2.8: Schematic drawing of an octree hierarchical subdivision. Left: Recursive subdivision

of a cube into octants. Right: The corresponding octree. Adapted from [?].

define an octree O and associate a function Fo to each node of the tree. The Octree

is defined over the dataset to be the minimal octree with the property that every point

sample falls into at a node that is no longer subdivided at depth D, also known as a

leaf node. The function space Fo is built in a multiresolution fashion for more efficient


computation: high resolution near the samples and coarser resolution at farther distances.

The selection of space functions Fo has additional advantages since the vector field ~V

can then be represented as the linear sum of the Fo and the Poission equation can be

solved as a matrix multiplication in terms of Fo. For every node o ∈ O, Fo is set to be

the ”node function” centered about the node o and stretched by the size of o:

Fo(p) ≡ F

(p− cw

)1

w3(2.8)

where c and w are the center and the width of node o respectively. The function F is

defined as the n-th convolution of a box filter with itself:

Fo(x, y, z) = (B(x)B(y)B(z))n with B(t) =

1 |t| < 0.5

0 otherwise(2.9)

as n increases Fo approximates a unit-variance Gaussian, in this work n = 3. Trilinear

interpolation is used to sample the point sample position across the eight nearest neighbor

nodes NgbrD(p). Finally, the gradient of the indicator funcion can be approximated as:

~V (p) =∑p∈∂M

∑o∈NgbrD(p)

αo,pFo(p) ~N(p) (2.10)

where αo,p are the trilinear interpolation weights for point p closest to the eight depth-D

nodes NgbrD(p), and ~N(p) is the estimated orientation of point p. Having obtained the

vector field ~N(p), the next step is to solve the Poisson equation to find the function χ such

that its gradient ∇χM is the closest to ~V . In this work, the breast boundary indicator

function χ is represented in an adaptive octree and the Poisson equation is solved in

successive well conditioned sparse liner systems at multiple octree depths. The method

can be summarized in three main steps: 1) The relationship between the gradient of the

indicator function and an integral of the field of surface normals is established. 2) The

surface integral is solved by a summation over the given oriented point samples. 3) The

indicator function is reconstructed from the gradient field as a Poisson problem.


2.4.4 Extracting the breast boundary as an isosurface

The indicator function for the breast boundary (χM) corresponds to a scalar field where

equal values define isolines in 2D or isosurfaces in 3D (see figure 2.9). In order to obtain

the breast boundary reconstructed surface ∂M , it is necessary to select an appropriate

isovalue and then extract the corresponding isosurface from the indicator function. An

optimal isovalue is one that extracts an isosurface that is closest to the edges of the

breast region (i.e the extracted surface should closely correspond to the position of the

input samples). A natural way to extract this isosurface is to evaluate χM at the sample

positions (s ∈ S) and used the average sampled value as the optimal isovalue:

∂M ≡ {q ∈ R3 | χM(q) = γ} with γ =1

|S|∑s∈S

χM(s) (2.11)

Figure 2.9: Initial breast boundary surface via Poisson surface reconstruction. a) Poisson

reconstruction of breast boundary indicator function with color-coded isolines in 2D at equal

isovalues. b) An example of a extracted breast boundary (∂M) obtained with equation 2.8. c)

A view on 2D of a slice of the closed surface.

Once an appropriate isovalue γ is found, the 3D mesh representing the surface is

extracted using a Marching Cubes isosurface generation algorithm [53] adapted to octree


representations [54] [52]. Figure 2.9 shows a surface reconstruction example that was

solved using an octree depth of 8, which corresponds to an octree resolution of 2563. Some

spurious clusters of the surface are captured on the chest region where edge features are

also detected. These elements however, do not necessarily interfere with the estimation

of the complete breast surface and can be removed with an additional refinement step

described in the next chapter.

2.5 Conclusion

Phase congruency edge detectors were used to extract features of edge significance. The

computation of phase congruency in this work has the advantage of avoiding the use of an

oriented filter bank, which entails the selection of multiple appropriate filter orientations.

Surface reconstruction based on a Poisson formulation using edge features identified with

the edge detector robustly extracts the boundary of the breast as a surface. The results

obtained indicate that both methods are appropriate for building upon the next phase

of the segmentation pipeline, which attempts to redefine the surface of the breast region

to finalize the segmentation.

Chapter 3

Atlas-based segmentation of the

breast in 3D

3.1 Introduction

The extraction of an atlas of the breast in 3D using a population is described in this chap-

ter. Section 3.2 reports the feasibility of combining groupwise and pairwise registration

for establishing subject correspondences as well as the landmark-based representation of

the average volume. Generating a shape representation of the breast in 3D can be sum-

marized into two main sections: 1) Deriving the average breast volume in the population

using groupwise registration, and 2) generating an automatic shape representation based

on a 3D mesh.

The rationale for using an atlas for segmentation is to incorporate high-level or expert

knowledge about the shape of the breast in 3D. This approach is more robust to local

image artifacts and noise than low-level algorithms. In general, adapting an atlas or

model to an image suffers from a common problem to all local search algorithms, and

that is their vulnerability to stop at local minima and not necessarily find the global

optimum. However, if the search is properly initialized in proximity to the desired global

34

Chapter 3. Atlas-based segmentation of the breast in 3D 35

optimum often times the found local minimum will be close to the optimal solution and

the segmentation will be successful in matching the atlas to the desired image features.

For breast segmentation in MRI, thresholding operations to remove the background of

images have been proposed to obtain a scalar map of labels to guide the segmentation [27]

[55]. However, as explained on chapter 1, Breast MRI is often affected by low contrast,

field inhomogeneities and coil artifacts that render selective thresholding techniques weak

candidates to correctly separate tissues in the chest-wall. In this context, a more robust

approach is to use the edge map (phase-congruency, see section 2.3) and the estimation of

the breast region surface (Poisson reconstruction, see section 2.4) for proper initialization

of the atlas. The basic idea for proper initialization is to draw the atlas towards the edges

of the image so that its initial position is close to the desired boundaries.

This chapter describes the final step in the segmentation pipeline, which consist on

adapting the population atlas to the found target surface during initialization. Section

3.3 describes an approach to automatically map 3D landmarks from the atlas surface

to the initialization surface by treating landmark mapping as a correspondence problem

that can be solved using a Laplacian framework. Section 3.4 describes the validation

results of the segmentation and the evaluation of the algorithm performance.

3.2 Extracting a shape atlas of the breast in 3D

3.2.1 Groupwise registration and average volume extraction

An entropy-based groupwise registration using the stack entropy cost function (Balci et

al. [56]) and a multi-resolution B-spline free form deformation (FFD) introduced by

Rueckert et al. [57] was used to register the total set of available breast cases. Since

all breast datasets were acquired with the same acquisition protocol, they are similar in

terms of contrast, size, and orientation. Therefore, it is suitable to perform volumetric

grey-level registration. The choice of groupwise registration was motivated on finding a


population-based correspondence.

Figure 3.1: Population-based construction of the average volume. Selected sagittal slices of

mean volume and standard deviation volume in the population: a) before groupwise registration

and b) after groupwise registration. (Top row) mean and (Bottom row) standard deviation.

A common approach when registering a population of images is to register every sub-

ject in the population to a reference subject or template, using pairwise registration.

The problem with this approach is that some bias is introduced with the arbitrary se-

lection of the reference image. A strategy to overcome this bias is to avoid using an

arbitrary subject as an anatomical reference but rather construct a template from the

joint statistics of the population. Groupwise registration schemes have been developed

for this purpose. Balci and colleagues [56] used a multi-resolution registration approach

that considers the sum of univariate entropies along pixel stacks as a joint alignment

criterion. The registration starts with global affine transformation at a coarse scale and

proceeds with successive nonrigid deformations at finer scales in an attempt to capture

anatomical variability while minimizing the entropy in the population. The idea is that

if the images are aligned properly, intensity values at corresponding image locations will


form a low entropy distribution. The results of the registration were visually assessed

using the mean and the standard deviation volumes before (see Figure 3.1-a) and after

(see Figure 3.1-b) registration. The mean image appears to get sharper and the standard

deviation volume gets narrower, indicating that the alignment of the images increases

after performing groupwise registration.

3.2.2 Population shape average and 3D shape representation

The purpose of performing groupwise registration is to obtain the average breast volume

of the population. In this work, the average breast volume is further thresholded to

obtain a binary volume. The marching cubes algorithm [53] on the binary volume was

used to generate a dense surface representation of the shape of the breast in 3D. Following

surface generation, a series of local operations on geometry and topology on the surface

triangulation (Schroeder et al. [58]) were performed to reduce the number of surface

elements.

Figure 3.2: Average shape representation with varying decimation factors: a) 25% decimation,

b) 75% decimation and c) 95% decimation.

The decimation algorithm works by removing excess triangles where the curvature of


the surface is low but preserves edges by keeping dense triangulations where the curvature

of the surface is high. After decimation, the spatial coordinates of the decimated mesh

vertices were selected as landmark points. Table 3.1 summarizes the number of landmarks

generated by using different decimation factors from an initial triangulation consisting on

Mesh decimation

Decimation factor 25% 50% 75% 95% 98%

# Landmarks 5743 3829 1915 384 348

# Triangles 11482 7654 3826 764 692

Table 3.1: Number of landmarks and triangles generated after surface mesh decimation.

7657 vertices and 15309 triangles. The density of triangle elements in the representation

of the average shape is reduced with an increasing decimation percentage (see figure 2.2).

The decimation percentage indicates the percent ratio between the triangle elements

that are eliminated with respect to the initial set of 7657 triangles. Different decimation

percentages (25%, 50%, 75%, 95% and 98%) were investigated. In addition, connectivity

data between landmark points was generated for further representation of structures.

3.3 Mapping 3D atlas landmarks via a Laplacian frame-

work

After obtaining the optimal isosurface to initialize the segmentation as explained on the

previous chapter, the next step in the segmentation pipeline is to incorporate shape

information contained on an atlas. For this purpose, the atlas is initially aligned to

the found isosurface by means of a rigid landmark transformation. The remaining step

before matching the atlas to the extracted surface is to map the atlas points to the

found optimal surface. This is in nature a correspondence problem and can be solved


using the mathematical properties of the Laplace equation. Figure 3.3 illustrates how

the correspondence can be solved and can be summarized as follows:

Figure 3.3: Initializing model points into optimal isosurface. a) binary versions of surfaces. b)

area enclosed between surfaces. c) Solution to Laplace equation. d) Normalized gradient.

Using the optimal isosurface on a binary image representation (Siso) and a binary

version of atlas (Sm), it is possible to find the region enclosed between the two aligned

surfaces (see figure 3.3-b) and solve a Laplacian equation with two boundary conditions,

each one corresponding to two fixed and different potentials as follows:

∇2ψ =δ2ψ

δx2+δ2ψ

δy2+δ2ψ

δz2= 0 (3.1)

with boundary conditions: ψ = ψ1 on Sm and ψ = ψ2 on Siso, where (ψ1, ψ2) are two

different fixed potentials. The solution to the Laplace equation is a scalar field ψ that

provides a transition from surface Sm to Siso as defined by set of nested surfaces [59] with

constant values between ψ1 and ψ2. Furthermore, given the geometric properties of the

Laplace equation, computing the normalized negative gradient of the Laplace solution,


N:

N =−∇ψ‖ −∇ψ ‖

(3.2)

returns a unit vector field that defines streamlines or field lines connecting both surfaces.

These streamlines are defined as being everywhere orthogonal to all equipotential surfaces

defined between the two surfaces. The resulting vector field can be used to map the path

between two corresponding points (i.e p1 on Sm to p2 on Siso). The path connecting p1

to p2 can be parameterized so the path is described by a vector function C(s) as follows:

∫ p2

p1

N(C(s))ds (3.3)

Initializing a given 3D landmarked point involves solving the above integral between the

atlas surface and the initialization iso-surface. This algorithm is implemented as follows:

Algorithm 3.1 Initializing model points into optimal isosurface Siso

Require: Binary representations of isosurface Siso and atlas Sm

Find region ∼ Sm ∩ Siso

Solve Laplace eq. 3.1 over ∼ Sm ∩ Siso

Find N = −∇ψ‖−∇ψ‖

Find∫ p2p1

N(C(s))ds

3.4 Validation: Segmentation of population cases

To quantitatively assess the performance of the 3D atlas-based segmentation approach,

segmentation errors with respect to target surfaces were determined for the total popu-

lation (n = 409) on a leave-one-out approach. The test can be summarized in 7 steps

which are illustrated on figure 3.4. First, a shape atlas is built using all but one case

of the training population. Second, the omitted case is treated as an unseen case, and


therefore a phase congruency edge map is extracted followed by the boundary surface via

Poisson surface reconstruction, and finally, the atlas is mapped to the breast boundaries

using a Laplacian approach explained on section 3.3.

Build population

atlas

1

Stage 2:

For omitted case on stage 1:

Phase congruency,

edge points selection

2

Sample point

orientations

3

Solve Poisson

system from V

4

Stage 1:

For all-but-one cases on population:

5Extract

isosurface

6map atlas points

to isosurface

Stage 3:

Final atlas-based segmentation

Figure 3.4: Experimental design to assess performance of segmentation using a leave-one-out

approach.

The test provided fully automatically segmented breast boundary surfaces. We com-

pared a total of 367 breast volumes, since 42 cases were excluded due to failures in the

initial surface alignment (steps b) and f) in fig. 3.6). The goal of our experiments was to

evaluate how accurate was the segmentation of the breast obtained with our algorithm

with respect to the expert segmentation. We therefore looked at distance errors from

surface to surface, volumetric errors in cm3 as well as standard overlap metrics such as

the the total volume overlap, the mean overlap and the Dice coefficient.


3.4.1 Distance errors

Distances were measured from the shape atlas to the breast boundary iso-surface after the

initial Poisson boundary reconstruction (see table 3.2) and after finding corresponding

points using the Laplacian equation (see table 3.3). Segmentation errors were computed

as the Euclidean distance (in mm) between each atlas point (xi) and the closest point

(xtargeti ), located on the surface of the original manually segmented shapes:

Serror = d(xi,xtargeti ) =

√√√√ N∑i=1

|xi − xtargeti |2 (3.4)

This distance reflects the proximity of the segmented surface to the target surface, since

a very accurate segmentation is attained when all atlas points lie on the target surface

and the segmentation error is zero.

After Poisson Quartiles of distances (mm)

Boundary region 0.25 0.50(median) 0.75 IQR

air-breast 0.63 1.41 3.61 2.97

breast-chest 0.63 1.40 3.93 3.30

Table 3.2: Distances from manually annotated surfaces discriminated by boundary region after

Poisson surface reconstruction. IQR=Interquantile range.

After Laplacian Quartiles of distances (mm)

Boundary region 0.25 0.50(median) 0.75 IQR

air-breast 0.79 1.36 2.93 2.15

breast-chest 1.12 2.68 4.82 3.70

Table 3.3: Distances from manually annotated surfaces discriminated by boundary region after

Laplacian correspondences. IQR=Interquantile range.

By labelling the relative position of each individual landmark in the shape atlas, one

can discriminate the mapping accuracy among specific regions of the breast boundary,


such as the chest-wall or the air-breast boundary regions. Table 3.2 and 3.3 summarizes

the mapping accuracy by the algorithm. Figure 3.5 shows a comparison between the

target distance errors for the two stages of segmentation, after Poisson surface recon-

struction and after Laplacian mapping of atlas points. The distance errors given by the

outliers in the distribution are larger for the Poisson surface reconstruction results. This

is likely the result from the spurious detection of edge features in the thoracic region,

which are mostly removed with the used of the 3D atlas Laplacian mapping stage. Due

to the fact that the Laplacian is a correspondence mapping procedure, the accuracy is

reduced in the chest wall as indicated by the median from 1.4 mm to 2.68 mm. This loss

in accuracy could be reversed by using the result of the Laplacian as a mask to remove

the spurious elements in the chest region.

Figure 3.6 shows representative results of the breast boundary reconstructions and

the atlas mapping segmentation. For illustrating the technique two cases are shown. The

first case (top row) is a large sized breast with a manually measured volume of 1229.21

cm3 and an atlas-based segmented volume of 1136.59 cm3. The second case (bottom

row) is a smaller sized breast with a manually measured volume of 572.8 cm3 and an

atlas-based segmented volume of 601.5 cm3. The color-coded surfaces on figure 3.6-d and

3.6-h illustrate the distance to target surfaces after mapping atlas points (figures 3.6-c

and 3.6-g) to the Poisson reconstructed surface (figures 3.6-a and 3.6-e).

3.4.2 Volume errors

In order to test the impact of the segmentation accuracy on the breast volume assessment,

in addition to distance errors, volume errors between manually annotated volumes and

the total volume enclosed by our segmentation boundaries were also computed according

to Alyassin et al. [60]. For the total breast volumes measured in cm3, the median of the

absolute percent measurement error was 6.75%. Figure 3.6 presents the Bland-Altman

plot that compares the volumes obtained automatically with the volumes measured from


0

10

20

30

40

50

60

70

80

90

100

Comparison of distance distributions after Poisson Reconstruction and Laplacian atlas mapping.

Breast-air region

after Poisson

Clo

sest dis

tance to s

urf

ace (

mm

)

Breast-air region

after Laplacian

Chest-wall region

after Poisson

Chest-wall region

after Laplacian

Figure 3.5: Box-and-Whisker plots: Comparison of target distance errors for the two stages

of segmentation, after Poisson surface reconstruction and after Laplacian mapping of atlas

points. Points are drawn as outliers if they are larger than q3 + 1.5(q3 − q1) or smaller than

q1− 1.5(q3− q1), where q1 and q3 are the 25th and 75th percentiles, respectively.

the manual segmentations. The magnitude of the disagreement (in terms of bias and

limits of agreement) was 29.8 cm3 (-141.5 to 201.1) cm3. The scatter of the values tends

to increase for larger volumes. The fact that some values are more scattered on the

negative side of the plot (with respect to zero difference line), indicates that in some

cases the automatic segmentation tends to underestimate the true volume of large breast

volumes (<900 cm3).


Figure 3.6: Representative results of 2 breast boundary reconstructions (top and bottom):

a) and e) Breast iso-surface (output of Poisson reconstruction). b) and f) Aligned mean of

population showing initial position of atlas points before Laplacian mapping. c) and g) Mapping

results to target surface via Laplace equation. d) and h) Color-coded closest distances to manual

annotated surfaces.

3.4.3 Evaluation of standard overlap metrics

The evaluation of standard overlap metrics [61] was performed to test the agreement of

our segmentation with the manual results. Overlapping metrics can be explained with a

Venn diagram (see figure 3.7). Basically, the idea is to quantify the overlapping fraction

of two partially overlapping objects: a source S (model-based) and target T (manual)

segmentation volumes. In this work, two agreement measures and two error measures

were explored. The two agreement measures consists of total volume overlap (To) and

mean overlap also known as the Dice coefficient (Mo), which are defined as follows:

To =|S ∩ T ||T |

(3.5)


Figure 3.7: Bland-Altman plot showing the difference between automatic and manual measure-

ments of total breast volume (in cm3) against the average of the two methods.

Standard overlap metrics results

Metric mean ± std

Total overlap 0.89 ± 0.05

Mean overlap (Dice) 0.88 ± 0.04

False-negatives 0.11 ± 0.05

False-positives 0.13 ± 0.07

Table 3.4: Agreement and error overlap results (n=367).

Mo = 2

(|S ∩ T ||S|+ |T |

)(3.6)

where the operator || indicates volume computed as number of voxels. The two error

measures were metrics indicating the false negative (FN) and false positive (FP ) frac-

tions. The false negative error for a given volume can be measured as the fraction of the

target that falls outside of the source volume. Similarly, the false positive error can be


Figure 3.8: Overlap metric assessment for segmentation results. Venn diagram illustrating the

relationship between partially overlapping objects.

measured by how much of the source volume falls outside of the target volume:

FN =|T \ S||T |

(3.7)

FP =|S \ T ||S|

(3.8)

The table 3.3 summarizes the results obtained with the aforementioned metrics. Re-

sults are expressed in fractions as mean ± std. Figure 3.8 shows the distribution of

target overlap, mean overlap (Dice), false-negative and false-positive values as well as

the volume and surface areas obtained for the population of segmented examples. Figure

3.9 shows the distribution of percentage and absolute percentage error for total breast

volume and total surface area.

3.5 Conclusion

In this chapter, a method for obtaining the average of the population and an automatic

breast volume segmentation were described. The boundary localization results demon-

strate error distances from manual surfaces with an IQR of 2.15 mm for the air-breast


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distribution of Overlap metrics (n=367)

total mean union FN FP

(a) (b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

total mean FN FP

Distribution of overlap metric (n=367)

Figure 3.9: Boxplots: Distribution of overlap metrics, breast volumes and surface areas of

the manually segmented datasets compared with the atlas-based segmentation results. a) total:

total overlap, mean: Mean overlap (Dice), FN: false negative fraction, FP: false positive fraction.

b) Distribution of volumes and surface areas.

boundary and 3.7 mm for the breast-chest wall boundary. This indicates that the error

for localizing the breast-air boundary is within sub-resolution accuracy. Errors are rel-

atively higher for localizing the chest-wall boundary, probably due to the low contrast

between adjacent tissues to the breast in this region. However, the results obtained are

encouraging for using phase congruency edge detection and Poisson surface estimation to

segment the total volume of the breast. Furthermore, incorporating a manual override to

the automatic segmentation framework will be a feasible approach for computing total

breast volume during breast density assessment using MRI.


% Volume

error

70

60

50

40

30

20

10

0

-10

-20

-30

-40

% Absolute

Volume

error

% Surface

Area

error

% Absolute

Surface

Area error

Percentage and Absolute percentage Volume and Surface errors

Figure 3.10: % Error and Absolute % error for Volume and Surface areas when compared with

manual segmentations.

Chapter 4

Conclusions and future directions

4.1 Contribution of this work

4.1.1 Segmentation for breast density assessment

As stated on chapter 1, breast segmentation is a prerequisite task for breast density as-

sessment using MRI. In general, prior to the quantification of the proportion of water

content in the breast it is necessary to determine the total breast volume. This thesis

presents an automatic segmentation framework that allows the computation of the total

volume of the breast while excluding non-breast tissues, which facilitates the quantifi-

cation of breast density. This work attempts to first identify features of local phase

information in the image and then adapt a shape atlas to the found features. The use of

invariant features to contrast and illumination for edge detection as well as the use of a

shape atlas are attractive characteristics of the method. This signifies that the segmenta-

tion framework is not dependent on image intensity and may be applicable to other MRI

sequences or imaging modalities such as 3D mammography with breast tomosynthesis.

Several breast segmentation methods and density assessment methods for breast MRI

have appeared in the literature. Khazen et. al [30] used non-uniformity correction based

on the proton density data before performing threshold-based segmentation to exclude

50

Chapter 4. Conclusions and future directions 51

the background. Nie et al. [27] separated fat and fibroglandular tissue by applying the

fuzzy C-means algorithm, which requires manual adjustments of the number of clusters

depending on the fat content in the breast, a reason why this method has been criticized.

The segmentation performance results agree well with other methods. Ertas et al. [62]

proposed extracting the breast region using two cellular neural networks (CNNs). The

first CNN performs grey-level thresholding, the second erases small objects by perform-

ing morphological operators. Although the authors compared two cases with manual

segmentations, no numerical error analysis were performed. A recently published work

by Wang et al. [63] uses Hessian-based filters to segment the Pectoralis muscle boundary

in breast DCE-MRI. The main idea is to differentiate the boundary based on sheet-like

local structure of neighbouring pixels. The authors reported a mean distance error of

1.43 mm with a standard deviation of 0.46 mm for 30 datasets.

4.1.2 Segmentation in the context of CAD systems

In recent years, magnetic resonance imaging (MRI) of the breast has received consider-

able attention as an emerging tool for the detection and diagnosis of breast tumors [64].

Dynamic contrast-enhanced MRI (DCE-MRI) provides information about increased vas-

cularity and capillary permeability in lesions which adds sensitivity for lesion differenti-

ation. Breast MRI has been shown to have higher sensitivity than mammography and

ultrasound [65] in high risk populations, particularly for detecting cancers in women with

genetic risk factors for breast cancer [65],[66] and with dense breasts [67]. Despite its

well known advantages, translation of MRI into clinical practice has encountered some

difficulties, especially the lack of standardization and interpretation guidelines. Some of

the main drawbacks of MR mammography are the high inter-observer variability [68], as

the interpretation of scans still relies on subjective visual assessment. Given the large

volume of data produced per exam, evaluation of these images by the radiologist is a

very time-consuming task and a experience dependent process. On the other hand, there


is growing evidence to suggest that quantitative computerized interpretation in the con-

text of computer-aided diagnosis (CAD), may improve the accuracy [69], specificity [70],

objectivity and consistency [70],[71],[72] of breast MRI interpretation.

Current FDA approved breast MRI CAD workstations provide post-processing image

features (e.g color-maps of contrast kinetic parameters, time-intensity curves), which

facilitates the reviewing process, by making MRI data more comprehensible. However,

breast MRI CAD’s full potential has not been reach yet in providing a computer-aided

evaluation of the degree of suspicion in a lesion. The need for computer-aided diagnostic

(CAD) systems is relevant when it comes to improve diagnostic accuracy while retaining

high sensitivity and efficiency for breast cancer diagnosis. One of the fundamental ideas

for DCE-MRI automatic lesion detection is that surrounding non-breast regions such as

the chest wall muscles, thoracic cavity, lungs and heart must be previously segmented

out to prevent false-positive detections on these regions. Therefore, a prerequisite for

computer-aided lesion detection is the segmentation of the breast region. For this reason,

the 3D automatic breast segmentation presented in this work has important implications

in the context of CAD development and validation.

4.2 Future directions

Although no quantitative measurement of breast density was performed in order to com-

pare the automatically segmented results with the manually segmented ones, a feasible

extension of the stage of this work is the implementation of a graphical user interface

(GUI). With a GUI the automatic segmentation can be incorporated with a manual

override to the breast density assessment workflow.

The integration of the breast boundary localization described in this thesis as the

starting point for a subsequent model-based segmentation is a logical extension of this

work. The advantage of incorporating an additional step on the segmentation pipeline is


to perform model-based segmentation, which offers potentially a more robust segmenta-

tion. Statistical shape models (SSM) incorporate prior knowledge about the structure of

the object to be segmented and are able to learn patterns of biological variability. This

variability is compactly represented with a model of variation and can be of shape or

appearance. Hence, a logical direction of this work is to build an statistical shape model

(SSM) from the groupwise registered shapes. Preliminary results towards this end were

obtained and are reported on Appendix A. The performance of the model-based seg-

mentation was tested on the total population of shapes that were initialized successfully

(n=367). It is expected that an additional model-based adjustment to the boundaries

will refine the 3D representation of the breast surface and reduce the distance errors.

In contrast, the comparison of results shown in figure 4.1 indicate that the chest-wall

distance errors increase with the SSM model fit to the boundary points as well as the

volume discrepancy, demonstrating that further improvement is necessary.

This increase on errors can be due to a number of pitfalls with the current model

fitting algorithm:

1) The least-square fit of the model to the points can be strongly affected by the pres-

ence of outliers during the search for landmark positions. Other optimization schemes

for matching the model to the found points during search could be optimal. Gradient

descent optimization could be implemented if the partial derivatives of an error function,

such as a weighting function that penalizes the fit to an outlier, is added to the min-

imization of the sum of squares. To this end Lekadir et al. [73] proposed a weighted

fitness measure using the ratio of interlandmark distances as a local shape dissimilarity

measure for each outlier based on a tolerance model.

2) Another problematic area is the use of PCA for dimensionality reduction, since

there is an explicit assumption that the data distribution forms a linear subspace. Fur-

thermore, by manipulating the shape parameter vector b on equation 9 (see appendix A),

different shapes can be generated. Now the allowed variation from the mean of the popu-


(a) (b)

Volume

error %

Absolute

Volume error %

Volume

error %

Absolute

Volume error %

Figure 4.1: Model fitting compared to boundary localization distance error results. a) Distance

and Volume errors after Laplacian initialization of the atlas. b) Distance and Volume errors

after SSM model fitting.


lation is limited by hard shape constraints imposed with the radial basis function matrix.

The quality of the radial basis functions or modes of variation used in the model is highly

relevant. It is clear that the captured variability should ideally only capture variation

due to shape differences among the training cases, an not due to correlated global differ-

ence (e.g changes in position). Therefore global linear transformation components such

as translations or rotations should be completely excluded from the deformations used

to aligned the cases (conventionally, Procrustes analysis [74]). If similarity deformations

are not removed, the first modes of variation will not necessarily capture the truly largest

variation in shape. In this work, it is possible that residual variation could be preventing

the modes of variations to broadly capture shape variability.

3) A third area of potential improvement is decreasing the shape constraints imposed

by the model. Relaxing the hard shape constraints and allowing additional flexibility

during model fitting has been previously proposed. One traditional approach on this

paradigm is to fit a normal SSM until convergence and then include an additional re-

finement stage that improves the fitness to edge-based features, usually using a free

deformation. Heimann et al. [75] combines a statistical shape model and a deformable

surface using posterior probabilities given by the data.

Appendix A

Statistical shape model of the breast

in 3D

56

Automatic model-based 3D segmentation of the breast in MRI

Cristina Gallegoa,b, Anne L. Martela,b aDepartment of Medical Biophysics, University of Toronto, Toronto, Canada

bDepartment of Imaging Research, Sunnybrook Health Sciences Centre, Toronto, Canada

ABSTRACT

A statistical shape model (SSM) is constructed and applied to automatically segment the breast in 3D MRI. We present an approach to automatically construct a SSM: first, a population of 415 semi-automatically segmented breast MRI volumes is groupwise registered to derive an average shape. Second, a surface mesh is extracted and further decimated to reduce the density of the shape representation. Third, landmarks are obtained from the averaged decimated mesh, which are non-rigidly deformed to each individual shape in the training set, using a set of pairwise deformations. Finally, the resulting landmarks are consistently obtained in all cases of the population for further statistical shape model (SSM) generation. A leave-one-out validation demonstrated that near sub-voxel resolution reconstruction (2.5mm) error is attainable when using a minimum of 15 modes of variation. The model is further applied to automatically segment the anatomy of the breast in 3D. We illustrate the results of our segmentation approach in which the model is adjusted to the image boundaries using an iterative segmentation scheme.

Keywords: Breast Segmentation, population-based statistical shape model, Breast MRI.

1. INTRODUCTION In recent years, Magnetic Resonance Imaging (MRI) has received considerable attention as an emerging diagnostic tool for detection and characterization of breast tumors as well as for breast cancer risk assessment. As a consequence, the number of MRI breast examinations carried out is expanding rapidly and computational tools to analyze these studies are becoming widely demanded. In particular, excluding any non-breast regions is a prerequisite for many image analysis tasks such as in the case of computing image metrics for breast density assessment or for computer-aided diagnosis. In the development of computer-aided diagnosis (CAD) systems for breast MRI, automatically excluding non-breast regions particularly in the chest wall has become an important initial step to leave out all the potential false positive detections that occur in the chest wall during training and validation of the CAD.

Model-based segmentation based on statistical shape models (SSM) has been proposed as a suitable technique for locating objects with known structure. One of the most attractive aspects of SSM is their ability to incorporate prior knowledge about the anatomy to be segmented that accounts for biological variability and is introduced with the annotation of training cases. Coots et al.1 demonstrated the applicability of Active Shape Models (ASM) for segmenting structures in 2D images. However, constructing a 3D statistical shape model requires dense representation of surfaces in a training population as well as establishing correspondences among all surfaces. We have focused on automatically generating landmark correspondences using a population-based approach based on 3D deformable groupwise registration.

This work aims to develop a 3D SSM of the breast for fully automatic 3D breast segmentation. Our approach can be broken down into four main sections: 1) automatic landmark-based representation of the average shape in a training population, 2) automatic identification of landmark correspondences between the average shape and each individual shape in the population, 3) construction of a SSM and 4) adaptation of the SSM for 3D breast MRI segmentation.

In sections 2.2 and 2.3, we report the feasibility of combining groupwise and pairwise registration for establishing landmark correspondences, and test the ability of the SSM to reconstruct all shapes in the population in section 3.2. In sections 2.4 and 2.5, we present our approach to derive a SSM and match it to the anatomical boundaries of the breast for segmentation. Section 3 describes the experimental setup and validation of our methodology using a population of 415 training cases. Sections 4 and 5 present discussion of the significance of this work and conclusions.

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2. MATERIALS AND METHODS 2.1 Segmented Breast Cases

A population of breast segmentations (n = 415) was used from a previous study by Boyd et al.2. Breast images were acquired in the sagital plane with a slightly modified version of the GE FSE Dixon sequence. A 28 cm field of view was used with a 256 x 128 acquisition matrix. The slice thickness was 7 mm interleaved with TE 14x8 ms, ETL 8 and TR 2500.

Each segmented breast consists of semi-automatically 2D delineated contours using an active contour approach with manual correction2. The delineated contours were stacked together in adjacent cross-sections. The resulting stacked volume was resampled to isotropic voxel size of 2.56 mm. Surface meshes representing the 3D breast anatomy were finally obtained from the isotropic volumes and these surfaces were used as the “gold-standard” for testing the accuracy of our automatic selection of landmarks (section 3.2) and the performance of our model-based segmentation approach (section 3.4).

2.2 Groupwise registration and 3D shape representation

An entropy-based groupwise registration using the stack entropy cost function (Balci et al.3) and a multi-resolution B-spline free form deformation (FFD) introduced by Rueckert et al.4 was used to register the total set of available breast cases. Grey-level isotropic volumes were used to derive the average breast volume of the population which was thresholded to obtain the final binary volume. The marching cubes algorithm on the final binary volume was used to generate a dense surface representation of the average shape in the population. Following surface generation, a series of local operations on geometry and topology (Schroeder et al.5) were performed to reduce the number of triangles. The decimation works by removing excess triangles where the curvature of the surface is low but preserves edges by keeping dense triangulations where the curvature of the surface is high. After decimation, the spatial coordinates of the decimated mesh vertices were selected as landmark points. Different decimation percentages (25%, 50%, 75%, 95% and 98%) were investigated. In addition, connectivity data between landmark points was generated for further representation of structures. Table 1 summarizes the number of landmarks generated by using different decimation factors from an initial set of 7657 landmarks. Results of the decimation are illustrated in Figure 1.

Figure 1. Decimation results with varying decimation factor of a) 25% decimation, b) 75% decimation, c) 95% decimation.



Table 1. Number of landmarks generated after triangle mesh decimation with different decimation factors.

Triangle mesh decimations Decimation factor 25% 50% 75% 95% 98%

Total Landmarks 5743 3829 1915 384 348

2.3 Automatic selection of landmarks

Having obtained a set of landmarks on the average shape of the population, corresponding landmarks on each individual shape were generated using volumetric grey-level registration between individual cases and the average volume data. The result of this registration is a deformation field that can be use to map landmarks on the average shape in the registered space, to corresponding landmarks on each individual case back into the unregistered space, a method that is called “landmark propagation”. We assessed the accuracy of landmark propagations using both pairwise transformations between individual cases and the average volume data as well as inverted groupwise transformations already employed during the average volume extraction. A comparison between the groupwise and the pairwise generated deformation fields and their ability to produce suitable landmarks was performed. The landmark propagation error (equation 1) was computed as the Euclidean distance (in mm) between the ith

propagated landmark pix and the closest point surface

ix located on the surface of the original meshes described in

section 2.1. The closest point surfaceix to a landmark point p

ix is found using a spatial search structure 6.

( ) ∑=

−==N

i

surfaceiisurfacep xxXXdgElandmarkin

1

2p, (1)

2.4 Statistical shape model construction

After obtaining a set of corresponding landmarks among N shapes, a statistical shape model (SSM) is generated as introduced by Cootes et al.1. A single vector of concatenated 3D landmark coordinates represents each shape (equation 2) and principal component analysis (PCA) of the covariance matrix (equation 4) describes the principal modes of variation φk that can be captured in the population. Here φk is referred as the “kth eigenvector” and λk is the corresponding “kth eigenvalue” with the kth largest variance (equation 5). After sorting the eigenvectors φk and eigenvalues λk in descending order (i.e 1+≥ kk λλ ), the number of modes of variation is chosen so that the total variance of the model (equation 6) is at least 98%. As a result, the eigenvector matrix Φ is obtained by concatenating the first t eigenvectors (equation 7). { }111111000 ,,,...,,,,...,,,,,, −−−= imimimikikikiiiiiii zyxzyxzyxzyxX (2)

X =1N

Xii=1

n∑ (3)



( )( )∑=

−−−

=N

i

Tii XXXX

ND

111

(4)

Dφk = λkφk (5)

∑=

=t

kkTV

1λ (6)

Φ = φ1, | ... |φk | ... |φt( ) (7)

With a statistical shape model (SSM), all shapes in the training population can be approximated using the mean shape (equation 3) and a linear combination of modes that account for the majority of the variation Φ weighted by appropriate shape parametersb (equation 8). Given the assumption that eigenvector matrix Φ is orthogonal, parameter b is a t-dimensional vector given by equation 9. bXX Φ+≈model (8)

( )XXb T −Φ= (9)

The ability of the model to accurately reconstruct each of the training cases was evaluated on a leave-one-out test. Before obtaining a reconstructed shape with the model a set of b parameters is computed using equation 9, where X are

the propagated landmarks (obtained in section 2.3), X is the mean shape and, TΦ is the transposed eigenvector matrix. The average reconstruction error (equation 10) was computed as the Euclidean distance (in mm) between the ith

predicted landmark ( modelix ) by the model of the omitted case and the ith propagated landmark ( p

ix ) of the same case.

( ) ( )∑=

−==N

iiip xxXXdtionEreconstruc

1

2pmodelmodel, (10)

2.5 Model-based segmentation of the breast

In this framework, a model-based segmentation is based on adapting the shape model to the boundaries of the breast. Points located in the air-breast boundary airy , are obtained in the normal direction of each model point as they intersect a boundary mesh product of a thresholding operation excluding the background and the chest wall region. Points located in the chest wall boundary chesty , are obtained in the normal direction of current model points as well but using the magnitude of the grey-level gradient of the image to locate the strongest edge. The edge strength is quantified as the image intensity in the magnitude gradient image. It is assumed that the strongest edge corresponds to the boundary of the breast. For locating the breast-air boundary this assumption is highly correct, since the maximum gradient magnitude corresponds to the step edge between the skin and the background. However, for locating the chest-wall boundary this assumption can be wrong since in clinical breast MRI the contrast between the chest muscles and the breast might be not sufficient and the strongest edge does not necessarily correspond to the chest-wall boundary. This could lead to a local minima and the model would be fitted to an inaccurate boundary in the chest wall. To overcome this limitation, a close initialization to the boundaries of interest is performed prior to the model-based segmentation. To tackle the initialization of the model in the image we proposed to use a rigid landmark registration that finds a rigid transformation (translation, rotation, scaling) needed to minimize the distance between model points and points obtained at the image boundary by simple thresholding operations and surface reconstructions.



Starting with an optimal initialization in the image, our model-based segmentation approach continues to look for locations of maximum edge strength along a normal direction and within a 50 to 10 mm radius. The found points during search form an incomplete vector of candidate model points chestairsparse yyy += . The robustness of the statistical

shape model approach allows us to obtain a non-sparse update of the shape parameters b , using a sparse matrix of coefficients that correspond to the sparse candidate model points (equation 11).

( )Xyb Tsparse −=Φ sparse (11)

Finally, a new shape instance is computed with the found shape parameters using equation 8. This iterative process is repeated until there is no relevant change (less than 5%) on the shape parameters.

3. RESULTS 3.1 Groupwise registration and 3D shape representation

We successfully applied a population-based atlas construction to obtain the average shape of the population. The results of the registration were visually assessed using the mean and the standard deviation volumes before (Figure 2-a) and after (Figure 2-b) registration. The mean image appears to get sharper and the standard deviation volume gets narrower, indicating that the alignment of the images increases after performing groupwise registration.

Figure 2. Population-based atlas construction of the average shape. Mean volume selected sagital slices a) before groupwise registration and b) after groupwise registration.

3.2 Automatic selection of landmarks

We successfully applied a population-based atlas construction to obtain the average shape of the population. The results of the registration were visually assessed using the mean and the standard deviation volumes. Furthermore, 3D shape representation based on the marching cubes reconstruction resulted on a set of 7657 landmarks that were subsequently decimated to only 384 landmark points, using 95% decimation factor. The comparison between the groupwise and the pairwise generated deformation fields and their ability to produce suitable landmarks, (i.e. landmarks in close proximity to the target surface) yielded smaller registration errors for pairwise generated deformation fields. Over 80% of the



generated landmarks using the pairwise generated deformation fields were located below 5 mm distance from the surface, as shown in figure 3-a) and b). Figure 3-c) shows the mean, median and standard deviation of distances to target surface when propagating landmarks using groupwise and pairwise corresponding deformation fields according to equation 1. Given this advantage of the pairwise over the groupwise deformations, we generated corresponding landmarks using the pairwise deformation fields.

Figure 3. Propagation of landmark points: a) Cumulative distribution of distances to target surface for generated landmarks. b) Frequency of distances to target surface for total generated landmarks using pairwise registration. c) Summary statistics of landmark generation errors. c) Mean, median and standard deviation of distances to target surface for generated landmarks using registration (groupwise and pairwise).

3.3 Shape variability and ability of the model the reconstruct training cases

The first 10 principal modes of variation account for over 90% of the variation explained by the model as shown in figure 4-a). The ability of the model to represent the shapes in the training population was investigated using a leave-one-out test. The average reconstruction error was computed. The error (in mm) between the predicted landmarks using optimal model parameters and the original landmarks was calculated according to equation 10. The average reconstruction error as a function of the number of principal modes is plotted in figure 4-b). Near sub-voxel resolution reconstruction (2.5mm) error is attainable when using a minimum of 15 main modes of variation.

Figure 4. Model construction results: a) Cumulative variance explained by the model as a function of modes of variation. b) Average reconstruction error for the leave-one-out validation as a function of modes of variation.



3.4 Iterative segmentation: Matching the shape model to the anatomical boundaries of the breast.

Optimal adjustments of the model are those that relocate the model surface points to the edges of the breast. This adaptation is performed in an iterative fashion. Figure 5 illustrates cross-sections of the segmented image in two segmentation cases. In figure 5-a) the breast-air boundaries as well as the chest-wall boundary have been correctly found and therefore the segmentation of the breast is correct. In figure 5-b) the chest-wall boundary localization fails to identify the true chest-wall boundary so the iterative search falls in local minima.

Figure 5. 2D slices over the 3D segmentation result: a) Correct segmentation example. b) Segmentation fails to locate exact boundary corresponding to the breast tissue boundary in the chest wall.

4. DISCUSSION The results indicate that our approach of average shape construction using groupwise registration followed by pairwise propagation of landmarks from the average shape to each individual shape in the population is suitable for automatic identification of landmark correspondences. One of the advantages of using groupwise registration for this purpose is the reduction of the bias introduced with the selection of the reference image and therefore the landmarked average shape is not biased towards any specific shape representation in the population. Testing the ability of the model to reconstruct training cases using a leave-one-out validation demonstrated that near sub-voxel resolution reconstruction (2.5mm) error is attainable when using a minimum of 15 main modes of variation, higher accuracy in the reconstruction of 1mm is attainable when using up to 200 modes of variation. In this aspect, our work



relates well with previous research done in this area. Frangi et al.7 reported an average placement error of 2.2 mm between landmarks manually placed and those propagated from a mean atlas. Further work is needed to improve the localization of the chest wall given that the model matching is prone to fail in local minima.

5. CONCLUSIONS Our approach is a population-based automatic construction of statistical models of shape for 3D breast that can be applied to 3D automatic breast segmentation in MRI. Our approach is not grey-level dependent so it can potentially be applied to different MRI acquisitions. Results of the generation of landmarks using groupwise and pairwise registration have been presented. A leave-one-out validation test demonstrated that on average the reconstruction error is of sub-voxel accuracy is attainable. Full validation of the segmentation performance with improved chest wall localization will be a matter of further work.

6. ACKNOWLEDGMENTS This research was supported by the Canadian Breast Cancer Research Alliance. The authors would like to thank Dr. Norman Boyd at the Campbell Family Institute for Breast Cancer Research, Ontario, Canada, for facilitating the breast MRI data for this study and also thanks to Anoma Gunasekara and Sofia Chavez for their valuable contributions.

REFERENCES

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[2] Boyd, N., Martin, L., Chavez, S., Gunasekara, A., Salleh, A., Melnichouk, O., Yaffe, M., Friedenreich, C., Minkin S., Bronskill, M., "Breast-tissue composition and other risk factors for breast cancer in young women: a cross-sectional study," Lancet Oncology 10(6), 569-580 (2009).

[3] Balci, SK., Golland, P., Shenton, M., Wells, WM., "Free-Form B-spline Deformation Model for Groupwise Registration", Statistical Registration Workshop: MICCAI 2007, 23-30 (2007).

[4] Rueckert, D., Sonoda, L., Hayes, C., Hill, D., Leach, M., Hawkes, DJ., "Nonrigid registration using free-form deformations: application to breast MR images" IEEE Trans. Med. Imag. 18(8), 712-721 (1999).

[5] Schroeder, W. J., "Decimation of Triangle Meshes," Computer Graphics 26, 65 - 70 (1992). [6] Meagher, D. J., "Efficient synthetic image generation of arbitrary 3D objects." Pattern Recognition and Image

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dimensional statistical shape models: Application to cardiac modeling," IEEE Trans. Med. Imag. 21, 1151-1166 (2002).



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Documents

Automatic 3D segmentation of the breast in MRI · Automatic 3D segmentation of the breast in MRI Cristina Gallego Ortiz Master of Science Graduate Department of Medical Biophysics