Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 8)

Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 8)

8 A MRF-Based Approach for the Measurement

of Skin Thickness in Mammography

Antonis Katartzis, Hichem Sahli, Jan Cornelis, Lena Costaridou, and George Panayiotakis

CONTENTS

8.1 Introduction 8.2 Background8.2.1 MRF Labeling8.2.2 MRF-Based Mammographic Image Analysis8.3 Data and Scene Model

8.3.1 Image Acquisition8.3.2 Radiographic and Geometrical Properties of the Skin8.4 Estimation and Extraction Methods

8.4.1 Skin Feature Estimation8.4.1.1 External Border of the Skin8.4.1.2 Exclusion of the Region of the Nipple Estimation

of the Normals to the Breast Border8.4.1.3 Estimation of Gradient Orientation8.4.2 Skin-Region Extraction — MRF Framework8.4.2.1 Selection of a Region of Interest 8.4.2.2 Markovian Skin Model Labeling Scheme 8.5 Results

8.5.1 Measurement of Skin Thickness8.5.2 Clinical Evaluation

8.6 ConclusionsReferences

8.1 INTRODUCTION

Breast skin changes are considered by physicians as an additional sign of breastpathology They can be divided into two major categories, namely skin retraction2089_book.fm copy Page 315 Monday, May 16, 2005 11:41 AM

316 Medical Image Analysis

and localized or generalized skin thickening, which can be either benign or nant The skin can attain a thickness of 10 to 20 times normal before it can beperceived as abnormal by palpation [1, 2] Both retraction and thickening may beevident mammographically before they can be clinically detected The existingtechniques for the measurement of breast skin thickness are based on manual esti-mations on the mammograms, using simple measuring devices [3, 4] Consideringthe continuous evolution of computer-aided diagnostic systems, the aforementionedmanual methods appear quite obsolete As far as time and accuracy are concerned,the quantitative analysis of breast skin changes can be substantially improved with

malig-a computer-malig-assisted memalig-asurement technique

We have developed a computerized method for the measurement of breast skinthickness from digitized mammograms that involves a salient feature (hereinafterdenoted as a skin feature) that captures the radiographic properties of the skin regionand a dedicated Markovian model that characterizes its geometry [5] During a firstprocessing stage, we apply a combination of global and local thresholding operationsfor breast border extraction The estimation of the skin feature comprises a methodfor the exclusion of the region of the nipple and an estimation of the gray-levelgradient orientation, based on a multiscale wavelet decomposition of the image.Finally, the region of the skin is identified based on two anatomical properties,namely its shape and its relative position with respect to the surrounding mammo-graphic structures This a priori knowledge can be easily modeled in the form of aMarkov random field (MRF), which captures the contextual constraints of the skinpixels The proposed MRF model is defined on a binary set of interpretation labels(skin, no skin), and the labeling process is carried out using a maximum a posteriori

probability (MAP) estimation rule The method is tested on a series of mammogramswith enhanced contrast at the breast periphery, obtained by an exposure-equalizationtechnique during image acquisition The results are compared with manual measure-ments performed on each of the films

The chapter is organized as follows In Section 8.2 we present the main principles

of Markov random field theory and its application to labeling problems and provide

an overview of related work on mammographic image analysis In Section 8.3 wedescribe the image-acquisition process and state the main properties of the skin asviewed in a typical mammogram Section 8.4 initially refers to the extraction of thesalient feature that discriminates the skin from other anatomical structures at thebreast periphery The section concludes with a description of the proposed Markovianmodel and the labeling scheme for the extraction of skin region The validation ofour method, which includes representative results for the measurement of skinthickness, is presented in Section 8.5 Finally, a discussion and suggested directionsfor future research are given in Section 8.6

8.2 BACKGROUND 8.2.1 MRF L ABELING

The use of contextual constraints is indispensable for every complex vision system

A scene is understood through the spatial and visual context of the objects in it; the

A MRF-Based Approach for the Measurement of Skin Thickness 317

objects are recognized in the context of object features at a lower level representation;

the object features are identified based on the context primitives at an even lowerlevel; and the primitives are extracted in the context of image pixels at the lowestlevel of abstraction

Markov random field theory provides a convenient and consistent way of eling context-dependent entities, constituting the nodes of a graph [6] This isachieved through characterizing mutual influences among such entities using MRFprobabilities Theory tells us how to model the a priori probability of context-dependent patterns A particular MRF model favors its own class of patterns byassociating them with larger probabilities than other pattern classes Such models,defined on regular lattices of image pixels, have been effectively used in texturedescription and segmentation [7], as well as in image restoration and denoising [8,9] In higher levels of abstraction, MRF models are able to encode the spatialdependencies between object features, giving rise to efficient schemes for perceptualgrouping and object recognition [10]

mod-We will briefly review the concept of MRF defined on graphs Let G = {S,N}

be a graph, where S = {1, 2, …, m} is a discrete set of nodes, representing eitherimage pixels or structures of higher abstraction levels, and N = {Ni|∀i∈ S} is agiven neighborhood system on G Ni is the set of all nodes in S that are neighbors

be numeric as well as symbolic, e.g., interpretation labels) The family L is called

a MRF, with respect to the neighborhood system N, if and only if

1 P(L = l) > 0, for all realizations l of L

a subset of S such that it contains either a single node or several nodes that are allneighbors of each other If we denote the collection of all the cliques of G, withrespect to the neighborhood system N, as C(G,N), then the general form of arealization of P(l) can be expressed as the following Gibbs distribution

318 Medical Image Analysis

where is called the Gibbs energy function and V c(l) the clique

potential functions defined on the corresponding cliques c∈C(G,N) The functional

form of these potentials conveys the main properties of the Markovian model Finally,

is a normalizing constant called the partition function

In the case of a labeling problem, where L represents a set of interpretation

labels and d = {d1, …, d m} a set of physical measurements that correspond to the

realization of an observation field D on S, the most optimal labeling of the graph

G can be obtained based on a maximum a posteriori probability (MAP) criterion.

According to the Bayes rule, the posterior probability can be computed using the

following formulation

(8.2)

where P(L = l) is the prior probability of labeling l, p(D = d|L = l) is the conditional

probability distribution function (PDF) of the observations d, also called the

likeli-hood function of l for d fixed, and p(D = d) is the density of d, which is constant

when d is given In a more simplified form, Equation 8.2 can be written as

By associating an energy function to p(d|l) and P(l), the posterior probability obtains

the following form

P(l|d) ∝ e −U(l|d) , U(l|d) = U(d|l) + U(l) (8.4)Following this formulation, the optimal labeling is then accomplished via the

minimization of the posterior energy function U(l|d) [6] The combinatorial problem

of finding the global minimum of U(l|d) is generally solved using one of the

following relaxation algorithms: (a) simulated annealing (SA) [8], or (b) iterated

conditional modes (ICM) [12]

8.2.2 MRF-B ASED M AMMOGRAPHIC I MAGE A NALYSIS

Several mammographic image analysis techniques, based on MRF models, have

been proposed in the literature These models are capable of representing explicit

knowledge of the spatial dependence between different anatomical structures and

can lead to very efficient image-segmentation schemes The segmentation process

is performed by defining either a MRF on the original lattice of image pixels or a

cascade of MRF models on a multiresolution, pyramidal structure of the image In

both cases, the parameter estimation of the Markovian priors is carried out either

empirically or using selected training data

In the early work of Karssemeijer [13], a stochastic Bayesian model was used

for segmenting faint calcifications from connective-tissue structures The method

was based on local contrast and orientation observation measures and a resolution MRF describing both spatial tissue dependencies and the clustering char-acteristics of microcalcifications Comer et al [14] proposed a statistical algorithmfor the segmentation of mammograms into homogeneous texture regions In theirapproach, both the mammographic image and the underlying label field (representing

single-a finite number of tissue clsingle-asses) single-are modeled single-as discrete-psingle-arsingle-ameter rsingle-andom fields.The labeling is performed via a maximization of the posterior marginals (MPM)process [11], where the unknown likelihood parameters are estimated using theexpectation-maximization (EM) algorithm

In recent years, the need to reduce the complexity of MRF models on image lattices gave rise to a series of hierarchical/multiresolution analysis methods

large-Li et al [15] developed a technique for tumor detection based on an initial tation using a multiresolution MRF model and a postprocessing classification stepbased on fuzzy, binary decision trees With a pyramidal image representation and apredefined set of tissue labels, the segmentation is carried out in a top-down fashion,starting from the lowest spatial resolution and considering the label configurations

segmen-as the realizations of a dedicated MRF The segmentation at each resolution levelcomprises a likelihood-parameter estimation step and a MAP labeling scheme usingthe ICM algorithm, initialized with the result of the previous resolution In theapproach of Zheng et al [16], a similar hierarchical segmentation scheme is applied

on a multiresolution tower constructed with the use of the discrete wavelet transform

At each resolution, the low-frequency subband is modeled as a MRF that represents

a discrete set of spatially dependent image-intensity levels (tissue signatures) taminated with independent Gaussian noise Finally, Vargas-Voracek and Floyd [17]introduced a hierarchical MRF model for mammographic structure extraction usingboth multiple spatial and intensity resolutions The authors presented qualitativeresults for the identification of the breast skin outline, the breast parenchyma, andthe mammographic image background

con-All of the aforementioned labeling techniques consider the image labels (tissuetypes) as being mutually exclusive, without taking into account the projective nature

of the mammographic image modality McGarry and Deriche [18] presented a hybridmodel that describes both anatomical tissue structural information and tissue-mixturedensities, derived from the mammographic imaging process Spatial dependenciesamong anatomical structures are modeled as a MRF, whereas image observations,which represent the mixture of several tissue components, are expressed in terms oftheir linear attenuation coefficients These two sources of information are combinedinto a Bayesian framework to segment the image and extract the regions of interest.The MRF-based method presented in this chapter falls in the scope of imagesegmentation/interpretation for the identification of an anatomical structure situated

at the breast periphery (skin region) It uses (a) an observation field that encompassesthe projective, physical properties of the mammographic image modality and (b)

a MRF model, defined on the full-resolution image lattice, that describes thegeometric characteristics of the skin in relation to its neighboring anatomicalstructures The following sections present in detail the different modules of theproposed approach

8.3 DATA AND SCENE MODEL

8.3.1 I MAGE A CQUISITION

In general, the effect of overexposure at the region of the film corresponding to thebreast periphery results in a poor visualization of the skin region, hampering itsidentification Contrast enhancement at the breast periphery can be accomplishedwith a series of exposure or density-equalization techniques Exposure equalizationcan be performed using either anatomical filters [19, 20] or more sophisticatedtechniques that modulate the entrance exposure, based on feed-back of the regionalvariations in X-ray attenuation [21, 22] The existing methods for density equaliza-tion mainly employ computer-based procedures for the matching of the opticaldensity between the periphery and the central part of the breast [23–27]

In our study, during the acquisition of each mammogram, we used the anatomicalfilter-based exposure-equalization (AFEE) technique of Panayiotakis et al [20] Thistechnique utilizes a set of solid anatomical filters made of Polyamide 6, as thismaterial meets the basic requirements of approximately unit density, homogeneity,and ease of manufacture The anatomical filters have a semicircular band shape withincreasing thickness toward the periphery The AFEE technique produces images ofimproved contrast characteristics at the breast periphery, ensuring minimization ofthe total dose to the breast through the elimination of a secondary exposure to patientswith an indication of peripheral breast lesions Its performance has been extensivelyevaluated using both clinical and phantom-based evaluation methods [28, 29].The mammographic images used in this study were digitized using an AgfaDuoScan digitizer (Agfa Gevaert, Belgium) at 12-bit pixel depth and a spatialresolution of 100 µm/pixel According to quality control measurements, this filmdigitizer is suitable for mammogram digitization, as the optical-density range of thecases used for validation falls into the linear range of its input/output response curve[30] Figure 8.1 shows an example from our test set of mammograms

8.3.2 R ADIOGRAPHIC AND G EOMETRICAL P ROPERTIES OF THE S KIN

Our approach for breast skin thickness extraction involves the construction of aphysical model of the skin region that describes both its radiographic and geometricproperties This model is based on the following three assumptions

1 Anatomically, if we consider an axial section of the breast, the skin is athin stripe of soft tissue situated at its periphery At its vicinity, there isthe subcutaneous fat, which radiographically is viewed as a structure withhigher optical density than the one of the skin This anatomical informa-tion, together with the fact that mammography is a projection imagingmodality, will be the basis of our model The region of the image that thephysicians indicate as skin does not correspond to the real one at any ofthe breast sections, and it is always bigger than the skin thickness that a

histological examination might give In fact, this virtual skin, indicated by

the physicians, is the superposition of thin stripes of soft tissue that

corre-spond to the real skin at several axial sections of the breast (Figure 8.2).

2 The shape of the skin’s external border should coincide with the shape ofthe breast Most of the time, this appears to be regular, and it can beapproximated by a circle or an ellipse In an effort to make the shapeestimation more accurate and reliable, we will not consider the breastborder as a whole Instead, we make the assumption that it can be dividedinto smaller segments, each of them corresponding to an arc of a circle.

3 From the configuration of Figure 8.2, we can infer that the external border

of the skin in a mammographic image is mainly formed by the projection

of the central section of the breast As we move inward, starting from thebreast periphery, we notice also the projections of the skin segments thatbelong to breast sections situated above and below the central one In thedigitized gray-level image, this results in a gradient at the periphery ofthe breast (where the skin is located), oriented perpendicularly to thebreast border

All the previously described assumptions are the main components of our model.Their combination leads to the following conclusion:

The salient feature (skin feature) that reveals the skin layer of the breast (as this is viewed on the mammogram) is the angle formed by the gradient vector and the normals

to the breast border Deeper structures, underneath the skin layer, do not conform to the previously mentioned radiographic and geometrical skin model.

FIGURE 8.1 Original image.

8.4 ESTIMATION AND EXTRACTION METHODS

8.4.1 S KIN F EATURE E STIMATION

8.4.1.1 External Border of the Skin

The external border of the skin separates the breast from the surrounding background,thus it coincides with the breast border Several computerized schemes have beendeveloped for the automatic detection of the breast region Most of them make use

of the gray-level histogram of the image Yin et al [31] have developed a method

to identify the breast region on the basis of a global histogram analysis Bick et al.[32] suggested a method based on the analysis of the local gray-value range toclassify each pixel in the image Davies and Dance [33] used a histogram-derivedthreshold in conjunction with a mode filter to exclude uniform background areasfrom the image Chen et al [34] proposed an algorithm that detects the skin-lineedge on the basis of a combination of histogram analysis and a Laplacian edgedetector Mendez et al [35] used a fully automatic technique to detect the breastborder and the nipple based on the gradient of the gray-level values

Our approach initially employs a noise-suppression median-filtering step (with

a filter size equal to five pixels), followed by an automated histogram thresholdingtechnique We assume that the histogram of each mammogram exhibits a certainbimodality: each pixel in the image belongs either to the directly exposed region

FIGURE 8.2 Geometrical representation of the imaging process of the skin.

X-ray source

Breast sections

Film

(image background) or to the potential object of interest (breast) For this purpose,

we have chosen the minimum-error thresholding technique proposed by Kittler andIllingworth [36] The principal idea behind this method is the minimization of acriterion function related to the average pixel classification error rate, under theassumption that the object and the background gray-level values are normally dis-tributed

Unfortunately, the presence of the anatomical filter, used for exposure tion, disturbs the bimodality of the image histogram A threshold selection, usingthe histogram of the whole image, results in an inaccurate identification of the breastborder More specifically, the gray values corresponding to the anatomical filterinduce a systematic error that increases the value of the threshold compared withthe optimal one The size of the resulting binary region will always be smaller thanthe real size of the breast

equaliza-To overcome this problem, we try to combine both local and global information.Initially, an approximation of the breast’s border is estimated by performing a globalthresholding on the histogram of the whole image using the method of Kittler and

Illingworth After thresholding, the breast border is extracted by using a logical opening operator with a square flat structuring element of size 5, followed

morpho-by a 4-point connectivity tracking algorithm We then define overlapping square

windows along the previously estimated border, where we apply local thresholdingusing the same approach as before (Figure 8.3) All the pixels situated outside theunion of the selected windows keep the label attributed to them by the initial global

FIGURE 8.3 Application of local thresholding for the extraction of the skin’s external border.

Background

Anatomical filter

Breast

1st Approximation

of the border

thresholding process The size of each window is empirically set to a physical length

of approximately 1.5 cm (150 × 150 pixels)

The histogram of each window can now be considered as bimodal, containingonly pixels from the breast and the filter For each of them, a threshold is estimatedusing the method of Kittler and Illingworth [36] Its final value is the average betweenthe threshold found in the current region and the ones of its two neighbors Because

of the overlap between neighboring windows, the resulting binary image is smooth,with no abrupt changes in curvature Finally, the rectified breast border is obtained

by applying, once again, a morphological opening operator with a square flat turing element of size 5, followed by a tracking algorithm The final result of ourapproach, applied to the image of Figure 8.1, is presented in Figure 8.4

struc-8.4.1.2 Exclusion of the Region of the Nipple Estimation of the

Normals to the Breast Border

Based on the second assumption of our skin model (see Section 8.3.2), we can dividethe breast border into several segments with equal lengths and consider each of them

as belonging to a circular arc The parameters of these circles (namely their radiiand the coordinates of their centers) are estimated by using the Levenberg-Marquardtiterated method for curve fitting [37] A χ2 merit function is defined that reflects theagreement between the data and the model In our case, the data are the coordinates

FIGURE 8.4 External border of the skin.

of the border points, and the model is a circle The optimal solution corresponds tothe minimum of the merit function.

Unfortunately, this circular model of the breast border is disturbed by the ence of the nipple Moreover, when the doctors examine a mammogram, they usuallysearch for possible skin changes along the breast border, except of the region behindthe nipple, mainly because of the presence of other anatomical structures that havesimilar densities as the skin (e.g breast areola) For these reasons, we first excludethe region of the nipple and then work only with the remaining part of the breastborder

pres-Nipple detection and localization is an ongoing research topic in mammographicimage analysis [35, 38] In our scheme, the exclusion of the nipple is performed inthree steps [5]:

1 The breast border is divided in three equal segments

2 We choose the central border segment (nipple included) and estimate thecoordinates of the circle that corresponds to it using the method of Lev-enberg-Marquardt [37]

3 We consider the profile of distances between the center of the circle andeach point of the central border segment The border points that corre-spond to the nipple are situated between the two most significant extrema

of the first derivative of the profile of distances

This technique works well in practice, except for extreme cases where the nipple

is not visible in the mammogram because of possible retraction or other types ofdeformation In these cases, manual intervention is needed

The removal of the nipple allows an efficient fitting of circular arcs to theremaining breast border and an accurate estimation of the directions normal to it.Experiments have shown that a number of five circles is sufficient for this purpose.The directions normal to the breast border can be found by simply connecting everypoint of each border segment to the center of the circle that corresponds to it

8.4.1.3 Estimation of Gradient Orientation

Most of the time, the image gradient is considered as a part of the general framework

of edge detection The basic gradient operators of Sobel, Prewitt, or Roberts [39]are very sensitive to noise, are not flexible, and cannot respond to a variety of edges

To cope with these types of problems, several multiscale approaches for edgedetection are proposed in the literature, such as the Gaussian scale-space approach

of Canny [40] or methods based on the wavelet transform [41, 42] In our study, theestimation of the multiscale gradient is performed using the wavelet approach pre-sented by Mallat and Zhong [42], which is equivalent to the multiscale operator ofCanny However, due to the pyramidal algorithm involved in the calculation of thewavelet transform, its computational complexity is significantly lower than thecomputational complexity of Canny’s approach In wavelet-based gradient estima-tion, the length of the filters involved in the filtering operation is constant, while thenumber of coefficients of the Canny filters increases as the scale increases

The method of Mallat and Zhong [42] is based on wavelet filters that correspond

to the horizontal and vertical components of the gradient vector Let ƒ(x, y) ∈ L2(R2)

be a two-dimensional (2-D) function representing the image, and φ(x, y) a smoothing function that becomes zero at infinity and whose integral over x and y is equal to

1 If we define two wavelet functions ψ1(x, y) and ψ2(x, y) such as

(8.5)then the wavelet transform of ƒ(x, y) at a scale s has two components defined by

(8.6)

By , i = {1, 2}, we denote the dilation of ψi (x, y) by the scale factor s, so

that: Following these notations, the orientation of thegradient vector is given in Equation 8.7

(8.7)

In the case of a discrete 2-D signal, the previously described wavelet model does

not keep a continuous scale parameter s Instead, it takes the form of a discrete dyadic wavelet transform, which imposes the scale to vary only along the dyadic

sequence 2j , j ∈ Z When we pass from the finest scale (j = 1) to coarser ones (j >

1), the signal-to-noise ratio in the image is increased This results in the elimination

of random and spurious responses related to the presence of noise On the otherhand, as the scale increases, the gradient computation becomes less sensitive to smallvariations of the gray-level values, resulting in a low precision of edge localizationand blurring of the image boundaries The selection of the optimal scale depends

on the spatial resolution of the digitized mammograms For our images (spatialresolution of 100 µm/pixel), we found that the third decomposition scale (j = 3)

gives a good approximation of the image gradient, as far as our region of interest

is concerned (breast periphery) An empirical study showed that the second and thefourth scale of the wavelet decomposition are optimal for mammograms digitizedwith 200 µm/pixel and 50 µm/pixel, respectively In our application, the waveletdecomposition and the estimation of the gradient orientation (Equation 8.7) wereperformed using the Wave2 source code [43] developed by Mallat and Zhong [42]

By knowing the gradient orientation and the normals to the breast border, wecan produce a transformed image that represents the values of our skin feature andhighlights the region of the skin At each point of the original image, the skin feature(as this is defined in Section 8.3.2) can be derived by estimating the angular differ-ence between the gradient vector and the normals to the breast border Figure 8.5shows the transformed image that represents the estimated angular difference forthe example of Figure 8.1, where black represents a difference of zero degrees andwhite a difference of 180° The dark stripe along the breast periphery corresponds

to the region of the skin Note that the middle part of the image, where the nipple

is situated, has been removed

x y y