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2089_book.fm copy Page 315 Monday, May 16, 2005 11:41 AM 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 8.2 Introduction Background 8.2.1 MRF Labeling 8.2.2 MRF-Based Mammographic Image Analysis 8.3 Data and Scene Model 8.3.1 Image Acquisition 8.3.2 Radiographic and Geometrical Properties of the Skin 8.4 Estimation and Extraction Methods 8.4.1 Skin Feature Estimation 8.4.1.1 External Border of the Skin 8.4.1.2 Exclusion of the Region of the Nipple Estimation of the Normals to the Breast Border 8.4.1.3 Estimation of Gradient Orientation 8.4.2 Skin-Region Extraction — MRF Framework 8.4.2.1 Selection of a Region of Interest 8.4.2.2 Markovian Skin Model Labeling Scheme 8.5 Results 8.5.1 Measurement of Skin Thickness 8.5.2 Clinical Evaluation 8.6 Conclusions References 8.1 INTRODUCTION Breast skin changes are considered by physicians as an additional sign of breast pathology They can be divided into two major categories, namely skin retraction Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 316 Monday, May 16, 2005 11:41 AM 316 Medical Image Analysis and localized or generalized skin thickening, which can be either benign or malignant The skin can attain a thickness of 10 to 20 times normal before it can be perceived as abnormal by palpation [1, 2] Both retraction and thickening may be evident mammographically before they can be clinically detected The existing techniques for the measurement of breast skin thickness are based on manual estimations on the mammograms, using simple measuring devices [3, 4] Considering the continuous evolution of computer-aided diagnostic systems, the aforementioned manual methods appear quite obsolete As far as time and accuracy are concerned, the quantitative analysis of breast skin changes can be substantially improved with a computer-assisted measurement technique We have developed a computerized method for the measurement of breast skin thickness from digitized mammograms that involves a salient feature (hereinafter denoted as a skin feature) that captures the radiographic properties of the skin region and a dedicated Markovian model that characterizes its geometry [5] During a first processing stage, we apply a combination of global and local thresholding operations for breast border extraction The estimation of the skin feature comprises a method for the exclusion of the region of the nipple and an estimation of the gray-level gradient 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 mammographic structures This a priori knowledge can be easily modeled in the form of a Markov random field (MRF), which captures the contextual constraints of the skin pixels 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 mammograms with enhanced contrast at the breast periphery, obtained by an exposure-equalization technique during image acquisition The results are compared with manual measurements 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 we describe the image-acquisition process and state the main properties of the skin as viewed in a typical mammogram Section 8.4 initially refers to the extraction of the salient feature that discriminates the skin from other anatomical structures at the breast periphery The section concludes with a description of the proposed Markovian model and the labeling scheme for the extraction of skin region The validation of our method, which includes representative results for the measurement of skin thickness, is presented in Section 8.5 Finally, a discussion and suggested directions for future research are given in Section 8.6 8.2 BACKGROUND 8.2.1 MRF LABELING 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 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 317 Monday, May 16, 2005 11:41 AM 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 lower level; and the primitives are extracted in the context of image pixels at the lowest level of abstraction Markov random field theory provides a convenient and consistent way of modeling context-dependent entities, constituting the nodes of a graph [6] This is achieved through characterizing mutual influences among such entities using MRF probabilities Theory tells us how to model the a priori probability of contextdependent patterns A particular MRF model favors its own class of patterns by associating them with larger probabilities than other pattern classes Such models, defined on regular lattices of image pixels, have been effectively used in texture description 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 spatial dependencies between object features, giving rise to efficient schemes for perceptual grouping and object recognition [10] 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 either image pixels or structures of higher abstraction levels, and N = {Ni|∀i ∈ S} is a given neighborhood system on G Ni is the set of all nodes in S that are neighbors of i, such that i ∈ Ni if j ∈ Ni, then i ∈ Nj Let L = {L1, L2, …, Lm} be a family of random variables defined on S, in which each random variable Li takes a value li in a given set (the random variables Li’s can 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 P(L = l) > 0, for all realizations l of L P(li|lj,∀j ≠ i) = P(li|lj), j ∈ Ni where P(L = l) = P(L1 = l1, L2 = l2, …, Lm = lm) (abbreviated by P(l)) and P(li|lj) are the joint and conditional probability functions, respectively Intuitively, the MRF is a random field with the property that the statistics at a particular node depend on that of its neighbors An important feature of the MRF model defined above is that its joint probability density function has a general functional form, known as Gibbs distribution, that is defined based on the concept of cliques A clique c, associated with the graph G, is a subset of S such that it contains either a single node or several nodes that are all neighbors of each other If we denote the collection of all the cliques of G, with respect to the neighborhood system N, as C(G,N), then the general form of a realization of P(l) can be expressed as the following Gibbs distribution P (l ) = Copyright 2005 by Taylor & Francis Group, LLC −U (l ) e Z (8.1) 2089_book.fm copy Page 318 Monday, May 16, 2005 11:41 AM 318 Medical Image Analysis where U (l ) = ∑c∈C Vc ( I ) is called the Gibbs energy function and Vc(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, Z = ∑l ∈L e −U (l ) 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, …, dm} 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 P (L = l | D = d ) = p (D = d | L = l ) P (L = l ) p (D = d ) (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 likelihood 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 P(l|d) ∝ p(d|l)P(l) (8.3) 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-BASED MAMMOGRAPHIC IMAGE ANALYSIS 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 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 319 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 319 was based on local contrast and orientation observation measures and a singleresolution MRF describing both spatial tissue dependencies and the clustering characteristics of microcalcifications Comer et al [14] proposed a statistical algorithm for the segmentation of mammograms into homogeneous texture regions In their approach, both the mammographic image and the underlying label field (representing a finite number of tissue classes) are modeled as discrete-parameter random fields The labeling is performed via a maximization of the posterior marginals (MPM) process [11], where the unknown likelihood parameters are estimated using the expectation-maximization (EM) algorithm In recent years, the need to reduce the complexity of MRF models on largeimage lattices gave rise to a series of hierarchical/multiresolution analysis methods Li et al [15] developed a technique for tumor detection based on an initial segmentation using a multiresolution MRF model and a postprocessing classification step based on fuzzy, binary decision trees With a pyramidal image representation and a predefined 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 as the realizations of a dedicated MRF The segmentation at each resolution level comprises a likelihood-parameter estimation step and a MAP labeling scheme using the ICM algorithm, initialized with the result of the previous resolution In the approach 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) contaminated with independent Gaussian noise Finally, Vargas-Voracek and Floyd [17] introduced a hierarchical MRF model for mammographic structure extraction using both multiple spatial and intensity resolutions The authors presented qualitative results for the identification of the breast skin outline, the breast parenchyma, and the mammographic image background All of the aforementioned labeling techniques consider the image labels (tissue types) as being mutually exclusive, without taking into account the projective nature of the mammographic image modality McGarry and Deriche [18] presented a hybrid model that describes both anatomical tissue structural information and tissue-mixture densities, derived from the mammographic imaging process Spatial dependencies among anatomical structures are modeled as a MRF, whereas image observations, which represent the mixture of several tissue components, are expressed in terms of their linear attenuation coefficients These two sources of information are combined into 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 image segmentation/interpretation for the identification of an anatomical structure situated at the breast periphery (skin region) It uses (a) an observation field that encompasses the projective, physical properties of the mammographic image modality and (b) a MRF model, defined on the full-resolution image lattice, that describes the geometric characteristics of the skin in relation to its neighboring anatomical structures The following sections present in detail the different modules of the proposed approach Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 320 Monday, May 16, 2005 11:41 AM 320 Medical Image Analysis 8.3 DATA AND SCENE MODEL 8.3.1 IMAGE ACQUISITION In general, the effect of overexposure at the region of the film corresponding to the breast periphery results in a poor visualization of the skin region, hampering its identification Contrast enhancement at the breast periphery can be accomplished with a series of exposure or density-equalization techniques Exposure equalization can be performed using either anatomical filters [19, 20] or more sophisticated techniques that modulate the entrance exposure, based on feed-back of the regional variations in X-ray attenuation [21, 22] The existing methods for density equalization mainly employ computer-based procedures for the matching of the optical density between the periphery and the central part of the breast [23–27] In our study, during the acquisition of each mammogram, we used the anatomical filter-based exposure-equalization (AFEE) technique of Panayiotakis et al [20] This technique utilizes a set of solid anatomical filters made of Polyamide 6, as this material meets the basic requirements of approximately unit density, homogeneity, and ease of manufacture The anatomical filters have a semicircular band shape with increasing thickness toward the periphery The AFEE technique produces images of improved contrast characteristics at the breast periphery, ensuring minimization of the total dose to the breast through the elimination of a secondary exposure to patients with an indication of peripheral breast lesions Its performance has been extensively evaluated using both clinical and phantom-based evaluation methods [28, 29] The mammographic images used in this study were digitized using an Agfa DuoScan digitizer (Agfa Gevaert, Belgium) at 12-bit pixel depth and a spatial resolution of 100 µm/pixel According to quality control measurements, this film digitizer is suitable for mammogram digitization, as the optical-density range of the cases 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 RADIOGRAPHIC AND GEOMETRICAL PROPERTIES OF THE SKIN Our approach for breast skin thickness extraction involves the construction of a physical model of the skin region that describes both its radiographic and geometric properties This model is based on the following three assumptions Anatomically, if we consider an axial section of the breast, the skin is a thin stripe of soft tissue situated at its periphery At its vicinity, there is the subcutaneous fat, which radiographically is viewed as a structure with higher optical density than the one of the skin This anatomical information, together with the fact that mammography is a projection imaging modality, will be the basis of our model The region of the image that the physicians indicate as skin does not correspond to the real one at any of the 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 correspond to the real skin at several axial sections of the breast (Figure 8.2) Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 321 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 321 FIGURE 8.1 Original image The shape of the skin’s external border should coincide with the shape of the breast Most of the time, this appears to be regular, and it can be approximated by a circle or an ellipse In an effort to make the shape estimation more accurate and reliable, we will not consider the breast border as a whole Instead, we make the assumption that it can be divided into smaller segments, each of them corresponding to an arc of a circle 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 the breast periphery, we notice also the projections of the skin segments that belong to breast sections situated above and below the central one In the digitized gray-level image, this results in a gradient at the periphery of the breast (where the skin is located), oriented perpendicularly to the breast 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, not conform to the previously mentioned radiographic and geometrical skin model Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 322 Monday, May 16, 2005 11:41 AM 322 Medical Image Analysis X-ray source Breast sections Film FIGURE 8.2 Geometrical representation of the imaging process of the skin 8.4 ESTIMATION AND EXTRACTION METHODS 8.4.1 SKIN FEATURE ESTIMATION 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 been developed 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 to classify each pixel in the image Davies and Dance [33] used a histogram-derived threshold in conjunction with a mode filter to exclude uniform background areas from the image Chen et al [34] proposed an algorithm that detects the skin-line edge on the basis of a combination of histogram analysis and a Laplacian edge detector Mendez et al [35] used a fully automatic technique to detect the breast border 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 thresholding technique We assume that the histogram of each mammogram exhibits a certain bimodality: each pixel in the image belongs either to the directly exposed region Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 323 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 323 (image background) or to the potential object of interest (breast) For this purpose, we have chosen the minimum-error thresholding technique proposed by Kittler and Illingworth [36] The principal idea behind this method is the minimization of a criterion function related to the average pixel classification error rate, under the assumption that the object and the background gray-level values are normally distributed Unfortunately, the presence of the anatomical filter, used for exposure equalization, disturbs the bimodality of the image histogram A threshold selection, using the histogram of the whole image, results in an inaccurate identification of the breast border More specifically, the gray values corresponding to the anatomical filter induce a systematic error that increases the value of the threshold compared with the optimal one The size of the resulting binary region will always be smaller than the real size of the breast 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 global thresholding on the histogram of the whole image using the method of Kittler and Illingworth After thresholding, the breast border is extracted by using a morphological opening operator with a square flat structuring element of size 5, followed by a 4-point connectivity tracking algorithm We then define overlapping square windows along the previously estimated border, where we apply local thresholding using the same approach as before (Figure 8.3) All the pixels situated outside the union of the selected windows keep the label attributed to them by the initial global Background Breast 1st Approximation of the border Anatomical filter FIGURE 8.3 Application of local thresholding for the extraction of the skin’s external border Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 324 Monday, May 16, 2005 11:41 AM 324 Medical Image Analysis FIGURE 8.4 External border of the skin 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, containing only pixels from the breast and the filter For each of them, a threshold is estimated using the method of Kittler and Illingworth [36] Its final value is the average between the 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 structuring element of size 5, followed by a tracking algorithm The final result of our approach, applied to the image of Figure 8.1, is presented in Figure 8.4 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 divide the 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 radii and the coordinates of their centers) are estimated by using the Levenberg-Marquardt iterated method for curve fitting [37] A χ2 merit function is defined that reflects the agreement between the data and the model In our case, the data are the coordinates Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 325 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 325 of the border points, and the model is a circle The optimal solution corresponds to the minimum of the merit function Unfortunately, this circular model of the breast border is disturbed by the presence of the nipple Moreover, when the doctors examine a mammogram, they usually search for possible skin changes along the breast border, except of the region behind the nipple, mainly because of the presence of other anatomical structures that have similar densities as the skin (e.g breast areola) For these reasons, we first exclude the region of the nipple and then work only with the remaining part of the breast border Nipple detection and localization is an ongoing research topic in mammographic image analysis [35, 38] In our scheme, the exclusion of the nipple is performed in three steps [5]: The breast border is divided in three equal segments We choose the central border segment (nipple included) and estimate the coordinates of the circle that corresponds to it using the method of Levenberg-Marquardt [37] We consider the profile of distances between the center of the circle and each point of the central border segment The border points that correspond 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 of deformation In these cases, manual intervention is needed The removal of the nipple allows an efficient fitting of circular arcs to the remaining 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 every point 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 edge detection 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, the estimation of the multiscale gradient is performed using the wavelet approach presented by Mallat and Zhong [42], which is equivalent to the multiscale operator of Canny However, due to the pyramidal algorithm involved in the calculation of the wavelet transform, its computational complexity is significantly lower than the computational complexity of Canny’s approach In wavelet-based gradient estimation, the length of the filters involved in the filtering operation is constant, while the number of coefficients of the Canny filters increases as the scale increases Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 326 Monday, May 16, 2005 11:41 AM 326 Medical Image Analysis 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 If we define two wavelet functions ψ1(x, y) and ψ2(x, y) such as ψ1 = ∂φ( x, y ) ∂φ( x, y ) , ψ2 = ∂x ∂y (8.5) then the wavelet transform of ƒ(x, y) at a scale s has two components defined by Ws1 f ( x, y ) = f * ψ 1s ( x, y ), Ws2 f ( x, y ) = f * ψ s2 ( x, y ) (8.6) By ψ is ( x, y ) , i = {1, 2}, we denote the dilation of ψi(x, y) by the scale factor s, so that: ψ is ( x , y) = s ψ i x s , y s Following these notations, the orientation of the gradient vector is given in Equation 8.7 ( ) ( ) As f ( x, y ) = arg[Ws1 f ( x, y ) + iWs2 f ( x, y )] (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 other hand, as the scale increases, the gradient computation becomes less sensitive to small variations of the gray-level values, resulting in a low precision of edge localization and blurring of the image boundaries The selection of the optimal scale depends on the spatial resolution of the digitized mammograms For our images (spatial resolution 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 the fourth scale of the wavelet decomposition are optimal for mammograms digitized with 200 µm/pixel and 50 µm/pixel, respectively In our application, the wavelet decomposition and the estimation of the gradient orientation (Equation 8.7) were performed using the Wave2 source code [43] developed by Mallat and Zhong [42] By knowing the gradient orientation and the normals to the breast border, we can produce a transformed image that represents the values of our skin feature and highlights 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 difference between the gradient vector and the normals to the breast border Figure 8.5 shows the transformed image that represents the estimated angular difference for the example of Figure 8.1, where black represents a difference of zero degrees and white 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 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 327 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 327 FIGURE 8.5 Spatial distribution of the skin feature throughout the whole image 8.4.2 SKIN-REGION EXTRACTION — MRF FRAMEWORK The knowledge of the spatial distribution of the skin feature (Figure 8.5) is the starting point for the identification of the skin This is carried out with a labeling process based on a Markovian skin model The following two subsections present the basic principles of our labeling process 8.4.2.1 Selection of a Region of Interest To reduce the computational burden of the labeling algorithm, we extract a region of interest (ROI), situated at the breast periphery, containing the skin and a part of the inner structures of the breast The ROI is a stripe with length equal to the length of the breast border Its width is approximately cm and corresponds to the maximum of the clinically observed thicknesses for the region that contains the skin and the subcutaneous fat Figure 8.6(a) shows an example of our region of interest, situated at the lower part of Figure 8.5 After the extraction of the ROI, we perform a transformation of the coordinates of its pixels to facilitate the skin identification process Let Ny be the number of pixels that corresponds to the width of our ROI, and Nx the number of pixels of the breast border The result of the spatial transformation is a Nx × Ny array with the following properties: Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 328 Monday, May 16, 2005 11:41 AM 328 Medical Image Analysis (a) (b) FIGURE 8.6 (a) ROI corresponding to the lower part of Figure 8.5 (b) Stretched version of the selected ROI (array A) • • • The first row represents the Nx pixels of the skin’s external border The following rows correspond to the Ny layers of pixels, situated behind the skin, toward the breast parenchyma Every column contains the Ny pixels found by scanning the ROI along a line perpendicular to the breast border The resulting array (denoted by A) can be considered as a stretched version of our ROI (Figure 8.6(b)) 8.4.2.2 Markovian Skin Model Labeling Scheme We consider the image formed by the array A of Figure 8.6(b) and represent its rectangular lattice as a graph G = {S, N}, where S = {1, 2, …, m} is the discrete set of pixels and N a particular neighborhood system At each node i we associate an observation measure di that represents the value of the skin feature at the current Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 329 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 329 position, and a binary label li, where li = if i belongs to the skin and li = otherwise Every configuration of the labels l = {l1, …, lm} is considered as the realization of a Markov random field denoted by L = {L1, …, Lm} Following a MAP estimation criterion, as described in Section 8.2.1, the optimal labeling of G is found by minimizing the posterior energy function U(l|d) (see Equation 8.4) In our application, the conditional energy term U(l|d) associates a Gaussian distribution to the observations of skin and no-skin classes The prior energy U(l) is expressed in terms of clique potential functions that describe contextual dependencies between the labels The selection of the neighborhood system and the potential functions are driven by our a priori knowledge about the geometrical characteristics of the skin region The following three subsections describe the explicit form of U(l|d) and the optimization procedure for its minimization 8.4.2.2.1 Conditional Probability Distribution We assume that each observation di is only conditioned by the corresponding label li, and that the dependencies between the different observations are exclusively determined by the dependencies between the labels li In this case, the conditional probability distribution p(d|l) can be defined as m p(d | l ) = ∏ p(d | l ) ∝ e i −U ( d |l ) (8.8) i i =1 This type of probability density function can be deduced from the observation field d and reflects the likelihood of every pixel as either belonging or not belonging to the skin We assume that the observation values d of both skin and no-skin regions are normally distributed This implies that p(di | li ) = 2π σ li − e ( di − µ li )2 σ l2 (8.9) i where µ li and σ li are the mean value and standard deviation of the class designated by li From Equation 8.8 and Equation 8.9, we obtain the following expression for the conditional energy term U(d|l): m U (d | l ) = ∑ i =1 (d − µ ) i li − log 2 2σ li 2πσ li (8.10) The mean value and standard deviation of the skin (li = 1) and no-skin (li = 0) classes (µ1, σ1 and µ0, σ0, respectively) can be estimated using the skin-feature values at the first and last row of the array A, respectively, as both are good representatives of the two classes Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 330 Monday, May 16, 2005 11:41 AM 330 Medical Image Analysis 8.4.2.2.2 Prior Probability of Labelings Our a priori knowledge about the geometrical characteristics of the skin generates the following two assumptions: A pixel i belongs to the skin if: • All pixels between i and the external border of the skin (outer layer of our ROI), situated on the same perpendicular to the border line as i, also belong to the skin • There are neighboring pixels, situated at the same breast layer as i, belonging to the skin A pixel i does not belong to the skin if: • All pixels between i and the inner layer of our ROI, situated on the same perpendicular to the border line as i, not belong to the skin • There are neighboring pixels, situated at the same breast layer as i, that not belong to the skin To express these contextual dependencies, we define a neighborhood system N = {Ni|∀i ∈ S}, where the neighbors Ni of a pixel i are all the pixels, except of i, situated in the same column of the array A, together with its V closest horizontal neighbors (V/2 at each side) The parameter V can be considered as a quantization factor that depends on the resolution of the digitized mammograms and represents the minimum expected length along the skin, where no variations of its thickness are present If we consider only pairwise cliques of the form c(i, j), ∀j ∈ Ni, the prior probability of labelings P(l) can be expressed in terms of a prior energy function U(l) and a set of clique potentials Vc(li, lj) P (l ) ∝ e −U (l ) , U (l ) = ∑ ∑ V (l l ) c i j (8.11) i∈S j ∈N i where Vc(li, lj) is a clique potential function associated with each clique c(i, j) For each pixel i (with coordinates (xi, yi)) the clique potential Vc(li, lj) depends on the label li and on the relative position of its neighbor j (with coordinates (xj, yj)) In particular, the potential function has the following form: aij , if li = 1, y j ≤ yi 0, if li = 1, y j > yi Vc (li , l j ) = 0, if li = 0, y j < yi aij , if li = 0, y j ≥ yi where Copyright 2005 by Taylor & Francis Group, LLC (8.12) 2089_book.fm copy Page 331 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 331 0, if li = l j aij = w > 0, otherwise (8.13) These types of potential functions penalize inconsistent configurations of labels with respect to the assumptions and High values of the penalization factor w favor more uniform representations of the skin region but at the same time suppress small variations of the skin thickness The optimal value of w should satisfy both requirements of uniformity and accuracy 8.4.2.2.3 MAP Estimation From the combination of Equations 8.4, 8.10, and 8.11, the posterior probability P(l|d) can be expressed in terms of a global energy function U(l|d), where (d i − µ l ) m U (l | d ) = ∑ i =1 i 2σ li − log 2πσ li + ∑ ∑V (l , l ) c i ∈S i j (8.14) j∈N i The MAP configuration of the label field is estimated by minimizing the energy function U(l|d) For the minimization of U(l|d), we follow a simulated annealing scheme based on a polynomial-time cooling schedule [44] Figure 8.7 shows the evolution of the labeling process toward the minimum energy state, using as example the array A of Figure 8.6(b) In this particular case, the parameters V and w were set to 20 and 2, respectively Finally, the last step of our approach consists of the mapping of the labeled pixels of A back to the coordinates of the original image 8.5 RESULTS 8.5.1 MEASUREMENT OF SKIN THICKNESS In our study, the measurements of the skin thickness are taken in regular intervals along the breast border Starting from each border point, we consider a perpendicular to the border line segment, which extends up to the internal border of the skin The skin thickness at the particular border point corresponds to the length of this line segment For the representation of the measurement results, we use the position of the nipple as a reference point We consider a polar representation of the breast border points using the orthogonal coordinate system of Figure 8.8 The x-axis corresponds to the image border, occupied by the largest part of the breast, and the y-axis is a vertical line that passes through the middle of the nipple The measurement position of the skin thickness in a given border point P is adequately defined by the polar coordinate θ of this particular point Following these notations, angle θ takes values in the interval [−90°, +90°], depending on the relative position of the measuring point P with respect to the nipple Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 332 Monday, May 16, 2005 11:41 AM 332 Medical Image Analysis (a) (b) (c) FIGURE 8.7 Energy minimization using simulated annealing The parameters V and w are equal to 20 and 2, respectively (a) Temperature T = 100 (b) Temperature T = 50 (c) Final result after convergence at temperature T = 0.01 y P − θ + x FIGURE 8.8 Polar representation of the breast border points 8.5.2 CLINICAL EVALUATION Our approach was tested on ten different cases of mammographic images with craniocaudal (CC) views of the breasts, two of them exhibiting advanced skin thickening at the breast periphery The normal range of breast skin thickness in CC views, as reported in the survey of Pope et al [4], is between 0.5 and 2.4 mm, with Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 333 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 333 (a) Manual Measurements Automatic Measurements Skin Thickness (µm) 2500 2000 1500 1000 500 −87 −68 −46 −23 23 44 Angle θ (degrees) 63 79 (b) FIGURE 8.9 (a) The detected skin region that corresponds to the mammogram of Figure 8.1 (b) Skin thickness along the breast border a standard deviation of approximately ±0.3 mm Figure 8.1, Figure 8.10(a), and Figure 8.11(a) present three examples of normal cases, with no severe skin changes along the breast periphery Figure 8.12(a) corresponds to a pathological case, with advanced skin thickening, which is clearly visible at the upper part of the mammogram The skin-detection results for these four examples are presented in Figure 8.9(a, b), Figure 8.10(b, c), Figure 8.11(b, c), and Figure 8.12(b, c), respectively Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 334 Monday, May 16, 2005 11:41 AM 334 Medical Image Analysis (a) (b) Manual Measurements Automatic Measurements Skin Thickness (µm) 2500 2000 1500 1000 500 −83 −58 −31 −4 20 44 Angle θ (degrees) 65 83 (c) FIGURE 8.10 (a) Original image (b) Detected skin region (c) Skin thickness along the breast border The results were obtained using the same values for the parameters V and w Given the resolution of our images, V has been set to a value equal to 20 pixels The penalization factor w in Equation 8.13 has been empirically set to On the other hand, the parameters µ li and σ li in Equation 8.10 are estimated on each image separately, as explained in Section 8.4.2.2 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 335 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness (a) (b) Manual Measurements Automatic Measurements 3000 Skin Thickness (µm) 335 2500 2000 1500 1000 500 −89 −70 −46 −18 34 Angle θ (degrees) 56 74 (c) FIGURE 8.11 (a) Original image (b) Detected skin region (c) Skin thickness along the breast border The validation of our method is performed by comparing the detected skin thickness values with the ones obtained after a manual measurement on each film at several predefined points along the breast periphery This process resulted in an average root mean square (RMS) error of 0.3 mm for normal cases, reaching a maximum value of 0.5 mm in pathological cases with skin thickening The maximum RMS error was observed in the case of Figure 12(a), in which the exact borders of Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 336 Monday, May 16, 2005 11:41 AM 336 Medical Image Analysis Skin Thickness (µm) (a) (b) 9000 8000 7000 6000 5000 4000 3000 2000 1000 −61 −49 −37 −25 −11 Manual Measurements Automatic Measurements 17 30 42 55 67 80 Angle θ (degrees) (c) FIGURE 8.12 (a) Original image (b) Detected skin region (c) Skin thickness along the breast border the skin are not clearly defined because of its advanced deformation Compared with the normal range of breast skin thickness, the estimated errors are relatively small and not influence the clinical assessments The computational time of our approach is rather demanding, mainly because of the optimization step (simulated annealing) Nevertheless, the optimization scheme is stable and converges to a good approximation of the global minimum solution, independently of the initial realization of labelings For a 2300 × 1400 image on a Pentium III at 500 MHz, the estimation of the spatial distribution of the skin feature lasts around min, whereas the labeling process takes approximately 15 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 337 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 337 8.6 CONCLUSIONS We present a model-based method for the measurement of skin thickness in mammography and, at the same time, tackle secondary issues emerging from the solution of this problem, like the identification of the breast border and the extraction of the region of the nipple The skin model is based on physical and geometrical a priori knowledge about the skin to reveal the feature that discriminates it from the other anatomical structures of the breast The MRF framework is used to endow this a priori knowledge to a labeling scheme, which identifies the skin structure Experimental results illustrate the efficiency of our method, which produced results comparable with manual measurements performed on each film The estimation of the proposed saliency skin feature requires a good visualization of the breast periphery The employed anatomical filter for exposure equalization at the breast periphery currently limits the application of the technique to craniocaudal (CC) views A potential alternative could be a digital density-equalization technique [25–27] that allows the use of both CC and mediolateral (ML) views Finally, future work will involve the extension of our method toward a hierarchical/multiresolution Markovian approach The multiresolution pyramid can be created via the dyadically subsampled counterpart of the wavelet transform of Section 8.4.1.3 Based on such hierarchy, the skin feature is estimated at each resolution level separately, without the empirical choice of any particular decomposition scale, and the labeling process can be performed using a computationally efficient top-down hierarchical scheme as presented by Li et al [15] REFERENCES Putman, C.E and Ravin, C.E., Textbook of Diagnostic Imaging, W B Saunders Co., 1994 Tabar, L and Dean, P.B., Anatomy of the breast, in Teaching Atlas of Mammography, 2nd ed., Frommhold, W and Thurn, P., Eds., Thieme, New York, 1985 Willson, S.A., Adam, A.J., and Tucker, A.K., Patterns of breast skin thickness in normal mammograms, Clinical Radiol., 33, 691, 1982 Pope, T.L et al., Breast skin thickness: normal range and causes of thickening shown on film-screen mammography, J Can Assoc Radiologists, 85, 365, 1984 Katartzis, A et al., A model-based technique for the measurement of skin thickness in mammography, IEEE Medical Biological Eng Computing, 40, 153, 2002 Li, S.Z., Markov Random Field Modeling in Computer Vision, Computer Science Workbench, Springer-Verlag, Heidelberg, 1995 Derin, H and 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Francis Group, LLC 2089_book.fm copy Page 339 Monday, May 16, 2005 11:41 AM A MRF-Based Approach for the Measurement of Skin Thickness 339 34 Chen, J., Flynn, M.J., and Rebner, M., Regional contrast enhancement and data compression for digital mammographic images, Proc Soc Photo-Op Instrum Eng., 1905, 752, 1993 35 Mendez, A et al., Automatic detection of breast border and nipple in digital mammograms, Comput Methods Programs Biomed., 49, 253, 1996 36 Kittler, J and Illingworth, J., Minimum error thresholding, Pattern Recognition, 19, 41, 1986 37 Press, W.H et al., Numerical Recipes in C: the Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, U.K., 1992 38 Chandrasekhar, R and Attikiouzel, Y., A simple method for automatically locating the nipple on mammograms, IEEE Trans Medical Imaging, 16, 483, 1997 39 Gonzalez, R.C and Woods, R.E., Digital Image Processing, Addison-Wesley, Reading, MA, 1992 40 Canny, J., A computational approach to edge detection, IEEE Trans Pattern Anal Mach Intell., 8, 679, 1986 41 Costaridou, L et al., Quantifying image quality at breast periphery vs mammary gland in mammography using wavelet analysis, Br J Radiol., 74, 913, 2001 42 Mallat, S and Zhong, S., Characterization of signals from multiscale edges, IEEE Trans Pattern Anal Mach Intell., 14, 710, 1992 43 Wave2 software; available on-line at ftp://cs.nyu.edu/pub/software, last accessed 6/10/2003 44 Aarts, E.H.L and Korst, J.H.M., Simulated Annealing and Boltzmann Machines, John Wiley & Sons, New York, 1989 Copyright 2005 by Taylor & Francis Group, LLC [...]... 16, 2005 11:41 AM 338 Medical Image Analysis 11 Marroquin, J., Miter, S., and Poggio, T., Probabilistic solution of ill-posed problems in computational vision, J Am Stat Assoc., 82, 76, 1987 12 Besag, J., On the statistical analysis of dirty images, J R Stat Soc B, 48, 259, 1986 13 Karssemeijer, N., Stochastic model for automated detection of calcifications in digital mammograms, Image Vision Computing,... 2005 11:41 AM 334 Medical Image Analysis (a) (b) Manual Measurements Automatic Measurements Skin Thickness (µm) 2500 2000 1500 1000 500 0 −83 −58 −31 −4 20 44 Angle θ (degrees) 65 83 (c) FIGURE 8.10 (a) Original image (b) Detected skin region (c) Skin thickness along the breast border The results were obtained using the same values for the parameters V and w Given the resolution of our images, V has been... 72, 997, 1999 30 Kocsis, O et al., A tool for designing digital test objects for module performance evaluation in medical digital imaging, Medical Informatics, 24, 291, 1999 31 Yin, F.F et al., Computerized detection of masses in digital mammograms: analysis of bilateral subtraction images, Medical Phys., 18, 955, 1991 32 Bick, U et al., Automated segmentation of digitized mammograms, Academic Radiol.,... distributions, and Bayesian restoration of images, IEEE Trans Pattern Anal Mach Intell., 6, 721, 1984 9 Pizurica, A et al., A joint inter- and intrascale statistical model for Bayesian waveletbased image denoising, IEEE Trans Image Processing, 11, 545, 2002 10 Katartzis, A et al., A model-based approach to the automatic extraction of linear features from airborne images, IEEE Trans Geoscience and Remote... Francis Group, LLC 2089_book.fm copy Page 326 Monday, May 16, 2005 11:41 AM 326 Medical Image Analysis 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... equalization mammography, Medical Phys., 23, 887, 1996 23 Bick, U et al., Density correction of peripheral breast tissue on digital mammograms, RadioGraphics, 16, 1403, 1996 24 Byng, J.W., Critten, J.P., and Yaffe, M.J., Thickness-equalization processing for mammographic images, Radiology, 203, 564, 1997 25 Highnam, R.P., Brandy, M., and Stepstone, B.J., Mammographic image analysis, Eur J Radiol., 24,... Taylor & Francis Group, LLC 2089_book.fm copy Page 336 Monday, May 16, 2005 11:41 AM 336 Medical Image Analysis Skin Thickness (µm) (a) (b) 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 −61 −49 −37 −25 −11 3 Manual Measurements Automatic Measurements 17 30 42 55 67 80 Angle θ (degrees) (c) FIGURE 8.12 (a) Original image (b) Detected skin region (c) Skin thickness along the breast border the skin are... on mammograms, IEEE Trans Medical Imaging, 16, 483, 1997 39 Gonzalez, R.C and Woods, R.E., Digital Image Processing, Addison-Wesley, Reading, MA, 1992 40 Canny, J., A computational approach to edge detection, IEEE Trans Pattern Anal Mach Intell., 8, 679, 1986 41 Costaridou, L et al., Quantifying image quality at breast periphery vs mammary gland in mammography using wavelet analysis, Br J Radiol., 74,... depending on the relative position of the measuring point P with respect to the nipple Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm copy Page 332 Monday, May 16, 2005 11:41 AM 332 Medical Image Analysis (a) (b) (c) FIGURE 8.7 Energy minimization using simulated annealing The parameters V and w are equal to 20 and 2, respectively (a) Temperature T = 100 (b) Temperature T = 50 (c) Final... Trans Medical Imaging, 14, 565, 1995 16 Zheng, L et al., Detection of cancerous masses for screening mammography using DWT-based multiresolution Markov random field, J Digital Imaging, 12 (Suppl 1), 18, 1999 17 Vargas-Voracek, R and Floyd, C.E., Hierarchical Markov random field modeling for mammographic structure segmentation using multiple spatial and intensity image resolutions, SPIE Conf Image Proc.,