Segmentation of the pectoral muscle edge on mammograms by tunable parametric edge detection R CHANDRASEKHAR and Y ATTIKIOUZEL Australian Research Centre for Medical Engineering (ARCME) The University of Western Australia 35 Stirling Highway, Crawley, WA 6009 AUSTRALIA Abstract: Mammograms are used to screen for breast cancer and computerized analysis of these images can aid radiologists in detecting the disease. Any computerized method to analyze digitized mammograms must first partition the image into its visually and anatomically distinct regions. The pectoral muscle appears on mediolateral oblique views of mammograms and needs to be identified and segmented out before further analysis. This paper presents an algorithm for segmenting the edge of the pectoral muscle. A modification of the conventional edge detection paradigm gives rise to families of tunable parametric edge detectors, one of which has been used to extract the pectoral edge simply, controllably and reliably from mammograms. Keywords: tunable parametric edge detection, segmentation, mammogram, pectoral muscle edge Introduction Mammograms are used to screen for breast cancer. Computerized analysis of these images can aid radiologists in detecting abnormalities or lesions. Any method to analyze digitized mammograms by computer must first partition the image into its visually and anatomically distinct regions. Such segmentation includes identifying the breast border [] and locating the nipple [2]. This paper addresses the segmentation of the edge of the pectoral muscle using tunable parametric edge detection. To aid in visualizing the task being described, two typical mammogram views with their distinct regions are shown in Fig.. Because the pectoral muscle occupies a triangular region at the top left corner of Fig (a), it suffices to delineate the edge of the muscle, i.e., determine the hypotenuse, to accomplish segmentation of the pectoral muscle itself. The pectoral muscle, if it appears on a view, is unique enough to be identified easily even by the This research was partially supported by Australian Research Council (ARC) Large Grant No. A0000074 and by the Western Australian Government through funding of AR- CME as part of its Centres of Excellence programme. untrained observer. Although the edge of this muscle is visually continuous, detecting that edge accurately and automatically by computer is demanding because of natural variations in intensity, linearity, edge strength and visual texture. The pectoral edge is of interest for at least four reasons:. It has been suggested that in an adequate mediolateral oblique view, the pectoralis muscle should be visualized to approximately the level of the nipple [3, p 62]. 2. In their method for systematic viewing of mammograms, Tabár and Dean [4, pp 6 9] suggest oblique masking in which the edge of the pectoral muscle is used to define each arm of a V when paired mammograms are placed back-to-back for comparison matching. 3. Wallis, Walsh and Lee [5, p 5] have identified the loss of outline of the pectoral margin as a subtle or indirect sign of malignancy. 4. When detecting a mammographic lesion by computer it is desirable to distinguish whether it is located in the glandular tissue or superposed on the pectoral muscle.
(a) (b) (a) (b) Figure : Two typical mammogram views of different patients: (a) left mediolateral oblique and (b) right craniocaudal. The letters on the images refer to the following: b: background; s: skin boundary; n: nipple; g: glandular tissue; f: fat; and p: pectoral muscle, which is normally visible only in the mediolateral oblique view. Image (a) is from the MIAS database and (b) is from the UCSF/LLNL database. 2 Segmenting the pectoral edge using standard edge detectors The pectoral edge may be segmented by a variety of methods including intensity thresholding, edge detection, the Hough transform for detecting lines, and texture segmentation [6, chapter 8]. The difficulties arising from a direct application of standard edge detectors to the pectoral edge segmentation problem are now illustrated. MIAS image mdb8rl and its Sobel and Canny edge images are shown in Figure 2(a) to (d) respectively. The problems of selecting suitable thresholds to display the pectoral edge in the first place, and then isolating the pectoral edge from all other displayed edges, must be solved before the pectoral edge can be segmented. These problems are difficult to solve reliably and simply. We have previously examined the fundamentals of the edge detection paradigm, and advocated a fresh approach [7]. Here we extend that work and apply it to pectoral edge segmentation. By relaxing two implicit constraints in the conventional model, we obtain a family of tunable, parametric edge detectors, that in turn yield a simple but effective algorithm for detecting the pectoral edge on mammograms. (c) (d) Figure 2: (a) MIAS image mdb8rl orientated so that pectoral muscle occupies top left corner of image. (b) Scaled Sobel analog edge image with the pectoral edge hardly visible. (c) Thresholded and binarized Sobel edge image. (d) Canny edge image with pectoral edge lost in a sea of other edges. The selection of thresholds at which the analog edge image is binarized is critical, as is the isolation of the pectoral edge from all other edges, and both are difficult problems to solve simply and reliably. 3 Tunable, parametric edge detectors In conventional edge detection, orthogonally directed digital intensity gradients are used as components in an edge vector. These components are then combined by a norm function to yield an edgemagnitude image. This image is then thresholded in the final step to give the binary edge image as shown in Fig. 3. Two implicit constraints in this model are relaxed so:. We relax the constraint that the components of the edge vector should only be directed digital gradients, and instead allow any combi-
nation of edge sensitive features, including directed digital gradients, that are suited to the task at hand. 2. We relax the constraint that the function that combines vector components to yield a real scalar magnitude must satisfy the properties of a norm, and instead allow a blending function [7] that maps a vector to a real scalar, possibly combining the norming and thresholding operations into a single step, to yield an almost binary image suited to our purpose. (Note particularly that norms are not excluded from being blending functions.) The edge detectors generated by these modifications may be parametrized both by the nominated features and the selected blending function and its parameters, giving rise to families of edge detectors that may be tuned by their parameters. This approach affords a systematic framework for selecting the most suitable edge detector for any specific task. Original Image Edge Features Horizontal Digital Gradient Vertical Digital Gradient Blending Function Norm Threshold Binary Edge Image are as defined in equations () to (4) respectively. ϕ h (w) = (w 9 + 2w 8 + w 7 ) (w + 2w 2 + w 3 ) () ϕ v (w) = (w 3 + 2w 6 + w 9 ) (w + 2w 4 + w 7 ) (2) ϕ r (w) = max [w i] min [w i] (3) i 9 i 9 [ ] [ ] 2 ϕ s (w) = 9 9 w 2 i w i (4) 9 9 i= i= In order to ensure compatibility between the ranges of the different features, each is normalized so that the range of all four is [0, 4]. 3.2 Blending function There are many possibilities for choosing the blending function. For detecting the pectoral edge, we have found empirically that a sigmoidal blending function, like the logistic function [6] is suitable. The logistic function in one dimension has the form b L (x) = (5) + exp( λ(x β)) where λ and β are real, positive constants. The shapes of the logistic curves for fixed β and varying λ are shown in Fig. 4. We generalize the logistic One dimensional logistic functions with β = 0.5 and varying λ 0.9 Figure 3: Conventional and tunable parametric edge detection. We relax two implicit constraints in the traditional model: (i) any edge sensitive features (not only digital intensity gradients) may be used as components to the edge vector and (ii) a blending function need not necessarily have the properties of a norm and may combine the norming and thresholding operations into a single step to yield the desired edge image. 3. Edge Features Four neighbourhood-based edge features, two directed digital gradients and two statistical descriptors, were investigated. The pixels in a 3 3 window around a candidate pixel are strung out as a vector w of dimension 9, indexed from to 9 in raster scan fashion, from top to bottom and left to right, in the original neighbourhood. Then the absolute values of the horizontal and vertical Sobel digital gradients, and the range and standard deviation, y =.0/(.0 + exp( λ(x β))) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 λ = 00 0. λ = 0 λ = 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 4: Three one-dimensional logistic functions are graphed above. The value of β is constant at 0.5 while λ is varied from 00 through 0 to. blending function to accept a p-dimensional input vector ϕ and yield a real scalar by defining b L (t, ϕ, λ, β) = (6) + exp( λ( ϕ t β))
where the Minkowski t-norm t is defined by ϕ t = [ p i= ] ϕ i t t (7) 4 Mammograms experiments A sequence of experiments was performed in which. the components of the edge feature vector were varied while the -norm was used as the constant blending function; then (ϕ r ) (ϕ s ) 2. the edge feature vector was kept constant while the blending function and its parameters were varied; then 3. based on the results from these, the blending function was chosen and the feature vector varied to identify the most suitable set of features; and finally 4. the feature vector, blending function and its parameters were fixed, and the method was tested across different images. One image from the MIAS database, mdb8r, shown in Fig. 2(a) was used in all these experiments. It was histogram equalized and reduced in size and resolution to 800 micrometres per pixel for all experiments. The results of pectoral edge segmentation were evaluated visually by the authors and the best combination of features, logistic parameters and norm were chosen for the pectoral edge detection algorithm. After the parameters had been thus fixed, the algorithm was tested on six image pairs from the MIAS database. 5 Results Due to space constraints only salient results are recorded here. Figure 5 shows the results of varying the features keeping the -norm as the blending function. Neither the breast border nor the pectoral edge appears as an edge of constant strength. Different Minkowski norms were tested as blending functions and the feature vector was kept constant as ϕ = (ϕ h, ϕ v, ϕ r, ϕ s ) T. The resulting edge images were all similar although the -norm images seemed brighter. Consequently, this norm was used in the logistic blending function, i.e., t was set to (ϕ h, ϕ v ) (ϕ h, ϕ v, ϕ r, ϕ s ) Figure 5: Variation of edge features blended with the -norm. The features used are shown under each image. Both the range and standard deviation are good edge features, along with the horizontal and vertical Sobel components. The pectoral muscle edge starts out visible near the top but fades out as it approaches the left edge, in keeping with its diffuse appearance in that region in the original in Fig. 2(a). in equation (6), while its other two parameters λ and β were varied in the next set of experiments. The results are shown in Fig. 6. It is clear that varying β has a more profound effect on the output than varying λ. The value β = 0.5 appears suitable for revealing the full edge of the pectoral muscle. For the blending function b L, the choice of t =, λ = 00 and β = 0.5 seems a sensible one for delineating the edge of the pectoral muscle. The blending function was frozen as b L (, ϕ, 00, 0.5) and the feature vector ϕ was varied. Results from these experiments are shown in Fig. 7 for the feature vectors (ϕ h, ϕ v ) and (ϕ h, ϕ v, ϕ r, ϕ s ). It is clear from these experiments that the feature vector of choice for pectoral edge detection should be ϕ = (ϕ h, ϕ v, ϕ r, ϕ s ).
(λ, β) = (0, ) (λ, β) = (0, 0.5) (ϕ h, ϕ v ) (ϕ h, ϕ v, ϕ r, ϕ s ) Figure 7: The blending function was fixed to be b L (, ϕ, 00, 0.5) and the feature vector was varied. The features are shown below each figure. It is clear that the edge vector ϕ = (ϕ h, ϕ v, ϕ r, ϕ s ) gives the closest to the desired response. (λ, β) = (00, ) (λ, β) = (00, 0.5) Figure 6: Variation of parameters of logistic function with t =. Changing β had a more pronounced effect than changing λ. Based on these results, we setλ = 00 and β = 0.5 and froze the blending function to be b L (, ϕ, 00, 0.5) for experiments in which ϕ was varied. The preferred edge vector/blending function combination for the pectoral muscle edge detector is therefore ϕ = (ϕ h, ϕ v, ϕ r, ϕ s ) T and b L (, ϕ, 00, 0.5). This was tested out on six MIAS image pairs. The results were encouraging for 0 of the 2 test images and results for four images, including one exception, are shown in Figure 8. 6 Discussion The pectoral muscle does not show the crisp definition that the skin boundary presents in Figs. 5 to 7. The muscle edge is an internal edge defined by transitions in intensity and texture, whereas the skin edge is an external edge between two objects, and indeed, two phases. Therefore, the pectoral muscle presents the more difficult task for edge detection. The combination of the directed digital gradients and the range and standard deviation performs better than any feature alone or in pairs and perhaps illustrates the complementary nature of the information extracted by them. The function of the logistic curve is now discussed. Recall that b L is applied to the feature vector after some norm has been evaluated on it. Figure 4 shows the graphs of three logistic functions (defined by equation (5)) for which β is constant at 0.5 while λ varies from 00 through 0 to. These curves look like the familiar transfer characteristics encountered in electronic circuits, and may be so interpreted. The curve for λ = 00 is reminiscent of the transfer curve of a transistor, operating as a digital switch. That for λ =, on the other hand performs linear compression of the input, like an amplifier with fractional gain. The two salient features on these curves are the onset and steepness of transition. The value of β, set at 0.5 here, determines the onset of the transition. Its value, in turn, depends on the range of each feature, ϕ, which in this case lies in [0, 4]. If β is close to zero, the entire image is subjected to the sigmoidal modulation of the logistic. The appearance of the output image then depends on the value of λ, whose effect is discussed next. The steepness of the transition is determined by λ; the larger it is, the steeper the slope. If we use the analogy of electronic circuits, large values of λ lead to saturation behaviour and a binary image, whereas low values like a λ of (see the relevant
curve in Figure 4) lead to linear operation and a grey image. The two parameters of b L, β and λ, thus offer a simple but effective means of generating a wide range of edge images from which the most suitable values for any application may be determined by inspection of the outputs or other, more sophisticated criteria. Except for image mdb263lm and its pair, the edge of the pectoral muscle may be trivially located automatically using a priori knowledge of its position in the orientated image and seeking the right extreme of the black region when scanning from left to right and top to bottom, starting at the origin. If the black triangle at the upper left corner is noisy, median filtering could be used. Although our results stop short of demonstrating the pectoral edge alone on an image, a modest amount of post-processing (e.g., median filtering, morphological operations, etc.) would yield a binary pectoral edge image. mdb026rl mdb056rm 7 Conclusions We have developed an algorithm for segmenting the edge of the pectoral muscle that uses a family of tunable parametric edge detectors. These edge detectors incorporate edge sensitive features, like digital gradients and local statistics, and a logistic blending function with two parameters, that when properly chosen, give simple, controllable and reliable segmentation of the edge of the pectoral muscle. References mdb263lm mdb3ll Figure 8: Results of pectoral edge with different MIAS images. The method fails with image mdb263lm and its pair but is perhaps excusable, given the appearance of the original image which is not shown above. The other three images show results similar to the other ten images in the test set. [] R. Chandrasekhar and Y. Attikiouzel, Segmenting the Breast Border and Nipple on Mammograms, Australian Journal of Intelligent Information Processing Systems, vol. 6, no., pp. 24 29, 2000. [2] R. Chandrasekhar and Y. Attikiouzel, A Simple Method for Automatically Locating the Nipple on Mammograms, IEEE Transactions on Medical Imaging, vol. 6, pp. 483 494, Oct. 997. [3] M. E. Peters, D. R. Voegeli, and K. A. Scanlan, eds., Handbook of Breast Imaging. Handbooks of Diagnostic Imaging, New York: Churchill Livingstone, 989. [4] L. Tabár and P. B. Dean, Teaching Atlas of Mammography. New York: Thieme-Stratton, second revised ed., 985. [5] M. G. Wallis, M. T. Walsh, and J. R. Lee, A Review of False Negative Mammography in a Symptomatic Population, Clinical Radiology, vol. 44, pp. 3 5, 99. [6] R. Chandrasekhar, Systematic Segmentation of Mammograms. PhD Thesis, Centre for Intelligent Information Processing Systems, Department of Electrical and Electronic Engineering, The University of Western Australia, Nedlands, WA 6907, Australia, Oct. 996. [7] J. C. Bezdek, R. Chandrasekhar, and Y. Attikiouzel, A Geometric Approach to Edge Detection, IEEE Transactions on Fuzzy Systems, vol. 6, pp. 52 75, Feb. 998.