## Symmlet Wavelet Approach

As stated, there are many wavelet bases that can accomplish the detection and segmentation of calcified areas in mammogram and set the stage for the classification task. In a first application, we used a nearly symmetric wavelet. Our choice was guided by the close similarity between the wavelet profile and the calcification profiles (recall the correlation idea discussed above). First, the image is expanded as in Eq. (13.1). Deciding which components to discard is the crucial decision with this approach. This choice is dependent upon the calcification size (in both pixel width and actual linear measure) and the digital resolution. The term size is used in the average or expected sense. From the clinical view of a suspicious abnormality, calcifications of up to 1/2 mm are important. This translates into about 8-16 pixels in an image generated with a 35 ^m/pixel digital resolution; our original work was performed at this high resolution and this experience will be discussed here [46]. In pilot studies, the d and d4 images demonstrated the largest calcification signatures relative to the background and were empirically selected for the process. Two pathways could be followed: (1) Add the two relevant expansion images together and perform the detection or (2) perform the detection in each image and combine the results afterwards. The latter option gave better sensitivity performance, since it gives the opportunity to detect some calcified areas twice. Specifically, small calcifications had a stronger signature in the d3 images and large calcifications had stronger signatures in the d4 images. Many calcifications had of course signatures spread across the two components.

Rather than impose a detection or decision rule on the process, we decided early on to see whether a parametric approach to decision making could be followed. Our pilot studies showed that the wavelet expansion images could be

Figure 13.5: Representative mammographic section (2048 x 2048 pixels) with a malignant calcification cluster indicated by arrow. Image resolution is 35 ^m and 12 bits/pixel.

approximated with parametric methods to characterize their empirical pixel distributions [46-48]. Calcifications, and calcification clusters, occupy a relatively small portion of the image when present; Fig. 13.5 shows a typical calcification cluster associated with cancer. Given the small area properties, the wavelet modeling produced essentially the parametric probability distribution functions (PDF) for normal tissue at multiple resolutions.

Theoretically, knowledge of the surrounding tissue PDF allows for the development of a statistical criterion to test against it using maximum likelihood analysis [49]. Our work indicated that the PDFs may be approximated from a family of parametric PDFs indexed by N; when N = 1, the PDF is Laplacian, and when N is large it tends to a normal distribution and the PDF is symmetric about the origin (zero mean).

We will not delve into this area of statistics here, but will indicate the approximations used. In this application, we ignored the pixel correlation within a given expansion image and used a low order N approximation. Namely, if N is in the neighborhood of 3, the N = 1 approximation was applied to simplify the techniques. Likewise, before applying the maximum likelihood analysis, the expansion images where transformed to all positive values by taking the absolute value. Figure 13.6 shows the first three wavelet expansion images for the mammogram of Fig. 13.5.

Knowing the form of the PDF allows for the development of a test statistic, better described as a summary statistic, which follows from the maximum likelihood approach. However, the technique does not indicate how exactly to apply the test to the problem at hand. The maximum likelihood analysis indicated that the average was the test statistic. Tailoring this to our problem translated into sliding a small search window across the image matched in size to the expected calcification size. At each location the average was calculated, and if the local average deviated from the expected overall normal tissue average, it was labeled as suspicious and marked. Otherwise, the local region was set to zero. Thus, by systematically analyzing each local image region, most of the image was discarded and the potential abnormalities were labeled. A different search size window was used for the different images: an 8 x 8 pixel window was applied to the d image and a 16 x 16 pixel window was applied to the d4 image. The detection result yielded two binary images that for the most part were zero but were equal to 1 in areas corresponding to calcifications. The union of the two detections formed the initial total binary detection output. But detecting isolated calcifications was not the end of the process. Calcifications had to be grouped into clusters for further analysis. The clinical rule was followed here for grouping calcifications. Namely, a cluster was defined as three or more calcifications within a 1 cm2 area [50]. Thus, in a second run, a larger search box was scanned across the binary-labeled detection image and isolated spots were set to zero.

The threshold(s) that give the best trade-off between labeling a normal area as suspicious (FP detection) and labeling a true calcified area as normal (false negative (FN) detection) must be probed with experimental methods. Generally, this requires a sample set of images with known calcification clusters and another sample set of images with no abnormalities at all. This image assemblage is processed repeatedly while varying the thresholds and calculating the performance rates: (1) the number of correctly identified calcification clusters (true positive (TP) detections) and (2) the number of FP clusters. In our work,

Figure 13.6: First three wavelet expansion images (di, d2, and d from top to bottom) corresponding to the raw image of Fig. 13.5. The probability modeling and empirical histograms are displayed on the right after taking the absolute value of the data. The theoretical curves are represented by solid lines and the empirical data by dashed lines. For viewing purposes, the images are 256 x 256 pixel sections cut form the original image of Fig. 13.5 but the probability modeling is derived form the entire image.

Figure 13.6: First three wavelet expansion images (di, d2, and d from top to bottom) corresponding to the raw image of Fig. 13.5. The probability modeling and empirical histograms are displayed on the right after taking the absolute value of the data. The theoretical curves are represented by solid lines and the empirical data by dashed lines. For viewing purposes, the images are 256 x 256 pixel sections cut form the original image of Fig. 13.5 but the probability modeling is derived form the entire image.

there were two thresholds associated with each detection stage that were varied independently. Figure 13.7 shows the output of this process for the cluster shown in Fig. 13.5.

As we shall see below, the classification algorithm we developed requires the analysis of calcification attributes that were not fully present in the binary detection representation of Fig. 13.7. Two options could lead to the desired representation: (a) The binary detection output may be used as a mask that points back to the calcification location in the raw image, or, more generally, to any other data representation. For example, classification analysis could be done on any combination of the d images in Eq. (13.1). (b) Perform an additional segmentation operation on the binary detection output that would extract the shape and distribution of the detected cluster(s) and allow their shape analysis necessary for the classification step. The second option was selected in this application and calcifications were segmented in the detection image of Fig. 13.7 by applying an adaptive threshold process that is described in section 13.3.4.

It should be noted that the original symmlet wavelet method was developed and optimized for 35 ^m and 12 bits/pixel image resolution. The algorithm was modified to be applicable to images of 60 ^m and 16 bits/pixel, a resolution that was identified in separate experiments as optimum for morphology-based classification [28, 29]. Specifically, image resolutions of 60-90 ^m/pixel were found to maintain the calcification shape and size characteristics on which to base feature selection for classification. Higher spatial resolutions, i.e. 30 or 35 ^m, did not improve classification results but significantly increased computational intensity and image manipulation. Lower spatial resolutions, i.e., equal to or greater than 90 ^m/pixel, degraded classification performance because of the losses in morphology and distribution of the detected calcifications. So, in the following section, we will discuss the limitations of the symmlet wavelet method for applications other than 35 ^m and what led us to the design of a new filter for calcification detection independent of image resolution that significantly improved classification performance and robustness of the results.

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