## 3 Experimental Results and Discussions
### GCD for Mass Detection  
A total of 431 mammograms (digitized by the LumiSys scanner) were actually analyzed to validate the GCD algorithm in mass detection, which included 159 normal images (from 80 healthy individuals) and 272 cancerous images (with at least one malignant mass present on each image, from 270 cancerous patients). To sufficiently use the dataset (of 431 samples) and to conduct reliable results, a _ten-fold cross validation_[25, 26] method was used to divide the dataset into independent training and testing groups. By this procedure, each one of the ten subsets, in turn, was held out to be used as an independent testing set while the other nine subsets were combined to form a training set. From the training set, the typical patterns (features) were selected according to their overlapping ratios (e.g., greater than 0.75, for cancerous class only). The selected features were the most representative ones and then used to form a certain number of clusters (\(R=40\)) for each class with the FCM method. With the KNN (\(k=50\), _i.e._, find out the 50 nearest neighbors from 80 clusters) classifier, the \(P_{\mathrm{C}}\) value of each case in the testing set was computed. All cases were tested after 10 runs and the performance of this GCD algorithm was evaluated by using the metrics defined in Equation (9). At the end of segmentation (generating alarm segments), we had TP (Segmentation) \(=92\%\) and FPI (Segmentation) \(=7.78\). At the end of classification (reducing false alarms), by specifying \(T_{\mathrm{P}}=0.45\), the performance was TP (classification) \(=90\%\) and FPI (classification) \(=1.21\). Four cancerous cases of malignant masses are shown in Figures 5-8. The detailed descriptions and case analyses were given in figure captions. In each figure, _Assessment_ and _Subtlety_ of the first lesion are given. There are three panels in each figure: (a) is the original mammogram (ROI only); (b) is the preprocessed image (diffused and enhanced) with the annotations (detected by GCD, in heavy black outlines) and overlays (ground truths, in light black lines); (c) is the map of alarm segments generated from segmentation, overlapped on the overlays, and the probability values (\(P_{\mathrm{C}}\)) of alarm segments are also presented. In the DDSM database, the situation of a cancerous area is described by _Assessment_, rating from 1 to 5 (most advanced) and _Subtlety_, rating from 1 to 5 (most detailed). So a mass lesion of (_Assessment_, _Subtlety_) \(=\) (5, 5) is most easily detected. An early-stage mass detection was demonstrated in Figure 6, where (_Assessment_, _Subtlety_) \(=\) (1, 2). Figure 5: GCD results of Case “3499_Right” (one malignant mass lesion with irregular-shape and spiculated margin, assessment = 5, subtlety = 5): (a) Original mammogram; (b) Preprocessed image with one annotation (\(P_{\rm C}\) = 0.55, detected by GCD, in heavy black outlines) and one overlay (ground truths, in light black line)—a very good match between annotation and overlay; (c) 4 alarm segments (\(P_{\rm C}\) = 0.55, 0.44, 0.28, 0.26) overlapped on one overlay. Figure 6: GCD results of Case “3087_Right” (one malignant mass with irregular shape and obscured-ill_defined margin, assessment = 1, subtlety = 2, a very early-stage mass): (a) Original mammogram; (b) Preprocessed image with one annotation (\(P_{\rm C}\) = 0.57, detected by GCD, in heavy black outline) and one overlay (ground truth, in light black line)—an excellent match between annotation and overlay; (c) 3 alarm segments ((\(P_{\rm C}\) = 0.57, 0.37, 0.37) overlapped on one overlay. Figure 8: GCD results of Case “3062_Left” (one malignant mass with lobulated-architectural_distortion shape and obscured-ill_defined margin, assessment = 4, subtlety = 2): (a) Original mammogram; (b) Preprocessed image with one annotation (\(P_{\rm C}\) = 0.71, detected by GCD, in heavy black outline) and one overlay (ground truth, in light black lines)—pretty good match between annotation and overlay; (c) One alarm segment (\(P_{\rm C}\) = 0.71) overlapped on one overlay.