## 2 GCD Algorithm
### Mammogram Preprocessing  
The goal of preprocessing is to prepare for next two steps, segmentation and classification. Down-sampling, quantization, ROI (region of interest) extraction, denoising and enhancement are done in the preprocessing step. To reduce computation load without losing much sensitivity, original mammograms are down-sampled by factor of 4 (_i.e._, the new image size is reduced to 1/16 of its original size) and quantized down to 8 bits per pixel (256 gray level). The digitized mammograms in the DDSM database are of high resolution and of high fidelity. For example, the mammograms scanned by LumiSys in the DDSM database are of 12 bits per pixel and 50 microns per pixel, a typical mammogram of which has the resolution of 6,000 by 4,000 pixels. The region of interest on a mammographic image (breast ROI) is extracted to reduce the processing time (by ignoring the dark areas). First, a mammogram (\(\mathbf{I_{S}}\)) is mapped onto a special target image (\(\mathbf{I_{M}}\)) with Equation (1) by specifying (\(\mathbf{\mu_{T}}\), \(\mathbf{\sigma_{T}}\)) = (128, 0.75). Then the mapped image (\(\mathbf{I_{M}}\)) is binarized with a threshold (\(\mathbf{\mu_{T}}\)). Next, the largest 8-connected object is considered as the binary mask of breast ROI. Finally, morphological _closing_ operations can fill holes inside the ROI:  
\[\mathbf{I_{M}}=(\mathbf{I_{S}}-\mathbf{\mu_{S}})\cdot\frac{\mathbf{\sigma_{T}}} {\mathbf{\sigma_{S}}}+\mathbf{\mu_{T}}\,, \tag{1}\]  
where \(\mathbf{I_{M}}\) is the mapped (target) image, \(\mathbf{I_{S}}\) is the source (original) mammographic image; \(\mathbf{\mu}\) and \(\mathbf{\sigma}\) denote mean and standard deviation, respectively; the subscripts 'S' and 'T' refer to the source (original) and target (mapped) images. Nonlinear diffusion is performed to suppress noise while retaining edges. Nonlinear diffusion methods have been proven as powerful methods in the denoising and smoothing of image intensities while retaining and enhancing edges. Such an image smoothing process can be summarized as a successive coarsening of any given image while certain structures in that image are retained on a fine scale. Nonlinear diffusion is closely connected to a specific kind of multiscale analysis referred to as scale-space [14, 15], and was first used for image smoothing with simultaneous edge enhancement [16]. Figure 1: The flow chart of GCD algorithm: Three steps, preprocessing, segmentation and classification, are shown in three columns, respectively. “Save/Read the Preprocessed image” (within dashed rectangles) may be omitted in a continuous process. \(P_{\mathrm{C}}\) is the cancerous probability of analyzing alarm segment. In addition, [PERSON] _et al._[17] have proven that nonlinear diffusion is equivalent to adaptive smoothing. Basically, diffusion is a PDE (partial differential equation) method that involves two operators, smoothing and gradient, in 2D image space. The diffusion process smoothes the regions with lower gradients whereas stops smoothing at region boundaries with higher gradients. In other words, the diffused result is a nonlinear function of local gradients. [PERSON] _et al._[18] presented a semi-implicit scheme with an \"additive operator splitting\" (AOS) implementation for nonlinear diffusion filtering, which is stable for all time steps (\(t>>0.25\)) and guarantees equal treatment of all coordinate axes. The AOS scheme is at least ten times more efficient than the widely used explicit schemes (with limited time step, \(t\leq 0.25\)). In our experiments, the AOS scheme is implemented for mammogram diffusion by empirically specifying parameters to achieve a good balance between noise removal and detail retaining. Image enhancement is intended to benefit the CAD algorithm but not necessarily to favorite visual inspection, which is achieved with a \"thresholded histogram matching\". _Histogram matching_ (also referred as histogram specification) is usually used to enhance an image when _histogram equalization_ fails [19].