## 2 Materials And Methods
### Texture Analysis
#### 2.2.1 Multi-Scale Wavelet Decomposition  
The first step of the proposed work was mainly based on a multi-scale wavelet analysis, an appropriate technique for detection and classification of microcalcifications. Indeed, in digital mammograms, the microcalcifications appear brighter than the neighboring tissue, and represent one of the main high frequency components of the image spectrum. Consequently, in order to catch this signal, the wavelet transform can be used to decompose the original image into different frequency sub-bands. In particular, the Haar wavelet decomposition rescales the high frequency sub-band and reconstructs the image using such a scaled high frequency sub-band [21; 22].  
In the 2D Haar wavelet decomposition of the image, the original image is first high-pass filtered, yielding the three detailed coefficients' sub-images (Figure 3a-a1; top right: horizontal, bottom left: vertical, and bottom right: diagonal), then low-pass filtered and down-scaled, yielding an approximation coefficient sub-image (Figure 3a-a1, top left). To compute the successive level of decomposition, the process was iterated on the approximation coefficient sub-image (Figure 3b-b1, top left). All of these coefficients describe the local geometry of the image in terms of scale and orientation [22; 26], allowing suppressing the sub-band of the Haar decomposed image that carries the lowest frequencies and contains smooth (background) information, before the reconstruction of the image [26].  
In particular, for each image, the 2D Haar transform at two levels of decomposition was performed, obtaining eight sub-images (\(LL1\), \(HL1\), \(LH1\), \(HH1\), \(LL2\), \(HL2\), \(LH2\), \(HH2\)), as shown in Figure 3b-b1. As for the textural features derived from such a decomposition, for each of the sub-images were obtained the computed mean, variance, skewness, kurtosis, entropy, and relative smoothness. The second order moment of the mean, also known as variance, describes the visual roughness of the image, while the third order and fourth order moment, i.e., skewness and kurtosis,  
Figure 3: Image decomposition schema (**a**,**b**) and on the image sample (**a1–b1**). In the 2D Haar wavelet decomposition of the image, the original image is first high-pass filtered, yielding the three sub-images (**(a–a1)**; top right: horizontal; bottom left: vertical; bottom right: diagonal), then low-pass filtered and down-scaled, yielding an approximation coefficient sub-image (**(a–a1)**, top left). To compute the successive level of decomposition, the process is iterated on the approximation coefficient sub-image (**(b–b1)** top, left).  
respectively, reflect the asymmetry and the uniformity of the histogram of the image. If \(K\) denotes the gray scale, \(\mu\) the gray average, and \(h(k)\), \(k=0,1,2,...,K-1\), the histogram of each sub-image, they are defined as follows:  
\[Variance=\frac{1}{N}\sum_{k=0}^{K-1}(k-\mu)^{2}h(k) \tag{1}\]  
\[Skewness=\frac{1}{N}\sum_{k=0}^{K-1}(k-\mu)^{3}h(k) \tag{2}\]  
\[Kurtosis=\frac{1}{N}\sum_{k=0}^{K-1}(k-\mu)^{4}h(k) \tag{3}\]  
\[Entropy=-\frac{1}{N}\sum_{k=0}^{K-1}h(k)\log[h(k)] \tag{4}\]  
\[RelativeSmoothness=1-\frac{1}{1+m_{2}} \tag{5}\]  
This way results in a set of 48 statistical features (in the following HAAR SFset) for each ROI.