a1: Region Growing Including Gradient Information

This algorithm is inspired by recent work of Petrick et al. [134]. Similar to their other earlier work [135,136], the algorithm starts by preprocessing the image using a Density-Weighted Contrast Enhancement (DWCE) filter. This filter is based on two filtered images of the original mammogram $F(x,y)$ :

• A density image $F_D(x,y)$ , which is an smoothed version of the mammogram obtained by applying a Gaussian filter.
• A contrast image $F_C(x,y)$ , which is found by subtracting the original image and another smoothed version of this image, obtained using a different $\sigma$ value.

The density image is filtered again using a non-linear filter $K_M$ and used to define a multiplication factor which modifies the corresponding pixel in the contrast image. This way, it allows the local density value of each pixel to be weighted by the local contrast. This intermediate image referred to as $F_{KC}$ can be analytically described as:

$$F_{KC}(x,y) = K_M(F_D(x,y))*F_C(x,y)$$ (2.1)

where $*$ denotes the convolution operation. This image is used to define a second multiplication value using another non-linear filter $K_{NL}$ , which is multiplied again by the weighted contrast of the corresponding pixels:

$$F_{E}(x,y) = K_{NL}(F_{KC}(x,y))*F_{KC}(x,y)$$ (2.2)

The resulting image $F_E(x,y)$ is the output of the DWCE filter.

The main aim of preprocessing is to enhance possible mass lesions in the image. Once the image is filtered, morphological erosion techniques [38] are used to identify local maxima, which are the seeds of a subsequent region growing algorithm which is used to expand them using grey-level and gradient information. The gradient image is obtained using Frequency-Weighted Gaussian (FWG) filtering, which is based on the following decomposition:

$$F(x,y) = F_F(x,y) + F_{sub^+}(x,y) + F_{sub^-}(x,y)$$ (2.3)

where $F_F(x,y)$ is a smoothed version of the image $F(x,y)$ resulting from the application of a Gaussian filter with mean 0 and standard deviation $10$ , and

$$F_{sub^+}(x,y) \begin{cases} F(x,y)-F_F(x,y), & F(x,y) > F_F(x,y) \ 0, & \text{Otherwise} \end{cases}$$ (2.4)

$$F_{sub^-}(x,y) \begin{cases} F_F(x,y)-F(x,y), & F(x,y) < F_F(x,y) \ 0, & \text{Otherwise} \ \end{cases}$$ (2.5)

This filtering is repeated twice. The first iteration reduces the gradients within the breast, whilst the second one eliminates gradients in the background. Hence, the result of this decomposition is an enhancement of the contrast between the breast structures and the background. Subsequently, applying a Sobel filter produces the gradient image of the original mammogram with a significant amount of background eliminated. Finally, as a result of this additional information, the region growing algorithm has a limited number of regions to grow.