a2: Probability Driven Region Growing

Kupinski and Giger [95] compared three region growing approaches: traditional region growing using only grey-level information, region growing using gradient information, and region growing using probability information based on grey-level. The latter approach provided the best results and therefore algorithm a2 is based on that approach.

In the a2 algorithm, the probability of pixel grey-levels given a partition $L_i$ is modeled as:

$$P(f(x,y) \mid {L_i ,\sigma _l^2} ) = \begin{cases} N(f(x,y); f(\... ...sigma_l^2), & (x,y) \in L_i \ z(f(x,y)), & (x,y) \notin L_i \ \end{cases}$$ (2.6)

where $N(f(x,y); f(\mu_x,\mu_y),\sigma_l^2)$ is a normal distribution centred at the seed point grey-level $f(\mu_x,\mu_y)$ and variance $\sigma_l^2$ , whereas $z(f(x,y))$ is a function estimated for each breast using the grey-levels of all its pixels. This is based on kernel density estimation, which is an extension of histogram analysis. An Epanechnikov kernel [119] is used to estimate the pixel distribution. The probability of the image given a partition $L_i$ is:

$$p(I\mid L_i,\sigma_l^2) =\prod_{(x,y) \in I} p(f(x,y) \mid L_i,\sigma_l^2)$$ (2.7)

The partition $L_i$ used will be the one that maximizes this probability, i.e:

$$p(I\mid L_{final},\sigma_l^2) = argmax_i \{p(I\mid L_i,\sigma_l^2)\}$$ (2.8)

Although Kupinski and Giger [95] applied this method to a Region of Interest (RoI) manually selected by an expert, we have slightly modified it with the aim to apply it to whole mammograms. We have automatically placed seeds over the image at high-intensity regions, following an approach proposed to detect micro-calcifications [112].