Skin-Line Detection in Scale Space

The scale-space representation [106] describes an image as its decomposition at different scales. This is achieved by the convolution of the image with a Gaussian smoothing function at various scales (given by the $\sigma$ value of the Gaussian function). This representation has been used in conjunction with edge detection in order to automatically extract edges at their optimum scale. If a small scale is used, the edge localization is accurate but results are sensitive to noise. On the other hand, edges at larger scales have a better tolerance to noise but poor edge localization. The motivation of using scale space edge detection is given by the nature of the breast skin-line: a low contrast edge often affected by noise. It is our assertion that using a robust edge detection methodology would lead to a better skin-line estimation. Various approaches to automatic scale selection have been proposed [106]. A simple and common approach is to select as the optimum scale the one which obtains a maximum response from scale invariant descriptors. This is in general given by normalized derivatives, for instance Lindeberg [106] defines

$$L_{norm} = \sigma^{\frac{\alpha}{2}} (L_{x}^2+ L_{y}^2)$$ (A.1)

as an edge strength measure for scale $\sigma$ . $L_x$ and $L_y$ are the convolution of the image function with a first derivative Gaussian function in $x$ and $y$ , respectively. Here $\alpha$ is a parameter used as an additional degree of freedom for edge and ridge detection. A typical value of $1$ is generally used in the definition of normalized derivatives for edge detection. Edge points are obtained detecting zero-crossing points of the second derivative in the scale-space representation. The final edge strength of a zero crossing will be given by the maximum normalized strength measure along the different scales. This maximum scale is regarded as the edge scale at that particular point.