Probabilistic Mass Contour Template

The first objective of the proposal is the construction of a general mass template, which should take the shape variations into account. The main aim is that pixels with a boundary morphology which has a major representation in the database have a higher probability than the rest of the pixels. Hence, as the template is represented as an image, the pixel brightness will be associated with the probability of belonging to a contour. Thus, the designed template will have intensity 0 for those pixels which certainly do not represent a contour, intensity $1$ for the pixels which in all images of the database are on a contour (if any), and intermediate values for the rest of the pixels.

An initial solution for the construction of this template consists in considering only the boundaries of manually segmented masses. Note, however, that this solution prefixes a set of contours, and contours different to them will probably be refused while, in contrast, the probability to find two masses with similar shape is very low. Thus, in order to obtain a more general template, it is constructed by looking for the sub-space that these boundaries define. This is achieved by adapting the eigenfaces approach described in Section . Moreover, using this approach only a rough manual segmentation is needed, just including the centre and size of the mass.

With the obtained eigenmasses, it is possible to construct a probabilistic template per size (note from the previous section that the masses have been clustered according to their size and different templates can be created). For constructing these templates, the $N$ eigenvectors containing $95\%$ of variation explanation were used, considering more probable shapes those with the greatest eigenvalue. Therefore, an initial template is constructed as :

$$\psi^0(x,y) = \frac{1}{N}\sum_{k=1}^N w_k W_k(x,y)$$ (4.3)

where $\psi^0(x,y)$ is the template, $W_k(x,y)$ is the $k$ -th eigenmass and $w_k$ its normalized eigenvalue (the corresponding eigenvalue divided by their sum). The contour of the eigenmasses is found by extracting the gradient from $\psi^0(x,y)$ :

$$\begin{split} \nabla \psi^0(x,y) &= \nabla \left\{ \frac{1}{... ... &= \frac{1}{N} \sum_{k=1}^N w_k \nabla W_k(x,y) \end{split}$$ (4.4)

This equation (image) represents the template as a weighted contours of the eigenmasses. In order to obtain a deformable template, it is necessary to specify the modes of deformation of such initial (rigid) template. Note that the object deformation in an image is an unknown parameter of the model which will be estimated during the template matching step.

Plausible shapes are those obtained from linear combinations of the eigenmass contours, and deformation will only affect the weight of the eigenvalues of each eigenmass. This is represented by a vector $\xi$ of size N:

$$\nabla \psi^d(x,y) = \kappa \sum_{k=1}^N \xi_k w_k \nabla W_k(x,y)$$ (4.5)

where $\psi^d(x,y)$ is the deformed template and $\kappa$ is just a normalization factor. With this definition, the vector $\xi$ is all ones when no variations from the template occur, and results in larger difference to the original template as it increases/decreases its values. Hence, assuming a Gaussian distribution, the probability of finding a template with such deformation is:

$$Pr(\xi) = \frac{1}{\sqrt{2\pi}\sigma}\exp\{-\frac{1}{2\sigma^2} \sum_{k=1}^N (\xi_k-1)^2\}$$ (4.6)

Note that with this definition a new parameter ($\sigma$ ) is included. Changes in the value of $\sigma$ represent a more rigid (small $\sigma$ ) or a more flexible (large $\sigma$ ) template. Figure  shows the templates for four classes representing the range of mass sizes in the database.

Figure 4.4: The probabilistic templates obtained by clustering the dataset in four clusters of various sizes. Lighter pixels represent a higher probability of a mass contour.

Moreover, Figure  shows the average intensity over a circle (y-axis) as a function of the radii to the mass centre (x-axis), detailed for each of the four sizes. The peak in each curve represents the radii with highest probability of being a contour of a mass. Note that larger templates are not only a translation of the smaller ones at different radii but also have a different profile, showing the need of having a different training set for each size.

Figure 4.5: Intensity profile of the templates as a function of the distance to the centre (in pixels).