Likelihood

The likelihood is a measurement of the similarity between the deformed template and the object(s) present in the image. The deformable template will be attracted and aligned to the salient edges in the input image via a directional edge potential field. For a pixel $(x,y)$ in the input image its edge potential can be defined as:

$$\phi_Y(x,y) = -\exp(-\rho \sqrt{\delta_x^2+\delta_y^2})$$ (4.8)

where $\delta_x$ is the displacement to the nearest edge point in the horizontal direction, $\delta_y$ in the vertical direction, and $\rho$ is a smoothing factor which controls the degree of smoothness of the potential field. This potential is modified by introducing a directional component relating the deformed template $\psi^{s,\xi,d}$ to the edges of the input image $Y$ :

$$\Upsilon(\psi^{s,\xi,d},Y) = \frac{1}{T} \sum_{x,y \in \psi^{s,\xi,d}}(1+\phi_Y(x,y)\vert cos(\beta(x,y))\vert)$$ (4.9)

where the summation is over all the pixels on the deformed template, $T$ is the number of pixels on the template, $\beta(x,y)$ is the angle between the tangent of the nearest edge and the tangent direction of the template at position $(x,y)$ , and the constant $1$ is added so that $\Upsilon(\psi^{s,\xi,d},Y)$ is positive and takes values between 0 and $1$ . This definition requires that the template boundary agrees with the image edges not only in position, but also in the tangent direction. Figure  shows three different mammograms and their respective potential images, where a lighter colour indicates a higher potential. The vertical and horizontal stripes comes from those points far away of either a vertical or a horizontal edge.

Figure 4.6: Three different mammograms containing clear masses and their potential images. Note that contours of the internal tissue are clearly defined.

Using the above energy function, the probability density of the likelihood of observing the input image, given the deformations of the template is:

$$Pr(Y\vert s,d,\xi) = \alpha \exp\{ -\Upsilon(\psi^{s,\xi,d},Y) \}$$ (4.10)

where $\alpha$ is a normalizing constant to ensure that the above function integrates to $1$ . The maximum likelihood is achieved when $\Upsilon(\psi^{s,\xi,d},Y) = 0$ i.e., when the deformed template $\psi^{s,\xi,d}$ exactly matches the edges in the input image Y.