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Learning to Fit the Model

This first problem can be solved by learning how the parameters $\mathbf{c}_{i}$ (see Equation ) affect the model and how they should be changed in order for the model to align with the image to be interpreted. Each parameter in $\mathbf{c}_{i}$ has an effect on model synthesis (both shape and intensities vary simultaneously). By changing the value of each such parameter and learning the difference between the model and the target image (using pixel-based comparison), an index can be built which indicates how model parameters should be varied to better fit the target. This type of index indicates which parameters should be changed and, if so, in what way and to what degree.

More formally, the algorithm can be described as follows. For the model parameters $\mathbf{c}_{i}$, where $1<i<n$, a parameter change $\delta\mathbf{c}$ (one parameter or more can be changed simultaneously) is applied to generate new model synthesis consisting shape and texture. This process is repeated for each mode of displacement where both shape and intensity are changed, but only intensity difference is learned. Sum-of-squares of the pixel differences^3.5 is then used.

In [], a similar formulation is used. Shape transformation parameters, t, are used to define point position in X, which is the image frame of the model. The pixels in the region of this image, g$_{im}$, can then be projected into the texture frame $\mathbf{g}=T_{u}^{-1}(\mathbf{g}_{im})$ and the model texture is $\mathbf{g_{\mathbf{m}}=\mathbf{\bar{g}}}+Q_{g}\mathbf{c}$. The image difference is $\mathbf{r}(\mathbf{p})=g-g_{im}.$ From this, the image residuals can be derived. This is made possible after a Taylor expansion that yields a convenient term to optimise over. This term is referred to as $\delta\mathbf{I}$ from this point onwards.

With this measure of intensity difference, the relation between the parameter change and this difference can be expressed conveniently. That information, which is merely a correlation, can be learned by using a pseudo-target image, such as the model in its `standard' form (the mean). It can be used as the basis for a comparison which facilitates learning about the model displacements and their corresponding effects in image space.

This quantitative measure of difference will approximate ``goodness'' of a parameter change (as indicated when calculating SSD or MSD), but not more localised effects which such a change has on the given image. This means that it will not be obvious what parts in the two entities (model and target) remain similar and which ones do not.

To address the need for localised difference measures, the parameter change, $\delta\mathbf{c}$, which is applied to the collection of parameters $\mathbf{c}_{i}$, will be accompanied by a high-dimensional representation of intensity differences in the image. This correlation can be made accessible through an index whose size is proportional to the image size. This relationship is dictated by the following:

$$\delta\mathbf{c}=\mathbf{A}\delta\mathbf{I}$$ (3.9)

where $\mathbf{A}$ is a matrix^3.6 that encapsulates the change in intensities due to the parameter/s change $\delta\mathbf{c}$. This matrix is correspondent to an n-dimensional vector. This vector expresses the change which was discovered `off-line', i.e. prior to a search in an image. It linearly defines (in a potentially high-dimensional space) the relationship between change to the parameters and change to the intensities (the difference image). It can be used to choose directions of change directly when performing a search and thereby avoid unnecessary re-computation.