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Learning the Correlations

The way in which this problem can be circumvented quickly involves learning how the parameters $\mathbf{c}_{i}$ affect the model^2.25 with respect to a typical target. Each parameter in $\mathbf{c}_{i}$has an unequalled effect on different regions in the model, e.g. its size, intensities and so on. By changing the value of each such parameter and recording the change that is perceived in an image (using pixel-based comparison of some kind), a type of deformation index can be maintained. This index indicates which parameters should be changed and if so in what way and to what degree in order to approach good overlap between a model and some target image.

More formally, the procedure works as follows:

For the model parameters $\mathbf{c}_{i}$ where $1<i<n$, a parameter change $\delta\mathbf{c}$ (where one parameter value or more can be readjusted) is applied to generate some new shape and texture. $\delta\mathbf{c}$ expresses in a vector-based representation the offsets that each of the original parameters $\mathbf{c}_{i}$ is subjected to. The exhaustive pixel-wise difference in intensity^2.26 is calculated in accordance with:

$$\delta\mathbf{I}=\mathbf{I}_{model}-\mathbf{I}_{image}$$ (2.7)

to produce a new vector of intensities (the differences). This vector can also be visualised to display this difference to a human eye. A simple measure of difference is used although this need not necessarily be the case. Sum-of-squares of the pixel differences is then used because larger quadratic differences will have a greater effect on the final measure and summation then only consists of positive values. For example, observe the values derived in Equation and in . The former shows how the values of the vector in , and particularly their summed difference, get accentuated, whereas in the later case makes them almost negligible^2.27.

$$\delta\mathbf{I}=sumofsquares(\{-1,3,5,2,6,-10,-1\})$$ (2.8)

then becomes

$$\delta\mathbf{I}=sum(\{1,9,25,4,36,100,1\})=176$$ (2.9)

as opposed to

$$\delta\mathbf{I}=sum(\{-1,3,5,2,6,-10,-1\})=4.$$ (2.10)

With this measure of intensity difference recorded, relational information can be expressed between the parameter change and this difference as it appears in image space where a model is superimposed on some target. That information (merely a correlation) can be learned by using a pseudo-target image which is the model in its mean form. It can be used for basic comparison that infers something about the model displacements and their corresponding effect^2.28.

This quantitative measure of difference obtained will however indicate solely the approximate ``goodness'' of the parameter change (as inferred from SSD or MSD) and not an overall focalised effect that it has on the given image. This means that it will not necessarily be obvious what parts in the two entities (model and target) remained similar and which ones did not^2.29. A type of a sequential data such as a vector is hence more useful as it retains the location of each computed difference value. Unsurprisingly, this also consumes far more space (and many vectors of this kind will in fact be necessary).

In either case, under the premise that space is more expendable than time complexity, a vector of difference is calculated and the correlation can be formulated as follows:

$$\mathbf{c}_{i}\rightarrow\mathbf{c}_{i}+\delta\mathbf{c}\rightarrow\delta\mathbf{I}$$ (2.11)

This type of offset $\delta\mathbf{c}$ that was applied to the collection of parameters $\mathbf{c}_{i}$ is accompanied by a global change in intensity values across the image frame. This correlation can now be stored aside and become accessible from an index as its size is proportional to the image size. The storage is dictated by the following (somewhat artificial) relation:

$$\delta\mathbf{c}=\mathbf{A}\delta\mathbf{I}$$ (2.12)

where $\mathbf{A}$ is a matrix^2.30 recording the change in intensities due to the parameter/s change $\delta\mathbf{c}$. This is a type of matrix which is correspondent to an n-dimensional vector that expresses the change which was discovered off-line. It linearly defines (in a possibly high-dimensional space) the linear relation between change to the parameters and change to the intensities, or more precisely the difference image. It can be used to choose directions of change directly when performing a search and thereby avoid re-computation in a virtually recurring and almost identical problem^2.31.

The most fundamental (and perhaps even compact) procedure will carry out the steps above for each of the modes of variation, as well as the linear geometrical transformations. This can be a very laborious and cumbersome process although it depends on the robustness prescribed. As the next stage illustrates, models that are not rich enough will fail to converge in difficult scenarios, a classic example of which is inappropriate initialisation.

The matrix A holds real-valued numbers (preferably of limited accuracy to decrease space requirements and access speed). The values in this matrix form a 'path-finding' map that guides exploration for good parameter changes; this will be of great use when fitting the model to a target. In practice, such matrices are visualised by showing negative values as dark shades and positive one as increasingly brighter values.