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The MDL-based Objective Function

As was explained in the previous subsection, objective functions define the means by which a solution is to be found. Efficiency is a reasonable concern so a sophisticated function that is prudent to construct the model more frequently than necessary must be employed. The function used in this context needs to drive the search for shape correspondences using a suitable parameterisation (in the case of image registration - transformations which increase similarity across all images). The different nature of the problem and the methods of solving it convey the ulterior goal somewhat differently than the vast majority of methods to date, resulting in the formulation below.

For the similar case of image registration, one can denote a transformation function $W(\bullet,params)$ and the construction of an appearance model to be $Model(\mathbf{x}_{1},\mathbf{x}_{2},..,\mathbf{x}_{n})$ where $\mathbf{x}_{i}$ are the images used to train that model. One seeks a model that is more compact using the following (simplified) function

$F_{obj}=MDL(Model(\mathbf{x}_{1}...,\mathbf{x}_{i}..,\mathbf{x}_{n}))-MDL(Model(\mathbf{x}_{1}...,W(\mathbf{x}_{i},params)..,\mathbf{x}_{n}))$

where $params$ should be found to minimise this expression for each image vector $\mathbf{x}_{i}$. A succinct description of this algorithm is as follows:

• Repeat
  • For each image vector $\mathbf{x}_{i}$,
    • Optimise $F_{obj}$ by altering the values of $params$.
• Until convergence.

In practice, to indirectly and quickly evaluate MDL what will be obtained is ${\displaystyle \begin{array}{c} n\\ \sum\\ i=1\end{array}}log(\lambda_{i})$ where $\lambda_{1<i<n}$ are the $n$ Eigen-values of the covariance matrix whose magnitudes are the greatest. This is similar to the formulation of Kotcheff [] where ${\displaystyle \begin{array}{c} n\\ \sum\\ i=1\end{array}}log(\lambda_{i}+\delta)$ is calculated to approximate

$$det(\mathbf{M+\delta)}\equiv\begin{array}{c} n\\ \prod\\ i... ...\end{array}}log(\lambda_{i}+\delta)\equiv log(det(\mathbf{M}))$$ (5.2)

where $\mathbf{M}$ is the covariance matrix under consideration.