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The Objective Function

Objective functions define the means by which a solution is to be found. As explained before, efficiency is a reasonable concern so a sophisticated function that is prudent to construct the model more frequently than necessary must be employed. Our function needs to drive the search for 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.

Let us 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. We seek 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 our algorithm is as follows:

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

A combination of different objective functions, e.g. that which we have implemented for MSD, MI, normalised MI, wavelets and probability density functions alongside the model-based approach, produced encouraging results as well. We found that the performance of different methods varies depending on the data and the distance from convergence and AART can autonomously try a mixture of methods, thereupon hybrid objective functions are introduced. In practice, to indirectly and quickly evaluate MDL we obtain ${\displaystyle \begin{array}{c} n\\ \sum\\ i=1\end{array}}log(\lambda_{i})$ where $\lambda_{1<i<n}$ are the $n$ eigenvalues of the covariance matrix whose magnitudes are the greatest [17].