Subsets

The motivation here was to explore more efficient approaches. The subsets approach was intended to `localise' the optimisation. Rather than construct models by optimising the entire set of images at the same time, smaller sets are stochastically drawn from the complete set and then treated as though they are a complete set, only smaller. A different subset is chosen at each stage and this has the advantage of being able to handle large sets without scalability issues. I wanted to see if this subsets-based methodology would lead to better results that are reached more quickly, so I varied the size of subsets from which models were built and estimates for the objective function derived from.

Given the set of parameters that seemed to work best in the previous experiments, I performed the following last set of experiments.

10 repeated registrations with 10 different random instantiations of 10 bumps (comprising 200 sample points each) were completed. It needed to be demonstrated that by considering a subset of all images, the determinant becomes smaller or the same value reached more quickly given an objective function which is model based. 20 modes of variation were considered when computing the determinant and 98% of the variation kept when applying PCA.

200 iterations were used to refine the model and subsets re-chosen in random, without repetition in selection, every 10 iterations. Figure $[*]$ shows some results from these experiments.

**Figure:** Top: MSD score (image to image) as a function of the size of the subset (lower is better); bottom: distance from the correct solution (shape to correct shape, where less is better)

Figure $[*]$ shows the effect of the size of the set on speedup. Statistically speaking (see the error bars), there was no significant difference, so the idea of dividing the problem and handling smaller parts of it did not improve efficiency.

Other experiments that may be interesting to learn from are ones which study more parameters in this multi-faceted problem. For instance, it would be useful to see how varying the number of model modes which are considered or the variation kept when applying PCA can affect overall performance. In a problem such as this, there are many different parameters that can be varied simultaneously, so fine-tuning of the algorithm is an interesting task. In practice, I found that changing the assignment of values to particular parameters `on the fly'^5.7 was a way of reaching satisfactory results.

**Figure:** time (in seconds) to complete one NRR