Optimiser Tolerance

The goal of this portion of the work was to explore the potential for avoiding optimisation in areas where significant improvements are no longer being made. In the following set of experiments, the aim was to compare results obtained not by considering different optimisers (although this is practically possible) but to look at how the refinement with single-point CPS warps is handled when various degrees of tolerance are considered.

This study involved running a series of experiments where the NRR was applied to a fixed set of bumps (as above) and the results then assessed at the end, in the very same way as before. Then, when varying the tolerance and rerunning the experiment, it was possible see the effect on the quality of the results. I repeated these experiments 10 times with a different set of bumps and then considered the average to demonstrate that one choice of tolerance is consistently more favourable than others.

The numbers of repetitions, images, and iterations used here were the same as before. 200 iterations, i.e. 200 passes through the whole set of images, were run until completion with different levels of optimiser tolerance. At each stage, a different scale of refinement was chosen such that the scale values increased orders of magnitude each time (the numbers used here were 3 decimal places, 5 decimal places, 7 decimal places, 10 decimal places, and 15 decimal places, corresponding to 0.001, 0.00001, and so on). These numbers define the accuracy level which the optimiser strives to reach at each iteration. 3 decimal places would means that the optimiser looked at large-scale refinements and accepted these, whereas at 15 decimal places the optimiser worked for a longer time to refine the results at a finer scale (which took longer).

Figure $[*]$ and Figure $[*]$ show the results from these experiments. With this general-purpose Nelder-Mead optimiser, the best results wrt the ground-truth solution are obtained when the accuracy required from the optimiser is 7 decimal places. As expected, the greater the accuracy which is required, the longer it takes for the NRR to be completed. Requiring too great an accuracy at each stage has the negative effect of not allowing the algorithm to explore distant (and potentially better) solutions, so choosing too high a tolerance can lead to deviation from the correct solution.

**Figure:** from the correct solution (less is better) as a function of optimiser tolerance

**Figure:** time (in seconds, with error bars) taken to complete one NRR as a function of optimiser tolerance

In summary, these results indicate that changes to the accuracy of the optimisation do matter and a significant speedup is possible without loss of accuracy.