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Monday, January 23rd, 2012, 4:09 pm

Coarse Correspondence in Riemannian Manifolds: Masks and Multi-Resolution Approach

THIS post is part of a series that explores the potential of comparing surfaces using GMDS, or generalised multidimensional scaling.

A 15-level multi-resolution approach has led to no classification errors being made, at least thus far. This may have helped prevent the convergence near local minima, but it is painfully slow, especially when the C++ implementation does not get used. I have looked into more ways to widen the separation between correct pairs (same person) and incorrect pairs. I have begun looking at the impact of two factors; one is the size of the mask used prior to GMDS (or dilation through iteration) and another is the number of multi-resolution levels. Based on an ongoing experiment, a very coarse correspondence/initialisation leads to something quite reasonable when the pairs belong to the same subject and everything is a lot quicker and a bit of a mess otherwise (see the first two images).

At 3 cumulative levels in the multi-resolution reproach, a false classification does not take long to occur, so I increased that to 15 and ran that at 3 levels of dilation from the three centres for half a day. In spite of the optimisation process taking a lot longer, performance was not good, peaking well below 90% recognition rate. Although the tested dataset is not large enough to draw conclusions from, the recent experiments seem to suggest that not that a multi-scale approach on its own cannot resolve frequent recurrence of misclassifications.

In order to better understand what weakens these measures I have taken a closer look at visual GMDS output. It seems as though the scores are heightened when real (correspondent) pairs of surfaces are not yielding the correct correspondence, even after a 15-level optimisation when the data is exactly the same except the mask size (as shown in the images).

In the past, taking the best fit among all matches was tried (and watched as secondary/surrogate in all of the recent experiments in fact), but it does not perform well as a discriminant on its own. If GMDS succeeds at finding the accurate correspondence 95% of the time in these circumstances, then in this case we need to rerun GMDS several times to exceed it in terms of recognition rates. The other FMM-based method (the one I created) achieved better recognition rates than that.

In order to test the mask type and its effect on performance I ended up setting all parameters to fixed values and running very coarse-scale experiments, first on the entire face and later on a masked subset of limited size, particularly focusing on rigid parts alone.

Results were interesting in the sense that they showed that, based on GMDS as assessment criterion, smaller mask around the fixed points do not clearly and unambiguously produce better results, at least not in the case of this dataset. The assumption we had about removing the non-rigid area may have been misplaced — inherited from other work.

In the next stage I will return to fine levels to get vastly better results.

Ideally, we should initiate these high-resolution triangulations by the result we get from the lower resolution. Currently, by default, there are 9 levels, adjusted according to m, the number of sample size (300 in the latest few experiments). It’s the number of levels in the multi-resolution hierarchy. At the finest level there are typically 15999 faces and 8101 vertices (in older experiments we had just about ~2000 and in recent ones there were ~4000). A low-to-high resolution is operated by default, but it is not sufficient for evading local minima. This should explain the decrease in performance and Initialisation with lower resolution result (iteratively) should solve part of the problems.

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