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Thursday, January 12th, 2012, 1:48 am

Geometry Gurus Needed (First Order Geodesics on Surface Pairs)

I COULD really use the advice of some people who are interested in computer graphics and vision — inverse but complementary fields, which are largely related to geometry and mathematics at large. The challenge is to compare surfaces based on their 3-D characteristics, using Euclidean, geodesic or another non-Euclidean metrics.

Here is what the surfaces look like.

Subject surfaces

The following image represents a performance ROC curve we have.

ROC curve dilation approach

In simple terms, this means that we’re able to classify correctly about 70% of the time. Having summarised a year’s work, culminating in mostly unsuccessful experiments around diffusion (around 97^ recognition rate, the above is poor, So I return to working with geodesic distances as the Swiss army knife for measurement of distances. But the approach being explored at the moment is different and with further enhancement it can hopefully yield performance higher than 97% (matching rate). The sources of limitation are generally well understood in the sense that they can be visualised and overlaid on top of the image pairs. There is no trivial and reliable way to establish multiple landmark points around which to measure distances consistently, so I am trying another way, which at a very coarse level has a matching rate of about 80% (can be significantly improved soon).

The approach attempted at this stage involves triangle comparison post- and pre-marking, but the performance attained so far is not satisfactory.

The limitation is likely inherent in the measuring of distances in FMM and the sub-sampling that results from triangulation (see image for the size of triangles to be fully appreciated).

One particular weakness of the diffusion approach to masking is that it leads to holes in the data, which invalidates some of the measures that were used routinely beforehand. In order to fuse together both geodesic and spectral measures, we now attempt to get a more symbiotic approach that carves out surfaces based on geodesic properties and then uses spectral features on these surfaces. Since ordering does not exist (e.g. point-to-point correspondences), the histogram of images is used to describe the sub-surfaces (carved out around a known correspondence). By increasing the number of rings and bins in the histogram, the performance can be varied somewhat. I was running overnight experiment to test this and got some other appalling ROC curves, such as this:

Curve dilation approach with more rings

The results from the last experiment were disappointing because they did not provide good separability. So the following morning I designed and started running an experiment that explores the potential of measuring diffusion distance between furthest points in the surface carved in accordance with geodesic boundaries (several rings). This too did not give good results. Generally speaking, diffusion distance as a measurement has not proven to be anywhere as useful as FMM so far (since December). It seems to be insensitive to small differences and it does not seem to degrade linearly, either. The next experimental design will explore another new approach, perhaps conceding the potential of diffusion being integrated into the framework’s pipeline.

So the question is this: given two surfaces that are geodetically craves around the surface, what approach would you use to compare them for similarity? We’ve tried a variety of known methods, but none seems to yield very encouraging results thus far. Thanks for any advice or pointers you may have.

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