[Work Log] Rethinking Likelihood

June 20, 2013
Project Tulips
Subproject Data Association v2
Working path projects/​tulips/​trunk/​src/​matlab/​data_association_2
Unless otherwise noted, all filesystem paths are relative to the "Working path" named above.

Testing and debugging clean_correspodnence2.m has revealed some significant problems with the current approach to constructing the likelihood function.

5%-10% of fragements have correspondences that are nonsensical.

I hypothesize that this is due to bad correspondences early on, when there is little evidence to drive a good correspodnence. These bad correspodnences are propagated as new curves are added that could suggest a better correspodnence.

Large lateral gaps in triangulation result in large axial gaps in posterior curve.

The problem is that the index set is computed from the triangulation. The current fix for this -- smooth, re-index, repeat -- is very limited in the severity it can overcome. In practice, most gaps are only partially reduced.

Rethinking "Correspondence to Likelihood"

Corresondence is good for constructing a decent-quality 3d curve, but isn't good for computing fine-grained pointwise likelihood, due to sporradic terrible correspondences and gaps.

Instead of continuously Band-Aiding these issues that keep arising, its time to re-think how the likelhood is constructed.

1. the mean of 3d gaussians should project to the 2D position of the corresponding data point

2. the depth should be based on the corresponding position in the unobserved curve

Two issues here. First, how to localize the unobserved curve without having the likelihood already (chicken and egg). Second, how to identify the corresponding point of the unobserved curve?

In the old method, the answer to both was "use the correspondence matrix."

In the new method, we still use the correspondence matrix to triangulate, but we smooth the result using the prior and then throw away correspondence information.

Killing the correspondence grid

The corresponence matrix artificially forces points from different views to correspond to the same point. This is out of necessity -- we need correspondence to achieve triangulation. But we don't need to adhere to this to compute the likelhood. Indeed, observed points may fall anywhere in the continuous index set, not into a discrete set of predefined cells.

Random thoughts

Posted by Kyle Simek
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