[Work Log] Branching prior covariance; implementing
Goals:
- ground truth - trace datasets 7 through 10
- build new attachment covariance
- reconstruct ground truth using new attachment covariance
- work attachment covariance function into ml
Building Full Attachment Covariance Matrix
See attachment/construct_attachment_covariance.m (filename still in flux).
Issue: The recursive definition of the attached curve kernel that we outlined yesterday is too expensive to implement naively. Can implement by re-using precomputed values, but the branch point isn't necessarilly one of the observed points, so it's not necesarilly represented in the curve's prior_K. This means we need to construct covariance entries between each of the attachment points and their parent and child curves. Then we can run the "memoized" recursive function.
Base self covariance
Key idea: each child curve has a "base self covariance" that is extra covariance inherited from the branch point on parent that is added to all points of the curve self-covariance matrix. That curve, in turn, has a base covariance, that it inherits, all the way to the root, whose base covariance is the "position variance" parameter. Thus, the leaves have a base covariance that is the sum of all the incremental variance of all the branches to the root.
The base covariance is an NxN matrix, where N is the number of views. We construct each in isolation, and then cumulatively sum them from root to leaf.
Every element of the curve's observed-point covariance matrix is increased by the base_covariance relative to it's entry; e.g. if element (k_{ij}) is the covariance between a point in view 3 and a point in view 5, you'll add the (3,5) element of the base covariance matrix.
Base Parent covariance
Each child curve also hase a "parent covariance" (for lack of a better term), which is the set of covariances between the branch point and each of the parent points. Again, this is inherited recursively; just as the curve's covariance has it's self-covariance added, the parent covariance has the paren'ts self-covariance added.
This will be used for the off-diagonal blocks of the full attachment covariance matrix.
Implementation Misc
Issues during implementation and testing of construct_attachment_covariance
replacing Cell-of-structs with struct-array:
Corrs = cell2mat(Corrs);
The trivial code path of calling blkdiag takes 500ms. Wow!
Running on the WACV dataset is giving bad results.
Looks like the covariance matrix is different from the one in the reference implementation.
TODO:
- Need to convert from "index of branch index" to simply "branch index".
- More debugging.