The flag of best fit as a representative for a collection of linear subspaces
Date
2013
Authors
Marrinan, Timothy P., author
Kirby, Michael, advisor
Bates, Dan, committee member
Draper, Bruce, committee member
Peterson, Chris, committee member
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Abstract
This thesis will develop a technique for representing a collection of subspaces with a flag of best fit, and apply it to practical problems within computer vision and pattern analysis. In particular, we will find a nested sequence of subspaces that are central, with respect to an optimization criterion based on the projection Frobenius norm, to a set of points in the disjoint union of a collection of Grassmann manifolds. Referred to as the flag mean, this sequence can be computed analytically. Three existing subspace means in the literature, the Karcher mean, the extrinsic manifold mean, and the L2-median, will be discussed to determine the need and relevance of the flag mean. One significant point of separation between the flag mean and existing means is that the flag mean can be computed for points that lie on different Grassmann manifolds, under certain constraints. Advantages of this distinction will be discussed. Additionally, results of experiments based on data from DARPA's Mind's Eye Program will be compared between the flag mean and the Karcher mean. Finally, distance measures for comparing flags to other flags, and similarity scores for comparing flags to subspaces will be discussed and applied to the Carnegie Mellon University, 'Pose, Illumination, and Expression' database.
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Subject
flag manifold
flag mean
Grassmann manifold
Karcher mean
subspace average
SVD