Progressive Multigrid Eigensolvers for Multiscale Spectral Segmentation

We reexamine the role of multiscale cues in image segmentation using an architecture that constructs a globally coherent scale-space output representation. This characteristic is in contrast to many existing works on bottom-up segmentation, which prematurely compress information into a single scale. The architecture is a standard extension of Normalized Cuts from an image plane to an image pyramid, with cross-scale constraints enforcing consistency in the solution while allowing emergence of coarse-to-fine detail. We observe that multiscale processing, in addition to improving segmentation quality, offers a route by which to speed computation. We make a significant algorithmic advance in the form of a custom multigrid eigensolver for constrained Angular Embedding problems possessing coarseto-fine structure. Multiscale Normalized Cuts is a special case. Our solver builds atop recent results on randomized matrix approximation, using a novel interpolation operation to mold its computational strategy according to crossscale constraints in the problem definition. Applying our solver to multiscale segmentation problems demonstrates speedup by more than an order of magnitude. This speedup is at the algorithmic level and carries over to any implementation target.