Estimation of Subspace Arrangements: Its Algebra and Statistics

Abstract

In the literature of computer vision and image processing, a fundamental difficulty in modeling visual data is that multivariate image or video data tend to be heterogeneous or multimodal. That is, subsets of the data may have significantly different geometric or statistical properties. For example, image features from multiple independently moving objects may be tracked in a motion sequence, or a video clip may capture scenes of different events over time. Therefore, it seems to be desirable to segment mixed data into unimodal subsets and then model each subset with a distinct model. Recently, subspace arrangements have become an increasingly popular class of math-ematical objects to be used for modeling a multivariate mixed data set that is (approx-imately) piecewise linear. A subspace arrangement is a union of multiple subspaces. Each subspace can be conveniently used to model a homogeneous subset of the data. Hence, all the subspaces together can capture the heterogeneous structure of the data set. Such hybrid subspace models have been successfully applied to modeling different types of image features for purposes such as motion segmentation, texture analysis, and