| 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 mathematical objects to be used for modeling a multivariate mixed data set that is (approximately) 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 3-D reconstruction.;In this work, we study the problem of segmenting subspace arrangements. The problem is sometimes called the subspace segmentation problem. The work was inspired by generalized principal component analysis (GPCA), an algebraic solution that simultaneously estimates the segmentation of the data and the parameters of the multiple subspaces. Built on past extensive study of subspace arrangements in algebraic geometry, we propose a principled framework that summarizes important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangement models both efficient and robust, even when the given data are corrupted with noise and/or contaminated by outliers.;Algebraically, we study the properties of polynomials vanishing on a union of subspaces; and statistically, we study how to estimate these polynomials robustly from real sample sets with noise and outliers. These new methods in many ways improve and generalize extant methods for modeling or clustering mixed data. Finally, we will show results of these methods applied to computer vision and image processing. |
