%0 Journal Article
%T A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation
%A Akram Aldroubi
%J ISRN Signal Processing
%D 2013
%R 10.1155/2013/417492
%X The subspace segmentation problem is fundamental in many applications. The goal is to cluster data drawn from an unknown union of subspaces. In this paper we state the problem and describe its connection to other areas of mathematics and engineering. We then review the mathematical and algorithmic methods created to solve this problem and some of its particular cases. We also describe the problem of motion tracking in videos and its connection to the subspace segmentation problem and compare the various techniques for solving it. 1. Introduction The subspace clustering problem is fundamental in many engineering and mathematics applications [1¨C11]. It can be described as follows: let be the nonlinear set consisting of a union of subspaces of a Hilbert or a Banach space . Let be a set of data points drawn from . The subspace segmentation (or clustering) problem is then to determine (equivalently determine for ), from the data , that is, to(1)determine the number of subspaces ; (2)find an orthonormal basis for each subspace , ;(3)group the data points belonging to the same subspace into the same cluster. The data is often corrupted by noise; it may have outliers or some of the data vectors may have missing entries. Therefore, any technique for solving the subspace segmentation problem above must be robust and stable for the aforementioned nonideal cases. Depending on the application, the space can be finite or infinite dimensional. For example, the set of all two dimensional images of a given face , obtained under different illuminations and facial positions, can be modeled as a set of vectors belonging to a low dimensional subspace living in a higher dimensional space [12¨C14]. For this case, a set of such images from different faces is a union . Another application in which a union of subspaces provides a good model is the problem of motion tracking of rigid objects in videos. For this situation (further developed below), a 4-dimensional subspace is assigned to each moving object in a space , where is the number of frames in the video. Examples where is infinite dimensional arise in sampling theory, and in learning theory [15¨C19]. For example, signals with finite rate of innovations are modeled by a union of subspaces that belongs to an infinite dimensional space such as [2, 3, 20, 21]. 1.1. Known Number of Subspaces and Dimensions In some subspace segmentation problems, the number of subspaces or the dimensions of the subspaces are known or can be estimated [1, 8, 22, 23]. In these cases, the subspace segmentation problem, for both the finite and
%U http://www.hindawi.com/journals/isrn.signal.processing/2013/417492/