1. Introduction

Various fundamental applications in computer vision and machine learning require finding the basis of a certain subspace. The obtained basis are then used to represent any observation in the subspace, often using only the major basis in order to reduce the dimensionality and suppress noise. Examples of such applications include face detection, motion estimation, activity recognition … etc. An increasing interest has been recently placed on this area as a result of significant advances in the mathematics of matrix rank optimization. Interestingly, as we will discuss in more detail in the coming sections, robust subspace estimation can be posed as a low rank-optimization problem, which can be solved efficiently using techniques such as the method of Augmented Lagrange Multiplier.In this book we discuss fundamental formulations and extensions for low-rank optimization-based subspace estimation and representation. By minimizing the rank of the matrix containing observations drawn from images, we demonstrate to the reader how to solve four fundamental computer vision problems, including video denosing, background subtraction, motion estimation, and activity recognition.