Outlier-Robust Moving Object and Background Decomposition via Structured ${\ell }_p$-Regularized Low-Rank Representation

Abstract

Low-rank and sparse decomposition is a powerful tool in background extraction and moving object detection problems since the static background usually lies in a low dimensional subspace while the rapidly changing object maintains a sparse property. Conventional techniques for low-rank and sparse decomposition can handle noise-free or Gaussian noise scenarios. However, their performance significantly degrades in challenging scenarios contaminated by impulsive noise or outliers. In this paper, a new model with a separable noise component, which we call unstructured <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><b>p</b></sub> -regularized low-rank representation (ULLR), is proposed. Instead of incorporating the Frobenius norm constraint on the noise component, the proposed ULLR incorporates flexible <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><b>p</b></sub> -regularized schemes with 0 <; p ≤ 2 to handle varied scenarios. For the moving object detection problem, we further incorporate the local group structure and homogeneous group structure to encode the underlying object. Then, a new model called the structured <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><b>p</b></sub> -regularized low-rank representation (SLLR) is proposed to enhance the detected foreground while suppressing the noise and dynamic background. Especially, when 0 <; p ≤ 1, the SLLR achieves robustness against outliers and hence handles challenging scenarios well. The resultant optimization problem is solved using the framework of alternating direction method of multipliers (ADMM). Each ADMM subproblem can be efficiently solved. More importantly, the solutions for the noise-based subproblem are rigorously derived with respect to different <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><b>p</b></sub> -regularized schemes. Simulation results demonstrate the efficiency, accuracy, and robustness to outliers of the proposed algorithms.