Factorized Robust Matrix Completion

In this book chapter, we discuss a factorization-based approach to robust matrix completion. We begin our discussion with a gauge optimization perspective to robust matrix completion. We then discuss how our approach replaces the solution over the low rank matrix with its low rank factors. In this context, we develop a gauge minimization algorithm and an alternating direction method of multipliers algorithm that take advantage of the factorized matrix decomposition. We then focus on the particular application of video background subtraction, which is the problem of finding moving objects in a video sequence that move independently from the background scene. The segmentation of moving objects helps in analyzing the trajectory of moving targets and in improving the performance of object detection and classification algorithms. In scenes that exhibit camera motion, we first extract the motion vectors from the coded video bitstream and fit the global motion of every frame to a parametric perspective model. The frames are then aligned to match the perspective of the first frame in a group of pictures (GOP) and use our factorized robust matrix completion algorithm to fill in the background pixels that are missing from the individual video frames in the GOP. We also discuss the case where additional depth information is available for the video scene and develop a depth-weighted group-wise PCA algorithm that improves the foreground/background separation by incorporating the depth information into the reconstruction.