Low-rank Factorization Of Matrices With Missing Components And Its Application In Image/Video Processing

Low-rank matrix factorization with missing elements has many applications in computer vision. However, the original model without taking any prior information, which is to minimize the total reconstruction error of all the observed matrix elements, sometimes provides a physically meaningless solution in some applications. In this paper, we propose a regularized low-rank factorization model for a matrix with missing elements, called Smooth Incomplete Matrix Factorization (SIMF), and exploit a novel image/video denoising algorithm with the SIMF. Since data in many applications are usually of intrinsic spatial smoothness, the SIMF uses a2D discretized Laplacian operator as a regularizer to constrain the matrix elements to be locally smoothly distributed. It is formulated as two optimization problems under the l1norm and the Frobenius norm, and two iterative algorithms are designed for solving them respectively. Then, the SIMF is extended to the tensor case (called Smooth Incomplete Tensor Factorization, SITF) by replacing the2D Laplacian by a high-dimensional Laplacian. Finally, an image/video denoising algorithm is presented based on the proposed SIMF/SITF. Extensive experimental results show the effectiveness of our algorithm in comparison to other six algorithms.This paper focuses on the generalized version of the problem of a low-rank approximation of a matrix with missing components, i.e. low-rank approximations of a set of matrices with missing components, since data used in many applications are intrinsically in matrix form rather than in vector form. This generalized problem is formulated as an optimization problem at first, which minimizes the total reconstruction error of the known components in these matrices. Then, an iterative algorithm is designed for calculating the generalized low-rank approximations of matrices with missing components, called GLRAMMC. Finally, detailed algorithm analysis is given. Experimental results on synthetic data as well as on real image data show the effectiveness of our proposed algorithm in comparison to several existing algorithms.