Smooth incomplete matrix factorization and its applications in image/video denoising

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 a 2D 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 l1 norm 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 the 2D 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.