Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery

Recently, tensor ring (TR) approximation has received increasing attention in multi-dimensional image processing. In TR approximation, the key backbone is the shallow matrix factorizations, which approximate the circular unfolding of the multi-dimensional image. However, the shallow matrix factorization limits the standard TR approximation’s ability to represent images with complex details and textures. To address this limitation, we propose a nonlinear hierarchical matrix factorization-based tensor ring (NHTR) approximation. Specifically, instead of the shallow matrix factorization, we introduce the nonlinear hierarchical matrix factorization in NHTR approximation to approximate circularly \(\lceil \frac{N}{2}\rceil \) $\lceil \frac{N}{2}\rceil$ -modes unfoldings of an N-th order tensor. Benefiting from the powerful expressive capability of the nonlinear hierarchical matrix factorization, the proposed NHTR approximation can faithfully capture fine details of the clean image compared to classical tensor ring approximation. Empowered with the proposed NHTR, we build a multi-dimensional image recovery model and establish a theoretical error bound between the recovered image and the clean image based on the proposed model. To solve the highly nonlinear and hierarchical optimization problem, we develop an efficient alternating minimization-based algorithm. Experiments on multispectral images and color videos conclusively demonstrate the superior performance of our method over the compared state-of-the-art methods in multi-dimensional image recovery.