Block Customized Topology Term Decomposition for High-Dimensional Image Reconstruction

Recently, the block-term decomposition with rank- $(L_{r}, L_{r}, 1)$ (termed as LL1 decomposition), which decomposes a third-order tensor into the sum of the outer products between vector and matrix factors, has received increasing attention for high-dimensional image reconstruction. However, the fixed low-rank matrix decomposition in LL1 is restricted to third-order tensors, which hinders its development for higher-order tensor data (i.e., order $N \gt 3$ ). To address this, we propose a Block Customized Topology Term Decomposition (BCTD), which represents an $N$ th-order tensor as a sum of outer products of basis vectors and customized $(N-1)$ th-order coefficient tensors with flexible internal topological structures. The proposed BCTD enjoys two advantages: Firstly, it allows tackling higher-order tensors beyond the third-order tensor setting of LL1, which can better preserve the high-dimensional structure of the tensor. Secondly, it allows each term to have a customized topological structure beyond the fixed topological structure (i.e., low-rank matrix decomposition) in LL1, which can better explore the intrinsic high-dimensional low-rank structures of the tensor. To evaluate the performance of the proposed BCTD, we build the corresponding high-dimensional image reconstruction model and provide a theoretical generalization error bound between the recovered tensor of the proposed model and the underlying tensor. To solve the resulting optimization problem, we apply a proximal alternating minimization (PAM)-based algorithm with a theoretical convergence guarantee. Extensive experimental results on high-dimensional image completion and compression tasks using real-world datasets (color videos and light field images) demonstrate the superiority of the proposed model over other baseline models.