Quaternion tensor left ring decomposition and application for color image inpainting

In recent years, tensor networks have emerged as powerful tools for solving large-scale optimization problems. One of the most promising tensor networks is the tensor ring (TR) decomposition, which achieves circular dimensional permutation invariance in the model through the utilization of the trace operation and equitable treatment of the latent cores. On the other hand, more recently, quaternions have gained significant attention and have been widely utilized in color image processing tasks due to their effectiveness in encoding color pixels by considering the three color channels as a unified entity. Therefore, in this paper, based on the left quaternion matrix multiplication, we propose the quaternion tensor left ring (QTLR) decomposition, which inherits the powerful and generalized representation abilities of the TR decomposition while leveraging the advantages of quaternions for color pixel representation. In addition to providing the definition of QTLR decomposition and an algorithm for learning the QTLR format, the paper further proposes a low-rank quaternion tensor completion (LRQTC) model and its algorithm for color image inpainting based on the defined QTLR decomposition. Finally, extensive experiments on color image inpainting demonstrate that the proposed LRQTC method is highly competitive.