Multidimensional Image Reconstruction via Deep Nonlinear Low-Rank Tensor Decomposition

Low-rank tensor decomposition (LRTD) has demonstrated significant efficacy in multidimensional image reconstruction. Indeed, LRTD driven by nonlinear relationship can capture the underlying low-rank structure more accurately, since real-world data often exhibits complex nonlinear interactions. However, the existing nonlinear LRTD methods do not to investigate the inherent nonlinear interactions in spatial neighborhoods and spectral or temporal models. To address these challenges, we propose a novel deep nonlinear low-rank tensor decomposition (DNLRTD). Specifically, we design a deep nonlinear transform network (DNTN) using multiple convolutional layers and channel attention modules to form a deep nonlinear transform (DNT). The custom-designed DNT effectively captures nonlinear interactions within spatial neighborhoods while paying attention to the nonlinear interactions of spectral or temporal dimensions, consequently achieving a lower-rank representation. By integrating DNT into the low-tubal-rank decomposition framework, we induce the deep tubal-rank and form the DNLRTD. Also, we design a customized DNLRTD optimization strategy to make it flexible for different multidimensional image reconstruction tasks. Based on DNLRTD, we construct two multidimensional image reconstruction models and develop corresponding algorithms based on the alternating direction method of multipliers (ADMM) to solve them. Extensive experimental results on spectral compressive imaging and dynamic magnetic resonance image (MRI) reconstruction verify the superior performance of the proposed method.