HLRTF: Hierarchical Low-Rank Tensor Factorization for Inverse Problems in Multi-Dimensional Imaging

Inverse problems in multi-dimensional imaging, e.g., completion, denoising, and compressive sensing, are challenging owing to the big volume of the data and the inherent ill-posedness. To tackle these issues, this work unsupervisedly learns a hierarchical low-rank tensor factorization (HLRTF) by solely using an observed multi-dimensional image. Specifically, we embed a deep neural network (DNN) into the tensor singular value decomposition framework and develop the HLRTF, which captures the underlying low-rank structures of multi-dimensional images with compact representation abilities. This DNN herein serves as a nonlinear transform from a vector to another to help obtain a better low-rank representation. Our HLRTF infers the parameters of the DNN and the underlying low-rank structure of the original data from its observation via the gradient descent using a non-reference loss function in an unsupervised manner. To address the vanishing gradient in extreme scenarios, e.g., structural missing pixels, we introduce a parametric total variation regularization to constrain the DNN parameters and the tensor factor parameters with theoretical analysis. We apply our HLRTF for typical inverse problems in multi-dimensional imaging including completion, denoising, and snapshot spectral imaging, which demonstrates its generality and wide applicability. Extensive results illustrate the superiority of our method as compared with state-of-the-art methods.