Multi-Dimensional Image Recovery via Fully-Connected Tensor Network Decomposition Under the Learnable Transforms

Abstract

Multi-dimensional image recovery from incomplete data is a fundamental problem in data processing. Due to its advantage of capturing the correlations between any modes of the multi-dimensional image, i.e., the target tensor, the fully-connected tensor network (FCTN) decomposition has recently shown promising performance on multi-dimensional image recovery. However, FCTN decomposition suffers from computational deficiency, especially for large-scale multi-dimensional images. To address this deficiency, we propose a learnable transform-based FCTN model (termed as T-FCTN), which enjoys the remarkable advantage of FCTN decomposition with cheap computational cost. More concretely, we learn the semi-orthogonal transforms along each mode of the target tensor to project the large-scale tensor ${\mathcal {X}}$ $\in$ ${\mathbb {R}}^{I\times {I}\times {\cdots }\times {I}}$ into a small-scale essential tensor ${\mathcal {E}}$ $\in$ ${\mathbb {R}}^{r\times {r}\times {\cdots }\times {r}}$, and then apply FCTN decomposition on the small-scale essential tensor. To tackle the proposed model, we develop an efficient proximal alternating minimization (PAM)-based algorithm with theoretical convergence guarantee. Moreover, the computational complexity of PAM for T-FCTN is ${\mathcal {O}}{(N\sum _{k=2}^N{r^k}{R^{k(N-k)+k-1}}}+{N{r^{N-1}}R^{2(N-1)}+N{R}^{3(N-1)}+N{\sum _{k=1}^N{{r^k}{I}^{N-k+1}}})}$ at each iteration, which is significantly lower than ${\mathcal {O}}{(N\sum _{k=2}^N{I^k}{R^{k(N-k)+k-1}}}+N{I^{N-1}}R^{2(N-1)}+{N{R}^{3(N-1)})}$ of PAM for FCTN when $r\ll I$. Extensive numerical experiments on color videos and light field images illustrate the superiority of the proposed method over other state-of-the-art methods in terms of quality metrics, visual quality, and running time.