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

摘要

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 Xdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {X}}$$end{document}∈documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$in $$end{document}RI×I×?×Idocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {R}}^{Itimes {I}times {cdots }times {I}}$$end{document} into a small-scale essential tensor Edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {E}}$$end{document}∈documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$in $$end{document}Rr×r×?×rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {R}}^{rtimes {r}times {cdots }times {r}}$$end{document}, 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 O(N∑k=2NrkRk(N-k)+k-1+NrN-1R2(N-1)+NR3(N-1)+N∑k=1NrkIN-k+1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {O}}{(Nsum _{k=2}^N{r^k}{R^{k(N-k)+k-1}}}+{N{r^{N-1}}R^{2(N-1)}+N{R}^{3(N-1)}+N{sum _{k=1}^N{{r^k}{I}^{N-k+1}}})}$$end{document} at each iteration, which is significantly lower than O(N∑k=2NIkRk(N-k)+k-1+NIN-1R2(N-1)+NR3(N-1))documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {O}}{(Nsum _{k=2}^N{I^k}{R^{k(N-k)+k-1}}}+N{I^{N-1}}R^{2(N-1)}+{N{R}^{3(N-1)})}$$end{document} of PAM for FCTN when r?Idocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$rll I$$end{document}. Extensive numerical experiments on color videos and light field images illustrate the superior