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Fast Unsupervised Tensor Restoration via Low-rank Deconvolution

David Reixach, Josep Ramon Morros

TL;DR

This work addresses tensor restoration with noise by extending Low-rank Deconvolution (LRD) with differential regularization in the $DFT$ domain. Squared Total Variation and integral priors are incorporated to yield a linear, frequency-domain solution that efficiently denoises and enhances details in multi-dimensional data. The method is demonstrated on image denoising and video enhancement, showing competitive PSNR performance against unsupervised and self-supervised baselines while offering substantial speed advantages. The approach provides a principled, unsupervised alternative to heavy DL pipelines, with practical implications for real-time tensor restoration.

Abstract

Low-rank Deconvolution (LRD) has appeared as a new multi-dimensional representation model that enjoys important efficiency and flexibility properties. In this work we ask ourselves if this analytical model can compete against Deep Learning (DL) frameworks like Deep Image Prior (DIP) or Blind-Spot Networks (BSN) and other classical methods in the task of signal restoration. More specifically, we propose to extend LRD with differential regularization. This approach allows us to easily incorporate Total Variation (TV) and integral priors to the formulation leading to considerable performance tested on signal restoration tasks such image denoising and video enhancement, and at the same time benefiting from its small computational cost.

Fast Unsupervised Tensor Restoration via Low-rank Deconvolution

TL;DR

This work addresses tensor restoration with noise by extending Low-rank Deconvolution (LRD) with differential regularization in the domain. Squared Total Variation and integral priors are incorporated to yield a linear, frequency-domain solution that efficiently denoises and enhances details in multi-dimensional data. The method is demonstrated on image denoising and video enhancement, showing competitive PSNR performance against unsupervised and self-supervised baselines while offering substantial speed advantages. The approach provides a principled, unsupervised alternative to heavy DL pipelines, with practical implications for real-time tensor restoration.

Abstract

Low-rank Deconvolution (LRD) has appeared as a new multi-dimensional representation model that enjoys important efficiency and flexibility properties. In this work we ask ourselves if this analytical model can compete against Deep Learning (DL) frameworks like Deep Image Prior (DIP) or Blind-Spot Networks (BSN) and other classical methods in the task of signal restoration. More specifically, we propose to extend LRD with differential regularization. This approach allows us to easily incorporate Total Variation (TV) and integral priors to the formulation leading to considerable performance tested on signal restoration tasks such image denoising and video enhancement, and at the same time benefiting from its small computational cost.
Paper Structure (13 sections, 1 theorem, 24 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 1 theorem, 24 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Image Denoising
  4. Detail Enhancement
  5. Low-rank Deconvolution with Differential Regularization
  6. Notation
  7. Revisiting LRD
  8. Differential Regularization in the DFT Domain
  9. Image Denoising & Detail Enhancement
  10. Experiments
  11. Image Denoising
  12. Video Enhancement
  13. Conclusion

Key Result

Proposition 3.1

The problem presented in eq. (eq:lrd_tv) with $\Psi(\{\mathbf{X}_m^{(n)}\}) = \sum_{m = 1}^{M}\sum_{n = 1}^{N}\frac{\alpha}{2} \left\lVert \mathbf{X}_m^{(n)} \right\rVert_2^2$ has a solution given by a linear expression in the DFT domain given by: where we have made use of $\hat{\mathbf{s}}^{(n)}$, $\hat{\mathbf{x}}^{(n)}$ and $\hat{\mathbf{W}}^{(n)}$ defined in section sec:lrd. And defining: wi

Figures (3)

  • Figure 1: Quality of reconstruction (PSNR) vs. execution time (s). We display overall results on the whole dataset for the four levels of input noise respectively. PSNR evolution as a function of time for a single image denoising for the chosen methods.
  • Figure 2: Qualitative evaluation on RGB video enhancement. In all cases, we show five consecutive video frames. Top. Original color frames. Bottom. Video Enhancement. As it can be seen, our method can be used to enhance details in tensors, such video-sequences. Best viewed in color.
  • Figure 3: Quality of reconstruction (PSNR) vs. $\gamma$. We display overall results on the whole dataset for our method and the four levels of input noise.

Theorems & Definitions (2)

  • Proposition 3.1
  • proof