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Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration

Shuang Xu, Chang Yu, Jiangjun Peng, Xiangyong Cao, Deyu Meng

TL;DR

The paper tackles remote sensing image restoration where missing or corrupted regions must be recovered, highlighting limitations of existing low-rank and smoothness priors. It introduces the Haar Nuclear Norm ($HNN$), defined via the $2$-D frontal slice-wise Haar DWT ($2$-D FHWT) and the four wavelet subbands, to enforce low-rankness across both coarse and fine frequencies; an ADMM solver is developed for HNN-based matrix completion and RPCA, accompanied by recovery guarantees under incoherence conditions. Empirical results across HSI denoising, inpainting, and multi-temporal cloud removal show $HNN$ achieving $1$–$4$ dB PSNR improvements and $10$–$28\times$ speedups over state-of-the-art methods, demonstrating both accuracy and efficiency benefits. The work contributes a principled, scalable regularizer for remote sensing tasks and lays groundwork for extensions to HSI unmixing and classification, with practical impact on quality and usability of remote sensing imagery.

Abstract

Remote sensing image restoration aims to reconstruct missing or corrupted areas within images. To date, low-rank based models have garnered significant interest in this field. This paper proposes a novel low-rank regularization term, named the Haar nuclear norm (HNN), for efficient and effective remote sensing image restoration. It leverages the low-rank properties of wavelet coefficients derived from the 2-D frontal slice-wise Haar discrete wavelet transform, effectively modeling the low-rank prior for separated coarse-grained structure and fine-grained textures in the image. Experimental evaluations conducted on hyperspectral image inpainting, multi-temporal image cloud removal, and hyperspectral image denoising have revealed the HNN's potential. Typically, HNN achieves a performance improvement of 1-4 dB and a speedup of 10-28x compared to some state-of-the-art methods (e.g., tensor correlated total variation, and fully-connected tensor network) for inpainting tasks.

Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration

TL;DR

The paper tackles remote sensing image restoration where missing or corrupted regions must be recovered, highlighting limitations of existing low-rank and smoothness priors. It introduces the Haar Nuclear Norm (), defined via the -D frontal slice-wise Haar DWT (-D FHWT) and the four wavelet subbands, to enforce low-rankness across both coarse and fine frequencies; an ADMM solver is developed for HNN-based matrix completion and RPCA, accompanied by recovery guarantees under incoherence conditions. Empirical results across HSI denoising, inpainting, and multi-temporal cloud removal show achieving – dB PSNR improvements and – speedups over state-of-the-art methods, demonstrating both accuracy and efficiency benefits. The work contributes a principled, scalable regularizer for remote sensing tasks and lays groundwork for extensions to HSI unmixing and classification, with practical impact on quality and usability of remote sensing imagery.

Abstract

Remote sensing image restoration aims to reconstruct missing or corrupted areas within images. To date, low-rank based models have garnered significant interest in this field. This paper proposes a novel low-rank regularization term, named the Haar nuclear norm (HNN), for efficient and effective remote sensing image restoration. It leverages the low-rank properties of wavelet coefficients derived from the 2-D frontal slice-wise Haar discrete wavelet transform, effectively modeling the low-rank prior for separated coarse-grained structure and fine-grained textures in the image. Experimental evaluations conducted on hyperspectral image inpainting, multi-temporal image cloud removal, and hyperspectral image denoising have revealed the HNN's potential. Typically, HNN achieves a performance improvement of 1-4 dB and a speedup of 10-28x compared to some state-of-the-art methods (e.g., tensor correlated total variation, and fully-connected tensor network) for inpainting tasks.
Paper Structure (26 sections, 3 theorems, 38 equations, 8 figures, 8 tables)

This paper contains 26 sections, 3 theorems, 38 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Low-rank models
  4. Low-rank decomposition models
  5. Low-rank regularizations
  6. Low-rank models combined with smoothness
  7. Preliminaries
  8. Tensor operations
  9. Haar wavelet transform
  10. Haar Nuclear Norm
  11. Motivation
  12. Haar nuclear norm
  13. HNN based matrix completion
  14. HNN based robust principal component analysis
  15. Theory of Recovery Guarantee
  16. ...and 11 more sections

Key Result

Theorem 6

Consider a 3-order tensor $\mathcal{A} \in \mathbb{R}^{M \times N \times S}$. Let $[\mathcal{B}_{1}, \mathcal{B}_{2}; \mathcal{B}_{3}, \mathcal{B}_{4}] = \mathrm{FHWT}_{2}(\mathcal{A})$, and then there is for $n = 1, 2, 3$ and $i = 1, 2, 3, 4$.

Figures (8)

  • Figure 1: (a) Typical remote sensing imagery $\mathcal{A}\in \mathbb{R}^{M\times N\times S}$ and its wavelet coefficients, which exhibit rapid decay in singular values. $\mathscr{F}(\cdot)$ represents the 2D FHWT. (b) The CE curves for the original image and wavelet coefficients. (c) Comparison of TNN, TCTV, and HNN, where $\|\cdot\|_{*}$ denotes the matrix nuclear norm and $\|\cdot\|_{\rm TNN}$ represents the TNN.
  • Figure 2: Phase transitions of (a) RPCA models and (b) MC models. The success rates are shown in the title of each panel.
  • Figure 3: The false-color denoised images (band 24-17-1) of compared methods on the PC dataset for case 6.
  • Figure 4: False-color images (band: 193-97-19) of all compared methods on the BA dataset.
  • Figure 5: The false-color decloud images (the 5th time node, band 1-4-7) of all compared methods on the Jizzakh mountain dataset.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 1: $n$-Rank n_Rank
  • Definition 2: Tucker-Rank tucker_rank
  • Definition 3: 1-D Haar Discrete Wavelet Transform
  • Definition 4: 2-D Haar Discrete Wavelet Transform
  • Definition 5: 2-D Frontal Slice-wise Haar Discrete Wavelet Transform
  • Theorem 6: Low-rank Properties for Wavelet Coefficients
  • Definition 7
  • Theorem 8: HNN-RPCA Theorem
  • Theorem 9: HNN-MC Theorem