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Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition

Yao Wang, Jiangjun Peng, Qian Zhao, Deyu Meng, Yee Leung, Xi-Le Zhao

arXiv:1707.02477v1cs.CV

TL;DR

HSIs suffer mixed acquisition noise that degrades visual quality and downstream precision. The paper proposes LRTDTV, combining Tucker decomposition, SSTV regularization, sparse-noise modeling, and an ALM solver; experiments report stable restoration behavior and solver convergence.

  • Problem

    Existing HSI restoration methods often remove only one or two noise types, whereas real-world HSIs are corrupted by mixtures of noise.

  • Method

    LRTDTV combines low-rank Tucker decomposition, spatial-spectral total variation, ℓ1 sparse-noise regularization, and an optional Frobenius term for heavy Gaussian noise, solved using ALM.

  • Results

    The proposed LRTDTV method produces the most stable restoration curve in the reported real-data experiment, and its MPSNR and MSSIM changes converge toward zero with iterations.

  • Takeaways & Limitations

    Direct tensor modeling exploits global spatial-spectral correlation while SSTV characterizes piecewise smooth structure across spatial and spectral domains.

Abstract

from arXiv · show

Hyperspectral images (HSIs) are often corrupted by a mixture of several types of noise during the acquisition process, e.g., Gaussian noise, impulse noise, dead lines, stripes, and many others. Such complex noise could degrade the quality of the acquired HSIs, limiting the precision of the subsequent processing. In this paper, we present a novel tensor-based HSI restoration approach by fully identifying the intrinsic structures of the clean HSI part and the mixed noise part respectively. Specifically, for the clean HSI part, we use tensor Tucker decomposition to describe the global correlation among all bands, and an anisotropic spatial-spectral total variation (SSTV) regularization to characterize the piecewise smooth structure in both spatial and spectral domains. For the mixed noise part, we adopt the $\ell_1$ norm regularization to detect the sparse noise, including stripes, impulse noise, and dead pixels. Despite that TV regulariztion has the ability of removing Gaussian noise, the Frobenius norm term is further used to model heavy Gaussian noise for some real-world scenarios. Then, we develop an efficient algorithm for solving the resulting optimization problem by using the augmented Lagrange multiplier (ALM) method. Finally, extensive experiments on simulated and real-world noise HSIs are carried out to demonstrate the superiority of the proposed method over the existing state-of-the-art ones.

I. INTRODUCTION

HSI restoration must address mixed noise while preserving spatial and spectral structure that matrix-based and bandwise methods may fail to exploit. The paper proposes direct tensor modeling with low-rank Tucker decomposition, SSTV regularization, sparse-noise modeling, and ALM optimization.

  • I. INTRODUCTION: Bandwise denoising ignores correlations across spectral bands or spatial pixels, often producing unsatisfactory restoration.
  • I. INTRODUCTION: HSIs can contain Gaussian noise, impulse noise, dead lines, stripes, and other noise types acquired together.Such mixed noise degrades visual quality and affects downstream application precision.
  • I. INTRODUCTION: Low-rank matrix models for mixed-noise removal lose finer spatial-and-spectral information by modeling the HSI cube as a matrix.The paper identifies this loss as a source of suboptimal restoration in some heavily noisy situations.
  • I. INTRODUCTION: The proposed direct tensor approach exploits spatial-and-spectral priors in the clean HSI while characterizing intrinsic structures of the heavy-noise part.
  • B. Our contributions: Low-rank Tucker decomposition captures global spatial-and-spectral correlations, while SSTV models piecewise smoothness in both spatial and spectral domains.The framework also uses ℓ1 regularization for sparse noise and an ALM method to solve the resulting model.
  • B. Our contributions: The paper evaluates the proposed method on simulated and real data using quantitative and visual comparisons with other restoration techniques.

C. Organization of The Paper

The paper models HSI restoration by exploiting tensor structure, mixed-noise decomposition, and spatial-spectral correlations. It introduces the tensor preliminaries and motivates low-rank modeling and total variation across HSI modes.

  • Paper organization: The paper develops an anisotropic SSTV-regularized low-rank tensor model and solves it with an ALM-based procedure before evaluating it on simulated and real datasets.The stated organization proceeds from tensor preliminaries to modeling, optimization, experiments, and conclusions.
  • Motivation: An observed HSI is modeled as a clean cube plus Gaussian noise and sparse noise containing stripes, impulse noise, and dead pixels.The clean cube is estimated from the observation by exploiting the structures of both the signal and noise.
  • Low-rank structure: Tucker decomposition captures low-rank structure because HSI spectral signatures use few endmembers and spatial modes also exhibit correlations.The mode-3 matricization has relatively few nonzero singular values, while spatial-mode singular values show decaying trends.
  • Total variation: Band-by-band total variation misses local smoothness along the spectral mode, where adjacent-band differences are often nearly zero.The paper therefore motivates total variation regularization in both spatial and spectral directions.

B. Low-rank Tensor Decomposition with Anisotropic SSTV

The LRTDTV model combines Tucker decomposition with anisotropic spatial-spectral total variation to represent global correlation and piecewise smoothness in clean HSIs. Its nonconvex optimization is addressed using an ALM method.

  • Model: LRTDTV combines low-rank tensor decomposition with TV regularization in both spatial and spectral modes.The model is introduced by combining low-rank and TV properties of HSIs.
  • Low-rank representation: The Tucker representation uses a core tensor and factor matrices to encode the HSI’s low-rank structure.The factor matrices are constrained to be orthogonal, with ranks specified along the three modes.
  • SSTV regularization: The SSTV regularizer uses weighted first-order differences along the HSI’s spatial and spectral directions.Its weights control the regularization strength for each mode.
  • Optimization: The proposed model is nonconvex because Tucker decomposition is nonconvex, so ALM is used to seek a good local solution.The paper presents ALM as the optimization strategy for the resulting formulation.

C. Optimization Procedure

The optimization procedure reformulates LRTDTV with auxiliary variables and alternately updates decomposition, restoration, regularization, noise, and multiplier variables. The resulting ALM solver uses closed-form or efficient subproblem updates until convergence.

  • ALM formulation: Auxiliary variables convert the LRTDTV formulation into an equivalent constrained minimization problem.The constraints include Y = X + S + N, X = Z, and Dw(Z) = F.
  • ALM formulation: The weighted three-dimensional difference operator combines horizontal, vertical, and spectral first-order differences.The mode weights determine each direction’s contribution to the SSTV regularization.
  • Alternating updates: The augmented Lagrangian alternately updates Tucker variables, X, auxiliary variables, sparse noise S, Gaussian noise N, and Lagrange multipliers.The update sequence is described through subproblems for C, Ui, X, Z, F, S, N, and the multipliers.
  • Efficient updates: The Z update uses 3D FFT diagonalization of the block-circulant difference operator for efficient solution of its linear system.The Fourier-domain expression uses fftn and ifftn with elementwise squared magnitudes and division.
  • Noise updates: Soft-thresholding solves the F and S subproblems, while the N subproblem updates the Gaussian-noise estimate.The multiplier variables are then updated according to the ALM scheme.
  • Algorithm: Algorithm 1 repeats the variable and multiplier updates, increases the penalty parameter, and stops when the convergence condition is satisfied.The penalty parameter is initialized at µ = 10^-2 and updated as µ = min(ρµ, µmax).

D. The LRTDTV Approximate Model

When Gaussian noise is not heavy, the model can be simplified because TV regularization already removes some Gaussian noise. The approximate model retains an ALM solution procedure with sequential subproblem updates.

  • Approximate model: The approximate LRTDTV model separates mixture noise into sparse and Gaussian components.For non-heavy Gaussian noise, TV regularization provides some Gaussian-noise removal, allowing model simplification.
  • Optimization: The simplified model is solved with an ALM-based optimization procedure using closed-form solutions for its subproblems.The paper gives the subproblem solutions directly for concision.
  • Optimization: The approximate solver sequentially updates the model variables until a convergence condition is satisfied.The update equations correspond to the approximate model’s numbered optimization steps.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The experiments evaluate LRTDTV for HSI restoration on simulated and real data, comparing it with seven popular restoration methods. Parameters for competing methods followed their default strategies, while LRTDTV parameter selection was discussed separately.

  • Both simulated and real HSI experiments evaluated the effectiveness of LRTDTV restoration.
  • Seven popular HSI restoration methods were implemented as comparison baselines.
  • Compared-method parameters were manually adjusted according to their default strategies.
  • All HSI bands were normalized to [0, 1] for numerical calculation and visualization, then stretched back after restoration.

A. Simulated Data Experiments

The simulated experiments used a 145×145×224 Indian Pines-derived HSI and tested restoration methods under six mixtures of Gaussian, impulse, deadline, and stripe noise.

  • The synthetic HSI was generated from Indian Pines ground truth and had size 145×145×224.
  • All synthetic reflectance values were linearly mapped to [0, 1].
  • Six noise cases tested methods under combinations of Gaussian noise, impulse noise, deadlines, and stripes.
  • Case 1: Case 1 applied zero-mean Gaussian noise with variance 0.1 uniformly across bands.
  • Cases 5–6: Cases 5–6 used band-dependent Gaussian and impulse noise, deadlines, and additional stripes in bands 161–190 for Case 6.

1) Visual quality comparison:

Visual comparisons examined restored bands from two representative simulated cases, showing that all methods removed some mixed noise while LRTDTV performed best among the compared methods.

  • Figure 5 compares band 36, corrupted by Gaussian and impulse noise in Case 3.
  • Figure 6 compares band 116, severely corrupted by Gaussian noise, impulse noise, and deadlines in Case 5.
  • All compared methods removed mixed noise to some extent in both representative cases.
  • LRTDTV performed best among the compared methods in the visual comparisons.

2) Quantitative comparison:

Quantitative evaluation used MPSNR, MSSIM, and ERGAS, with Table II and per-band curves showing superior LRTDTV performance and spectral signatures.

  • MPSNR, MSSIM, and ERGAS were used to objectively evaluate restoration quality.MPSNR and MSSIM average bandwise PSNR and SSIM values.
  • Table II reports restoration results for all compared methods using the three indices.
  • LRTDTV achieved superior performance over the other popular approaches in the quantitative comparison.
  • TV-regularized methods, including LRTV and LRTDTV, were examined through per-band PSNR and SSIM values across simulated cases.
  • For Case 2–6, the approximation model was used because Gaussian noise was not dominant; Case 1 used the Gaussian-dominant model.
  • In Case 5, LRTDTV produced the best spectral signature among compared methods, consistent with its ERGAS values.

B. Real Data Experiments

Real-data and simulated-data experiments show that LRTDTV handles heavy mixed noise while better preserving spatial and spectral structures than competing methods.

  • Real datasets: Two real HSI datasets were evaluated: HYDICE urban and AVIRIS Indian Pines.The HYDICE image has size 307 × 307 × 210, while AVIRIS Indian Pines has 145×145 pixels and 220 bands.
  • HYDICE Urban Data Set: On HYDICE bands 109 and 207, low-rank matrix methods could not effectively remove stripes because stripe locations repeated across neighboring band ranges.The repeated stripe and deadline locations made the corruptions more likely to be absorbed into the low-rank component.
  • HYDICE Urban Data Set: BM4D and TDL degraded under heavy sparse or mixed noise, while LRTDTV better removed complex noise and preserved spatial texture than LRTV.BM4D over-smoothed and distorted structures, whereas LRTV preserved edges and local details less effectively than LRTDTV.
  • Profile comparisons: LRTDTV produced the most stable horizontal mean profile for band 109 and the best restored vertical mean profile among compared methods.The restored LRTDTV curves were reported as most stable or best in the corresponding profile comparisons.
  • Overall comparison: Across simulated and real data, extensive experiments demonstrated clear superiority of LRTDTV over other popular methods.The paper attributes this result to exploiting more useful structures.

C. Discussion

The discussion examines model variants, parameter choices, spectral-TV weighting, and solver convergence. It favors the approximate model for typical real HSI noise while retaining the general model for very heavy Gaussian noise.

  • Model variants: The general LRTDTV model performs better than the approximate model under very heavy Gaussian noise because it includes a Frobenius norm term.For real HSIs, Gaussian noise is usually less severe, so the approximate model is preferred for simplicity.
  • Parameter analysis: The sparse-noise parameter λ is set to λ = 10 in simulated experiments because results remain relatively stable as C varies from 10 to 25.The analysis evaluates both MPSNR and MSSIM while varying the tuning parameter C.
  • Rank constraint: Tucker decomposition encodes the low-rank prior, with spatial ranks empirically set to eighty percent of the spatial dimensions.The spectral rank is selected by examining MPSNR and MSSIM across different rank constraints.
  • SSTV weighting: Spectral TV helps remove noise, especially structure noise whose distribution differs across adjacent spectral bands.The reported analysis finds a benefit from tuning the SSTV weight along the spectral mode.
  • Solver convergence: As the iteration number increases, relative changes in MPSNR and MSSIM converge to zero, indicating convergent behavior of the LRTDTV solver.This convergence supports using the solver in practical situations.

V. CONCLUSION

The paper proposes a tensor-based HSI restoration framework combining low-rank Tucker modeling, SSTV regularization, sparse-noise detection, and heavy-Gaussian-noise modeling. Experiments report superior quantitative and visual performance, while future work targets more complex noise and richer tensor priors.

  • Conclusion: The method models clean HSIs with low-rank Tucker decomposition and spatial-spectral total variation, while separating sparse and heavy Gaussian noise.The sparse component includes stripes, impulse noise, and dead pixels; the Frobenius norm models heavy Gaussian noise.
  • Conclusion: Experiments on simulated and real data report superior quantitative and visual performance over popular competing methods.The conclusion summarizes evidence from both quantitative evaluation and visual comparison.
  • Conclusion: Future work will incorporate more complex-noise modeling, deep tensor architectures, and SSTV regularizers exploiting gradient similarity across bands.These directions are intended to enhance noise removal and better characterize clean-HSI and noise structures.
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