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Denoising Hyperspectral Image with Non-i.i.d. Noise Structure

Yang Chen, Xiangyong Cao, Qian Zhao, Deyu Meng, Zongben Xu

arXiv:1702.00098v1cs.CV

TL;DR

Existing HSI denoising methods often assume i.i.d. Gaussian noise despite the more complex, band-dependent structures found in real images. The paper introduces NMoG-LRMF, combining band-specific mixture-of-Gaussians noise modeling with Bayesian low-rank factorization and variational Bayes inference. Experiments on synthetic and real HSIs report more robust denoising than previous techniques, while future work targets broader applications and additional priors.

  • Problem

    Existing HSI denoising methods commonly use i.i.d. Gaussian noise assumptions, although real HSI noise has more complicated non-i.i.d. structures that affect denoising robustness.

  • Method

    NMoG-LRMF models each HSI band with a distinct Gaussian mixture, links bands through shared priors, embeds the noise model in LRMF, and uses variational Bayes inference.

  • Results

    The proposed method performs more robustly than previous techniques on synthetic and real noisy HSIs, achieving the best PSNR and SSIM across almost all bands in complex simulated noise cases.

  • Takeaways & Limitations

    Explicitly modeling non-i.i.d. noise improves HSI recovery under complex noise, including better detail preservation and noise removal in real HSI experiments.

  • Takeaways & Limitations

    Future work includes extending the method to video and face images, adding more HSI priors, and redesigning the noise model for specific application contexts.

Abstract

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Hyperspectral image (HSI) denoising has been attracting much research attention in remote sensing area due to its importance in improving the HSI qualities. The existing HSI denoising methods mainly focus on specific spectral and spatial prior knowledge in HSIs, and share a common underlying assumption that the embedded noise in HSI is independent and identically distributed (i.i.d.). In real scenarios, however, the noise existed in a natural HSI is always with much more complicated non-i.i.d. statistical structures and the under-estimation to this noise complexity often tends to evidently degenerate the robustness of current methods. To alleviate this issue, this paper attempts the first effort to model the HSI noise using a non-i.i.d. mixture of Gaussians (NMoG) noise assumption, which is finely in accordance with the noise characteristics possessed by a natural HSI and thus is capable of adapting various noise shapes encountered in real applications. Then we integrate such noise modeling strategy into the low-rank matrix factorization (LRMF) model and propose a NMoG-LRMF model in the Bayesian framework. A variational Bayes algorithm is designed to infer the posterior of the proposed model. All involved parameters can be recursively updated in closed-form. Compared with the current techniques, the proposed method performs more robust beyond the state-of-the-arts, as substantiated by our experiments implemented on synthetic and real noisy HSIs.

I. INTRODUCTION

HSI noise is complex and non-i.i.d., while many denoising methods assume i.i.d. Gaussian noise. The paper models band-specific noise within an LRMF framework to improve robustness on realistic HSI denoising tasks.

  • Motivation: HSIs preserve spatial and spectral information across many bands but suffer sensor-related noise that degrades image quality and downstream processing.Noise sources include sensor sensitivity, photon effects, calibration error, shot noise, and thermal noise.
  • Existing methods: Conventional pixelwise or bandwise denoising ignores spatial and spectral correlations, while later methods encode HSI structural priors.The related methods include wavelet, Wiener-filtering, and low-rank approaches.
  • Problem: Using an L2 reconstruction loss implicitly assumes i.i.d. Gaussian noise, which poorly represents the more complicated non-i.i.d. noise found in real HSIs.This mismatch can degrade performance in complex, realistic noise conditions.
  • Noise characteristics: Real HSI noise is approximately i.i.d. within a band, differs across bands because of sensor sensitivity, and remains correlated between neighboring bands.The paper illustrates similar noise in bands 189–191.
  • Contribution: The paper models each band with a distinct MoG distribution, places shared higher-level priors across bands, and embeds this strategy in Bayesian LRMF.A variational Bayes algorithm is used, and the approach is evaluated on synthetic and real noisy HSIs.

II. NON-I.I.D. MOG METHOD FOR HSI DENOISING

The method first encodes non-i.i.d. HSI noise and then integrates that encoding into an LRMF model with variational inference.

  • Method overview: The paper introduces a non-i.i.d. noise encoding strategy, constructs an NMoG-LRMF model within LRMF, and designs a variational inference algorithm.The method is organized around noise encoding, model construction, and posterior inference.
  • Method overview: The resulting framework represents HSI structure through low-rank factorization while explicitly modeling non-i.i.d. noise.
  • Method overview: The algorithm is designed to infer the posterior distributions of the proposed Bayesian model.

A. Non-i.i.d. noise encoding

The noise encoder models each HSI band with its own Gaussian mixture while using shared higher-level priors to capture cross-band correlation.

  • Noise assumptions: HSI noise is modeled as i.i.d. within each band, non-identical across bands, and correlated between different bands.
  • Bandwise modeling: Each band uses an independent MoG with band-specific parameters, allowing different noise distributions across bands.MoG is selected for its universal approximation capability for continuous densities.
  • Graphical model: The graphical model identifies observed HSI elements, low-rank factors, mixture parameters, and latent component assignments within the NMoG-LRMF framework.The red-box portion of the graphical model represents the noise encoding component.
  • Cross-band correlation: The Gaussian-mixture parameters are generated from a two-level prior distribution so correlations among noise distributions across bands are encoded.The latent variable representation connects mixture components to the bandwise noise model.

B. NMoG-LRMF model

NMoG-LRMF represents the clean HSI as a low-rank factorization and uses Bayesian priors to learn its effective rank and infer model variables.

  • Low-rank representation: The clean HSI matrix is represented as L = UV^T, encoding an underlying low-rank structure.
  • Adaptive rank: The model can adaptively estimate rank by starting with a preset upper bound beyond the true rank.
  • Adaptive rank: Columns of U and V share precision variables that impose a common sparsity profile and can automatically suppress unnecessary factor pairs.Large precision values for some variables enable automatic low-rank estimation.
  • Bayesian model: Combining the low-rank and NMoG components yields a Bayesian model whose posterior includes factor matrices, mixture parameters, assignments, and precision variables.

C. Variational Inference

The method uses variational Bayes to approximate the posterior of the NMoG-LRMF model through a factorized variational distribution and tractable KL minimization. Its inference procedure recursively updates noise, low-rank, and hyperparameter distributions, while automatically pruning components to infer rank.

  • Variational Bayes formulation: Variational Bayes approximates the true posterior p(θ|D) with a variational distribution q(θ) by minimizing KL divergence over a tractable constraint set.The approximation assumes q(θ) can be factorized into distributions over disjoint parameter groups.
  • Posterior updates: The inference procedure analytically updates factorized posterior distributions for noise variables, low-rank components, and model parameters.Updates cover latent variables, mixture proportions, noise means and precisions, low-rank factors U and V, and rank-control parameters γ.
  • Rank determination: Large inferred γl values remove corresponding rows from U and V, automatically determining the rank of the reconstructed matrix.This pruning mechanism implements the low-rank objective without fixing the rank beforehand.
  • Algorithm: The algorithm iteratively updates noise posteriors, low-rank factors, and γ until convergence, producing optimized factors Uopt and Vopt.The listed update sequence alternates between noise components, low-rank components, and model parameters.
  • Hyperparameter setting: All hyperparameters are initialized non-informatively, with m0=0 and the remaining listed hyperparameters set to 10^-3.The authors report stable performance under these settings.

III. EXPERIMENTAL RESULTS

The experiments evaluate NMoG-LRMF on synthetic and real HSI data against low-rank and HSI-prior denoising methods. Comparisons cover methods modeling deterministic, sparse, Gaussian, mixture, and other i.i.d. noise assumptions, alongside state-of-the-art baselines.

  • Experimental setup: Experiments evaluate NMoG-LRMF on both synthetic and real hyperspectral image data.The evaluation compares the proposed method with multiple low-rank matrix analysis and HSI-prior denoising methods.
  • Compared methods: Compared low-rank methods include LRMR, LRTV, PMoEP, and MoG-R, representing deterministic, sparse, mixture, and other i.i.d. noise assumptions.These baselines cover different noise-distribution models within low-rank matrix analysis.
  • Compared methods: TDL and BM4D are also evaluated as state-of-the-art HSI denoising methods that incorporate HSI priors.The experiments were implemented in Matlab R2014b on a 4.0GHz CPU with 31.4GB RAM.

A. Simulated HSI denoising experiments

Synthetic experiments evaluate NMoG-LRMF on two hyperspectral images under six simulated noise settings, using MPSNR, MSSIM, and runtime. The method is strongest on complex non-i.i.d. noise, while TDL and BM4D perform better in the i.i.d. Gaussian case and NMoG-LRMF has comparable computational order to most competitors.

  • Experimental setup: Two HSIs, DCmall and RemoteImage, were cropped and resized to 200 × 200 × 160 and 200 × 200 × 89, respectively, for synthetic-noise evaluation.The experiments normalize each band’s gray values into [0, 1].
  • Experimental setup: Six settings simulate i.i.d. Gaussian, non-i.i.d. Gaussian, Gaussian plus stripe, deadline, or impulse noise, and randomly mixed combinations.The mixed case assigns each band at least one noise type from the preceding non-i.i.d. and structured-noise cases.
  • Evaluation protocol: Performance is measured by mean PSNR, mean SSIM, and denoising time across the hyperspectral bands.All competing methods use 20 random initializations per noise case, with average results reported.
  • Quantitative comparison: NMoG-LRMF is generally superior in the tabulated results, except under i.i.d. Gaussian noise, where methods using richer HSI priors such as TDL and BM4D perform relatively better.The advantage is reported as especially evident for more complex, practical non-i.i.d. noise cases.
  • Computational cost: The proposed method has almost the same computational order of magnitude as most competitors while fitting a wider range of noise types.SVD is identified as an exception because its Matlab implementation is highly efficient.
  • Band-wise comparison: Across almost all bands in complex noise cases, NMoG-LRMF achieves the best PSNR and SSIM, whereas TDL leads under i.i.d. Gaussian noise.The band-wise results are presented for DCmall and RemoteImage in Figs. 3 and 4.
  • Visual comparison: NMoG-LRMF more effectively removes complex mixed noise and preserves edges and textures, while most competing restorations remain noisy or blurry.This advantage is reported both visually and quantitatively in representative DCmall and RemoteImage bands.

B. Real HSI denoising experiments

Real HSI experiments evaluate NMoG-LRMF on Urban and EO-1 Hyperion data containing complex noise. Across visual and profile comparisons, the method preserves details, removes noise, and produces smoother restorations than competing methods.

  • Experimental setting: The experiments evaluate NMoG-LRMF on Urban and EO-1 Hyperion hyperspectral datasets using comparisons with multiple denoising methods.The Hyperion evaluation uses a cropped 256×256×198 subset after removing zero-signature bands.
  • Urban HSI: NMoG-LRMF better restores texture and edge details while preserving less structural noise than competing methods on Urban bands 103, 139, and 207.Competing LRMF methods with i.i.d. noise assumptions do not finely restore the clean HSI under structural stripe noise.
  • Urban HSI: NMoG-LRMF produces evidently smoother horizontal mean profiles for Urban band 207 after mixed-noise restoration.The profile plots use row number on the horizontal axis and mean digital number value for each row on the vertical axis.
  • EO-1 Hyperion: On EO-1 Hyperion bands 100, 144, and 197, MoG-RPCA and PMoEP outperform several earlier methods through better noise fitting, while competing restorations still miss details.The referenced comparisons include SVD, RegL1ALM, CWM, LRMR, LRTV, TDL, BM4D, and NMoG.

C. Effect of component number on denoising performance

The component-number study tests NMoG-LRMF with K from 1 to 10 under two noise settings. Denoising performance becomes stable beyond K=2, supporting the use of K=3 in the real experiments.

  • Sensitivity analysis: The sensitivity study varies Gaussian component number K from 1 to 10 using 20 initializations on DC Mall data under two noise cases.The evaluated cases are Gaussian + impulse noise and mixture noise, with MPSNR and MSSIM reported in Fig. 15.
  • Sensitivity analysis: After K exceeds 2, NMoG-LRMF denoising performance tends to remain stable and is not very sensitive to K.The stability is examined through MPSNR and MSSIM tendency curves.
  • Practical setting: Setting K=3 throughout the real experiments yields consistently good performance.The paper reports this setting as a simple choice across the real-data evaluations.

IV. CONCLUSION

The paper proposes Bayesian NMoG-LRMF denoising for non-i.i.d. HSI noise and evaluates its robustness on synthetic and real noisy HSIs. It also identifies extending the method and incorporating additional priors as future directions.

  • Conclusion: The paper proposes a non-i.i.d. noise modeling strategy embedded in low-rank matrix factorization under a Bayesian framework.A variational Bayes algorithm infers the posterior of the proposed model.
  • Conclusion: Experiments on synthetic and real noisy HSIs substantiate that the proposed method performs more robustly than current state-of-the-art techniques.The stated robustness is attributed to adapting to various noise shapes encountered in applications.
  • Future work: Future work includes extending the method to video and face image data and integrating additional HSI prior terms.The paper also notes that the noise modeling strategy could be redesigned for specific application contexts.
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