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Sparsity Based Poisson Denoising with Dictionary Learning

Raja Giryes, Michael Elad

arXiv:1309.4306v3cs.CVstat.ML

TL;DR

Poisson denoising is challenging when low photon counts make Gaussianizing transformations less accurate. This paper combines a Poisson sparse patch model with greedy pursuit and dictionary learning; its evaluation reports a 2.63 dB average PSNR gap between simple sparse coding and advanced dictionary learning with reclustering.

  • Problem

    At low peak values, Gaussianizing transformations are less accurate, motivating Poisson denoising methods that rely directly on the noise statistics.

    Source: §I · p. 1

  • Method

    The method combines a Poisson likelihood with exponential sparse dictionary representations of patches and uses pursuit with a bootstrapped stopping criterion.

    Source: §II · p. 3, B. Sparse Coding · p. 5

  • Results

    2.63 dB average PSNR separates simple sparse coding (Setup I) from advanced dictionary learning with reclustering (Setup V).

    Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12

  • Takeaways & Limitations

    Dictionary learning accounts for the main reported recovery improvement: five learning steps yield an average 2.43 dB improvement, with reclustering and five more steps adding 0.2 dB.

    Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12

Abstract

from arXiv · show

The problem of Poisson denoising appears in various imaging applications, such as low-light photography, medical imaging and microscopy. In cases of high SNR, several transformations exist so as to convert the Poisson noise into an additive i.i.d. Gaussian noise, for which many effective algorithms are available. However, in a low SNR regime, these transformations are significantly less accurate, and a strategy that relies directly on the true noise statistics is required. A recent work by Salmon et al. took this route, proposing a patch-based exponential image representation model based on GMM (Gaussian mixture model), leading to state-of-the-art results. In this paper, we propose to harness sparse-representation modeling to the image patches, adopting the same exponential idea. Our scheme uses a greedy pursuit with boot-strapping based stopping condition and dictionary learning within the denoising process. The reconstruction performance of the proposed scheme is competitive with leading methods in high SNR, and achieving state-of-the-art results in cases of low SNR.

I. INTRODUCTION

Poisson denoising estimates an image from independent observations whose mean and variance equal the underlying intensity. Gaussianizing transformations work less accurately at low peak values, motivating methods that use Poisson statistics directly.

I. INTRODUCTION

  • Poisson noise is signal-dependent: lower-intensity pixels have stronger noise, and the image peak is used as a measure of noise power.The peak is a good measure when pixel values span the dynamic range, an assumption stated to hold for most natural images.Source: §I · p. 1
  • Anscombe and Fisz transformations make Poisson noise approximately white Gaussian with unit variance, allowing the use of Gaussian-denoising methods.Denoising transformed data estimates the transformed image; an inverse transform is needed to estimate the original.Source: §I · pp. 1–2
  • For peak values below 3, Gaussianizing transformations are only approximate; the paper targets denoising images with peaks up to 4, where Anscombe and Fisz are less effective.The threshold is a rule of thumb, not a fixed boundary.Source: §I · p. 1
  • When peak value is smaller than 3, noisy-image structure differs from Gaussian, with many zero pixels and others having very small integer values; at peak 0.1, it is almost binary, mainly zeros or ones.An exact unbiased inverse for the Anscombe transform eventually leads to better recovery performance.Source: §I · pp. 1–2

A. Our Contribution

The paper replaces GMM-based patch modeling with sparse representations and dictionary learning for Poisson denoising. Its contributions include a greedy pursuit, a bootstrapped stopping criterion, and reported state-of-the-art performance, especially at very low SNR.

A. Our Contribution

  • The method models image patches with sparse representations and dictionary learning instead of the GMM approach, using an OMP-like greedy pursuit.The paper uses an image-patch modeling path similar to and a dictionary-learning strategy.Source: A. Our Contribution · p. 2
  • The authors identify Poisson deconvolution and inpainting as possible extensions of their greedy technique.They relate these possibilities to extensions of other greedy methods in, and.Source: A. Our Contribution · p. 2
  • A bootstrapped stopping criterion is introduced for iterative pursuit; the paper reports that incorporating it leads to much improved results.The authors state that, to their knowledge, this criterion first appears in their paper.Source: A. Our Contribution · p. 2
  • The integrated scheme reports state-of-the-art recovery performance, particularly at very low SNR.The conclusion says it outperforms existing algorithms by more than 1 dB in some scenarios.Source: A. Our Contribution · p. 2

B. Organization

The paper proceeds from the denoising problem and prior work to the proposed algorithm, its evaluation, and conclusions.

B. Organization

  • Section II describes the Poisson denoising problem and previous contributions.Section III then introduces the proposed denoising algorithm.Source: B. Organization · p. 2
  • Section III develops the pursuit, clustering for non-locality, dictionary learning, and overall scheme.These are the components used to present the proposed denoising algorithm.Source: B. Organization · p. 2
  • Section IV presents tests and comparisons of denoising performance; Section V discusses future work and conclusions.Source: B. Organization · p. 2

II. POISSON SPARSITY MODEL

The model combines a Poisson observation likelihood with sparse dictionary representations of image patches. An exponential parameterization removes the non-negativity constraint on recovered patches, while the paper contrasts narrow group dictionaries with its global-dictionary approach.

II. POISSON SPARSITY MODEL

  • A sparse patch model represents each patch as a dictionary multiplied by a k-sparse coefficient vector, with k much smaller than the patch dimension.The paper notes that this sparsity-inspired model is commonly used for Gaussian denoising.Source: §II · p. 2
  • The Poisson likelihood objective alone is insufficient because its minimizer is the noisy patch itself, so the model adds a sparse prior.The likelihood objective permits zero recovered entries only where the corresponding noisy entries are also zero.Source: §II · p. 3
  • Setting the recovered patch to exp(Dα) makes it nonnegative while retaining a k-sparse representation.The exponential is applied element-wise.Source: §II · p. 3
  • NLPCA clusters noisy patches into disjoint groups and gives each group a narrow dictionary; NLSPCA uses SPIRAL to calculate patch representations.The cited approaches adapt sparse modeling to Poisson statistics rather than relying on a Gaussianizing transform at low photon counts,,,.Source: §II · pp. 2–3
  • The proposed scheme uses one global dictionary, greedily finds patch representations, and forces similar patches to share atoms for joint processing.Dictionary learning improves the initial dictionary; the authors also describe binning as a way to raise SNR, especially when the peak is below 0.5.Source: §II · p. 3

III. SPARSE POISSON DENOISING ALGORITHM (SPDA)

SPDA models image patches with an exponential sparse representation and alternates greedy sparse coding with dictionary learning. It groups similar patches and averages their recovered values to reconstruct the image.

III. SPARSE POISSON DENOISING ALGORITHM (SPDA)

  • SPDA extracts overlapping noisy-image patches, groups them, applies Poisson greedy coding, and averages the recovered patches into an image estimate.Dictionary learning and repeated coding rounds refine the estimate.Source: §III · pp. 3–4
  • The method alternates dictionary and representation updates because their joint minimization is difficult.The pursuit returns k-sparse representations for each patch.Source: §III · pp. 3–4
  • Dictionary learning updates the dictionary and representations while keeping their supports fixed, then the updated dictionary is reused in subsequent rounds.The algorithm also reclusters using the current estimate and dictionary.Source: §III · p. 4

A. Patch Clustering

Patch clustering creates disjoint groups of similar patches for joint sparse coding. Its greedy procedure balances group size and count, with a filtering-based distance that emphasizes rare high-intensity pixels.

A. Patch Clustering

  • Patch distances use Euclidean distance after Gaussian filtering, giving greater weight to rare high-valued pixels associated with high-intensity locations.The authors describe this as an approximation to earth mover’s distance for very-low-SNR, nearly binary images.Source: A. Patch Clustering · p. 4
  • Groups are built sequentially around a pivot, with patches admitted by distance and a target minimum group size.The method seeks enough atoms per group for support selection while retaining many groups for dictionary updates; groups are disjoint for computational reasons.Source: A. Patch Clustering · p. 4
  • The greedy clustering method is knowingly suboptimal, but the authors report it is faster and sufficiently good-performing.Its speed is the stated reason for preferring it over off-the-shelf methods.Source: A. Patch Clustering · p. 5

B. Sparse Coding

Sparse coding chooses representations jointly within each patch group, enforcing a shared support. Bootstrapping uses prior reconstructions as proxies for the true patches to adapt stopping, and the authors report improved recovery.

B. Sparse Coding

  • The greedy pursuit adds the atom that most reduces the group cost, enforcing the same sparsity pattern across all patches in that group.With support fixed, the group minimization is convex.Source: B. Sparse Coding · p. 5
  • Bootstrapping stopping improves recovery significantly by assigning each patch group a representation cardinality better suited to its content.Averaging and dictionary updates between iterations prevent the procedure from returning the same patch.Source: B. Sparse Coding · p. 5
  • After each iteration, reconstructed patches serve as proxies for true patches; coding stops when their distance begins increasing and returns the previous iteration’s representation.This stopping rule is used only when proxy patches are available.Source: B. Sparse Coding · p. 5
  • The first coding pass uses a fixed cardinality because no reconstruction-based proxy is yet available; later passes use patches from the preceding estimate.The authors use ℓ2 error on the denoised image rather than a Poissonian divergence.Source: B. Sparse Coding · p. 5

C. Dictionary Learning

Dictionary learning updates both dictionary atoms and patch representations while holding their supports fixed. The approach connects to GMM learning, while its exponential model requires peak-specific initialization dictionaries.

C. Dictionary Learning

  • The advanced learning strategy updates the dictionary and representations jointly while keeping their supports fixed.Unused atoms are removed when no representation provides information for updating them.Source: C. Dictionary Learning · p. 6
  • The authors view this learning strategy as a GMM generalization in which groups can share atoms, giving the scheme more flexibility.Small groups and sparse coding after each learning step contribute to that flexibility.Source: C. Dictionary Learning · p. 6
  • Separate initialization dictionaries are trained offline by running the denoising algorithm on clean images scaled to different peak values.The exponential sparsity representation is scale-sensitive, and no intuitively known dictionary is available for it.Source: C. Dictionary Learning · p. 6

D. SPDA Summary

SPDA alternates pursuit and dictionary learning to estimate image patches, then reconstructs the image by averaging patches at their corresponding locations.

  • Pursuit and dictionary-learning stages are iterated to estimate the image patches.Source: D. SPDA Summary · p. 6
  • The whole image is recovered by reprojection of each patch to its corresponding location, followed by averaging.Source: D. SPDA Summary · p. 6
  • After reconstruction, patches may be re-clustered using the recovered image and the full process repeated.Source: D. SPDA Summary · p. 6

E. Comparison to NLPCA and NLSPCA

SPDA differs from NLPCA and NLSPCA in its joint-sparsity grouping, shared dictionary, and greedy pursuit. Its sparse coding is more computationally demanding, though parallelization and dictionary shrinkage can reduce running time.

E. Comparison to NLPCA and NLSPCA

  • SPDA uses joint sparsity to form many disjoint small groups and one shared dictionary, unlike NLPCA and NLSPCA’s few large clusters and cluster-specific bases.SPDA’s dictionary can change size during learning, and patch cardinalities are dynamic.Source: E. Comparison to NLPCA and NLSPCA · p. 7
  • SPDA uses a new greedy pursuit for decoding, whereas NLPCA uses a Newton step and NLSPCA uses SPIRAL.Source: E. Comparison to NLPCA and NLSPCA · p. 7
  • SPDA’s overall complexity is O(Nk^2d^2), compared with NLPCA’s O(Nkd^2); sparse coding is SPDA’s bottleneck.For SPDA, the stated worst-case complexity is O(k^2d^2) per patch.Source: E. Comparison to NLPCA and NLSPCA · p. 7
  • SPDA’s higher computational cost can be reduced through parallelization, and sparse-coding complexity is likely to decrease as its dictionary shrinks.Source: E. Comparison to NLPCA and NLSPCA · p. 7

IV. EXPERIMENTS

The experiments compare SPDA with NLSPCA and Anscombe-based BM3D across images and peak values, then examine binning and algorithm stages. Results show performance depends on peak intensity, with dictionary learning providing the main recovery improvement.

IV. EXPERIMENTS

  • SPDA performs better on average for 5 of 6 tested peak values; its only shortfall is an insignificant 0.02 dB at peak 1.The comparison reports recovery error in PSNR across the tested images and peak values.Source: §IV · p. 8, Table II · p. 11
  • Binning improves performance at lower peaks, while SPDA without binning is preferred at larger peaks; low-peak binning better captures image structure.For the flag at peak 1, SPDA recovers the image accurately and learned atoms capture stars and lines; across the examples, SPDA recovers details better.Source: §IV · pp. 8, 11, Figure 5 · p. 9, Figure 7 · p. 9
  • At peaks 0.1 and 0.2, NLSPCA has better PSNR than SPDA, but binning reverses the comparison; larger SPDA patches might help yet are computationally infeasible.Binning compensates for this computational barrier by processing a low-resolution image with the same patch size.Source: §IV · p. 11

AFTER RECLUSTERING IS USED; (IV) FIVE ADVANCED DICTIONARY

  • Advanced dictionary learning with reclustering improves average PSNR by 2.63 dB over simple sparse coding, with dictionary learning accounting for most of the gain.Five dictionary-learning steps yield 2.43 dB on average, and reclustering plus five additional steps adds 0.2 dB.Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12
  • The bootstrapping stopping criterion improves recovery by 0.3 dB without binning but has a negligible effect with binning.Binning processes coarser-resolution images, making recovery less sensitive to the number of patches used.Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12
  • Advanced learning steps run four times faster than simple steps, while showing peak-specific gains over simple dictionary learning.The reported gains are 0.23 dB for SPDAbin at peaks 0.1, 0.2, and 0.5, and 0.18 dB for SPDA at peaks 1, 2, and 4.Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12

(ANS+G). THE AVERAGE IN THE RESULTS IS OVER FIVE NOISE

  • SPDA outperforms its Anscombe-based Gaussian version at every tested peak except 0.1, where binned SPDA is preferable.The Gaussian version performs poorly with binning because fewer patches are available for dictionary updates; the Poisson-objective version does not share this behavior.Source: (ANS+G). THE AVERAGE IN THE RESULTS IS… · p. 12

V. DISCUSSION AND CONCLUSION

The paper presents SPDA, a Poisson denoising method combining sparse coding and dictionary learning, with state-of-the-art recovery performance and gains exceeding 1 dB in some scenarios. The authors identify shared dictionary initialization and fixed learning-iteration counts as areas for improvement.

  • SPDA achieves state-of-the-art recovery performance and outperforms existing algorithms by more than 1 dB in some scenarios.Source: §V · p. 12
  • Using the same initialization dictionary for all image types leaves room for improvement; the authors suggest task-specific offline dictionary training when image types are known.They cite space images and fluorescence microscopy cell images as examples.Source: §V · pp. 12–13
  • A fixed number of dictionary-learning iterations can reduce output quality, while manual tuning from visual feedback is feasible in some applications.The paper also suggests automated tuning as a direction to consider,.Source: §V · p. 13
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