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Poisson Inverse Problems by the Plug-and-Play scheme

Arie Rond, Raja Giryes, Michael Elad

arXiv:1511.02500v1cs.CVmath.OC

TL;DR

At lower signal peaks, variance-stabilizing transforms become less efficient, while direct Poisson treatment leaves Gaussian restoration algorithms less available. The paper introduces Plug-and-Play Poisson restoration, coupling a Poisson inverse-problem update with Gaussian denoising. It outperforms the Anscombe-transform framework at lower peaks and competes favorably with it in other cases.

  • Problem

    At lower image peaks, variance-stabilizing transforms become less efficient, motivating direct Poisson treatment while Gaussian restoration algorithms remain useful.

    Source: §1 · p. 2

  • Method

    The method uses Plug-and-Play to integrate Gaussian denoisers as implicit priors into Poisson inverse problems, including denoising and deblurring.

    Source: §2.1 · p. 3, §3.1 · p. 4, §5 · p. 13

  • Results

    P4IP outperforms the Anscombe-transform framework at lower peaks and competes favorably with it in other cases.

    Source: §5 · p. 13

  • Takeaways & Limitations

    The approach offers a black-box route for using existing Gaussian denoisers in Poisson denoising and deblurring.

    Source: §5 · p. 13

    The extension is convex only if l(x) is convex, and separable only if l(x) is separable.

    Source: §2.2 · p. 3

Abstract

from arXiv · show

The Anscombe transform offers an approximate conversion of a Poisson random variable into a Gaussian one. This transform is important and appealing, as it is easy to compute, and becomes handy in various inverse problems with Poisson noise contamination. Solution to such problems can be done by first applying the Anscombe transform, then applying a Gaussian-noise-oriented restoration algorithm of choice, and finally applying an inverse Anscombe transform. The appeal in this approach is due to the abundance of high-performance restoration algorithms designed for white additive Gaussian noise (we will refer to these hereafter as "Gaussian-solvers"). This process is known to work well for high SNR images, where the Anscombe transform provides a rather accurate approximation. When the noise level is high, the above path loses much of its effectiveness, and the common practice is to replace it with a direct treatment of the Poisson distribution. Naturally, with this we lose the ability to leverage on vastly available Gaussian-solvers. In this work we suggest a novel method for coupling Gaussian denoising algorithms to Poisson noisy inverse problems, which is based on a general approach termed "Plug-and-Play". Deploying the Plug-and-Play approach to such problems leads to an iterative scheme that repeats several key steps: 1) A convex programming task of simple form that can be easily treated; 2) A powerful Gaussian denoising algorithm of choice; and 3) A simple update step. Such a modular method, just like the Anscombe transform, enables other developers to plug their own Gaussian denoising algorithms to our scheme in an easy way. While the proposed method bares some similarity to the Anscombe operation, it is in fact based on a different mathematical basis, which holds true for all SNR ranges.

1 Introduction

The paper frames Poisson inverse problems as recovering a clean image from degraded measurements, with Poisson pixel noise especially relevant when measured signal intensities are weak. It motivates a method that bridges direct Poisson treatment and Gaussian denoising, since variance-stabilizing transforms become less effective at lower peaks.

1 Introduction

  • Poisson inverse reconstruction seeks a clean image from degraded measurements, modeled using a forward process and a clean-image prior.The paper studies Poisson contamination, in which image gray levels are modeled as Poisson random variables.Source: §1 · p. 1
  • In the Poisson model, each clean pixel value is both the mean and variance of its noisy measurement, and the setting includes tasks with weak measured signal intensity.Examples include low-light imaging, CT reconstruction, fluorescence microscopy, astrophysics, and spectral imaging.Source: §1 · p. 1
  • At high peak levels, variance-stabilizing transforms can convert Poisson noise approximately to Gaussian noise, enabling Gaussian restoration followed by inverse transformation.The paper cites effective Gaussian restoration methods and VST-based approaches,,,.Source: §1 · pp. 1–2
  • At lower peaks, these transforms become less efficient, motivating direct Poisson methods; the proposed approach bridges the families by applying Gaussian denoising iteratively.Source: §1 · p. 2

2 The Plug-and-Play (PaP) Approach

Plug-and-Play separates the inverse problem from its image prior, allowing a Gaussian denoiser to supply the prior without an explicit penalty formulation. The paper extends this scheme to multiple denoisers and describes when the resulting optimization remains convex or separable.

2.1 Standard Plug-and-Play

  • Plug-and-Play uses ADMM to separate the inverse problem from the prior, which can be supplied by a Gaussian denoiser without explicitly formulating a penalty.The denoiser is activated indirectly as the prior component of the scheme.Source: §2.1 · pp. 2–3
  • The standard scheme targets a MAP estimate through alternating optimization steps, including a Gaussian denoising step for the prior.For Poisson denoising, the first step is convex and separable, yielding a scalar formula resembling the Anscombe transform.Source: §2.1 · pp. 2–3

2.2 Plug-and-Play Extension

  • The extension introduces separate auxiliary variables and two Gaussian denoising steps, each using a different prior.Source: §2.2 · p. 3
  • The extended expression is convex when l(x) is convex and separable when l(x) is separable.Source: §2.2 · p. 3
  • The scheme can generalize to multiple priors that address different image features, which may complement one another and yield an overall benefit.Examples include self-similarity, local smoothness, structure, and scale-invariance.Source: §2.2 · p. 3

3 P4IP Algorithm

P4IP integrates the Poisson log-likelihood with Plug-and-Play, pairing a convex optimization step with a Gaussian denoiser. Its denoising and deblurring variants retain this structure while adding problem-specific updates and improvements.

3 P4IP Algorithm

  • P4IP inserts the Poisson log-likelihood into the Plug-and-Play formulation, yielding a convex first task that can be efficiently solved and extended to use several denoisers.Source: §3 · p. 3, §3.1 · p. 4

3.1.1 Poisson Denoising

  • For H = I, the first ADMM step is separable and has a closed-form per-pixel update resembling the Anscombe transform.Source: §3.1.1 · p. 4
  • At λ = 0.25, x has the same variance as under Anscombe’s transform because they differ only by a constant; PaP’s curve changes across iterations and uses no inverse transform after Gaussian denoising.Source: §3.1.1 · p. 4

3.1.2 Poisson Deblurring

  • For deblurring, the first ADMM step is generally nonseparable and lacks an analytical solution; its convex objective has a computable gradient, and L-BFGS uses a surrogate to avoid negative Hx values.Source: §3.1.2 · p. 5, §3.2 · p. 6

3.2 Details and Improvements

  • β and λ0 correlate strongly with the noise peak, whereas λstep depends on it weakly; poor β choices can degrade results by several dB.Source: §3.2 · p. 5
  • Binning merges pixels to raise SNR and reduce runtime, but applies only to denoising; parameter choices also matter, while averaging runs with nearby settings improves results at added runtime cost.Source: §3.2 · pp. 5–6
  • The denoising variant M-P4IP combines BM3D with a simplified multiscale denoiser as a multiple-prior approach.Source: §3.2 · p. 5

4 Experiments

The experiments cover Poisson denoising and deblurring; the reported comparisons favor P4IP over the Anscombe-transform framework in the tested deblurring scenario and describe favorable denoising performance.

4.1 Denoising

  • The denoising evaluation tested peak values 0.1, 0.2, 0.5, 1, 2, and 4, with and without binning.Comparators included BM3D with the refined inverse Anscombe transform and.Source: §4.1 · p. 6
  • For denoising, P4IP competes favorably with BM3D+Anscombe and state-of-the-art algorithms, while binning benefits all algorithms at low peak values.With binning, P4IP took roughly 30 seconds per image, versus 0.5 seconds for BM3D+Anscombe and 15–20 minutes for SPDA; without binning, P4IP took a few minutes and SPDA about 10 hours.Source: §4.1 · p. 6
  • At peak 0.2, the tested multiple-prior algorithm gained 0.2 dB, although finding good parameters was harder.The test combined BM3D with a simplified version of.Source: §4.1 · pp. 6–7

4.2 Deblurring

  • The deblurring evaluation tested peak values 1, 2, and 4 across three blur kernels and compared P4IP with the Anscombe-transform framework.The specified kernel (i) was Gaussian, size 25 by 25, with σ = 1.6.Source: §4.2 · p. 10
  • P4IP outperforms the Anscombe-transform framework in the tested deblurring scenario.The reported runtime was about 5 minutes per image, about twice slower than Anscombe, over 44 iterations.Source: §4.2 · pp. 10–11

5 Conclusion and discussion

The paper proposes using Plug-and-Play to integrate Gaussian denoisers into Poisson inverse problems and demonstrates it on denoising and deblurring. It reports better performance than the Anscombe-transform framework at lower peaks and competitive performance in other cases.

5 Conclusion and discussion

  • P4IP integrates Gaussian denoising algorithms into Poisson inverse problems through Plug-and-Play, using the denoiser as a black box.The approach is demonstrated for image denoising and deblurring; reported results outperform the Anscombe-transform framework at lower peaks and compete favorably in other cases.Source: §5 · p. 13

Appendix A Derivation of first denoising ADMM step

For Poisson denoising, the appendix derives the first ADMM update from the denoising loss. The update is obtained by differentiating the Lagrangian, and the selected root yields nonnegative values when the observations are nonnegative.

Appendix A Derivation of first denoising ADMM step

  • For denoising, the loss is −y^T ln(x) + 1^T ln(Γ(y + 1)) + 1^T x.This expression is given for H = I.Source: Appendix A · p. 13
  • The first ADMM x update is convex and separable, and its entries have a closed-form solution.The appendix obtains the update by differentiating the Lagrangian with respect to each x[i].Source: Appendix A · p. 14
  • When y is nonnegative, the selected root gives nonnegative x values; the other root is purely negative and uninformative.Source: Appendix A · p. 14
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