Technical Report MSC-2016-14

Title: Handling Poisson Inverse Problems by the Plug-and-Play Priors scheme
Authors: Arie Rond
Supervisors: Michael Elad
PDFCurrently accessibly only within the Technion network
Abstract: A common problem in the process of acquiring a signal is its corruption by noise. When dealing with photon counting, which is common when acquiring certain weak signals, such a noise can be modeled by the Poisson distribution, where each measurement is a Poisson random variable, with its mean equal to the clean signal value. Restoration of the clean signal out of its noisy measurement is essential in various fields, and thus effective and efficient algorithms are required for this task. In the case of high signal-to-noise-ratio (SNR), where the noise level is low, the Poisson statistics resemble the Gaussian one. In such cases it is possible to approximately transform the Poisson contaminated signal into a Gaussian additive noise form, with a variance independent of the mean. This is done by a variance stabilizing transform (VST), such as the well known and widely used Anscombe transform. This transform is important and appealing, as it is easy to compute, and becomes handy in various inverse problems with Poisson noise contamination. The 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 attractiveness of this approach is due to the abundance of high-performance restoration algorithms designed for white additive Gaussian noise. This process is known to work well for high SNR images, where the Anscombe transform provides a nearly constant variance. 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 the vastly available solvers for Gaussian noise. In this work we suggest a novel method for coupling Gaussian denoising algorithms to Poisson noisy inverse problems. Our proposed method is based on a recently proposed approach termed "Plug-and-Play Priors". Deploying the Plug-and-Play Prior approach to such problems leads to an iterative scheme that repeats several key steps: (i) A convex programing task of simple form that can be easily treated; (ii) A powerful Gaussian denoising algorithm of choice; and (iii) A simple update step. Such a modular method, just like the Anscombe transform based path, enables other developers to plug their own Gaussian denoising algorithms to this scheme in an easy way. While the proposed method bears some similarity to the Anscombe operation approach, it is in fact based on a different mathematical basis, which holds true for all SNR ranges. We demonstrate the effectiveness of the proposed scheme for Poisson image denoising and deblurring, showing that in both applications we outperform existing Anscombe transform- based methods in high noise regimes and compete favorably with leading direct method.

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