Arie Rond, M.Sc. Thesis Seminar
The Anscombe transform offers an approximate conversion of a Poisson random variable into unit-variance 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. This process is known to work well for high SNR images, where the Anscombe transform provides a rather accurate transformation, with 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 vastly available Gaussian-solvers.
In this work we suggest a novel method for coupling Gaussian solvers to Poisson noisy inverse problems, which is based on a general approach termed "Plug-and-Play Prior". Deploying the Plug-and-Play Prior 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 based path, enables other developers to plug their own Gaussian solvers to our 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 on Poisson image denoising and deblurring.