Anscombe transform

Abstract: Many digital imaging devices operate by successive photon-to-electron, electron-to-voltage, and voltage-to-digit conversions. These processes are subject to various signal-dependent errors, which are typically modeled as Poisson-Gaussian noise. The removal of such noise can be effected indirectly by applying a variance-stabilizing transformation (VST) to the noisy data, denoising the stabilized data with a Gaussian denoising algorithm, and finally applying an inverse VST to the denoised data. The generalized Anscombe transformation (GAT) is often used for variance stabilization, but its unbiased inverse transformation has not been rigorously studied in the past. We introduce the exact unbiased inverse of the GAT and show that it plays an integral part in ensuring accurate denoising results. We demonstrate that this exact inverse leads to state-of-the-art results without any notable increase in the computational complexity compared to the other inverses. We also show that this inverse is optimal in the sense that it can be interpreted as a maximum likelihood inverse. Moreover, we thoroughly analyze the behavior of the proposed inverse, which also enables us to derive a closed-form approximation for it. This paper generalizes our work on the exact unbiased inverse of the Anscombe transformation, which we have presented earlier for the removal of pure Poisson noise.

Abstract: We denoise Poisson images with an iterative algorithm that progressively improves the effectiveness of variance-stabilizing transformations (VST) for Gaussian denoising filters. At each iteration, a combination of the Poisson observations with the denoised estimate from the previous iteration is treated as scaled Poisson data and filtered through a VST scheme. Due to the slight mismatch between a true scaled Poisson distribution and this combination, a special exact unbiased inverse is designed. We present an implementation of this approach based on the BM3D Gaussian denoising filter. With a computational cost at worst twice that of the noniterative scheme, the proposed algorithm provides significantly better quality, particularly at low signal-to-noise ratio, outperforming much costlier state-of-the-art alternatives.

Abstract: 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.

Abstract: Fluorescence microscopy images are contaminated by photon and readout noises, and hence can be described by mixed-Poisson-Gaussian (MPG) processes In this paper, a new variance stabilizing transform (VST) is designed to convert a filtered MPG process into a near Gaussian process with a constant variance This VST is then combined with the isotropic undecimated wavelet transform leading to a multiscale VST (MS-VST) We demonstrate the usefulness of MS-VST for image denoising and spot detection in fluorescence microscopy In the first case, we detect significant Gaussianized wavelet coefficients under the control of a false discovery rate A sparsity-driven iterative scheme is proposed to properly reconstruct the final estimate In the second case, we show that the MS-VST can also lead to a fluorescent-spot detector, where the false positive rate of the detection in pure noise can be controlled Experiments show that the MS-VST approach outperforms the generalized Anscombe transform in denoising, and that the detection scheme allows efficient spot extraction from complex background