Optimal Inversion of the Generalized Anscombe Transformation for Poisson-Gaussian Noise

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.

Impact of detector-element active-area shape and fill factor on super-resolution

“…In the case of modest affine motion and typical PSF parameters, this model holds in an approximate sense, as analyzed in Hardie et al [10]. The noise in Figure 1 is assumed to be Poisson-Gaussian noise, with both a signal-dependent and signal-independent component [17,18]. Incorporating the noise gives rise to the observed pixels, denoted…”

“…Consider a Poisson-Gaussian noise model that accounts for the photon arrival distribution as well as noise in the electronics [17,18]. Applying this model, the observed data are given by…”

“…For high mean values, a Poisson distribution is known to be well approximated by a Gaussian. In this case, we can approximate g as a heteroskedastic Gaussian random vector, such that g ∼ N(f, G) [17,18]. This is equivalent to an additive Gaussian noise with a signal dependent variance, given by g = f + n, where n ∼ N(0, G).…”

A Dictionary Learning Approach for Poisson Image Deblurring

“…Major contributions fall in the following three categories: (a) simply recover the image using a method designed for Gaussian noise removal; (b) transform Poisson noise into near-Gaussian noise by applying an appropriate transform to the noisy image (called a variance-stabilizing transform, VST), then process the transformed image with an algorithm designed for Gaussian noise removal, and finally apply to it the inverse transform in order to get the result [1], [24], [25]; (c) remove Poisson noise directly via a data-fidelity term derived from Poisson noise statistics [14], [26]- [28]. The present contribution belongs to the third category.…”

“…The present contribution belongs to the third category. Let us also mention recent works dealing with mixed Poisson-Gaussian noise removal [1], [24], [29].…”