Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

Abstract

Abstract-The removal of Poisson noise is often performed through the following three-step procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest.The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied. We introduce optimal inverses for the Anscombe transformation, in particular the exact unbiased inverse, a maximum likelihood (ML) inverse, and a more sophisticated minimum mean square error (MMSE) inverse. We then present an experimental analysis using a few state-of-theart denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the low-count regime. This results in a very ef cient ltering solution that is competitive with some of the best existing methods for Poisson image denoising.

Advances in Image Pre-Processing to Improve Automated 3d Reconstruction

“…Wavelet-based denoising methods (Nowak & Baraniuk, 1997;Kolaczyk, 1999) propose adaptation of the transform threshold to the local noise level of the Poisson process. Recent papers on the Anscombe transform by Makitalo &Foi (2011) andFoi (2011), argue that, when combined with suitable forward and inverse variance-stabilizing transformations (VST), algorithms designed for homoscedastic Gaussian noise work just as well as ad-hoc algorithms based on signal-dependent noise models.…”

Photon-limited Sensing and Surveillance

“…The photon-limited image estimation problem is particularly challenging because it introduces intensitydependent Poisson statistics which require specialized algorithms and analysis for optimal performance. Simply transforming Poisson data to produce data with approximately Gaussian noise (via, for instance, the variance stabilizing Anscombe transform [1,12] or Fisz transform [9,10]) can be effective when the number of counts is sufficiently high [6,22]. However, applying these methods to foreground estimation is a difficult problem due to the non-linearities induced by the transforms.…”

Noise estimation and removal in MR imaging: The variance-stabilization approach

“…The problem of bias in variance-stabilized denoising is solved by the exact unbiased inverse [14], [7] that is defined by the mapping…”

“…Under the rather generic assumption that D − E{ f (z)|ν,σ } is distributed according to a unimodal distribution with mode at 0, it can be easily shown (with a proof and motivation analogous to that in [14]) that I f (D) and V f (D) are maximum-likelihood estimates of E{z|ν,σ } and ν, respectively. We refer the reader to [14] for further details about this form of inversion.…”