Multi-Scale Variance Stabilizing Transform for Multi-Dimensional Poisson Count Image Denoising

We propose in this paper a multi-scale variance stabilizing transform (MSVST) for approximately Gaussianizing and stabilizing the variance of a sequence of independent Poisson random variables (RVs) filtered by a low-pass linear filter. This approach is shown to be fast, very well adapted to extremely low-count situations and easily applicable to any dimensional data. It is shown that the RV transformed using Anscombe VST can be reasonably considered as stabilized for an intensity lambda gsim 10, using Fisz VST for lambda gsim 1 and using our VST (after low-pass filtering) for lambda gsim 0.1. We then use the MSVST technique to stabilize the detail coefficients of the isotropic undecimated wavelet transform (IUWT) of multi-dimensional Poisson count data. We use a hypothesis testing framework in the wavelet domain to denoise the Gaussianized and stabilized coefficients, and then apply the inverse MSVST-IUWT to get the estimated intensity image underlying the Poisson data. Finally, potential applicability of our approach is illustrated on an astronomical example where isotropic structures must be recovered