Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

Abstract

Abstract-In order to denoise Poisson count data, we introduce a variance stabilizing transform (VST) applied on a filtered discrete Poisson process, yielding a near Gaussian process with asymptotic constant variance. This new transform, which can be deemed as an extension of the Anscombe transform to filtered data, is simple, fast, and efficient in (very) low-count situations. We combine this VST with the filter banks of wavelets, ridgelets and curvelets, leading to multiscale VSTs (MS-VSTs) and nonlinear decomposition schemes. By doing so, the noise-contaminated coefficients of these MS-VST-modified transforms are asymptotically normally distributed with known variances. A classical hypothesis-testing framework is adopted to detect the significant coefficients, and a sparsity-driven iterative scheme reconstructs properly the final estimate. A range of examples show the power of this MS-VST approach for recovering important structures of various morphologies in (very) low-count images. These results also demonstrate that the MS-VST approach is competitive relative to many existing denoising methods.

Image Informatics Strategies for Deciphering Neuronal Network Connectivity

“…More advanced (e.g. wavelet-based 38 ) methods that correct for Poisson noise have been described as well. Heterogeneous illumination and nonlinearities in the acquisition path are usually corrected for by subtracting an image of an empty region (flat-field correction) or by local background subtraction (pseudo-flat field correction).…”

Multichannel Poisson denoising and deconvolution on the sphere: application to theFermiGamma-ray Space Telescope

“…We recall that }¨} 1 " ř j 1 , j 2 ,k r ,k t |d j 1 , j 2 rk r , k t s| is the 1 -norm playing the role of regularization and is well-known to promote sparsity (Donoho 2004). This problem can be solved efficiently using the hybrid steepest descent algorithm (Yamada 2001;Zhang et al 2008), and requires about ten iterations in practice. Transposed into our context, its main steps can be summarized as follows:…”

Source detection using a 3D sparse representation: application to the Fermi gamma-ray space telescope

“…In a recent paper, Zhang et al (2008a) have proposed to merge a variance stabilization technique and the multiscale decomposition, leading to the Multi-Scale Variance Stabilization Transform (MSVST). In the case of the isotropic undecimated wavelet transform, as the wavelet coefficients w j are derived by a simple difference of two consecutive dyadic scales of the input image (see Sect.…”

“…In this section, we review the MSVST method (Zhang et al 2008a), restricted to the Isotropic Undecimated Wavelet Transform (IUWT). Indeed, the MSVST can use other transforms such as the standard three-orientation undecimated wavelet transform, the ridgelet or the curvelet transforms; see Zhang et al (2008a).…”

“…The multiscale variance stabilization Transform (MSVST) has recently been proposed for Poisson data denoising (Zhang et al 2008a). This procedure, which is nonparametric, is based on thresholding wavelet coefficients.…”