Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

• Published in: IEEE transactions on image processing Vol. 17; no. 7; pp. 1093 - 1108
• Main Authors: Bo Zhang, Fadili, J.M., Starck, J.L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Asymptotic properties; Bayesian methods; Computer Science; Computer Simulation; Counting; Curvelets; Detection, estimation, filtering, equalization, prediction; Discrete wavelet transforms; Engineering Sciences; Exact sciences and technology; Filter bank; Filter banks; filtered Poisson process; Gaussian; Gaussian noise; Gaussian processes; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image Processing; Information, signal and communications theory; Mathematical models; Maximum likelihood estimation; Miscellaneous; Models, Statistical; multiscale variance stabilizing transform; Noise reduction; Poisson Distribution; Poisson intensity estimation; Reproducibility of Results; ridgelets; Sensitivity and Specificity; Signal and communications theory; Signal and Image processing; Signal processing; Signal, noise; Studies; Telecommunications and information theory; Testing; Transforms; Variance; Wavelet; Wavelet domain; wavelets; Wiener filter; Poisson intensity estimation; Noise; Noise reduction; Iterative method; ridgelets; filtered Poisson process; Poisson process; Hypothesis test; Curvelets; Gaussian process; Wavelet transformation; Multiscale method; wavelets; Signal processing; multiscale variance stabilizing transform; Filter bank; Open market; Radon transformation
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2008.924386

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.