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Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal
"... 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 t ..."
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Cited by 42 (2 self)
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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) lowcount situations. We combine this VST with the filter banks of wavelets, ridgelets and curvelets, leading to multiscale VSTs (MSVSTs) and nonlinear decomposition schemes. By doing so, the noisecontaminated coefficients of these MSVSTmodified transforms are asymptotically normally distributed with known variances. A classical hypothesistesting framework is adopted to detect the significant coefficients, and a sparsitydriven iterative scheme reconstructs properly the final estimate. A range of examples show the power of this MSVST approach for recovering important structures of various morphologies in (very) lowcount images. These results also demonstrate that the MSVST approach is competitive relative to many existing denoising methods. Index Terms—Curvelets, filtered Poisson process, multiscale variance stabilizing transform, Poisson intensity estimation, ridgelets, wavelets. I.
A proximal iteration for deconvolving Poisson noisy images using sparse representations
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Image Denoising in Mixed Poisson–Gaussian Noise
, 2011
"... We propose a general methodology (PURELET) to design and optimize a wide class of transformdomain thresholding algorithms for denoising images corrupted by mixed Poisson–Gaussian noise. We express the denoising process as a linear expansion of thresholds (LET) that we optimize by relying on a pur ..."
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Cited by 35 (2 self)
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We propose a general methodology (PURELET) to design and optimize a wide class of transformdomain thresholding algorithms for denoising images corrupted by mixed Poisson–Gaussian noise. We express the denoising process as a linear expansion of thresholds (LET) that we optimize by relying on a purely dataadaptive unbiased estimate of the meansquared error (MSE), derived in a nonBayesian framework (PURE: Poisson–Gaussian unbiased risk estimate). We provide a practical approximation of this theoretical MSE estimate for the tractable optimization of arbitrary transformdomain thresholding. We then propose a pointwise estimator for undecimated filterbank transforms, which consists of subbandadaptive thresholding functions with signaldependent thresholds that are globally optimized in the image domain. We finally demonstrate the potential of the proposed approach through extensive comparisons with stateoftheart techniques that are specifically tailored to the estimation of Poisson intensities. We also present denoising results obtained on real images of lowcount fluorescence microscopy.
Patchbased nonlocal functional for denoising fluorescence microscopy image sequences
, 2009
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Multiscale methods for data on graphs and irregular multidimensional situations,
, 2008
"... Summary. For regularly spaced 1D data, wavelet shrinkage has proven to be a compelling method for nonparametric function estimation. We create three new multiscale methods that provide waveletlike transforms for both data arising on graphs and for irregularly spaced spatial data in more than 1D. T ..."
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Cited by 20 (0 self)
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Summary. For regularly spaced 1D data, wavelet shrinkage has proven to be a compelling method for nonparametric function estimation. We create three new multiscale methods that provide waveletlike transforms for both data arising on graphs and for irregularly spaced spatial data in more than 1D. The concept of scale still exists within these transforms but as a continuous quantity rather than dyadic levels. Further, we adapt recent empirical Bayesian shrinkage techniques to enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular spatial data. We demonstrate that our methods perform very well when compared to several other methods for spatial regression for both real and simulated data. Although our article concentrates on multiscale shrinkage (regression) we present our new 'wavelet transforms' as generic tools intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data.
Fast interscale wavelet denoising of Poissoncorrupted images, signal Processing,
, 2010
"... a b s t r a c t We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistic ..."
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Cited by 19 (4 self)
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a b s t r a c t We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistics across scales within the Haar DWT. The minimization of the MSE estimate is performed independently in each wavelet subband, but this is equivalent to a global imagedomain MSE minimization, thanks to the orthogonality of Haar wavelets. This is an important difference with standard Poisson noiseremoval methods, in particular those that rely on a nonlinear preprocessing of the data to stabilize the variance. Our nonredundant interscale wavelet thresholding outperforms standard variancestabilizing schemes, even when the latter are applied in a translationinvariant setting (cyclespinning). It also achieves a quality similar to a stateoftheart multiscale method that was specially developed for Poisson data. Considering that the computational complexity of our method is orders of magnitude lower, it is a very competitive alternative. The proposed approach is particularly promising in the context of low signal intensities and/or large data sets. This is illustrated experimentally with the denoising of lowcount fluorescence micrographs of a biological sample.
GOES8 Xray sensor variance stabilization using the multiscale datadriven HaarFisz transform
, 2007
"... Summary. We consider the stochastic mechanisms behind the data collected by the solar Xray sensor (XRS) on board the the GOES8 satellite. We discover and justify a nontrivial meanvariance relationship within the XRS data. Transforming such data so that its variance is stable and its distribution ..."
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Cited by 14 (6 self)
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Summary. We consider the stochastic mechanisms behind the data collected by the solar Xray sensor (XRS) on board the the GOES8 satellite. We discover and justify a nontrivial meanvariance relationship within the XRS data. Transforming such data so that its variance is stable and its distribution is taken closer to the Gaussian is the aim of many techniques (e.g. Anscombe, BoxCox). Recently, new techniques based on the HaarFisz transform have been introduced that use a multiscale method to transform and stabilize data with a known meanvariance relationship. In many practical cases, such as the XRS data, the variance of the data can be assumed to increase with the mean, but other characteristics of the distribution are unknown. We introduce a method, the datadriven HaarFisz transform (DDHFT), which uses HaarFisz but also estimates the meanvariance relationship. For known noise distributions, the DDHFT is shown to be competitive with the fixed HaarFisz methods. We show how our DDHFT method denoises the XRS series where other existing methods fail.
Skellam shrinkage: Waveletbased intensity estimation for inhomogeneous Poisson data
"... The ubiquity of integrating detectors in imaging and other applications implies that a variety of realworld data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vectorvalued signal of interest. In this article, we first show how the socalled Skel ..."
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Cited by 10 (7 self)
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The ubiquity of integrating detectors in imaging and other applications implies that a variety of realworld data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vectorvalued signal of interest. In this article, we first show how the socalled Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the nearoptimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of HaarFisz variancestabilized data, along with accompanying lowcomplexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam
Multivariate nonparametric regression using lifting
, 2004
"... Summary For regularly spaced onedimensional data wavelet shrinkage has proven to be a compelling method for nonparametric function estimation. We argue that this is not the case for irregularly spaced data in two or higher dimensions. This article develops three methods for the multiscale analysis ..."
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Cited by 10 (5 self)
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Summary For regularly spaced onedimensional data wavelet shrinkage has proven to be a compelling method for nonparametric function estimation. We argue that this is not the case for irregularly spaced data in two or higher dimensions. This article develops three methods for the multiscale analysis of irregularly spaced data based on the recently developed lifting paradigm by "lifting one coefficient at a time". The concept of scale still exists within these transforms but as a continuous quantity rather than dyadic levels. We develop empirical Bayes methods that take account of the continuous nature of the scale. We apply our new methods to the problems of estimation of krill density and rail arrival delays. We demonstrate good performance in a simulation study on new twodimensional analogues of the wellknown Blocks, Bumps, Doppler and Heavisine and a new piecewise linear function called maartenfunc.
Data Mining of Early Day Motions and Multiscale Variance Stabilisation of Count Data
, 2008
"... A dissertation submitted to the University of Bristol in accordance with the requirements ..."
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Cited by 1 (0 self)
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A dissertation submitted to the University of Bristol in accordance with the requirements