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45
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.
Sparse poisson intensity reconstruction algorithms
 in Proc. IEEE Work. Stat. Signal Processing (SSP
, 2009
"... The observations in many applications consist of counts of discrete events, such as photons hitting a dector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or tempo ..."
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Cited by 40 (8 self)
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The observations in many applications consist of counts of discrete events, such as photons hitting a dector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f) from Poisson data (y) cannot be accomplished by minimizing a conventional ℓ2 − ℓ1 objective function. The problem addressed in this paper is the estimation of f from y in an inverse problem setting, where (a) the number of unknowns may potentially be larger than the number of observations and (b) f admits a sparse approximation in some basis. The optimization formulation considered in this paper uses a negative Poisson loglikelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). This paper describes computational methods for solving the constrained sparse Poisson inverse problem. In particular, the proposed approach incorporates key ideas of using quadratic separable approximations to the objective function at each iteration and computationally efficient partitionbased multiscale estimation methods. Index Terms—Photonlimited imaging, Poisson noise, wavelets, convex optimization, sparse approximation, compressed sensing
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.
Compressed Sensing Performance Bounds Under Poisson Noise
"... Abstract—This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied ..."
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Cited by 27 (6 self)
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Abstract—This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signalindependent and/or bounded noise models do not apply to Poisson noise, which is nonadditive and signaldependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical `2 0 `1 minimization leads to overfitting in the highintensity regions and oversmoothing in the lowintensity areas. In this paper, we describe how a feasible positivityand fluxpreserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition. Index Terms—Complexity regularization, compressive sampling, nonparametric estimation, photonlimited imaging, sparsity. I.
Lasso and probabilistic inequalities for multivariate point processes
 SUBMITTED TO THE BERNOULLI
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Minimax optimal level set estimation
 IN PROC. SPIE, WAVELETS XI, 31 JULY  4
, 2005
"... This paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data. The identification of the boundaries of such sets is an important theoretical problem with applications for digital elevation maps, ..."
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Cited by 23 (4 self)
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This paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data. The identification of the boundaries of such sets is an important theoretical problem with applications for digital elevation maps, medical imaging, and pattern recognition. This problem is significantly different from classical segmentation because level set boundaries may not correspond to singularities or edges in the underlying function; as a result, segmentation methods which rely upon detecting boundaries would be potentially ineffective in this regime. This issue is addressed in this paper through a novel error metric sensitive to both the error in the location of the level set estimate and the deviation of the function from the critical level. Hoeffding’s inequality is used to derive a novel regularization
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.
Conditional density estimation by penalized likelihood model selection and applications. ArXiv 1103.2021
, 2011
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 10 (3 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Near optimal thresholding estimation of a Poisson intensity on the real line
 Elec. J. Statist
, 2010
"... Abstract: The purpose of this paper is to estimate the intensity of a Poisson process N by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of N with respect to ndx where n is a fixed parameter, is assumed to be noncompactly supported. The estim ..."
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Cited by 10 (3 self)
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Abstract: The purpose of this paper is to estimate the intensity of a Poisson process N by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of N with respect to ndx where n is a fixed parameter, is assumed to be noncompactly supported. The estimatorfn,γ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties offn,γ on Besov spaces B α p,q are established. Under mild assumptions, we prove that sup f ∈B α p,q ∩L∞
Socioscope: SpatioTemporal Signal Recovery from Social Media
"... Many realworld phenomena can be represented by a spatiotemporal signal: where, when, and how much. Social media is a tantalizing data source for those who wish to monitor such signals. Unlike most prior work, we assume that the target phenomenon is known and we are given a method to count its occ ..."
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Cited by 7 (1 self)
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Many realworld phenomena can be represented by a spatiotemporal signal: where, when, and how much. Social media is a tantalizing data source for those who wish to monitor such signals. Unlike most prior work, we assume that the target phenomenon is known and we are given a method to count its occurrences in social media. However, counting is plagued by sample bias, incomplete data, and, paradoxically, data scarcity – issues inadequately addressed by prior work. We formulate signal recovery as a Poisson point process estimation problem. We explicitly incorporate human population bias, time delays and spatial distortions, and spatiotemporal regularization into the model to address the noisy count issues. We present an efficient optimization algorithm and discuss its theoretical properties. We show that our model is more accurate than commonlyused baselines. Finally, we present a case study on wildlife roadkill monitoring, where our model produces qualitatively convincing results.