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42
A review of curvelets and recent applications
 IEEE Signal Processing Magazine
, 2009
"... Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing ..."
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Cited by 128 (10 self)
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Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the past years. In this paper, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and 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.
Optimal inversion of the Anscombe transformation in lowcount Poisson image denoising
 IEEE TRANSACTIONS
"... The removal of Poisson noise is often performed through the following threestep procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the n ..."
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Cited by 30 (4 self)
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The removal of Poisson noise is often performed through the following threestep procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest. The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied. We introduce optimal inverses for the Anscombe transformation, in particular the exact unbiased inverse, a maximum likelihood (ML) inverse, and a more sophisticated minimum mean square error (MMSE) inverse. We then present an experimental analysis using a few stateoftheart denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the lowcount regime. This results in a very efficient filtering solution that is competitive with some of the best existing methods for Poisson image denoising.
Patchbased nonlocal functional for denoising fluorescence microscopy image sequences
, 2009
"... ..."
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.
Bayesian inference on multiscale models for Poisson intensity estimation: applications to photonlimited image denoising
 IEEE Trans. Image Process
, 2009
"... Abstract—We present an improved statistical model for analyzing Poisson processes, with applications to photonlimited imaging. We build on previous work, adopting a multiscale representation of the Poisson process in which the ratios of the underlying Poisson intensities (rates) in adjacent scales ..."
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Cited by 16 (2 self)
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Abstract—We present an improved statistical model for analyzing Poisson processes, with applications to photonlimited imaging. We build on previous work, adopting a multiscale representation of the Poisson process in which the ratios of the underlying Poisson intensities (rates) in adjacent scales are modeled as mixtures of conjugate parametric distributions. Our main contributions include: 1) a rigorous and robust regularized expectationmaximization (EM) algorithm for maximumlikelihood estimation of the rateratio density parameters directly from the noisy observed Poisson data (counts); 2) extension of the method to work under a multiscale hidden Markov tree model (HMT) which couples the mixture label assignments in consecutive scales, thus modeling interscale coefficient dependencies in the vicinity of image edges; 3) exploration of a 2D recursive quadtree image representation, involving Dirichletmixture rateratio densities, instead of the conventional separable binarytree image representation involving betamixture rateratio densities; and 4) a novel multiscale image representation, which we term PoissonHaar decomposition, that better models the image edge structure, thus yielding improved performance. Experimental results on standard images with artificially simulated Poisson noise and on real photonlimited images demonstrate the effectiveness of the proposed techniques. Index Terms—Bayesian inference, expectationmaximization (EM) algorithm, hidden Markov tree (HMT), photonlimited imaging, PoissonHaar decomposition, Poisson processes. I.
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∞
Fast Haarwavelet denoising of multidimensional fluorescence microscopy data
 in: Proceedings of the Sixth IEEE international conference on Symposium on Biomedical Imaging: From Nano to Macro, IEEE
, 2009
"... We propose a novel denoising algorithm to reduce the Poisson noise that is typically dominant in fluorescence microscopy data. To process large datasets at a low computational cost, we use the unnormalized Haar wavelet transform. Thanks to some of its appealing properties, independent unbiased MSE ..."
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Cited by 5 (0 self)
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We propose a novel denoising algorithm to reduce the Poisson noise that is typically dominant in fluorescence microscopy data. To process large datasets at a low computational cost, we use the unnormalized Haar wavelet transform. Thanks to some of its appealing properties, independent unbiased MSE estimates can be derived for each subband. Based on these Poisson unbiased MSE estimates, we then optimize linearly parametrized interscale thresholding. Correlations between adjacent images of the multidimensional data are accounted for through a sliding window approach. Experiments on simulated and real data show that the proposed solution is qualitatively similar to a stateoftheart multiscale method, while being orders of magnitude faster. Index Terms — Poisson noise, Fluorescence, Haar Wavelet, MSE estimation
A Dictionary Learning Approach for Poisson Image Deblurring
, 2013
"... The restoration of images corrupted by blur and Poisson noise is a key issue in medical and biological image processing. While most existing methods are based on variational models, generally derived from a Maximum A Posteriori (MAP) formulation, recently sparse representations of images have shown ..."
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Cited by 5 (0 self)
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The restoration of images corrupted by blur and Poisson noise is a key issue in medical and biological image processing. While most existing methods are based on variational models, generally derived from a Maximum A Posteriori (MAP) formulation, recently sparse representations of images have shown to be efficient approaches for image recovery. Following this idea, we propose in this paper a model containing three terms: a patchbased sparse representation prior over a learned dictionary, the pixelbased total variation regularization term and a datafidelity term capturing the statistics of Poisson noise. The resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting. Extensive experimental results suggest that in terms of visual quality, PSNR value and the method noise, the proposed algorithm outperforms stateoftheart methods.
Sparsity Based Poisson Denoising with Dictionary Learning
"... The problem of Poisson denoising appears in various imaging applications, such as lowlight photography, medical imaging and microscopy. In cases of high SNR, several transformations exist so as to convert the Poisson noise into an additive i.i.d. Gaussian noise, for which many effective algorithm ..."
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Cited by 5 (2 self)
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The problem of Poisson denoising appears in various imaging applications, such as lowlight photography, medical imaging and microscopy. In cases of high SNR, several transformations exist so as to convert the Poisson noise into an additive i.i.d. Gaussian noise, for which many effective algorithms are available. However, in a low SNR regime, these transformations are significantly less accurate, and a strategy that relies directly on the true noise statistics is required. A recent work by Salmon et al let@tokeneonedot[1], [2] took this route, proposing a patchbased exponential image representation model based on GMM (Gaussian mixture model), leading to stateoftheart results. In this paper, we propose to harness sparserepresentation modeling to the image patches, adopting the same exponential idea. Our scheme uses a greedy pursuit with bootstrapped stopping condition and dictionary learning within the denoising process. The reconstruction performance of the proposed scheme is competitive with leading methods in high SNR, and achieving stateoftheart results in cases of low SNR.