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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 17 (1 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.
Minimax optimal level set estimation
 in Proc. SPIE, Wavelets XI, 31 July  4
, 2005
"... Abstract — 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 elevat ..."
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Cited by 13 (4 self)
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Abstract — 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
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 12 (2 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 9 (1 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.
Toward a Model for Source Address of Internet Background Radiation
 In Proceedings of Passive and Active Measurement Conference (PAM 2006
, 2006
"... Abstract. Internet background radiation, the fundamentally unproductive traffic that arises from misconfigurations and malicious activities, is pervasive and has complex characteristics. Understanding the network locations of hosts that generate background radiation can be helpful in the development ..."
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Cited by 5 (1 self)
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Abstract. Internet background radiation, the fundamentally unproductive traffic that arises from misconfigurations and malicious activities, is pervasive and has complex characteristics. Understanding the network locations of hosts that generate background radiation can be helpful in the development of new techniques aimed at reducing this unwanted traffic. This paper presents an initial investigation of the network locations of hosts that generate malicious background radiation using source addresses in packet traces from network telescopes, firewalls and intrusion detection systems distributed throughout the Internet. We characterize background radiation source addresses across the IPv4 address space for /8, /16 and /24 aggregates. Using a conservative multiscale density estimation method, we find that source addresses of background radiation form a relatively small number of tight clusters – i.e., that the distribution of source addresses exhibits characteristics of a highly irregular multifractal with a broad spectrum that is consistent over all of our data. We verify that the distributional properties are consistent with multifractals, and propose a multiscale multiplicative innovations (MMI) model for host locations that can be used to generate random variates with the same distributional properties as our empirical data. This model is targeted for use in analytic, simulation and emulation evaluations of methods for reducing unwanted traffic as well as potential real time monitoring and detection applications. 1
Near optimal thresholding estimation of a Poisson intensity on the real line
, 810
"... 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 estimat ..."
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Cited by 4 (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 estimator ˜ fn,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 of ˜ fn,γ on Besov spaces B α p,q are established. Under mild assumptions, we prove that sup f∈Bα E( p,q∩L∞ ˜ fn,γ − f  2 2) ≤ C log n
Socioscope: SpatioTemporal Signal Recovery from Social Media
"... Abstract. 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 ..."
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Cited by 4 (1 self)
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Abstract. 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. 1
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 3 (1 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.
Multiscale intensity estimation for marked poisson processes
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing — ICASSP
, 2007
"... Inference on severely datastarved Poisson processes can be dramatically improved by using auxiliary information about measured discrete events in the form of “marks”. Marks are widely available in many applications, and can take the form of photon energy, time delay information, packet size, or oth ..."
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Cited by 2 (2 self)
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Inference on severely datastarved Poisson processes can be dramatically improved by using auxiliary information about measured discrete events in the form of “marks”. Marks are widely available in many applications, and can take the form of photon energy, time delay information, packet size, or other forms of characterizations. Effectively using marks results in innovative signal processing methods and dramatic error reductions for Poisson intensity estimation. The efficacy of the proposed method is demonstrated in the context of photonlimited spatiospectral intensity estimation.
THE MDL PRINCIPLE, PENALIZED LIKELIHOODS, AND STATISTICAL RISK
"... ABSTRACT. We determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f) such that the optimizer ˆ f of the penalized log likelihood criterion log 1/likelihood(f) + pen(f) has statistical risk not more than the index of resolvability co ..."
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Cited by 1 (1 self)
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ABSTRACT. We determine, for both countable and uncountable collections of functions, informationtheoretic conditions on a penalty pen(f) such that the optimizer ˆ f of the penalized log likelihood criterion log 1/likelihood(f) + pen(f) has statistical risk not more than the index of resolvability corresponding to the accuracy of the optimizer of the expected value of the criterion. If F is the linear span of a dictionary of functions, traditional descriptionlength penalties are based on the number of nonzero terms of candidate fits (the ℓ0 norm of the coefficients) as we review. We specialize our general conclusions to show the ℓ1 norm of the coefficients times a suitable multiplier λ is also an informationtheoretically valid penalty. 1.