Nonparametric Estimation of Intensity Maps Using Haar Wavelets and Poisson Noise Characteristics

Kolaczyk, Eric D.; Dixon, David

doi:10.1086/308718

Cited by 42 publications

(36 citation statements)

References 22 publications

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“…For the standard UWT case, direct reconstruction procedure is unavailable since the convolution (by ) operator and the nonlinear VST operator do not commute in (13). For the IUWT case, the estimate can be reconstructed by (11).…”

Section: E Iterative Reconstructionmentioning

confidence: 99%

“…2) Wavelet wiener filtering: Nowak and Baraniuk [11] and Antoniadis and Sapatinas [12] proposed a wavelet domain filter, which can be interpreted as a data-adaptive wiener filter in a wavelet basis. 3) Hypothesis testing: Kolaczyk first introduced a Haar domain threshold [13], which implements a hypothesis testing procedure controlling a user-specified false positive rate (FPR). The hypothesis tests have been extended to the biorthogonal Haar domain [14], leading to more regular reconstructions for smooth intensities.…”

mentioning

confidence: 99%

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Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

Zhang

Fadili

Starck

2008

IEEE Trans. on Image Process.

320

239

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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) 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.

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Section: E Iterative Reconstructionmentioning

confidence: 99%

mentioning

confidence: 99%

Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

Zhang

Fadili

Starck

2008

IEEE Trans. on Image Process.

320

239

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show abstract

“…The Á 2 additions exclude models with many breaks, while the final fit without the Á 2 contributions has 2 % and typically 3Y5 power-law segments. So that noise fluctuations in the data do not generate spurious light-curve breaks, we denoise the soft-and hard-channel lightcurve with Haar wavelets (see Kolaczyk & Dixon 2000) prior to fitting. We fit the count rate and hardness simultaneously so that spectrally distinct regions are separated.…”

Section: Light-curve Region Selection and Fittingmentioning

confidence: 99%

X‐Ray Hardness Variations as an Internal/External Shock Diagnostic

Butler

Kocevski

2007

ApJ

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The early, highly time-variable X-ray emission immediately following gamma-ray bursts (GRBs) exhibits strong spectral variations that are unlike the temporally smoother emission that dominates after t $ 10 3 s. The ratio of hardchannel (1.3Y10.0 keV ) to soft-channel (0.3Y1.3 keV ) counts in the Swift X-ray telescope provides a new measure delineating the end time of this emission. We define T H as the time at which this transition takes place and measure for 59 events a range of transition times that spans 10 2 to 10 4 s, on average 5 times longer than the prompt T 90 duration observed in the gamma-ray band. It is very likely that the mysterious light-curve plateau phase and the later powerlaw temporal evolution, both of which typically occur at times greater than T H and hence exhibit very little hardness ratio evolution, are both produced by external shocking of the surrounding medium and not by the internal shocks thought responsible for the earlier emission. We use the apparent lack of spectral evolution to discriminate among proposed models for the plateau phase emission. We favor energy injection scenarios with a roughly linearly increasing input energy versus time for six well-sampled events with nearly flat light curves at t % 10 3 Y10 4 s. Also, using the transition time T H as the delineation between the GRB and afterglow emission, we calculate that the kinetic energy in the afterglow shock is typically a factor of 10 lower than that released in the GRB. Three very bright events suggest that this presents a missing X-ray flux problem rather than an efficiency problem for the conversion of kinetic energy into the GRB. Lack of hardness variations in these three events may be due to a very highly relativistic outflow or due to a very dense circumburst medium. There are a handful of rare cases of very late time t > 10 4 s hardness evolution, which may point to residual central engine activity at very late time.

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“…Although denoising algorithms speci cally designed for Poisson noise have been proposed (e.g., [1], [2], [3], [4], [5], [6], [7]), often the removal of Poisson noise is performed through the following three-step procedure. First, the noise variance is stabilized by applying the Anscombe root transformation [8] f : z 7 !…”

Section: Introductionmentioning

confidence: 99%

Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

Mäkitalo

Foi

2011

IEEE Trans. on Image Process.

307

258

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Abstract-The removal of Poisson noise is often performed through the following three-step 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 state-of-theart denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the low-count regime. This results in a very ef cient ltering solution that is competitive with some of the best existing methods for Poisson image denoising.

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Nonparametric Estimation of Intensity Maps Using Haar Wavelets and Poisson Noise Characteristics

Cited by 42 publications

References 22 publications

Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

X‐Ray Hardness Variations as an Internal/External Shock Diagnostic

Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

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