## Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

Citations: | 17 - 1 self |

### BibTeX

@MISC{Zhang_wavelets,ridgelets,,

author = {Bo Zhang and Jalal M. Fadili and Jean-luc Starck},

title = {Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal},

year = {}

}

### OpenURL

### Abstract

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. Index Terms—Curvelets, filtered Poisson process, multiscale variance stabilizing transform, Poisson intensity estimation, ridgelets, wavelets. I.

### Citations

2850 |
Controlling the false discovery rate: a practical and powerful approach to multiple testing
- Benjamini, Hochberg
- 1995
(Show Context)
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- Donoho, Johnstone
- 1994
(Show Context)
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804 | DL(1995). “De-noising by soft-thresholding
- Donoho
(Show Context)
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- Benjamini, Hochberg
- 1995
(Show Context)
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- Donoho, Michael
- 2003
(Show Context)
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- Donoho
(Show Context)
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- Candes, Donoho
- 2002
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- Candes
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- Coifman, Donoho
- 1995
(Show Context)
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- Mallat
- 1999
(Show Context)
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- Shensa
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- Bauschke, Borwein
- 1996
(Show Context)
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- Nason, Silverman
- 1995
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- Candès, Demanet, et al.
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- Varadhan
- 1988
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- Abramovich, Benjamini, et al.
- 2006
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- Starck, Murtagh, et al.
- 1998
(Show Context)
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- Donoho
- 1993
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- Bickel, A
- 1977
(Show Context)
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- Anscombe
- 1948
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- Starck, Elad, et al.
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- 2003
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- Antoniadis, Bigot, et al.
- 2001
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67 |
A real-time algorithm for signal analysis with the help of the wavelet transform
- Holschneider, Kronland-Martiner, et al.
- 1989
(Show Context)
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- Timmermann, Nowak
- 1999
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Citation Context ...symptotic approximation adopted by [16] may not allow reasonable solutions in lowcount situations. 4) empirical Bayesian and penalized ML estimations: empirical Bayesian estimators are studied in [17]=-=[18]-=-[19][10]. The low-intensity case apart, Bayesian approaches generally outperform the direct wavelet filtering [11][12] (see also [20] for a comparative review). Poisson denoising has also been formula... |

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- Fryz´lewicz, Nason
- 2004
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Citation Context ...o the Gaussian denoising problem, standard approaches such as wavelet thresholding [5][6] are used before the VST is inverted to get the final estimate. Haar-Fisz transform is another widely used VST =-=[7]-=-[8], which combines the Fisz transform [9] within the Haar transform. Jansen [10] introduced a conditional variance stabilization (CVS) approach which can be applied in any wavelet domain resulting in... |

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- 2001
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Citation Context ...(·, θ). Different digital ridgelet transforms can be derived depending on the choice of both the Radon algorithm and the wavelet decomposition [46]. For example, the Slant Stack Radon (SSR) transform =-=[47]-=-[48] is a good candidate, which has the advantage of being geometrically accurate, and is used in our experiments. The inverse SSR has however the drawback to be iterative. If computation time is an i... |

46 |
Ridgelets: The key to high dimensional intermittency
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- 1999
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Citation Context ...sely representing different kinds of information. For example, to represent regular structures with point singularities, a qualified candidate is the wavelet transform [25][1]. The ridgelet transform =-=[26]-=- is very effective in representing global lines in an image. The curvelet system [27][28] is highly suitable for representing smooth (C 2 ) images away from C 2 contours. These transforms are also com... |

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- 2007
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Citation Context ...re. This is however not trivial and requires further investigations. Second, new multiscale transforms have been recently proposed such as the fast curvelet transform [52] and the wave atom transform =-=[53]-=-, and it would also be very interesting to investigate how our MS-VST could be linked to them. Finally, here we have considered the denoising with a single multiscale transform only. If the data conta... |

45 |
Bayesian multi-scale models for Poisson processes
- Kolaczyk
(Show Context)
Citation Context ...e asymptotic approximation adopted by [16] may not allow reasonable solutions in low-count situations. 4) Empirical Bayesian and penalized ML estimations: Empirical Bayesian estimators are studied in =-=[17]-=-, [18], [10], [19]. The low-intensity case apart, Bayesian approaches generally outperform the direct wavelet filtering [11], [12] (see also [20] for a comparative review). Poisson denoising has also ... |

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- Nowak, Kolaczyk
- 2000
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39 | Multiresolution support applied to image filtering and restoration
- Starck, Murtagh, et al.
- 1995
(Show Context)
Citation Context ...the following, we will concentrate on the 1D case for clarity. We suppose that the underlying intensity function Λ is sparsely represented in the wavelet domain. We define the multiresolution support =-=[40]-=- M, which is determined by the set of detected significant coefficients at each scale j and location k, i.e., M := {(j, k) | if dj[k] is significant} (17) The estimation is then formulated as a constr... |

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- Donoho, Duncan
- 1999
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- Nowak, Baraniuk
- 1999
(Show Context)
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34 |
Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds
- Kolaczyk
- 1999
(Show Context)
Citation Context ...nsity function (pdf ) of any wavelet coefficient, which allows HTs in an arbitrary wavelet basis. However, as the pdf has no closed forms, [15] is more computationally complex than Haarbased methods. =-=[16]-=- proposed “corrected” versions of the usual Gaussian-based thresholds for Poisson data. However, the asymptotic approximation adopted by [16] may not allow reasonable solutions in lowcount situations.... |

31 |
The limiting distribution of a function of two independent random variables and its statistical application
- Fisz
- 1955
(Show Context)
Citation Context ... approaches such as wavelet thresholding [5][6] are used before the VST is inverted to get the final estimate. Haar-Fisz transform is another widely used VST [7][8], which combines the Fisz transform =-=[9]-=- within the Haar transform. Jansen [10] introduced a conditional variance stabilization (CVS) approach which can be applied in any wavelet domain resulting in stabilized coefficients. 2) wavelet wiene... |

24 | Multiscale poisson intensity and density estimation. Submitted to
- Willett, Nowak
(Show Context)
Citation Context ... shown in Fig. 3(b). We present the restoration results given by Anscombe [4] [Fig. 3(c)], Haar–Fisz [7] [Fig. 3(d)], CVS [10] [Fig. 3(e)], Haar hypothesis tests [13] [Fig. 3(f)], platelet estimation =-=[23]-=-, [45], [24] [Fig. 3(g)], and the MS-VST denoiser using iterative [Fig. 3(h)] and direct [Fig. 3(i)] reconstructions. IUWT has been used to produce the results in Fig. 3(c), (d), (e), (h), and (i); st... |

23 |
Wavelet shrinkage for natural exponential families with quadratic variance functions,” Biometrika 88
- Antoniadis, Sapatinas
- 2001
(Show Context)
Citation Context ...riance stabilization (CVS) approach which can be applied in any wavelet domain resulting in stabilized coefficients. 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 im... |

23 |
A Comparative Simulation Study of Wavelet Shrinkage Estimators For Poisson Counts
- Besbeas, Feis, et al.
- 2004
(Show Context)
Citation Context ...d ML estimations: empirical Bayesian estimators are studied in [17][18][19][10]. The low-intensity case apart, Bayesian approaches generally outperform the direct wavelet filtering [11][12] (see also =-=[20]-=- for a comparative review). Poisson denoising has also been formulated as a penalized maximum likelihood (ML) estimation problem [21][22][23][24] within wavelet, wedgelet and platelet dictionaries. We... |

22 | The undecimated wavelet decomposition and its reconstruction
- Starck, Fadili, et al.
(Show Context)
Citation Context ...ence, we propose to reformulate the reconstruction as a convex sparsity-promoting optimization problem and solve it by an iterative steepest descent algorithm (Section III-E). B. MS-VST+IUWT The IUWT =-=[33]-=- uses the filter bank (h, g = δ − h, ˜ h = δ, ˜g = δ) where h is typically a symmetric low-pass filter such as the B3-Spline filter. The particular structure of the analysis filters (h, g) leads to th... |

21 |
Very high quality image restoration by combining wavelets and curvelets
- Starck, Donoho, et al.
- 2001
(Show Context)
Citation Context ... features with different morphologies, it could be better to introduce several multiscale transforms in the denoising algorithm. This could be done in a very similar way as in the Gaussian noise case =-=[54]-=-. November 5, 2007 DRAFT 23s50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 (a) (d) (g) 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 (b) (e) (h) Fig. 7. Poisson denoising of fluor... |

20 | Multiscale Poisson data smoothing
- Jansen
(Show Context)
Citation Context ...ng [5][6] are used before the VST is inverted to get the final estimate. Haar-Fisz transform is another widely used VST [7][8], which combines the Fisz transform [9] within the Haar transform. Jansen =-=[10]-=- introduced a conditional variance stabilization (CVS) approach which can be applied in any wavelet domain resulting in stabilized coefficients. 2) wavelet wiener filtering: Nowak and Baraniuk [11], a... |

20 |
Gaussian approximations of fluorescence microscope point-spread function models
- Zhang, Zerubia, et al.
- 2007
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Citation Context ... to the rightmost one, source radii increase from 50nm to 350nm. This image has been convolved with a Gaussian function with a standard deviation of 103nm which approximates a confocal microscope PSF =-=[44]-=-. The source amplitudes range from 0.08 to 4.99, and the background level is 0.03. This spot grid can be deemed as a model for cellular vesicles of different sizes and intensities. A realization of th... |

19 | Automatic smoothing with wavelets for a wide class of distributions
- Sardy, Antoniadis, et al.
- 2004
(Show Context)
Citation Context ...nerally outperform the direct wavelet filtering [11][12] (see also [20] for a comparative review). Poisson denoising has also been formulated as a penalized maximum likelihood (ML) estimation problem =-=[21]-=-[22][23][24] within wavelet, wedgelet and platelet dictionaries. Wedgelet (platelet-) based methods are more efficient than wavelet-based estimators in denoising piecewise constant (smooth) images wit... |

18 |
Flesia, Digital ridgelet transform based on true ridge functions
- Donoho, G
- 2003
(Show Context)
Citation Context ...θ). Different digital ridgelet transforms can be derived depending on the choice of both the Radon algorithm and the wavelet decomposition [46]. For example, the Slant Stack Radon (SSR) transform [47]=-=[48]-=- is a good candidate, which has the advantage of being geometrically accurate, and is used in our experiments. The inverse SSR has however the drawback to be iterative. If computation time is an issue... |

15 |
Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numerical Functional Analysis and Optimization 25
- Yamada, Ogura
- 2004
(Show Context)
Citation Context ...lassical projected (sub-)gradient method is also difficult to apply here since the projector on the feasible set is unknown. Below we propose an alternative based on the hybrid steepest descent (HSD) =-=[43]-=-. The HSD approach allows minimizing convex functionals over the intersection of fixed point sets of nonexpansive mappings. It is much faster than LP, and in our problem, the nonexpansive mappings do ... |

14 |
nonparametric estimation of intensity maps using Haar wavelets and Poisson noise characteristics
- Kolaczyk, Dixon
- 2000
(Show Context)
Citation Context ...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 (HTs) have been extended to the biorthogonal Haar domain [14], leading to... |

14 | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data
- DONOHO, J
- 2006
(Show Context)
Citation Context ...on power; 2) it can easily handle correlated data [37]. The latter point allows the FDR control in non-orthogonal wavelet domains. Minimaxity of FDR has also been studied in various settings (see [38]=-=[39]-=- for details). E. Iterative reconstruction Following the detection step, we have to invert the MS-VST scheme to reconstruct the estimate. For the standard UWT case, direct reconstruction procedure is ... |