## Image Denoising in Mixed Poisson–Gaussian Noise (2011)

Citations: | 10 - 1 self |

### BibTeX

@MISC{Luisier11imagedenoising,

author = {Florian Luisier and Thierry Blu and Michael Unser},

title = {Image Denoising in Mixed Poisson–Gaussian Noise},

year = {2011}

}

### OpenURL

### Abstract

We propose a general methodology (PURE-LET) to design and optimize a wide class of transform-domain 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 data-adaptive unbiased estimate of the mean-squared error (MSE), derived in a non-Bayesian framework (PURE: Poisson–Gaussian unbiased risk estimate). We provide a practical approximation of this theoretical MSE estimate for the tractable optimization of arbitrary transform-domain thresholding. We then propose a pointwise estimator for undecimated filterbank transforms, which consists of subband-adaptive thresholding functions with signal-dependent thresholds that are globally optimized in the image domain. We finally demonstrate the potential of the proposed approach through extensive comparisons with state-of-the-art techniques that are specifically tailored to the estimation of Poisson intensities. We also present denoising results obtained on real images of low-count fluorescence microscopy.

### Citations

702 | Adapting to unknown smoothness via wavelet shrinkage
- DONOHO, JOHNSTONE
- 1995
(Show Context)
Citation Context ...antage of the Skellam distribution of the unnormalized Haar wavelet coefficients to derive a so-called SkellamShrink [16], [17], which can be viewed as a Poisson variant of Donoho’s et al. SUREshrink =-=[18]-=-. Recently, we proposed a non-Bayesian framework to estimate Poisson intensities in the unnormalized Haar wavelet domain (PURE-LET [19], see Fig. 1 for an illustration of its principle). The qualitati... |

573 |
Ideal spatial adaptation via wavelet shrinkage. Biometrika
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...s for arbitrary wavelet shrinkage of “burst-like” Poisson intensities [21]. This pair of thresholds can be seen as an adapted version of Donoho’s et al. universal threshold that was designed for AWGN =-=[22]-=-. This approach was generalized to arbitrary kinds of Poisson-distributed data by Charles and Rasson [23]. Based on the statistical method of cross validation, Nowak et al. derived a wavelet shrinkage... |

370 | Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain
- Portilla, Strela, et al.
- 2003
(Show Context)
Citation Context ...mixed Poisson-Gaussian noise. Among various image-denoising strategies, the transform-domain approaches in general, and in particular the multiscale ones, are very efficient for AWGN reduction (e.g., =-=[1]-=-–[3]). As many natural images can be represented by few significant coefficients in a suitable basis/frame, the associated transform-domain processing amounts to a (possibly multivariate) thresholding... |

302 |
M.: Image denoising via sparse and redundant representations over learned dictionaries
- Elad, Aharon
- 2006
(Show Context)
Citation Context ...al. in [38] and Fadili et al. in [39]. The use of an overcomplete dictionary, either fixed in advance (as in our case) or trained, is at the core of the K-SVD-based denoising algorithm of Elad et al. =-=[40]-=-.704 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 3, MARCH 2011 Fig. 8. (a) Part of the original Moon image at peak intensity 20. (b) Noisy version, dB. (c) Denoised with a translation-invaria... |

300 | Estimation of the Mean of a Multivariate Normal Distribution,” The Annals of Statistics 9 - Stein - 1981 |

106 | Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data
- Donoho
- 1993
(Show Context)
Citation Context ...ally performed by applying a nonlinear mapping (e.g., a square root) to the raw data, an approach that has been theorized by Anscombe in [7] and first exploited in denoising applications by Donoho in =-=[8]-=-. The so-called Anscombe variance-stabilizing transform (VST) has been later generalized by Murtagh et al. to stabilize the variance of a Poisson random variable corrupted by AWGN [9]. After stabiliza... |

93 |
The transformation of Poisson, binomial and negative-binomial data
- Anscombe
- 1948
(Show Context)
Citation Context ...nsists in “Gaussianizing” the Poisson measurements. This is usually performed by applying a nonlinear mapping (e.g., a square root) to the raw data, an approach that has been theorized by Anscombe in =-=[7]-=- and first exploited in denoising applications by Donoho in [8]. The so-called Anscombe variance-stabilizing transform (VST) has been later generalized by Murtagh et al. to stabilize the variance of a... |

57 | Multiscale modeling and estimation of poisson processes with application to photon-limited imaging
- Timmermann, Nowak
- 1999
(Show Context)
Citation Context ... child (scaling coefficient at the next finer scale) is very simple; the distribution of a child conditioned on its parent is binomial. These properties have been exploited in a Bayesian framework in =-=[11]-=-–[14], as well as in a user-calibrated hypothesis testing [15]. Hirakawa 1 See Fig. 1 for a filterbank implementation of the unnormalized Haar wavelet transform. 1057-7149/$26.00 © 2011 IEEELUISIER e... |

57 | Multiscale likelihood analysis and complexity penalized estimation,” to appear in Annals of Stat.. Available at http://www.ece.wisc.edu/∼nowak/pubs.html
- Kolaczyk, Nowak
(Show Context)
Citation Context ... concept of multiscale likelihood factorizations, Kolaczyk and Nowak introduced complexity-penalized estimators that can also handle a wide class of distributions (Gaussian, Poisson, and multinomial) =-=[27]-=-. This methodology was further exploited by Willett et al. who proposed a platelet-based penalized likelihood estimator that has been demonstrated to be particularly efficient for denoising piecewise-... |

54 |
A Haar-Fisz algorithm for Poisson intensity estimation
- Fryzlewicz, Nason
- 2004
(Show Context)
Citation Context ...l. Fryzlewicz et al. have proposed a VST based on the observation that the scaling coefficients at a given scale are good local estimates of the noise variances of the same-scale wavelet coefficients =-=[10]-=-. Their approach gave state-of-the-art denoising results (in the minimum mean-squared error sense) at the time of its publication (2004). Another interesting property of the unnormalized Haar transfor... |

51 | A new SURE approach to image denoising: Interscale orthonormal wavelet thresholding - Luisier, Blu, et al. - 2007 |

47 | Bayesian multiscale models for Poisson processes - Kolaczyk - 1999 |

36 | Wavelet domain filtering for photon imaging systems
- Nowak, Baraniuk
- 1999
(Show Context)
Citation Context ...uted data by Charles and Rasson [23]. Based on the statistical method of cross validation, Nowak et al. derived a wavelet shrinkage, whose threshold is locally adapted to the estimated noise variance =-=[24]-=-. The modulation estimator devised by Antoniadis and Spatinas [25], which is based on cross-validation as well, covers all univariate natural exponential families with quadratic variance functions, of... |

35 |
Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds
- Kolaczyk
- 1999
(Show Context)
Citation Context ...Poisson-intensity estimators applicable to arbitrary multiscale transforms. Kolaczyk has developed (a pair of) soft/hard-thresholds for arbitrary wavelet shrinkage of “burst-like” Poisson intensities =-=[21]-=-. This pair of thresholds can be seen as an adapted version of Donoho’s et al. universal threshold that was designed for AWGN [22]. This approach was generalized to arbitrary kinds of Poisson-distribu... |

31 |
The SURE-LET approach to image denoising
- Blu, Luisier
- 2007
(Show Context)
Citation Context ...d Poisson-Gaussian noise. Among various image-denoising strategies, the transform-domain approaches in general, and in particular the multiscale ones, are very efficient for AWGN reduction (e.g., [1]–=-=[3]-=-). As many natural images can be represented by few significant coefficients in a suitable basis/frame, the associated transform-domain processing amounts to a (possibly multivariate) thresholding of ... |

25 | Image restoration with noise suppression using a multiresolution support
- Murtagh, Starck, et al.
(Show Context)
Citation Context ...ons by Donoho in [8]. The so-called Anscombe variance-stabilizing transform (VST) has been later generalized by Murtagh et al. to stabilize the variance of a Poisson random variable corrupted by AWGN =-=[9]-=-. After stabilization, any high-quality AWGN denoiser can be applied (e.g., [1]). Several works take advantage of the fact that the unnormalized Haar wavelet transform 1 has the remarkable property of... |

25 |
Wavelet shrinkage for natural exponential families with cubic variance functions. Sankhy a
- Antoniadis, Besbeas, et al.
- 2001
(Show Context)
Citation Context ...hod of cross validation, Nowak et al. derived a wavelet shrinkage, whose threshold is locally adapted to the estimated noise variance [24]. The modulation estimator devised by Antoniadis and Spatinas =-=[25]-=-, which is based on cross-validation as well, covers all univariate natural exponential families with quadratic variance functions, of which Gaussian and Poisson distributions are two particular cases... |

24 | Multiscale poisson intensity and density estimation. Submitted to
- Willett, Nowak
(Show Context)
Citation Context ...ogy was further exploited by Willett et al. who proposed a platelet-based penalized likelihood estimator that has been demonstrated to be particularly efficient for denoising piecewise-smooth signals =-=[28]-=-. Recently, the VST-based approach has been revitalized thanks to the contributions of Jansen, who combined VST with multiscale Bayesian models [29], and Zhang et al. who proposed a multiscale VST tha... |

24 |
A comparative simulation study of wavelet shrinkage estimators for Poisson counts
- Besbeas, Feis, et al.
- 2004
(Show Context)
Citation Context ...elets) [30]. This latter VST solution can also stabilize Poisson data embedded in AWGN [31]. Most of the denoising algorithms discussed in this section have been carefully evaluated by Besbeas et al. =-=[32]-=- using 1-D data. Most of them are specifically designed for pure Poisson noise. To the best of our knowledge, there are very few denoising algorithms that can properly handle mixed Poisson-Gaussian no... |

22 | Multiscale Poisson data smoothing
- Jansen
- 2006
(Show Context)
Citation Context ...ly efficient for denoising piecewise-smooth signals [28]. Recently, the VST-based approach has been revitalized thanks to the contributions of Jansen, who combined VST with multiscale Bayesian models =-=[29]-=-, and Zhang et al. who proposed a multiscale VST that can better stabilize very low intensity signals, and showed how it can be efficiently used with the latest multiresolution transformations (e.g., ... |

22 |
Very high quality image restoration by combining wavelets and curvelets
- Starck, Donoho, et al.
- 2001
(Show Context)
Citation Context ...dictionary of bases which sparsely represents a wide class of natural images. The idea of combining several complementary transforms was exploited in the context of AWGN reduction by Starck et al. in =-=[38]-=- and Fadili et al. in [39]. The use of an overcomplete dictionary, either fixed in advance (as in our case) or trained, is at the core of the K-SVD-based denoising algorithm of Elad et al. [40].704 I... |

21 | A variational approach to reconstructing images corrupted by poisson noise - Le, Chartran, et al. - 2007 |

20 | Automatic smoothing with wavelets for a wide class of distributions
- Sardy, Antoniadis, et al.
- 2004
(Show Context)
Citation Context ...ributions are two particular cases. Sardy et al. proposed a generalization of Donoho and Johnstone’s wavelet shrinkage for a broad class of exponential noise distributions, including the Poisson case =-=[26]-=-. Their estimator is the solution of a log-likelihood problem, regularized by the addition of a wavelet-domain -penalty. Using the concept of multiscale likelihood factorizations, Kolaczyk and Nowak i... |

19 | ridgelets and curvelets for Poisson noise removal
- Zhang, Fadili, et al.
(Show Context)
Citation Context ... et al. who proposed a multiscale VST that can better stabilize very low intensity signals, and showed how it can be efficiently used with the latest multiresolution transformations (e.g., curvelets) =-=[30]-=-. This latter VST solution can also stabilize Poisson data embedded in AWGN [31]. Most of the denoising algorithms discussed in this section have been carefully evaluated by Besbeas et al. [32] using ... |

19 | Learning to be Bayesian without supervision
- Raphan, Simoncelli
- 2007
(Show Context)
Citation Context ...have that (9) Putting back (11), (12), and (13) into (6) finally leads to the desired result (8). E. The PURE-LET Strategy Similarly to what has been proposed for SURE-based denoising [2], [3], [33], =-=[37]-=-, we describe the denoising function as the linear expansion of thresholds (LET) defined as (14) (10) Thanks to this linear parameterization, PURE becomes quadratic in the ’s. Therefore, the search fo... |

19 |
The colored revolution of bioimaging
- Vonesch, Aguet, et al.
- 2006
(Show Context)
Citation Context ...: IMAGE DENOISING IN MIXED POISSON–GAUSSIAN NOISE 707 channel and Alexa568 in the red channel). The average of the 100 images provides a “ground truth” for visual comparison [Fig. 10(b)]. We refer to =-=[41]-=- for a general introduction to fluorescence microscopy and to [42] for a detailed investigation on the various sources of noise in confocal microscopy. In order for our data to fit the noise model des... |

18 |
A natural identity for exponential families with applications in multiparameter estimation
- Hudson
- 1978
(Show Context)
Citation Context ...ly of vectors be the canonical basis of . Then where . This property can be thought of as the “Poisson’s equivalent” of Stein’s lemma. A proof of a similar result can be found, for instance, in [19], =-=[34]-=-, and [35]. B. PURE: An Unbiased MSE Estimate for Poisson Data Degraded by AWGN In practice, we obviously do not have access to the original noise-free signal . Therefore, we cannot compute the actual... |

14 |
nonparametric estimation of intensity maps using Haar wavelets and Poisson noise characteristics
- Kolaczyk, Dixon
- 2000
(Show Context)
Citation Context ...imple; the distribution of a child conditioned on its parent is binomial. These properties have been exploited in a Bayesian framework in [11]–[14], as well as in a user-calibrated hypothesis testing =-=[15]-=-. Hirakawa 1 See Fig. 1 for a filterbank implementation of the unnormalized Haar wavelet transform. 1057-7149/$26.00 © 2011 IEEELUISIER et al.: IMAGE DENOISING IN MIXED POISSON–GAUSSIAN NOISE 697 Fig... |

10 |
Multiframe SURE-LET denoising of timelapse fluorescence microscopy images
- Delpretti, Luisier, et al.
- 2008
(Show Context)
Citation Context ...as well as the variance of the AWGN, are determined from the data. In practice, a robust linear regression is first performed on a collection of local estimates of the sample mean and sample variance =-=[43]-=-, [44]; then, the slope of the fitted line gives the amplification factor . The parameters and can be estimated independently in signal-free regions of the image. We have applied our UWT PURE-LET algo... |

9 | Improved poisson intensity estimation: Denoising application using poisson data.IEEE Trans - Lu, Kim, et al. |

7 | Optimal denoising in redundant representations
- Raphan, Simoncelli
- 2008
(Show Context)
Citation Context ...ases. Because we are dealing with transformations that are not necessarily orthonormal anymore, the denoising process has to be optimized in the image domain, to ensure a global MSE minimization [3], =-=[33]-=-. To make tractable the optimization of arbitrary transform-domain processing, we also need a practical approximation of the PURE. All these extensions allow us to get the best (in the minimum PURE se... |

6 | Multilevel algorithms for a poisson noise removal model with total variation regularization - Chan, Chen - 2007 |

6 | Bayesian inference on multiscale models for Poisson intensity estimation: applications to photonlimited image denoising
- Lefkimmiatis, Maragos, et al.
- 2009
(Show Context)
Citation Context ...d (scaling coefficient at the next finer scale) is very simple; the distribution of a child conditioned on its parent is binomial. These properties have been exploited in a Bayesian framework in [11]–=-=[14]-=-, as well as in a user-calibrated hypothesis testing [15]. Hirakawa 1 See Fig. 1 for a filterbank implementation of the unnormalized Haar wavelet transform. 1057-7149/$26.00 © 2011 IEEELUISIER et al.... |

6 |
Multiscale variance-stabilizing transform for mixed-Poisson-Gaussian processes and its applications in bioimaging
- Zhang, Fadili, et al.
- 2007
(Show Context)
Citation Context ...ty signals, and showed how it can be efficiently used with the latest multiresolution transformations (e.g., curvelets) [30]. This latter VST solution can also stabilize Poisson data embedded in AWGN =-=[31]-=-. Most of the denoising algorithms discussed in this section have been carefully evaluated by Besbeas et al. [32] using 1-D data. Most of them are specifically designed for pure Poisson noise. To the ... |

6 | Patch-based nonlocal functional for denoising fluorescence microscopy image sequences
- Boulanger, Kervrann, et al.
(Show Context)
Citation Context ...l as the variance of the AWGN, are determined from the data. In practice, a robust linear regression is first performed on a collection of local estimates of the sample mean and sample variance [43], =-=[44]-=-; then, the slope of the fitted line gives the amplification factor . The parameters and can be estimated independently in signal-free regions of the image. We have applied our UWT PURE-LET algorithm ... |

5 |
Fast interscale wavelet denoising of Poisson-corrupted images
- Luisier, Vonesch, et al.
- 2010
(Show Context)
Citation Context ...AGE DENOISING IN MIXED POISSON–GAUSSIAN NOISE 697 Fig. 1. Filterbank implementation of the unnormalized discrete Haar wavelet transform and principle of the class of denoising algorithms described in =-=[19]-=-. The superscript indicates the level of decomposition; is the vector of noisy scaling coefficients ( is thus the noisy input); is the vector of noisy wavelet coefficients; is the subband-dependent th... |

5 |
Sources of Noise in Three-Dimensional Microscopical Data Sets
- PAWLEY
- 1994
(Show Context)
Citation Context ...Alexa568 in the red channel). The average of the 100 images provides a “ground truth” for visual comparison [Fig. 10(b)]. We refer to [41] for a general introduction to fluorescence microscopy and to =-=[42]-=- for a detailed investigation on the various sources of noise in confocal microscopy. In order for our data to fit the noise model described in (1), we need to first subtract the offset value of the C... |

4 |
Simultaneous estimation of several Poisson parameters under k-normalized squared error loss, Ann
- Tsui, Press
- 1982
(Show Context)
Citation Context ...ors be the canonical basis of . Then where . This property can be thought of as the “Poisson’s equivalent” of Stein’s lemma. A proof of a similar result can be found, for instance, in [19], [34], and =-=[35]-=-. B. PURE: An Unbiased MSE Estimate for Poisson Data Degraded by AWGN In practice, we obviously do not have access to the original noise-free signal . Therefore, we cannot compute the actual MSE and m... |

3 | SkellamShrink: Poisson intensity estimation for vector-valued data
- Hirakawa, Wolfe
- 2009
(Show Context)
Citation Context ...he estimated noise-free wavelet (resp. scaling) coefficients. et al. have taken advantage of the Skellam distribution of the unnormalized Haar wavelet coefficients to derive a so-called SkellamShrink =-=[16]-=-, [17], which can be viewed as a Poisson variant of Donoho’s et al. SUREshrink [18]. Recently, we proposed a non-Bayesian framework to estimate Poisson intensities in the unnormalized Haar wavelet dom... |

3 |
Wavelet denoising of Poisson-distributed data and applications
- Charles, Rasson
(Show Context)
Citation Context ...e seen as an adapted version of Donoho’s et al. universal threshold that was designed for AWGN [22]. This approach was generalized to arbitrary kinds of Poisson-distributed data by Charles and Rasson =-=[23]-=-. Based on the statistical method of cross validation, Nowak et al. derived a wavelet shrinkage, whose threshold is locally adapted to the estimated noise variance [24]. The modulation estimator devis... |

3 |
Morphological diversity and sparse image denoising
- Fadili, Starck, et al.
(Show Context)
Citation Context ...sparsely represents a wide class of natural images. The idea of combining several complementary transforms was exploited in the context of AWGN reduction by Starck et al. in [38] and Fadili et al. in =-=[39]-=-. The use of an overcomplete dictionary, either fixed in advance (as in our case) or trained, is at the core of the K-SVD-based denoising algorithm of Elad et al. [40].704 IEEE TRANSACTIONS ON IMAGE ... |

2 |
Efficient Multivariate Skellam Shrinkage for Denoising Photon-Limited Image Data: An Empirical Bayes Approach
- Hirakawa, Wolfe
- 2009
(Show Context)
Citation Context ...imated noise-free wavelet (resp. scaling) coefficients. et al. have taken advantage of the Skellam distribution of the unnormalized Haar wavelet coefficients to derive a so-called SkellamShrink [16], =-=[17]-=-, which can be viewed as a Poisson variant of Donoho’s et al. SUREshrink [18]. Recently, we proposed a non-Bayesian framework to estimate Poisson intensities in the unnormalized Haar wavelet domain (P... |

2 |
Deconvolution of 3d fluorescence micrographs with automatic risk minimization
- Ramani, Vonesch, et al.
- 2008
(Show Context)
Citation Context ...ved for AWGN only, and the PURE we recently exposed in [19], which was devised for pure Poisson noise reduction in the unnormalized Haar wavelet domain. Note that the result of Theorem 1 was given in =-=[36]-=- without proof, where it was applied to linear image deconvolution. The variance of the PURE estimator notably depends on the number of samples , as well as on the number of parameters (degrees of fre... |

1 |
Computer analysis of images and patterns
- Sawatzky, Brune, et al.
- 2009
(Show Context)
Citation Context ...egies, we discuss hereafter the main multiscale techniques that have been considered for Poisson intensity estimation. Note that there are also non-multiscale methods for Poisson denoising, e.g., [4]–=-=[6]-=-. A. Related Work Since the Poisson statistics are generally more difficult to track in a transformed domain than the traditional Gaussian ones, a natural solution consists in “Gaussianizing” the Pois... |