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## Image denoising using a scale mixture of Gaussians in the wavelet domain (2003)

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Venue: | IEEE TRANS IMAGE PROCESSING |

Citations: | 509 - 17 self |

### Citations

3448 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...is is reflected in the observed marginal distributions of bandpass filter responses, which show a large peak at zero, and tails that fall significantly slower than a Gaussian of the same variance [5]�=-=��[7]-=- [see Fig. 1(a)]. When one seeks a linear transformation that maximizes the non-Gaussianity1 of the marginal responses, the result is a basis set of bandpass oriented filters of different sizes spanni... |

1511 |
Embedded image coding using zerotrees of wavelet coefficients
- Shapiro
- 1993
(Show Context)
Citation Context ...es the denoising performance, on average. Inclusion of parent coefficient has been found to provide a significant improvement in performance in a number of applications, e.g., [21], [22], [25], [26], =-=[46]-=-. Note that since the parent subband is sampled at half the density of the reference subband, it must be upsampled and interpolated in order to obtain values for neighborhoods at every choice of refer... |

1287 |
Emergence of simple-cell receptive field properties by learning a sparse code for natural images
- Olshausen, Field
- 1996
(Show Context)
Citation Context ... transformation that maximizes the non-Gaussianity1 of the marginal responses, the result is a basis set of bandpass oriented filters of different sizes spanning roughly an octave in bandwidth, e.g., =-=[8]-=-, [9]. Due to the combination of these qualitative properties, as well as an elegant mathematical framework, multiscale oriented subband decompositions have emerged as the representations of choice fo... |

1236 | Ideal spatial adaptation by wavelet shrinkage
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...g literature (where it is known as “coring,” e.g., [10]), and specific shrinkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condit=-=ion [11], -=-[12], [57], and Bayesian estimation with non-Gaussian priors, e.g., [13]–[19], [58]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussi... |

1072 | The design and use of steerable filters
- Freeman, Adelson
- 1991
(Show Context)
Citation Context ... increased selectivity in orientation [16], [19], [34], [38]. For the current paper, we have used a particular variant of an overcomplete tight frame representation known as a steerable pyramid [38], =-=[39]-=-. The basis functions of this multiscale linear decomposition are spatially localized, oriented, and span roughly one octave in bandwidth. They are polar-separable in the Fourier domain, and are relat... |

817 | Relations between the statistics of natural images and the response properties of cortical cells
- Field
- 1987
(Show Context)
Citation Context .... This is reflected in the observed marginal distributions of bandpass filter responses, which show a large peak at zero, and tails that fall significantly slower than a Gaussian of the same variance =-=[5]��-=-�[7] [see Fig. 1(a)]. When one seeks a linear transformation that maximizes the non-Gaussianity1 of the marginal responses, the result is a basis set of bandpass oriented filters of different sizes sp... |

719 |
Bayesian Inference in Statistical Analysis
- Box, Tiao
- 1992
(Show Context)
Citation Context ...cally. One advantage of the ML solution is that it is easily extended for use with the noisy observations, by replacing with the noisy observation. A fourth choice is a so-called noninformative prior =-=[44]-=-, which has the advantage that it does not require the fitting of any parameters to the noisy observation. Such solutions have been used in establishing marginal priors for image denoising [45]. We ha... |

612 | The "independent components" of natural scenes are edge filters
- Bell, Sejnowski
- 1997
(Show Context)
Citation Context ...sformation that maximizes the non-Gaussianity1 of the marginal responses, the result is a basis set of bandpass oriented filters of different sizes spanning roughly an octave in bandwidth, e.g., [8], =-=[9]-=-. Due to the combination of these qualitative properties, as well as an elegant mathematical framework, multiscale oriented subband decompositions have emerged as the representations of choice for man... |

554 | Shiftable multi-scale transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...l densitysPORTILLA et al.: IMAGE DENOISING USING SCALE MIXTURES OF GAUSSIANS IN THE WAVELET DOMAIN 1349 Fig. 7. (a) System diagram for the extended version of the steerable pyramid used in this paper =-=[38]-=-. The input image is first split into a lowpass band and a set of highpass oriented bands. The lowpass band is then split into a lower-frequency band and a set of oriented subbands. The pyramid recurs... |

413 | Wavelet-based statistical signal processing using hidden markov models. to appear
- Crouse, Nowak, et al.
- 1998
(Show Context)
Citation Context ... images. For example, Baraniuk and colleagues used a 2-state hidden multiplier variable to characterize the two modes of behavior corresponding to smooth or low-contrast textured regions and features =-=[25]-=-, [26]. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable [2], [3], [27], [28]. This model can capture the strongly leptokurtotic... |

407 | A parametric texture model based on joint statistics of complex wavelet coefficients
- Portilla, Simoncelli
(Show Context)
Citation Context ... important structural properties of local image features, by including additional dependencies such as phase congruency between the coefficients of complex multiscale oriented transforms, e.g., [55], =-=[56]-=-. APPENDIX A STEERABLE PYRAMID We use a transform known as a steerable pyramid [38], [39] to decompose images into frequency subbands. The transform is implemented in the Fourier domain, allowing exac... |

404 | ARCH Models
- Bollerslev, Engle, et al.
- 1994
(Show Context)
Citation Context ...a random field, these kinds of models have been found useful in the speech-processing community [23]. A related set of models, known as autoregressive conditional heteroskedastic (ARCH) models, e.g., =-=[24], ha-=-ve proven useful for many real signals that suffer from abrupt fluctuations, followed by relative “calm” periods (stock market prices, for example). These kinds of ideas have also been found effec... |

390 | The curvelet transform for image denoising
- Starck, Candès, et al.
- 2002
(Show Context)
Citation Context ...e functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], [57], and Bayesian estimation with non-Gaussian priors, e.g., [13]�=-=��[19]-=-, [58]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when they are secondorder de... |

378 | Complex wavelets for shift invariant analysis and filtering of signals
- Kingsbury
(Show Context)
Citation Context ...e in bandwidth. They are polar-separable in the Fourier domain, and are related by translation, dilation, and rotation. Other authors have developed representations with similar properties [19], [40]�=-=��[42]-=-. Details of the steerable pyramid representation are provided in Appendix A. A. Gaussian Scale Mixtures Consider an image decomposed into oriented subbands at multiple scales. We denote as the coeffi... |

318 |
Digital image enhancement and noise filtering by use of local statistics
- Lee
- 1980
(Show Context)
Citation Context ...mplitudes, as illustrated in Fig. 1.s1340 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 11, NOVEMBER 2003 B. Empirical Bayes Denoising Using Variance-Adaptive Models More than 20 years ago, Lee =-=[29]-=- suggested a two-step procedure for image denoising, in which one first estimates the local signal variance from a neighborhood of observed pixels, and then (proceeding as if this were the true varian... |

259 | Wavelet thresholding via a Bayesian approach
- Abramovich, Sapatinas, et al.
- 1998
(Show Context)
Citation Context ...ge functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], and Bayesian estimation with non-Gaussian priors [e.g. 13], [14], =-=[15]-=-, [16], [17], [18], [19]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when they ... |

255 | Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage
- Chambolle, DeVore, et al.
- 1998
(Show Context)
Citation Context ... (where it is known as “coring,” e.g., [10]), and specific shrinkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [1=-=2], [57], -=-and Bayesian estimation with non-Gaussian priors, e.g., [13]–[19], [58]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint sta... |

237 | Image compression via joint statistical characterization in the wavelet domain
- Buccigrossi, Simoncelli
- 1999
(Show Context)
Citation Context ...ur in clusters. The conditional histograms of pairs of coefficients indicates that the standard deviation of a coefficient scales roughly linearly with the amplitude of nearby coefficients [2], [21], =-=[22]-=- [see Fig. 1(c)]. The dependency between local coefficient amplitudes, as well as the associated marginal behaviors, can be modeled using a random field with a spatially fluctuating variance. A partic... |

230 | Noise removal via bayesian wavelet coring
- Simoncelli, Adelson
- 1996
(Show Context)
Citation Context ...inkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], [57], and Bayesian estimation with non-Gaussian priors, e.g., =-=[13]��-=-�[19], [58]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when they are secondord... |

225 |
Scale mixtures of normal distributions
- Andrews, Mallows
- 1974
(Show Context)
Citation Context ...dden Markov models have become widely used, for example, in speech processing. In this article, we develop a model for neighborhoods of oriented pyramid coefficients based on a Gaussian scale mixture =-=[1]-=-: the product of a Gaussian random vector, and an independent hidden random scalar multiplier. We have previously demonstrated that this model can account for both marginal and pairwise joint distribu... |

217 | Spatially adaptive wavelet thresholding with content modeling for image denoising
- Chang, Yu, et al.
- 1998
(Show Context)
Citation Context ...omparison of denoising performance of several recently published methods. Curves depict output PSNR as a function of input PSNR. Square symbols indicate our results, taken from Table I. (a,b) circles =-=[32]-=-; crosses [35]; asterisk [52] 3 ; (c,d) crosses [31]; diamonds [51]. undecimated separable wavelet transform. The decrease is substantial, however, in the case of the critically sampled representation... |

214 | Analysis of multiresolution image denoising schemes using generalized-gaussian and complexity priors, in
- Moulin, Liu
- 1999
(Show Context)
Citation Context ... the multiplier. Several authors have suggested the generalized Gaussian (stretched exponential) family of densities as an appropriate description of wavelet coefficient marginal densities [7], [13], =-=[17]-=-: where the scaling variable controls the width of the distribution, and the exponent controls the shape (in particular, the heaviness of the tails), and is typically estimated to lie in the range for... |

203 |
Adaptive bayesian wavelet shrinkage
- Chipman, Kolaczyk, et al.
- 1997
(Show Context)
Citation Context ...hrinkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], and Bayesian estimation with non-Gaussian priors [e.g. 13], =-=[14]-=-, [15], [16], [17], [18], [19]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when... |

201 | Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency
- Sendur, Selesnick
- 2002
(Show Context)
Citation Context ...me subband have a similar effect on the estimation. Sendur and Selesnick have recently developed a MAP estimator based on a circular-symmetric Laplacian density model for a coefficient and its parent =-=[49], [5-=-0]. Their resulting shrinkage function is qualitatively similar to that of Fig. 2(b), except that ours is smoother and, due to covariance adaptation, its “dead zone” is not necessary aligned with ... |

200 |
The statistics of natural images
- Ruderman
- 1993
(Show Context)
Citation Context ...pposed to considering only spatial neighbors within each subband. This modeling choice is consistent with the empirical findings of strong statistical dependence across scale in natural images, e.g., =-=[4]-=-, [46]. Note, however, that the inclusion of the parent results in only a modest increase in performance compared to the other elements shown in Table II. We believe the impact of including a parent i... |

185 | Low-complexity image denoising based on statistical modeling of wavelet coefficients
- Mihcak, Kozintsev, et al.
- 1999
(Show Context)
Citation Context ...ooth or low-contrast textured regions and features [25], [26]. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable [2], [3], [27], =-=[28]-=-. This model can capture the strongly leptokurtotic behavior of the marginal densities of natural image wavelet coefficients, as well as the correlation in their local amplitudes, as illustrated in Fi... |

182 | Bayesian tree-structured image modeling using Wavelet-domain hidden Markov models
- Romberg, Choi, et al.
- 2001
(Show Context)
Citation Context ...s. For example, Baraniuk and colleagues used a 2-state hidden multiplier variable to characterize the two modes of behavior corresponding to smooth or low-contrast textured regions and features [25], =-=[26]-=-. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable [2], [3], [27], [28]. This model can capture the strongly leptokurtotic behav... |

168 | Scale mixtures of Gaussians and the statistics of natural images. Advances in neural information processing systems
- Wainwright, Simoncelli
(Show Context)
Citation Context ...ndom vector, and an independent hidden random scalar multiplier. We have previously demonstrated that this model can account for both marginal and pairwise joint distributions of wavelet coefficients =-=[2]-=-, [3]. Here, we develop a local denoising solution as a Bayesian least squares estimator, and demonstrate the performance of this method on images corrupted by simulated additive white Gaussian noise ... |

167 |
The cortex transform: rapid computation of simulatedneuralimages,”Computer Vision
- Watson
- 1987
(Show Context)
Citation Context ...octave in bandwidth. They are polar-separable in the Fourier domain, and are related by translation, dilation, and rotation. Other authors have developed representations with similar properties [19], =-=[40]��-=-�[42]. Details of the steerable pyramid representation are provided in Appendix A. A. Gaussian Scale Mixtures Consider an image decomposed into oriented subbands at multiple scales. We denote as the c... |

166 | Ridgelets: a key to higher-dimensional intermittency
- Candès, Donoho
- 1999
(Show Context)
Citation Context ...octave in bandwidth. They are polar-separable in the Fourier domain, and are related by translation, dilation, and rotation. Other authors have developed representations with similar properties [40], =-=[41]-=-, [42], [19]. Details of the steerable pyramid representation are provided in Appendix I. A. Gaussian scale mixtures Consider an image decomposed into oriented subbands at multiple scales. We denote a... |

160 | Statistical models for images: Compression, restoration and synthesis
- Simoncelli
- 1997
(Show Context)
Citation Context .... In particular, even when they are secondorder decorrelated, the coefficients corresponding to pairs of basis functions of similar position, orientation and scale exhibit striking dependencies [20], =-=[21]-=-. Casual observation indicates that large-amplitude coefficients are sparsely distributed 1 Different authors have used different measures of non-Gaussianity, but have obtained similar results. (a) (b... |

160 | Image features from phase congruency. Videre - Kovesi - 1999 |

131 | Nonlinear wavelet shrinkage with bayes rules and bayes factors
- Vidakovic
- 1998
(Show Context)
Citation Context ...tions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], [57], and Bayesian estimation with non-Gaussian priors, e.g., [13]–[19]=-=, [58]-=-. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when they are secondorder decorrel... |

121 | Bivariate shrinkage with local variance estimation
- Sendur, Selesnick
- 2002
(Show Context)
Citation Context ...and have a similar effect on the estimation. S¸endur and Selesnick have recently developed a MAP estimator based on a circular-symmetric Laplacian density model for a coefficient and its parent [51], =-=[52]-=-. Their resulting shrinkage function is qualitatively similar to that of Fig. 2(b), except that ours is smoother and, due to covariance adaptation, its ”dead zone” is not necessary aligned with the in... |

116 | Sparse code shrinkage: Denoising of nongaussian data by maximum likelihood estimation
- Hyvärinen
- 1999
(Show Context)
Citation Context ...been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], and Bayesian estimation with non-Gaussian priors [e.g. 13], [14], [15], [16], [17], =-=[18]-=-, [19]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joint statistical behavior. In particular, even when they are second-order d... |

110 | Origins of scaling in natural images
- Ruderman
- 1997
(Show Context)
Citation Context ...els, the spatial extent of the neighborhood depends on the scale of the subband (the basis functions grow in size as ) as is appropriate under the assumption that image statistics are scale-invariant =-=[47]-=-, [48]. In our implementation, the integral of (8) is computed numerically. The range and sample spacing for this integration are chosen as a compromise between accuracy and computational cost. Specif... |

107 | Bayesian Denoising of Visual Images in the Wavelet Domain
- Simoncelli
- 1999
(Show Context)
Citation Context ...e estimated the local variance from a collection of wavelet coefficients at nearby positions, scales, and/or orientations, and then used these estimated variances in order to denoise the coefficients =-=[16], -=-[21], [28], [31]–[33]. Solutions based on GSM models, with different prior assumptions about the hidden variables, have produced some of the most effective methods for removing homogeneous additive ... |

96 | Random cascades on wavelet trees and their use in analyzing and modeling natural images
- Wainwright, Simoncelli, et al.
(Show Context)
Citation Context ...vector, and an independent hidden random scalar multiplier. We have previously demonstrated that this model can account for both marginal and pairwise joint distributions of wavelet coefficients [2], =-=[3]-=-. Here, we develop a local denoising solution as a Bayesian least squares estimator, and demonstrate the performance of this method on images corrupted by simulated additive white Gaussian noise of kn... |

93 |
Translation-invariant de-noising,” in Wavelet and
- Coifman, Donoho
- 1995
(Show Context)
Citation Context ...” or “ringing”). A widely followed solution to this problem is to use basis functions designed for orthogonal or biorthogonal systems, but to reduce or eliminate the decimation of the subbands, =-=e.g., [37]-=-. Once the constraint of critical sampling has been dropped, however, there is no need to limit oneself to these basis functions. Significant improvement comes from the use of representations with a h... |

75 | Image features from phase congruency
- Kovesi
- 1999
(Show Context)
Citation Context ...apture important structural properties of local image features, by including additional dependencies such as phase congruency between the coefficients of complex multiscale oriented transforms, e.g., =-=[55]-=-, [56]. APPENDIX A STEERABLE PYRAMID We use a transform known as a steerable pyramid [38], [39] to decompose images into frequency subbands. The transform is implemented in the Fourier domain, allowin... |

73 | The empirical Bayes approach to statistical decision problems - Robbins - 1964 |

72 | Wavelet-based image estimation: An empirical Bayes approach using Jeffrey’s noninformative prior
- Figueiredo, Nowak
- 2001
(Show Context)
Citation Context ... prior [44], which has the advantage that it does not require the fitting of any parameters to the noisy observation. Such solutions have been used in establishing marginal priors for image denoising =-=[45]. -=-We have examined the most widely used solution, known as Jeffrey’s prior (see [44]). In the context of estimating the multiplier from coefficients , this takes the form: where is the Fisher informat... |

65 |
Wavelet-based image denoising using a Markov random field a priori model
- Malfait, Roose
- 1997
(Show Context)
Citation Context ...cal variance from a collection of wavelet coefficients at nearby positions, scales, and/or orientations, and then used these estimated variances in order to denoise the coefficients [16], [21], [28], =-=[31]��-=-�[33]. Solutions based on GSM models, with different prior assumptions about the hidden variables, have produced some of the most effective methods for removing homogeneous additive noise from natural... |

65 | A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising
- Pizurica, Philips, et al.
- 2002
(Show Context)
Citation Context ...ince there are many different versions of the test images available on the Internet, whenever it was possible we have verified directly with the authors that we are using the same images (Refs. [35], =-=[51]-=-, [52], [50]), or have used other authors’ data included in previous comparisons from those authors (Refs. [31], [32]) (see Appendix II for more details about the origin of the images). 910 20 30 40 5... |

56 |
Entropy reduction and decorrelation in visual coding by oriented neural receptive fields
- Daugman
- 1989
(Show Context)
Citation Context ...s is reflected in the observed marginal distributions of bandpass filter responses, which show a large peak at zero, and tails that fall significantly slower than a Gaussian of the same variance [5], =-=[6]-=-, [7] (see Fig. 1(a)). When one seeks a linear transformation that maximizes the non-Gaussianity1 of the marginal responses, the result is a basis set of bandpass oriented filters of different sizes s... |

49 |
Description and generation of spherically invariant speech-model signals
- Brehm, Stammler
- 1987
(Show Context)
Citation Context ... which are those that can be defined as functions of a quadratic norm of the random vector. Embedded in a random field, these kinds of models have been found useful in the speech-processing community =-=[23]-=-. A related set of models, known as autoregressive conditional heteroskedastic (ARCH) models, e.g., [24], have proven useful for many real signals that suffer from abrupt fluctuations, followed by rel... |

47 | Adaptive Wiener Denoising using a gaussian scale mixture model in the wavelet domain - Portilla, Strela, et al. - 2001 |

46 | Image denoising using Gaussian scale mixtures in the wavelet domain
- Portilla, Strela, et al.
- 2003
(Show Context)
Citation Context ...We are currently working on several extensions of the estimator presented here. First, we have begun developing a variant of this method to denoise color images taken with a commercial digital camera =-=[55]-=-. We find that the sensor noise of such cameras has two important features that must be characterized through calibration measurements: spatial and cross-channel correlation, and signal-dependence. We... |

41 |
Spatially adaptive image denoising under overcomplete expansion
- Li, Orchard
- 2000
(Show Context)
Citation Context ...tial work in this area developed a maximum likelihood (ML) estimator [34]. Mihçak et al. used a maximum a posteriori (MAP) estimator based on an exponential marginal prior [28], as did Li and Orchard=-= [35]-=-, whereas Portilla et al. used a lognormal prior [36]. Wainwright et al. developed a tree-structured Markov model to provide a global description for the set of multiplier variables [3]. Despite these... |

29 |
Statistical dependence between orientation filter outputs used in an human vision based image code
- Wegmann, Zetzsche
- 1990
(Show Context)
Citation Context ...havior. In particular, even when they are secondorder decorrelated, the coefficients corresponding to pairs of basis functions of similar position, orientation and scale exhibit striking dependencies =-=[20]-=-, [21]. Casual observation indicates that large-amplitude coefficients are sparsely distributed 1 Different authors have used different measures of non-Gaussianity, but have obtained similar results. ... |

27 | Image denoising using a local gaussian scale mixture model in the wavelet domain
- Strela, Portilla, et al.
- 2000
(Show Context)
Citation Context ...bles, have produced some of the most effective methods for removing homogeneous additive noise from natural images to date. Our initial work in this area developed a maximum likelihood (ML) estimator =-=[34].-=- Mihçak et al. used a maximum a posteriori (MAP) estimator based on an exponential marginal prior [28], as did Li and Orchard [35], whereas Portilla et al. used a lognormal prior [36]. Wainwright et ... |

25 | Empirical Bayes approach to block wavelet function estimation
- Abramovich, Besbeas, et al.
- 2002
(Show Context)
Citation Context ...ariance from a collection of wavelet coefficients at nearby positions, scales, and/or orientations, and then used these estimated variances in order to denoise the coefficients [16], [21], [28], [31]�=-=��[33]-=-. Solutions based on GSM models, with different prior assumptions about the hidden variables, have produced some of the most effective methods for removing homogeneous additive noise from natural imag... |

20 | Denoising via Block Wiener Filtering in Wavelet Domain
- Strela
- 2000
(Show Context)
Citation Context ...s considerably simplified by partitioning the coefficients into nonoverlapping neighborhoods. One can then specify either a marginal model for the multipliers (treating them as independent variables) =-=[43]-=-, or specify a joint density over the full set of multipliers [3]. Unfortunately, the use of disjoint neighborhoods leads to noticeable denoising artifacts at the discontinuities introduced by the nei... |

18 |
A joint interand intrascale statistical model for Bayesian wavelet based image denoising
- Pizurica, Philips, et al.
(Show Context)
Citation Context ...thods. Curves depict output PSNR as a function of input PSNR. Square symbols indicate our results, taken from Table I. (a,b) circles [32]; crosses [35]; asterisk [52] 3 ; (c,d) crosses [31]; diamonds =-=[51]-=-. undecimated separable wavelet transform. The decrease is substantial, however, in the case of the critically sampled representation. From the comparison of the outcomes of both sets of experiments, ... |

17 | Image restoration using gaussian scale mixtures in the wavelet domain
- Portilla, Simoncelli
- 2003
(Show Context)
Citation Context ...s: spatial and cross-channel correlation, and signal-dependence. We are also extending the denoising solution to address the complete image restoration problem, by incorporating a model of image blur =-=[54]-=-. Finally, we are developing an ML estimator for the noise variance, when the normalized power spectral densitysPORTILLA et al.: IMAGE DENOISING USING SCALE MIXTURES OF GAUSSIANS IN THE WAVELET DOMAIN... |

13 |
Very high quality image restoration
- Starck, Donoho, et al.
- 2001
(Show Context)
Citation Context ...mance of several recently published methods. Curves depict output PSNR as a function of input PSNR. Square symbols indicate our results, taken from Table I. (a,b) circles [32]; crosses [35]; asterisk =-=[52]-=- 3 ; (c,d) crosses [31]; diamonds [51]. undecimated separable wavelet transform. The decrease is substantial, however, in the case of the critically sampled representation. From the comparison of the ... |

13 | Bayesian wavelet domain image modeling using hidden Markov trees
- Romberg, Choi, et al.
- 1999
(Show Context)
Citation Context ...s. For example, Baraniuk and colleagues used a 2-state hidden multiplier variable to characterize the two modes of behavior corresponding to smooth or low-contrast textured regions and features [25], =-=[26]-=-. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable [27], [2], [28], [3]. This model can capture the strongly leptokurtotic behav... |

10 |
Digital techniques for reducing television noise
- Rossi
- 1978
(Show Context)
Citation Context ... operation, suppressing low-amplitude values while retaining high-amplitude values. The concept was developed originally in the television engineering literature (where it is known as “coring,” e.=-=g., [10]-=-), and specific shrinkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [11], [12], [57], and Bayesian estimation with non-Gaus... |

10 | Duality of log-polar image representations in the space and the spatial-frequency domains
- Tabernero, Portilla, et al.
- 1999
(Show Context)
Citation Context ...he spatial extent of the neighborhood depends on the scale of the subband (the basis functions grow in size as ) as is appropriate under the assumption that image statistics are scale-invariant [47], =-=[48]-=-. In our implementation, the integral of (8) is computed numerically. The range and sample spacing for this integration are chosen as a compromise between accuracy and computational cost. Specifically... |

9 |
Wavelet image coding based on a new generalized gaussian mixture model
- LoPresto, Ramchandran, et al.
- 1997
(Show Context)
Citation Context ... to smooth or low-contrast textured regions and features [25], [26]. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable [2], [3], =-=[27]-=-, [28]. This model can capture the strongly leptokurtotic behavior of the marginal densities of natural image wavelet coefficients, as well as the correlation in their local amplitudes, as illustrated... |

8 |
Buccigrossi and E P Simoncelli. Image compression via joint statistical characterization in the wavelet domain
- W
- 1999
(Show Context)
Citation Context ...o occur in clusters. The conditional histograms of pairs of coefficients indicates that the standard deviation of a coefficient scales roughly linearly with the amplitude of nearby coefficients [23], =-=[24]-=-, [2] (see Fig. 1(c)). The dependency between local coefficient amplitudes, as well as the associated marginal behaviors, can be modeled using a random field with a spatially fluctuating variance. A p... |

5 |
Multiscale regularization in Besov spaces
- Leporini, Pesquet
- 1998
(Show Context)
Citation Context ...rature (where it is known as “coring,” e.g., [10]), and specific shrinkage functions have been derived under a variety of formulations, including minimax optimality under a smoothness condition [1=-=1], [12], -=-[57], and Bayesian estimation with non-Gaussian priors, e.g., [13]–[19], [58]. In addition to the non-Gaussian marginal behavior, the responses of bandpass filters exhibit important non-Gaussian joi... |

5 |
P Box and C Tiao, Bayesian Inference in Statistical Analysis
- E
- 1992
(Show Context)
Citation Context ...ly. One advantage of the ML solution is that it is easily extended for use with the noisy observations, by replacing xm with the noisy observation. A fourth choice is a so-called noninformative prior =-=[46]-=-, which has the advantage that it does not require the fitting of any parameters to the noisy observation. Such solutions have been used in establishing marginal priors for image denoising [47]. We ha... |

4 | A Olshausen and D J Field. Emergence of simple-cell receptive eld properties by learning a sparse factorial code - unknown authors - 1996 |

3 |
Image Denoising Using Gaussian Scale
- Portilla, Strela, et al.
- 2002
(Show Context)
Citation Context ...We are currently working on several extensions of the estimator presented here. First, we have begun developing a variant of this method to denoise color images taken with a commercial digital camera =-=[53]-=-. We find that the sensor noise of such cameras has two important features that must be characterized through calibration measurements: spatial and cross-channel correlation, and signal-dependence. We... |

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Wavelet image cod based on a new generalized Gaussian mixture model
- LoPresto, Ramchandran, et al.
- 1997
(Show Context)
Citation Context ...responding to smooth or low-contrast textured regions and features [27], [28]. Our own work, as well as that of others, assumes that the local variance is governed by a continuous multiplier variable =-=[29]-=-, [2], [30], [3]. This model can capture the strongly leptokurtotic behavior of the marginal densities of natural image wavelet coefficients, as well as the correlation in their local amplitudes, as i... |

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endur and I W Selesnick, “Bivariate shrinkage functions for waveletbased denoising exploiting interscale dependency
- Ş
- 2002
(Show Context)
Citation Context ...e subband have a similar effect on the estimation. Şendur and Selesnick have recently developed a MAP estimator based on a circular-symmetric Laplacian density model for a coefficient and its parent =-=[49]-=-, [50]. Their resulting shrinkage function is qualitatively similar to that of Fig. 2(b), except that ours is smoother and, due to covariance adaptation, its ”dead zone” is not necessary aligned with ... |

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endur and I W Selesnick, “Bivariate shrinkage with local variance estimation
- Ş
- 2002
(Show Context)
Citation Context ...and have a similar effect on the estimation. Şendur and Selesnick have recently developed a MAP estimator based on a circular-symmetric Laplacian density model for a coefficient and its parent [49], =-=[50]-=-. Their resulting shrinkage function is qualitatively similar to that of Fig. 2(b), except that ours is smoother and, due to covariance adaptation, its ”dead zone” is not necessary aligned with the in... |