A Unified View on Patch Aggregation

Saint‐Dizier, Alexandre; Delon, Julie; Bouveyron, Charles

doi:10.1007/s10851-019-00921-z

Cited by 4 publications

(3 citation statements)

References 37 publications

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“…Concerning our denoising approach, fine tuning steps as discussed in [28] such as aggregations of patches [42], the use of an oracle image or a variable patch size to better cope with textured and homogeneous image regions may improve the denoising results. Further, in all our examples we used uniform weights, but weights based for instance on spatial distance or similarity would make sense as well.…”

Section: Resultsmentioning

confidence: 99%

Multivariate Myriad Filters Based on Parameter Estimation of Student-$t$ Distributions

Laus¹,

Steidl²

2019

SIAM J. Imaging Sci.

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The contribution of this study is twofold: First, we propose an efficient algorithm for the computation of the (weighted) maximum likelihood estimators for the parameters of the multivariate Student-t distribution, which we call generalized multivariate myriad filter. Second, we use the generalized multivariate myriad filter in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle heavy-tailed noise such as Cauchy noise, as well as the other extreme, namely Gaussian noise. Furthermore, we detail how the limiting case ν → 0 of the projected normal distribution in two dimensions can be used for the robust denoising of periodic data, in particular for images with circular data corrupted by wrapped Cauchy noise. distribution we propose to start with the family of more general Student-t distributions, which possesses an additional degree of freedom parameter ν > 0 that allows to control the robustness of the resulting filter. While the Cauchy distribution is obtained as the special case ν = 1, the Student-t distribution converges for ν → ∞ to the normal distribution, so that in the limit also mean filters are covered.The multivariate Student t-distribution is frequently used in statistics [22], whereas the multivariate Cauchy distribution is far less common and in contrast to the one-dimensional case usually not considered separately from the Student-t distribution. The parameter(s) of a multivariate Student t-distribution are usually estimated via the Maximum Likelihood (ML) method in combination with the EM algorithm [5,7,8,10,34]. The EM algorithm for the Student-t distribution has been derived, e.g. in [23], For an overview of estimation methods for the multivariate Student t-distribution, in particular the EM algorithm and its variants, we refer to [37] and the references therein.Recently, the Student-t distribution and the closely related Student-t mixture models (SMM) have found interesting applications in various image processing tasks. One of the first papers which suggested a variational approach for denoising of images corrupted by Cauchy noise was [1]. In [24], the authors proposed a unified framework for images corrupted by white noise that can handle (range constrained) Cauchy noise as well. Other recent approaches that consider also the task of deblurring include [11,54]. Concerning mixture models, in [48] it has been shown that Student-t mixture models are superior to Gaussian mixture models for modeling image patches and the authors proposed an application in image compression. Further applications include robust image segmentation [2,38,45] as well as robust registration [16,55]. In both cases, the SMM is estimated using the EM algorithm derived in [39].In this paper, we propose an application of the Student-t distribution to robust denoising of images corrupted by different kinds of noise. The initial motivation for this work were the recent papers [26,35,43] for Cauchy noise removal. In [35,43] the authors propos...

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Section: Resultsmentioning

confidence: 99%

Multivariate Myriad Filters Based on Parameter Estimation of Student-$t$ Distributions

Laus¹,

Steidl²

2019

SIAM J. Imaging Sci.

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“…A concrete application where a distribution must be reconstructed from a set of marginals appears in image processing with patch-based aggregation [28]. Patches are small overlapping image pieces and it is usual to infer stochastic models (for example a Gaussian or GMM distribution) on these patches [12].…”

Section: Introductionmentioning

confidence: 99%

Generalized Wasserstein barycenters between probability measures living on different subspaces

Delon,

Gozlan,

Saint-Dizier

2021

Preprint

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In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of R d . We study the existence and uniqueness of this barycenter, we show how it is related to a larger multimarginal optimal transport problem, and we propose a dual formulation. Finally, we explain how to compute numerically this generalized barycenter on discrete distributions, and we propose an explicit solution for Gaussian distributions.

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“…A large number of works build on the original EPLL formulation to deal with more general prior or go beyond the denoising problem [12,37,6,33,41,52,11,44]. EPLL uses a GMM prior learned from a very large set of patches extracted from clean images.…”

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confidence: 99%

Compressive Learning for Patch-Based Image Denoising

Shi¹,

Traonmilin²,

Aujol³

2022

SIAM J. Imaging Sci.

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The Expected Patch Log-Likelihood algorithm (EPLL) and its extensions have shown good performances for image denoising. The prior model used by EPLL is usually a Gaussian Mixture Model (GMM) estimated from a database of image patches. Classical mixture model estimation methods face computational issues as the high dimensionality of the problem requires training on large datasets. In this work, we adapt a compressive statistical learning framework to carry out the GMM estimation. With this method, called sketching, we estimate models from a compressive representation (the sketch) of the training patches. The cost of estimating the prior from the sketch no longer depends on the number of items in the original large database. To accelerate further the estimation, we add another dimension reduction technique (low-rank modeling of the covariance matrices) to the compressing learning framework. To demonstrate the advantages of our method, we test it on real large-scale data. We show that we can produce denoising performances similar to performances obtained with models estimated from the original training database using GMM priors learned from the sketch with improved execution times.

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A Unified View on Patch Aggregation

Cited by 4 publications

References 37 publications

Multivariate Myriad Filters Based on Parameter Estimation of Student-$t$ Distributions

Multivariate Myriad Filters Based on Parameter Estimation of Student-$t$ Distributions

Generalized Wasserstein barycenters between probability measures living on different subspaces

Compressive Learning for Patch-Based Image Denoising

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