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Publications (13)6.5 Total impact

  • Vincent Duval, Gabriel Peyré
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    ABSTRACT: We focus on support recovery for signal deconvolution with sparsity assumption. We adopt the continuous setting defined by several recent works and we try to reconstruct a sum of Dirac masses from its low frequencies (possibly perturbed by some noise), by using a total variation prior for Radon measures (i.e. the generalization to measures of the ℓ1 norm). We show that, under a non degenerate source condition, there exists a small noise regime in which the model recovers exactly the same number of spikes as the original signal, and the spikes converge to those of the original signal as the noise vanishes. This continuous setting, by allowing the spikes to “move”, provides robust support recovery for signals composed of well separated spikes. In a discrete setting, where the spikes are reconstructed on a grid, similar low noise regimes which guarantee the exact recovery of the support also exist (see [3]). Yet, this property only concerns a small class of signals. Considering the asymptotics of the discrete problems as the size of the grid tends to zero, we show that the support of the original signal cannot be stable on thin grids, and that the discrete models actually reconstruct pairs of spikes near each original spike. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
    PAMM 12/2014; 14(1).
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    ABSTRACT: Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $\Gamma$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials.
    10/2014;
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    Vincent Duval, Gabriel Peyré
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    ABSTRACT: This paper studies sparse spikes deconvolution over the space of measures. For non-degenerate sums of Diracs, we show that, when the signal-to-noise ratio is large enough, total variation regularization (which the natural extension of L1 norm of vector to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. Finally we draw connections between the performances of sparse recovery on a continuous domain and on a discretized grid.
    06/2013;
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    ABSTRACT: We propose in this paper an extension of the Non-Local Means (NL-Means) denoising algorithm. The idea is to replace the usual square patches used to compare pixel neighborhoods with various shapes that can take advantage of the local geometry of the image. We provide a fast algorithm to compute the NL-Means with arbitrary shapes thanks to the fast Fourier transform. We then consider local combinations of the estimators associated with various shapes by using Stein's Unbiased Risk Estimate (SURE). Experimental results show that this algorithm improve the standard NL-Means performance and is close to state-of-the-art methods, both in terms of visual quality and numerical results. Moreover, common visual artifacts usually observed by denoising with NL-Means are reduced or suppressed thanks to our approach.
    Journal of Mathematical Imaging and Vision 06/2012; · 1.77 Impact Factor
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    ABSTRACT: This paper deals with the parameter choice for the nonlocal means (NLM) algorithm. After basic computations on toy models highlighting the bias of the NLM, we study the bias-variance trade-off of this filter so as to highlight the need of a local choice of the parameters. Relying on Stein’s unbiased risk estimate, we then propose an efficient algorithm to locally set these parameters, and we compare this method with the NLM with optimal global parameter.
    SIAM Journal on Imaging Sciences 01/2011; 4:760-788. · 2.97 Impact Factor
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    ABSTRACT: This paper is about extending the classical Non-Local Means (NLM) denoising algorithm using general shapes instead of square patches. The use of various shapes enables to adapt to the local geometry of the image while looking for pattern redundancies. A fast FFT-based algorithm is proposed to compute the NLM with arbitrary shapes. The local combination of the different shapes relies on Stein's Unbiased Risk Estimate (SURE). To improve the robustness of this local aggregation, we perform an anistropic diffusion of the risk estimate using a properly modified Perona-Malik equation. Experimental results show that this algorithm improves the NLM performance and it removes some visual artifacts usually observed with the NLM.
    Scale Space and Variational Methods in Computer Vision - Third International Conference, SSVM 2011, Ein-Gedi, Israel, May 29 - June 2, 2011, Revised Selected Papers; 01/2011
  • Charles-Alban Deledalle, Vincent Duval
    01/2011;
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    ABSTRACT: In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling.
    Journal of Mathematical Imaging and Vision 01/2010; 37:232-248. · 1.77 Impact Factor
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    ABSTRACT: This paper deals with the parameter choice for the NL-Means algorithm. Starting with basic computations on toy models, we study the bias-variance trade-off of this filter using a simple notion of regularity in the patch space. We show that this regularity is necessarily local and so should be the parameters of the filter. Relying on Stein's Unbiased Risk Estimate, we then propose a way to locally set these parameters, and we compare this method with the Non-Local Means with optimal global parameter.
    01/2010;
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    ABSTRACT: This work deals with color image processing, with a focus on color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v = f − u. u contains the geometric information of the original image, while v is made of the oscillating patterns of f, such as textures. We propose a numerical scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. A direct convergence proof of the scheme is provided, and some analysis on color texture modeling is given.
    Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings; 01/2009
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    ABSTRACT: The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated Rudin-Osher-Fatemi image restoration model, yielding important advances in both fields. We provide the reader with a geometrical characterization of the TVL1 model. We show that, in the convex case, exact solutions of the TVL1 problem are given by an opening followed by a simple test over the ratio perimeter/area. Shapes remain or suddenly vanish depending on this test. As a result of our theoritical study, we suggest a new and efficient numerical scheme to apply the model to digital images. As a by-product, we justify the use of the TVL1 model for image decomposition, by establishing a connection between the model and morphological granulometry. Eventually, we propose an extension of TVL1 into an adaptive framework, in which we derive some theoretical results.
    01/2009;
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    ABSTRACT: In this paper, we are interested in color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling.
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    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we are interested in color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling.