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Documents  94A08 | enregistrements trouvés : 8

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In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine learning.
imaging - image processing - sparsity - convex optimization - inverse problem - super-resolution
In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine ...

62H35 ; 65D18 ; 94A08 ; 68U10 ; 90C31 ; 80M50 ; 47N10

Post-edited  25+ years of wavelets for PDEs
Kunoth, Angela (Auteur de la Conférence) | CIRM (Editeur )

Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality and optimality of approximations and related concepts from approximation the- ory. We will see that wavelet bases can serve as a basic ingredient, both for the theory as well as for algorithmic realizations. Particularly for situations where solutions exhibit singularities, wavelet concepts enable adaptive appproximations for which convergence and optimal algorithmic complexity can be established. I will describe corresponding implementations based on biorthogonal spline-wavelets.
Moreover, wavelet-related concepts have triggered new developments for efficiently solving complex systems of PDEs, as they arise from optimization problems with PDEs.
Ingrid Daubechies' construction of orthonormal wavelet bases with compact support published in 1988 started a general interest to employ these functions also for the numerical solution of partial differential equations (PDEs). Concentrating on linear elliptic and parabolic PDEs, I will start from theoretical topics such as the well-posedness of the problem in appropriate function spaces and regularity of solutions and will then address quality ...

65T60 ; 94A08 ; 65N12 ; 65N30 ; 49J20

The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, Poissonian, and multiplicative, and with several types of regularizers (TV, frame-based analysis, synthesis, or combinations thereof). I will then describe more recent work on using ADMM for other problems, namely blind deconvolution and image segmentation, as well as very recent work where ADMM is used with plug-in learned denoisers to achieve state-of-the-art results in class-specific image deconvolution. Finally, on the theoretical front, I will describe very recent work on tackling the infamous problem of how to adjust the penalty parameter of ADMM. The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, ...

65J22 ; 65K10 ; 65T60 ; 94A08

One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This finding triggered an enormous research activity in recent years both in signal processing applications as well as their mathematical foundations. The present talk discusses connections of compressive sensing and time-frequency analysis (the sister of wavelet theory). In particular, we give on overview on recent results on compressive sensing with time-frequency structured random matrices.

Keywords: compressive sensing - time-frequency analysis - wavelets - sparsity - random matrices - $\ell_1$-minimization - radar - wireless communications
One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This ...

94A20 ; 94A08 ; 42C40 ; 60B20 ; 90C25

In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view of the world, and the sparsity property of wavelet expansions, and present two recent results. First, we show new bounds for the time-frequency spread of sequences, and construct maximally compact sequences. Interestingly they differ from sampled Gaussians. Next, we review work on sampling of finite rate of innovation signals, which are sparse continuous-time signals for which sampling theorems are possible. We conclude by arguing that the interface of signal processing and applied harmonic analysis has been both fruitful and fun, and try to identify lessons learned from this experience.

Keywords: wavelets ­ filter banks - subband coding ­ uncertainty principle ­ sampling theory ­ sparse sampling
In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view ...

94A08 ; 94A12 ; 65T60 ; 42C40

Multi angle  Minicourse shape spaces and geometric statistics
Pennec, Xavier (Auteur de la Conférence) ; Trouvé, Alain (Auteur de la Conférence) | CIRM (Editeur )

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