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Approximation of Integral Operators Using Product-Convolution Expansions

Escande, Paul and Weiss, Pierre Approximation of Integral Operators Using Product-Convolution Expansions. (2017) Journal of Mathematical Imaging and Vision. 1-16. ISSN 0924-9907

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Official URL: http://dx.doi.org/10.1007/s10851-017-0714-8

Abstract

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.

Item Type:Article
Additional Information:Thanks to Springer editor. The definitive version is available at http://link.springer.com/article/10.1007/s10851-017-0714-8
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Laboratory name:
Funders:
PRES of Toulouse University and Midi-Pyrenees Region
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Deposited By: Paul Escande
Deposited On:24 Apr 2017 14:58

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