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Preprint 23-2014

Cartoon Approximation with α-Curvelets

Source file is available as :   Portable Document Format (PDF)

Author(s) : Philipp Grohs , Sandra Keiper , Gitta Kutyniok , Martin Schäfer

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 23-2014

MSC 2000

42C40 Wavelets
41A25 Rate of convergence, degree of approximation

Abstract :
It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defi ned as piecewise C2-functions, separated by a C2 singularity curve. In this paper, we consider the more general case of piecewise Cβ-functions, separated by a Cβ singularity curve for β (1;2]. We fi rst prove a benchmark result for the possibly achievable best N-term approximation rate for this more general signal model. Then we introduce what we call α-curvelets, which are systems that interpolate between wavelet systems on the one hand (α = 1) and curvelet systems on the other hand (α= 1/2). Our main result states that those frames achieve this optimal rate for α = 1/β, up to log-factors.

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