Cartoon Approximation with $$\alpha $$-Curvelets |
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Authors: | Philipp Grohs Sandra Keiper Gitta Kutyniok Martin Schäfer |
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Institution: | 1.University of Vienna,Oskar Morgenstern Platz 1,Vienna,Austria;2.Seminar for Applied Mathematics,ETH Zürich,Zurich,Switzerland;3.Department of Mathematics,Technische Universitaet Berlin,Berlin,Germany |
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Abstract: | It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise \(C^2\)-functions, separated by a \(C^2\) singularity curve. In this paper, we consider the more general case of piecewise \(C^\beta \)-functions, separated by a \(C^\beta \) singularity curve for \(\beta \in (1,2]\). We first 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 \(\alpha \)-curvelets, which are systems that interpolate between wavelet systems on the one hand (\(\alpha = 1\)) and curvelet systems on the other hand (\(\alpha = \frac{1}{2}\)). Our main result states that those frames achieve this optimal rate for \(\alpha = \frac{1}{\beta }\), up to \(\log \)-factors. |
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