Cartoon Approximation with $\alpha$-Curvelets
Philipp Grohs, Sandra Keiper, Gitta Kutyniok, Martin Sch\"afer

TL;DR
This paper extends the theory of optimal approximation for cartoon images to functions with less smoothness, introducing $eta$-dependent $ ext{alpha}$-curvelets that interpolate between wavelets and curvelets, achieving near-optimal rates.
Contribution
It generalizes approximation results to $C^eta$-functions and introduces $ ext{alpha}$-curvelets that adapt to the smoothness parameter $eta$, achieving optimal approximation rates.
Findings
Optimal $N$-term approximation rates for $C^eta$-functions.
$ ext{alpha}$-curvelets achieve these rates for $ ext{alpha} = 1/eta$.
Results hold up to logarithmic factors.
Abstract
It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise -functions, separated by a singularity curve. In this paper, we consider the more general case of piecewise -functions, separated by a singularity curve for . We first prove a benchmark result for the possibly achievable best -term approximation rate for this more general signal model. Then we introduce what we call -curvelets, which are systems that interpolate between wavelet systems on the one hand () and curvelet systems on the other hand (). Our main result states that those frames achieve this optimal rate for , up to -factors.
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Taxonomy
TopicsMedical Image Segmentation Techniques · Image and Signal Denoising Methods · Advanced Numerical Analysis Techniques
