Cartoon Approximation with $\alpha$-Curvelets

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 $C^2$-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 ($\alpha = \frac12$). Our main result states that those frames achieve this optimal rate for $\alpha = \frac{1}{\beta}$, up to $\log$-factors.