Cartoon Approximation with $α$-Curvelets

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 $C^β$-functions, separated by a $C^β$ singularity curve for $β\in (1,2]$. 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 $α$-curvelets, which are systems that interpolate between wavelet systems on the one hand ($α= 1$) and curvelet systems on the other hand ($α= \frac12$). Our main result states that those frames achieve this optimal rate for $α= \frac{1}β$, up to $\log$-factors.