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