The Role of $\alpha$-Scaling for Cartoon Approximation

The class of cartoon-like functions, classicly defined as piecewise $C^2$ functions consisting of smooth regions separated by $C^2$ discontinuity curves, is a well-established model for image data. The quest for optimal approximation of this class has among others led to the development of curvelets, contourlets, and shearlets. Due to parabolic scaling, these systems are able to provide a quasi-optimal $N$-term approximation rate of order $N^{-2}$. Replacing parabolic scaling by $\alpha$-scaling, one obtains $\alpha$-curvelets and $\alpha$-shearlets, which interpolate between wavelet-type systems ($\alpha=1$), parabolically scaled systems ($\alpha=\frac12$), and ridgelet-type systems ($\alpha=0$). Previous research shows that in the range $\alpha\in[\frac{1}{2},1)$ they provide quasi-optimal approximation for cartoons of regularity $C^{1/\alpha}$ with a rate of order $N^{-1/\alpha}$. In this work we continue to explore $\alpha$-scaled representation systems, with the aim to better understand the role of the parameter $\alpha$ for approximation. Concerning $\alpha$-curvelets with $\alpha<1$, we prove that the best possible $N$-term approximation rate achievable for cartoons with curved edges is limited to at most $N^{-1/(1-\alpha)}$, independent of the smoothness of the cartoons. The maximal rate achievable by simple thresholding of the frame coefficients is even bounded by $N^{-1/\max\{\alpha,1-\alpha\}}$. If the edges of the cartoons are straight the approximation performance of $\alpha$-curvelets is different: Assuming $C^\beta$ regularity, we establish an approximation rate of order $N^{-\min\{\alpha^{-1},\beta\}}$, which is quasi-optimal if $\alpha\in [0,\beta^{-1}]$. Finally, via the framework of $\alpha$-molecules, the obtained results are extended to other $\alpha$-scaled systems including in particular $\alpha$-shearlets.