Cartoon Approximation with α-Curvelets

It is well-known that curvelets provide optimal approximations for so-called cartoon images whichare defined as piecewise C-functions, separated by a C singularity curve. In this paper, we considerthe more general case of piecewise C-functions, separated by a C singularity curve for β ∈ (1, 2].We first prove a benchmark result for the possibly achievable best N -term approximation rate forthis more general signal model. Then we introduce what we call α-curvelets, which are systems thatinterpolate between wavelet systems on the one hand (α = 1) and curvelet systems on the otherhand (α = 12). Our main result states that those frames achieve this optimal rate for α = 1β, up tolog-factors.