Cartoon Approximation with α-Curvelets

2014/04/03 by Philipp Grohs, Sandra Keiper, Grohs, Philipp +5
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1404.1043

arxiv created 2014/04/03 · arxiv updated 2014/04/04

Abstract

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise C²-functions, separated by a C² 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∈ (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= \frac1β, up to log-factors.

Related

• The Role of α-Scaling for Cartoon Approximation
• An asymptotic analysis of separating pointlike and C^β-curvelike singularities
• Compactly Supported Shearlets are Optimally Sparse
• Some refined results on convergence of curvelet transform
• Shearlets and Optimally Sparse Approximations