Optimally Sparse Representations of Cartoon-Like Cylindrical Data

Abstract

Sparse representations of multidimensional data have received a significant attention in the literature due to their applications in problems of data restoration and feature extraction. In this paper, we consider an idealized class $${\mathcal {C}}^2(Z) \subset L^2({\mathbb {R}}^3)$$ of 3-dimensional data dominated by surface singularities that are orthogonal to the xy plane. To deal with this type of data, we introduce a new multiscale directional representation called cylindrical shearlets and prove that this new approach achieves superior approximation properties not only with respect to conventional multiscale representations but also with respect to 3-dimensional shearlets and curvelets. Specifically, the N-term approximation $$f_N^S$$ obtained by selecting the N largest coefficients of the cylindrical shearlet expansion of a function $$f \in {\mathcal {C}}(Z)$$ satisfies the asymptotic estimate $$ \Vert f - f_N^S\Vert _2^2 \le c \, N^{-2} \, (\ln N)^3, \quad \text {as } N \rightarrow \infty .$$ This is the optimal decay rate, up the logarithmic factor, outperforming 3d wavelet and 3d shearlet approximations which only yield approximation rates of order $$N^{-1/2}$$ and $$N^{-1}$$ (ignoring logarithmic factors), respectively, on the same type of data.