Optimally Sparse Representations of Cartoon-Like Cylindrical Data

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 𝒞^2(Z) ⊂ L^2(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 ∈𝒞(Z) satisfies the asymptotic estimate ‖ f - f_N^S‖ _2^2 ≤ c N^-2 (ln N)^3, as N →∞ . 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.