Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Abstract

Abstract. Anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data, in particular from image data. In previous work, we have proposed a locally adaptive method for efficient image approximation, called adaptive thinning, which relies on linear splines over anisotropic Delaunay triangulations. In this paper, we prove asymptotically optimal N -term approximation rates for linear splines over anisotropic Delaunay triangulations, where our analysis applies to relevant classes of target functions: (a) piecewise linear horizon functions across α Hölder smooth boundaries, (b) functions of W α,p regularity, where α > 2/p−1, (c) piecewise regular horizon functions of W α,2 regularity, where α > 1.

Optimally sparse image approximation by adaptive linear splines over anisotropic triangulations

“…Proof: Let us first remark that the asymptotic N -term approximation (4) can also be found in our previous work [6]. Nevertheless, it is quite instructive to recall the construction of T N from [6]. This is done in three steps as follows.…”

“…to obtain the desired triangulation T N (see [6]), which in turn yields the unique linear spline interpolant f N ∈ S T N to f with…”

“…Let {Q N } N denote the sequence of quadtree partitions from the Birman-Solomjak theorem, satisfying (6). By V Q N = {(x m , y m )} m=1,...,M we denote, for any N ∈ N, the vertex set of Q N , comprising all M vertices from the |Q N | ≤ N quadtree leaves, so that M ≤ 4 × N .…”

“…In [6], we have proven asymptotically optimal N -term decay rates of the form (1) for linear spline approximation over locally adaptive anisotropic Delaunay triangulations for relevant classes of target functions f , including…”

“…Our constructive approach in [6] essentially depends on the local regularity of the target function f , where the resulting adaptive approximation method applies in particular to piecewise Hölder continuous horizon functions f of order α ∈ (1, 2] with singularities along α Hölder smooth boundaries. The outline of this paper is as follows.…”