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.