Anisotropic triangulations provide efficient methods for sparse image representations. In previous work, we have proposed a locally adaptive algorithm for sparse image approximation, adaptive thinning, which relies on linear splines over anisotropic Delaunay triangulations. In this contribution, we address theoretical and practical aspects concerning image approximation by linear splines over anisotropic conformal triangulations. Our discussion includes asymptotically optimal N-term approximations on relevant classes of target functions, such as horizon functions across α Hölder smooth boundaries and regular functions of Wα, p regularity, for α > 2/p-1. Moreover, we demonstrate the good performance of our adaptive thinning algorithm by numerical examples and comparisons.