Natural images contain crucial information in sharp geometrical boundaries between objects. Therefore, their description by smooth isotropic function spaces (e.g. Sobolev or Besov spaces) is not sufficiently accurate. Moreover, methods known to be optimal for such isotropic spaces (tensor product wavelet decompositions) do not provide optimal nonlinear approximations for piecewise smooth bivariate functions. Among the geometrybased alternatives that were proposed during the last few years, adaptive thinning methods work with continuous piecewise affine functions on anisotropic triangulations to construct sparse representations for piecewise smooth bivariate functions. In this article, a customized compression method for coding the sparse data information, as output by adaptive thinning, is proposed. The compression method is based on contextual encoding of both the sparse data positions and their attached luminance values. To this end, the structural properties of the sparse data representation are essentially exploited. The resulting contextual image compression method of this article outperforms our previous methods (all relying on adaptive thinning) quite significantly. Moreover, our proposed compression method also outperforms JPEG2000 for selected natural images, at both low and middle bitrates, as this is supported by numerical examples in this article.