Images typically contain strong geometric features, such as edges, that impose a structure on pixel values and wavelet coefficients. Modeling the joint coherent behavior of wavelet coefficients is difficult, and standard image coders fail to fully exploit this geometric regularity. We introduce wedgelets as a geometric tool for image compression. Wedgelets offer piecewise-linear approximations of edge contours and can be efficiently encoded. We describe the fundamental challenges that arise when applying such a tool to image compression. To meet these challenges, we also propose an efficient rate-distortion framework for natural image compression using wedgelets.