Edges are of particular interest for image compression, as they communicate important information, contribute large amounts of high-frequency energy, and can generally be described with few parameters. Many of today's most competitive coders rely on wavelets to transform and compress the image, but modeling the joint behavior of wavelet coefficients along an edge presents a distinct challenge. In this thesis, we examine techniques for exploiting the simple geometric structure which captures edge information. Using a multiscale wedgelet decomposition, we present methods for extracting and compressing a cartoon sketch containing the significant edge information, and we discuss practical issues associated with coding the residual textures. Extending these techniques, we propose a rate-distortion optimal framework (based on the Space-Frequency Quantization algorithm) using wedgelets to capture geometric information and wavelets to describe the rest. At low bitrates, this method yields compressed images with sharper edges and lower mean-square error.