Natural images can be viewed as combinations of smooth regions, textures, and geometry. Wavelet-based image coders, such as the space-frequency quantization (SFQ) algorithm, provide reasonably efficient representations for smooth regions (using zerotrees, for example) and textures (using scalar quantization) but do not properly exploit the geometric regularity imposed on wavelet coefficients by features such as edges. In this paper, we develop a representation for wavelet coefficients in geometric regions based on the wedgelet dictionary, a collection of geometric atoms that construct piecewise-linear approximations to contours. Our wedgeprint representation implicitly models the coherency among geometric wavelet coefficients. We demonstrate that a simple compression algorithm combining wedgeprints with zerotrees and scalar quantization can achieve near-optimal rate-distortion performance D(R) ~ (log R)²/R² for the class of piecewise-smooth images containing smooth C² regions separated by smooth C² discontinuities. Finally, we extend this simple algorithm and propose a complete compression framework for natural images using a rate-distortion criterion to balance the three representations. Our Wedgelet-SFQ (WSFQ) coder outperforms SFQ in terms of visual quality and mean-square error.