<title>Multiscale geometric image processing</title>

Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the state-of-the-art for applications including image coding (JPEG2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and wavelet based processing algorithms often produce images with ringing around the edges. As a first step towards accounting for this structure, we will show how to explicitly capture the geometric regularity of contours in cartoon images using the wedgelet representation and a multiscale geometry model. The wedgelet representation builds up an image out of simple piecewise constant functions with linear discontinuities. We will show how the geometry model, by putting a joint distribution on the orientations of the linear discontinuities, allows us to weigh several factors when choosing the wedgelet representation: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structures. Finally, we will analyze a simple wedgelet coder based on these principles, and show that it has optimal asymptotic performance for simple cartoon images.