Edges play a dominant role in our perception of image structure. In this talk, we will examine the geometrical constraints that edges place on the wavelet coefficients of natural images. We approximate these constraints in terms of a statistical "edge grammar" of dyadic wedgelets -- multiscale piecewise linear edge segments -- that efficiently capture the regularity of edge contours. A Markov wedgelet tree model places a joint probability distribution on the orientations of the wedgelets and allows us to balance several competing factors: the error between the image and the wavelet/wedgelet representation, the parsimony of the representation, and whether the wedgelets form "natural" geometrical structures like edges along smooth contours. In the approximation of images consisting of smooth regions separated by smooth edges, the tandem of wavelet vocabulary and Markov wedgelet grammar can attain the optimal error decay rate, an order of magnitude faster than wavelet nonlinear approximation or thresholding. A prototype geometry/texture image coder based on these ideas shows promise in terms of both MSE and visual performance metrics.
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