Edges discontinuities play a dominant role in our perception of image structure. Extracting edges from background textures is an important, but difficult, inverse problem. Current wavelet-based "edge detection" algorithms are well-suited to analyzing isotropically smooth texture regions but ill-suited for the highly anisotropic geometric smoothness of edge contours. New tools like curvelets and ridgelets have been developed for edge estimation, but unfortunately they are ill-suited for smooth texture regions.
In this talk, we give wavelet-based approaches a second chance by examining the geometrical constraints that edges place on the wavelet coefficients of natural images. We approximate these constraints in terms of an "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.
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