We devise a statistical framework for edge detection by performing a statistical analysis of zero crossings of the second derivative of an image. This analysis enables us to estimate at each pixel of an image the probability that an edge passes through the pixel. We present a statistical analysis of the the Lindeberg operators that we use to compute image derivatives. We also introduce a confidence probability that tells us how reliable the edge probability is, given the image's noise level and the operator's scale. Combining the edge and confidence probabilities leads to a probabilistic scale selection algorithm. We present the results of experiments on natural images.

Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.

Relations between anisotropic diffusion and robust statistics are described. We show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in the image. We extend the framework to vector-valued images and show applications to robust image sharpening.

We describe a robust, dynamic algorithm to compute the arrangement of a set of line segments in the plane, and its implementation.The algorithm is robust because, following Greene [6] and Hobby [8], it rounds the endpoints and intersections of all line segments to representable points, but in a way that is globally topologically consistent.The algorithm is dynamic because, following Mulmttley [13], it uses a randomized hierarchy of vertical cell decompositions to make locating points, and inserting and deleting line segments, efficient.Our algorithm is novel because it marries the robustness of the Greene and Hobby algorithms with Mulmuley's dynamic algorithm in a way that preserves the desirable properties of each.

Rendering systems can produce images that include the entire range of visible colors. Imaging hardware, however, can reproduce only a subset of these colors: the device gamut. An image can only be correctly displayed if all of its colors lie inside of the gamut of the target device. Current solutions to this problem are either to correct the scene colors by hand, or to apply gamut mapping techniques to the final image. We propose a methodology called device-directed rendering that performs scene color adjustments automatically. Device-directed rendering applies classic minimization techniques to a symbolic representation of the image that describes the relationship of the scene lights and surfaces to the pixel colors. This representation can then be evaluated to produce an image that is guaranteed to be in gamut. Although our primary application has been correcting out-of-gamut colors, this methodology can be generally applied to the problem of adjusting a scene description to accommodate constraints on the output image pixel values.

We present several new geometric methods for relative stereo and motion reconstruction using a discrete set of point correspondences. We suppose that the epipoles are known but do not assume any knowledge of the cameras' intrinsic or extrinsic parameters. In each case, we choose a set of five points as a basis for projective space and perform reconstruction relative to these five points. We also present a new technique for reprojection without reconstruction. We have implemented the proposed methods and present several examples using real images.>