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We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis. The framework has five key components. The beamlet dictionary is a dyadically-organized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations.(More)
Consider a planar Brownian motion run for nite time. The frontier or \outer boundary" of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has innnite length. We improve this by showing that the Hausdorr dimension of the frontier is strictly greater than 1. (It has been conjectured that the(More)
The Lobula Giant Movement Detector (LGMD) is a higher-order visual interneuron of Orthopteran insects that responds preferentially to objects approaching on a collision course. It receives excitatory input from an entire visual hemifield that anatomical evidence suggests is retinotopic. We show that this excitatory projection activates calcium-permeable(More)
How higher-order sensory neurons generate complex selectivity from their simpler inputs is a fundamental question in neuroscience. The lobula giant movement detector (LGMD) is such a visual neuron in the locust Schistocerca americana that responds selectively to objects approaching on a collision course or their two-dimensional projections, looming stimuli(More)
Neurons in a variety of species, both vertebrate and invertebrate, encode the kinematics of objects approaching on a collision course through a time-varying firing rate profile that initially increases, then peaks, and eventually decays as collision becomes imminent. In this temporal profile, the peak firing rate signals when the approaching object's(More)
We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with C α metric). These coordinates are bi-Lipschitz on embedded balls of the domain or manifold, with distortion constants that depend only on natural geometric properties of the(More)
present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {x j } in R d , the algorithm attempts to find k nearest neighbors for each of x j , where k is a user-specified integer parameter. The algorithm is iterative, and its CPU time requirements are proportional to T · N · (d · (log d) + k(More)
This paper is devoted to the decomposition of an image f into u + v, with u a piecewise-smooth or " cartoon " component, and v an oscillatory component (texture or noise), in a variational approach. The cartoon component u is modeled by a function of bounded variation, while v, usually represented by a square integrable function, is now being modeled by a(More)