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- David L. Donoho, Xiaoming Huo, +13 authors Stephen Semmes
- 2001

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 dyadicallyorganized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations.… (More)

- Stefan Kindermann, Stanley Osher, Peter W. Jones
- Multiscale Modeling & Simulation
- 2005

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)

- Peter W Jones, Mauro Maggioni, Raanan Schul
- Proceedings of the National Academy of Sciences…
- 2008

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 (alpha) metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the… (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 Cα 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)

- Peter W. Jones
- 1991

- Peter Wilcox Jones, Andrei Osipov, Vladimir Rokhlin
- Proceedings of the National Academy of Sciences…
- 2011

We 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 running time requirements are proportional to T·N·(d·(log d) +… (More)

Article history: Received 17 April 2008 Revised 19 August 2008 Accepted 21 August 2008 Communicated by Charles K. Chui This paper is devoted to the study of local scales (oscillations) in images and use the knowledge of local scales for image decompositions. Denote by Kt(x)= ( e−2πt|ξ |2 )∨ (x), t > 0, the Gaussian (heat) kernel. Motivated from the… (More)

dimensional Euclidean space. Given N points {xj} in Rd, the algorithm attempts to find k nearest neighbors for each of xj , 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 · (log k) · (log N)) + N · k2 · (d + log k), with T the number of iterations performed.… (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)