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- Ronald R Coifman, Stéphane Lafon
- 2006

In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets. We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient representations of complex geometric structures. The associated family of diffusion distances,… (More)

Adapted waveform analysis uses a library of or-thonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets, and localized trigonometric functions, have… (More)

- R R Coifman, D L Donoho
- 1995

De-Noising with the traditional (orthogonal, maximally-decimated) wavelet transform sometimes exhibits visual artifacts; we attribute some of these { for example , Gibbs phenomena in the neighborhood of discontinuities { to the lack of translation invariance of the wavelet basis. One method to suppress such artifacts, termed \cycle spinning" by Coifman, is… (More)

We provide a framework for structural multiscale geometric organization of graphs and subsets of R(n). We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to… (More)

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adja-cency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the… (More)

We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. Classes of operators satisfying these… (More)

- A Averbuch, R R Coifman, D L Donoho, M Israeli, J Waldén
- 2001

We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically faithful: the lines exhibit no 'wraparound effects'.… (More)

We introduce intrinsic, nonlinearly invariant, parameterizations of empirical data, generated by a nonlinear transformation of independent variables. This is achieved through anisotropic diffusion kernels on observable data manifolds that approximate a Laplacian on the inaccessible independent variable domain. The key idea is a symmetrized second order… (More)

- A Averbuch, R R Coifman, D L Donoho, M Elad, M Israeli
- 2004

In a wide range of applied problems of 2-D and 3-D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However , the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high… (More)

- G Beylkin, R R Coifman, V Rokhlin, Comm

This paper describes exact and explicit representations of the differential operators, d n /dx n , n = 1, 2, · · ·, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution… (More)