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- Ronald R. Coifman, M. Victor Wickerhauser
- IEEE Trans. Information Theory
- 1992

Adapted waveform analysis uses a library of orthonormal 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)

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)

- Ronald R. Coifman, D . L . DonohoThis
- 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)

- Ronald R. Coifman, Stéphane Lafon, +4 authors Steven W. Zucker
- Proceedings of the National Academy of Sciences…
- 2005

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)

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)

For voice signals and images this procedure leads to remarkable compression algorithms; see the references W2] and W3] below. The best basis method may be applied to ensembles of vectors, more like classical Karrhunen-Lo eve analysis. The so-called \energy compaction function" may be used as an information cost to compute the joint best basis over a set of… (More)

- Stéphane Lafon, Yosi Keller, Ronald R. Coifman
- IEEE Transactions on Pattern Analysis and Machine…
- 2006

Data fusion and multicue data matching are fundamental tasks of high-dimensional data analysis. In this paper, we apply the recently introduced diffusion framework to address these tasks. Our contribution is three-fold: first, we present the Laplace-Beltrami approach for computing density invariant embeddings which are essential for integrating different… (More)

- Bradley K. Alpert, Gregory Beylkin, Ronald R. Coifman, Vladimir Rokhlin
- SIAM J. Scientific Computing
- 1993

A class of vector-space bases is introduced for the sparse representation of discretizations of integral operators. An operator with a smooth, nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix, to high precision. A method is presented that employs these bases for the… (More)

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)