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About: Wavelet is a(n) research topic. Over the lifetime, 78027 publication(s) have been published within this topic receiving 1386969 citation(s). more


Open accessJournal ArticleDOI: 10.1109/34.192463
Stéphane Mallat1Institutions (1)
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. > more

Topics: Wavelet (65%), Wavelet transform (65%), Orthogonal wavelet (65%) more

19,033 Citations

Open accessBook
D.L. Donoho1Institutions (1)
01 Jan 2004-
Abstract: Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0 more

Topics: Basis pursuit (56%), Orthonormal basis (55%), Linear combination (54%) more

18,593 Citations

Open accessBook
01 Jan 1998-
Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes. more

Topics: Wavelet packet decomposition (71%), Discrete wavelet transform (67%), Wavelet transform (67%) more

17,299 Citations

Open accessBook
Ingrid Daubechies1Institutions (1)
01 May 1992-
Abstract: Introduction Preliminaries and notation The what, why, and how of wavelets The continuous wavelet transform Discrete wavelet transforms: Frames Time-frequency density and orthonormal bases Orthonormal bases of wavelets and multiresolutional analysis Orthonormal bases of compactly supported wavelets More about the regularity of compactly supported wavelets Symmetry for compactly supported wavelet bases Characterization of functional spaces by means of wavelets Generalizations and tricks for orthonormal wavelet bases References Indexes. more

Topics: Spline wavelet (68%), Continuous wavelet (67%), Biorthogonal wavelet (64%) more

16,065 Citations

Abstract: A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change... more

Topics: Wavelet (69%), Wavelet transform (68%), Morlet wavelet (67%) more

11,219 Citations

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Topic's top 5 most impactful authors

Michael Unser

190 papers, 11.9K citations

Richard G. Baraniuk

103 papers, 10.2K citations

Jean-Luc Starck

64 papers, 3.5K citations

Stéphane Mallat

58 papers, 35.2K citations

Nick Kingsbury

55 papers, 3.9K citations

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