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Harmonic wavelet transform

About: Harmonic wavelet transform is a research topic. Over the lifetime, 9602 publications have been published within this topic receiving 247336 citations.


Papers
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Journal ArticleDOI
Olivier Rioul1, Pierre Duhamel1
TL;DR: The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts.
Abstract: Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm (1989), the 'a trous' algorithm, and their generalizations by Shensa. The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. Most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform. Although this general method is already efficient, it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques, such as the FFT (fast Fourier transform), in a suitable manner. The modified algorithms are termed 'fast' because of their ability to reduce the computational complexity per computed coefficient from L to log L (within a small constant factor) for large filter lengths L. For short filters, smaller gains are obtained: 'fast running FIR (finite impulse response) filtering' techniques allow one to achieve typically 30% savings in computations. >

639 citations

Journal ArticleDOI
01 Jan 1969
TL;DR: A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed, which provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.
Abstract: The introduction of the fast Fourier transform algorithm has led to the development of the Fourier transform image coding technique whereby the two-dimensional Fourier transform of an image is transmitted over a channel rather than the image itself. This devlopement has further led to a related image coding technique in which an image is transformed by a Hadamard matrix operator. The Hadamard matrix is a square array of plus and minus ones whose rows and columns are orthogonal to one another. A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed. Since only real number additions and subtractions are required with the Hadamard transform, an order of magnitude speed advantage is possible compared to the complex number Fourier transform. Transmitting the Hadamard transform of an image rather than the spatial representation of the image provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.

634 citations

Journal ArticleDOI
TL;DR: The main features of so-called wavelet transforms are illustrated through simple mathematical examples and the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds are presented.
Abstract: This paper starts with a brief discussion of so-called wavelet transforms, i.e., decompositions of arbitrary signals into localized contributions labelled by a scale parameter. The main features of the method are first illustrated through simple mathematical examples. Then we present the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds.

622 citations

Book
01 Jan 1980
TL;DR: In this paper, Fourier transform is used for spectral analysis of periodical signals and some properties of the spectrum are analyzed, and it is demonstrated that the spectrum is strongly depended of signal duration.
Abstract: This paper analyses Fourier transform used for spectral analysis of periodical signals and emphasizes some of its properties. It is demonstrated that the spectrum is strongly depended of signal duration that is very important for very short signals which have a very rich spectrum, even for totally harmonic signals. Surprisingly is taken the conclusion that spectral function of harmonic signals with infinite duration is identically with Dirac function and more of this no matter of duration, it respects Heisenberg fourth uncertainty equation. In comparison with Fourier series, the spectrum which is emphasized by Fourier transform doesn’t have maximum amplitudes for signals frequencies but only if the signal lasting a lot of time, in the other situations these maximum values are strongly de-phased while the signal time decreasing. That is why one can consider that Fourier series is useful especially for interpolation of nonharmonic periodical functions using harmonic functions and less for spectral analysis. Key-Words — signals, Fourier transform, continuous spectrum properties, Quantum Physics, Fourier series, discrete spectrum

609 citations

16 Sep 1998
TL;DR: A new implementation of the Discrete Wavelet Transform is presented, suitable for a range of signal and image processing applications, that employs a dual tree of wavelet lters to obtain the real and imaginary parts of complex wavelet coeecients.
Abstract: A new implementation of the Discrete Wavelet Transform is presented, suitable for a range of signal and image processing applications. It employs a dual tree of wavelet lters to obtain the real and imaginary parts of complex wavelet coeecients. This introduces limited redundancy (4:1 for 2-dimensional signals) and allows the transform to provide approximate shift in-variance and directionally selective lters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational eeciency. An application to texture synthesis is presented.

605 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202323
202274
20213
20207
20196
201831