Wurzeln Aus Der Hankel-, Fourier-Und Aus Anderen Stetigen Transformationen

Kober, H.

doi:10.1093/qmath/os-10.1.45

Cited by 27 publications

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“…The FrFT on L 2 (R) was originally described by Kober [1] and was later rediscovered by Namias in the context of quantum mechanics [2]. One equivalent definition is as follows (see [3, Definition F, p. 132]):…”

Section: A Reformulation Of the Fractional Fourier Transformmentioning

confidence: 99%

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The fractional Fourier transform and quadratic field magnetic resonance imaging

Irarrázaval

Lizama

Parot

et al. 2011

Computers & Mathematics with Applications

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a b s t r a c tThe fractional Fourier transform (FrFT) is revisited in the framework of strongly continuous periodic semigroups to restate known results and to explore new properties of the FrFT. We then show how the FrFT can be used to reconstruct Magnetic Resonance (MR) images acquired under the presence of quadratic field inhomogeneity. Particularly, we prove that the order of the FrFT is a measure of the distortion in the reconstructed signal. Moreover, we give a dynamic interpretation to the order as time evolution of a function. We also introduce the notion of ρ-α space as an extension of the Fourier or k-space in MR, and we use it to study the distortions introduced in two common MR acquisition strategies. We formulate the reconstruction problem in the context of the FrFT and show how the semigroup theory allows us to find new reconstruction formulas for discrete sampled signals. Finally, the results are supplemented with numerical examples that show how it performs in a standard 1D MR signal reconstruction.

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Section: A Reformulation Of the Fractional Fourier Transformmentioning

confidence: 99%

“…The fractional Fourier transform (FrFT) is an extension of the Fourier transform first developed by Kober [1] in the late thirties and rediscovered by Namias [2] in the late seventies. Namias used the FrFT in the context of quantum mechanics as a way to solve certain problems involving quantum harmonic oscillators.…”

Section: Introductionmentioning

confidence: 99%

The fractional Fourier transform and quadratic field magnetic resonance imaging

Irarrázaval

Lizama

Parot

et al. 2011

Computers & Mathematics with Applications

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“…First, because this fractional FT can easily be realized experimentally by using simple optical setups [22], and secondly, because it produces a mere rotation of the two fundamental phase-space distributions: the WD and the AF. The canonical fractional FT was introduced more than 60 years ago in the mathematical literature [19]; after that, it was reinvented for applications in quantum mechanics [20,21], optics [15,16,18], and signal processing [23]. After the main properties of the fractional FT were established, the perspectives for its implementations in filter design, signal analysis, phase retrieval, watermarking, and so forth became clear.…”

Section: Fractional Fourier Transform and Generalized Fractional Convmentioning

confidence: 99%

“…Nowadays, fractional transforms play an important role in information processing [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], and the obvious question is: why do we need fractional transformations if we successfully apply the ordinary ones? First, because they naturally arise under the consideration of different problems, for example, in optics and quantum mechanics, and secondly, because fractionalization gives us a new degree of freedom (the fractional order) which can be used for more complete characterization of an object (a signal, in general) or as an additional encoding parameter.…”

Section: Introductionmentioning

confidence: 99%

Fractional Transforms in Optical Information Processing

Alieva

Bastiaans

Calvo

2005

EURASIP J. Adv. Signal Process.

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We review the progress achieved in optical information processing during the last decade by applying fractional linear integral transforms. The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail. A general algorithm for the fractionalization of linear cyclic integral transforms is introduced and it is shown that they can be fractionalized in an infinite number of ways. Basic properties of fractional cyclic transforms are considered. The implementation of some fractional transforms in optics, such as fractional Hankel, sine, cosine, Hartley, and Hilbert transforms, is discussed. New horizons of the application of fractional transforms for optical information processing are underlined.

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“…In this paper, we generalize the concept of the fractional Fourier transform (FRFT) as introduced by Kober [1] and show its application for solving certain energy localization problems in phase space. In the sequential sections, we will deal with the FRFT; however, here we briefly recall the definition and some properties of the Wigner distribution.…”

Section: Introductionmentioning

confidence: 99%

The Fractional Fourier Transform and Its Application to Energy Localization Problems

Oonincx¹,

Morsche

2003

EURASIP J. Adv. Signal Process.

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Applying the fractional Fourier transform (FRFT) and the Wigner distribution on a signal in a cascade fashion is equivalent to a rotation of the time and frequency parameters of the Wigner distribution. We presented in ter Morsche and Oonincx, 2002, an integral representation formula that yields affine transformations on the spatial and frequency parameters of the n-dimensional Wigner distribution if it is applied on a signal with the Wigner distribution as for the FRFT. In this paper, we show how this representation formula can be used to solve certain energy localization problems in phase space. Examples of such problems are given by means of some classical results. Although the results on localization problems are classical, the application of generalized Fourier transform enlarges the class of problems that can be solved with traditional techniques

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Wurzeln Aus Der Hankel-, Fourier-Und Aus Anderen Stetigen Transformationen

Cited by 27 publications

References 0 publications

The fractional Fourier transform and quadratic field magnetic resonance imaging

The fractional Fourier transform and quadratic field magnetic resonance imaging

Fractional Transforms in Optical Information Processing

The Fractional Fourier Transform and Its Application to Energy Localization Problems

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