Inversion of exponential \(k\)-plane transforms
type: Item
instance of: scholarly article
title: Inversion of exponential \(k\)-plane transforms
zbMATH Open document ID: 0973.44001
DOI: 10.1007/BF02510660
author: Boris Rubin
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-11-22T00:00:00Z
full work available at URL: https://eudml.org/doc/59639
review text: Due to its non-invasiveness, magnetic resonance imaging (MRI) will replace X-ray computerized tomography (CT) in future medical diagnosis. Because the CT modality depends upon the inversion of specific versions of the Radon transform, this and related transforms nowadays are favorite examples of the application of transform analysis. In the paper under review, approximate and explicit inversion formulae are obtained for a class of exponential \(k\)-plane transforms. This class includes the windowed Radon transform. The case \(k =1\) corresponds to the exponential X-ray transform arising in the modality of single photon emission tomography. Two alternative methods are presented. In a general setup, the first one is based on the use of the generalized Calderón reproducing formula for a suitable wavelet measure and multidimensional fractional integrals with a Bessel function involved in the kernel [cf. \textit{B. Rubin}, Fractional integrals and potentials. Addison-Wesley (1996; Zbl 0864.26004)]. The second method employs interrelations between the exponential \(k\)-plane transform and the associated oscillatory potentials.
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 92C55
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1445621
zbMATH Keywords: magnetic resonance imaging
zbMATH Keywords: X-ray computerized tomography
zbMATH Keywords: medical diagnosis
zbMATH Keywords: Radon transform
zbMATH Keywords: explicit inversion formulae
zbMATH Keywords: exponential \(k\)-plane transforms
zbMATH Keywords: windowed Radon transform
zbMATH Keywords: expontial X-ray transform
zbMATH Keywords: single photon emission tomography
zbMATH Keywords: wavelet measure
zbMATH Keywords: fractional integrals
reviewed by: Walter Schempp
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1445621