Inversion of exponential \(k\)-plane transforms
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.
scientific article; zbMATH DE number 1445621
scientific article