Inversion of exponential k-plane transforms

Authors
Citation
B. Rubin, Inversion of exponential k-plane transforms, J FOURIER A, 6(2), 2000, pp. 185-205
Citations number
46
Categorie Soggetti
Mathematics,"Engineering Mathematics
Journal title
JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
ISSN journal
10695869 → ACNP
Volume
6
Issue
2
Year of publication
2000
Pages
185 - 205
Database
ISI
SICI code
1069-5869(2000)6:2<185:IOEKT>2.0.ZU;2-T
Abstract
Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by (P(mu)f)(x, Theta) = integral(R) k f(x + Theta xi)e(mu .xi) d xi where x epsilon R-n, Theta is a k-frame in R-n, 1 less than or equal to k less than or equal to n - 1, mu epsilon C-k is an arbitrary complex vector The case k = 1, mu epsilon R corresponds to the exponential X-ray transform arising in single photon emission tomograph y. Similar inversion formulas are established for the accompanying transfor m (P(mu)f)(x, V) = integral(R)k f(x + V xi)e(mu .xi) d xi where V is a real (n x k)-matrix. Two alternative methods, leading to the relevant continuou s wavelet transforms. are presented The first one is based on the use of th e generalized Calderon reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel. The second method employs i nterrelation between P-mu and the associated oscillatory potentials