Inversion of exponential k-plane transforms

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