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_μf)(x, Θ) = ∫_ℝk f(x + Θξ)e^μ·ξ where x ∈ ℝⁿ, Θ is a k-frame in ℝⁿ, 1 ≤ k ≤ n - 1, μ ∈ ℂ^k is an arbitrary complex vector. The case k = 1, μ ∈ ℝ corresponds to the exponential X-ray transform arising in single photon emission tomography. Similar inversion formulas are established for the accompanying transform (P_μf)(x, V) = ∫_ℝk f(x + Vξ)e^μ·ξ dξ where V is a real (n × k)-matrix. Two alternative methods, leading to the relevant continuous wavelet transforms, are presented. The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel. The second method employs interrelation between P_μ and the associated oscillatory potentials.