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Inversion of exponentialk-plane transforms

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Abstract

Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by\((\mathcal{P}_\mu f)(x,\Theta ) = \smallint _{\mathbb{R}^k } f(x + \Theta \xi )e^{\mu \cdot \xi } d\xi \) where x∈ℝn, Θ is a k-frame in ℝn, 1≤k≤n−1, μ∈ℂk 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\((P_\mu f)(x,V) = \smallint _{\mathbb{R}^k } f(x + V\xi )e^{\mu \cdot \xi } d\xi \) 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 Pμ and the associated oscillatory potentials.

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  1. Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Boris Rubin

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Communicated by Eric Todd Quinto

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Rubin, B. Inversion of exponentialk-plane transforms. The Journal of Fourier Analysis and Applications 6, 185–205 (2000). https://doi.org/10.1007/BF02510660

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