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Distributional Extension and Invertibility of the k-Plane Transform and Its Dual

R. Parhi, M. Unser

SIAM Journal on Mathematical Analysis, vol. 56, no. 4, pp. 4662–4686, 2024.

We investigate the distributional extension of the k-plane transform in ℝd and of related operators. We parameterize the k-plane domain as the Cartesian product of the Stiefel manifold of orthonormal k-frames in ℝd with ℝd−k. This parameterization imposes an isotropy condition on the range of the k-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual k-plane transform (the "backprojection" operator). We provide a systematic construction (via a completion process) to identify Banach spaces in which the backprojection operator is invertible and present some prototypical examples. These include the space of isotropic finite Radon measures and isotropic Lp-functions for 1 < p < ∞. Finally, we apply our results to study a new form of regularization for inverse problems. 

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