DISTRIBUTIONAL EXTENSION AND INVERTIBILITY OF THE k-PLANE TRANSFORM AND ITS DUAL

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.