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

TL;DR

This paper extends the $k$-plane transform to distributional settings, explores its invertibility, and introduces a systematic framework for regularization in inverse problems using these extended operators.

Contribution

It provides a distributional extension of the $k$-plane transform, characterizes invertibility of its dual, and develops a systematic method for regularization in inverse problems.

Findings

Distributional extension of the $k$-plane transform is achieved.

Invertibility of the dual transform is characterized in specific Banach spaces.

New regularization methods for inverse problems are proposed.

Abstract

We investigate the distributional extension of the $k$-plane transform in $\mathbb{R}^d$ and of related operators. We parameterize the $k$-plane domain as the Cartesian product of the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^d$ with $\mathbb{R}^{d-k}$. 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 $L^p$-functions for $1 < p < \infty$. Finally, we apply our results to study a new form of regularization for inverse problems.