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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.01233 |
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| _version_ | 1866910510409056256 |
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| author | Parhi, Rahul Unser, Michael |
| author_facet | Parhi, Rahul Unser, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01233 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual Parhi, Rahul Unser, Michael Functional Analysis 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. |
| title | Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2310.01233 |