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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.05372 |
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| _version_ | 1866913918752915456 |
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| author | Grewar, Murdock G. |
| author_facet | Grewar, Murdock G. |
| contents | We present two theorems describing analytic left-inverses of partial X-ray transforms. The first theorem concerns X-ray data collected with an arbitrary distribution of parallel projections; it contains a convolution-backprojection formula and a backprojection-convolution formula for recovering the transformed volume, provided the data is sufficient. The second theorem concerns X-ray data collected with a cone-beam; it contains a backprojection-convolution formula for recovering the transformed volume, provided the data is amenable to this method (for example: (n-1)-dimensional source loci that `surround' the reconstruction support; detectors of finite size are supported). These theorems are the outcome of a modestly general and rigorous investigation undertaken into the existence of backprojection-convolution methods in n-dimensional space. Necessary and sufficient conditions on the experiment geometry are established for the existence of such methods, as are the particular error metrics minimised by backprojection-convolution methods and the uniqueness of those minimum-error solutions. A major practical outcome of this work is the production of the first known exact inversion methods for cone-beam geometries where the X-ray source point loci are multidimensional, such as (in 3D) a cylinder or a sphere of X-ray source positions. A separate article describes a practical computer implementation for the case of a cylinder in 3D. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier-based Inversion of Partial X-ray Transforms in n Dimensions Grewar, Murdock G. Medical Physics Functional Analysis 45Q05, 92C55, 44A05 We present two theorems describing analytic left-inverses of partial X-ray transforms. The first theorem concerns X-ray data collected with an arbitrary distribution of parallel projections; it contains a convolution-backprojection formula and a backprojection-convolution formula for recovering the transformed volume, provided the data is sufficient. The second theorem concerns X-ray data collected with a cone-beam; it contains a backprojection-convolution formula for recovering the transformed volume, provided the data is amenable to this method (for example: (n-1)-dimensional source loci that `surround' the reconstruction support; detectors of finite size are supported). These theorems are the outcome of a modestly general and rigorous investigation undertaken into the existence of backprojection-convolution methods in n-dimensional space. Necessary and sufficient conditions on the experiment geometry are established for the existence of such methods, as are the particular error metrics minimised by backprojection-convolution methods and the uniqueness of those minimum-error solutions. A major practical outcome of this work is the production of the first known exact inversion methods for cone-beam geometries where the X-ray source point loci are multidimensional, such as (in 3D) a cylinder or a sphere of X-ray source positions. A separate article describes a practical computer implementation for the case of a cylinder in 3D. |
| title | Fourier-based Inversion of Partial X-ray Transforms in n Dimensions |
| topic | Medical Physics Functional Analysis 45Q05, 92C55, 44A05 |
| url | https://arxiv.org/abs/2505.05372 |