Convergence of Ray- and Pixel-Driven Discretization Frameworks in the Strong Operator Topology
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| Format: | Preprint |
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2025
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| _version_ | 1866912927181701120 |
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| author | Huber, Richard |
| author_facet | Huber, Richard |
| contents | Tomography is a central tool in medical applications, allowing doctors to investigate patients' interior features. The Radon transform (in two dimensions) is commonly used to model the measurement process in parallel-beam CT. Suitable discretization of the Radon transform and its adjoint (called the backprojection) is crucial. The most commonly used discretization approach combines what we refer to as the ray-driven Radon transform with what we refer to as the pixel-driven backprojection, as anecdotal reports describe these as showing the best approximation performance. However, there is little rigorous understanding of induced approximation errors. These methods involve three discretization parameters: the spatial-, detector-, and angular resolutions. Most commonly, balanced resolutions are used, i.e., the same (or similar) spatial- and detector resolutions are employed. We present an interpretation of ray- and pixel-driven discretizations as `convolutional methods', a special class of finite-rank operators. This allows for a structured analysis that can explain observed behavior. In particular, we prove convergence in the strong operator topology of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions, thus theoretically justifying this approach. In particular, with high enough resolutions one can approximate the Radon transform arbitrarily well. Numerical experiments corroborate these theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Ray- and Pixel-Driven Discretization Frameworks in the Strong Operator Topology Huber, Richard Numerical Analysis 44A12, 65R10, 94A08, 41A25 Tomography is a central tool in medical applications, allowing doctors to investigate patients' interior features. The Radon transform (in two dimensions) is commonly used to model the measurement process in parallel-beam CT. Suitable discretization of the Radon transform and its adjoint (called the backprojection) is crucial. The most commonly used discretization approach combines what we refer to as the ray-driven Radon transform with what we refer to as the pixel-driven backprojection, as anecdotal reports describe these as showing the best approximation performance. However, there is little rigorous understanding of induced approximation errors. These methods involve three discretization parameters: the spatial-, detector-, and angular resolutions. Most commonly, balanced resolutions are used, i.e., the same (or similar) spatial- and detector resolutions are employed. We present an interpretation of ray- and pixel-driven discretizations as `convolutional methods', a special class of finite-rank operators. This allows for a structured analysis that can explain observed behavior. In particular, we prove convergence in the strong operator topology of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions, thus theoretically justifying this approach. In particular, with high enough resolutions one can approximate the Radon transform arbitrarily well. Numerical experiments corroborate these theoretical findings. |
| title | Convergence of Ray- and Pixel-Driven Discretization Frameworks in the Strong Operator Topology |
| topic | Numerical Analysis 44A12, 65R10, 94A08, 41A25 |
| url | https://arxiv.org/abs/2503.03069 |