An explicit spectral decomposition of the ADRT
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917187360391168 |
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| author | Li, Weilin Otness, Karl Ren, Kui Rim, Donsub |
| author_facet | Li, Weilin Otness, Karl Ren, Kui Rim, Donsub |
| contents | The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An explicit spectral decomposition of the ADRT Li, Weilin Otness, Karl Ren, Kui Rim, Donsub Numerical Analysis Spectral Theory 44A12, 65R10, 92C55, 68U05, 15A04 The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime. |
| title | An explicit spectral decomposition of the ADRT |
| topic | Numerical Analysis Spectral Theory 44A12, 65R10, 92C55, 68U05, 15A04 |
| url | https://arxiv.org/abs/2412.11151 |