Exact image representation via a number‐theoretic Radon transform
This study presents an integer‐only algorithm to exactly recover an image from its discrete projected views that can be computed with the same computational complexity as the fast Fourier transform (FFT). Most discrete transforms for image reconstruction rely on the FFT, via the Fourier slice theore...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-08-01
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| Series: | IET Computer Vision |
| Subjects: | |
| Online Access: | https://doi.org/10.1049/iet-cvi.2013.0101 |
| _version_ | 1797684613571674112 |
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| author | Shekhar Chandra Imants Svalbe |
| author_facet | Shekhar Chandra Imants Svalbe |
| author_sort | Shekhar Chandra |
| collection | DOAJ |
| description | This study presents an integer‐only algorithm to exactly recover an image from its discrete projected views that can be computed with the same computational complexity as the fast Fourier transform (FFT). Most discrete transforms for image reconstruction rely on the FFT, via the Fourier slice theorem (FST), in order to compute reconstructions with low‐computational complexity. Consequently, complex arithmetic and floating point representations are needed, the latter of which is susceptible to round‐off errors. This study shows that the slice theorem is valid within integer fields, via modulo arithmetic, using a circulant theory of the Radon transform (RT). The resulting number‐theoretic RT (NRT) provides a representation of images as discrete projections that is always exact and real‐valued. The NRT is ideally suited as part of a discrete tomographic algorithm, an encryption scheme or for when numerical overflow is likely, such as when computing a large number of convolutions on the projections. The low‐computational complexity of the NRT algorithm also provides an efficient method to generate discrete projected views of image data. |
| first_indexed | 2024-03-12T00:33:11Z |
| format | Article |
| id | doaj.art-860953eac48f4c0f9d936ddfb1163a86 |
| institution | Directory Open Access Journal |
| issn | 1751-9632 1751-9640 |
| language | English |
| last_indexed | 2024-03-12T00:33:11Z |
| publishDate | 2014-08-01 |
| publisher | Wiley |
| record_format | Article |
| series | IET Computer Vision |
| spelling | doaj.art-860953eac48f4c0f9d936ddfb1163a862023-09-15T10:11:08ZengWileyIET Computer Vision1751-96321751-96402014-08-018433834610.1049/iet-cvi.2013.0101Exact image representation via a number‐theoretic Radon transformShekhar Chandra0Imants Svalbe1Australian e‐Health Research CentreDivision of Computational InformaticsCSIROAustraliaSchool of PhysicsMonash UniversityMelbourneAustraliaThis study presents an integer‐only algorithm to exactly recover an image from its discrete projected views that can be computed with the same computational complexity as the fast Fourier transform (FFT). Most discrete transforms for image reconstruction rely on the FFT, via the Fourier slice theorem (FST), in order to compute reconstructions with low‐computational complexity. Consequently, complex arithmetic and floating point representations are needed, the latter of which is susceptible to round‐off errors. This study shows that the slice theorem is valid within integer fields, via modulo arithmetic, using a circulant theory of the Radon transform (RT). The resulting number‐theoretic RT (NRT) provides a representation of images as discrete projections that is always exact and real‐valued. The NRT is ideally suited as part of a discrete tomographic algorithm, an encryption scheme or for when numerical overflow is likely, such as when computing a large number of convolutions on the projections. The low‐computational complexity of the NRT algorithm also provides an efficient method to generate discrete projected views of image data.https://doi.org/10.1049/iet-cvi.2013.0101exact image representationnumber-theoretic Radon transforminteger-only algorithmdiscrete projected viewsfast Fourier transformFFT |
| spellingShingle | Shekhar Chandra Imants Svalbe Exact image representation via a number‐theoretic Radon transform IET Computer Vision exact image representation number-theoretic Radon transform integer-only algorithm discrete projected views fast Fourier transform FFT |
| title | Exact image representation via a number‐theoretic Radon transform |
| title_full | Exact image representation via a number‐theoretic Radon transform |
| title_fullStr | Exact image representation via a number‐theoretic Radon transform |
| title_full_unstemmed | Exact image representation via a number‐theoretic Radon transform |
| title_short | Exact image representation via a number‐theoretic Radon transform |
| title_sort | exact image representation via a number theoretic radon transform |
| topic | exact image representation number-theoretic Radon transform integer-only algorithm discrete projected views fast Fourier transform FFT |
| url | https://doi.org/10.1049/iet-cvi.2013.0101 |
| work_keys_str_mv | AT shekharchandra exactimagerepresentationviaanumbertheoreticradontransform AT imantssvalbe exactimagerepresentationviaanumbertheoreticradontransform |