Algebraic Reconstruction of Piecewise-Smooth Functions of Two Variables from Fourier Data
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909524292534272 |
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| author | Levinov, Michael Yomdin, Yosef Batenkov, Dmitry |
| author_facet | Levinov, Michael Yomdin, Yosef Batenkov, Dmitry |
| contents | We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While many techniques have been proposed over the years to solve the problem, very few consider the accurate reconstruction of the discontinuities themselves.
In this work we develop an algebraic reconstruction technique for two-dimensional functions consisting of two continuity pieces with a smooth discontinuity curve. By extending our earlier one-dimensional method, we show that both the discontinuity curve and the function itself can be reconstructed with high accuracy from a finite number of Fourier measurements. The accuracy is commensurate with the smoothness of the pieces and the discontinuity curve. We also provide a numerical implementation of the method and demonstrate its performance on synthetic data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic Reconstruction of Piecewise-Smooth Functions of Two Variables from Fourier Data Levinov, Michael Yomdin, Yosef Batenkov, Dmitry Numerical Analysis Primary 65T40, Secondary 65D15 We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While many techniques have been proposed over the years to solve the problem, very few consider the accurate reconstruction of the discontinuities themselves. In this work we develop an algebraic reconstruction technique for two-dimensional functions consisting of two continuity pieces with a smooth discontinuity curve. By extending our earlier one-dimensional method, we show that both the discontinuity curve and the function itself can be reconstructed with high accuracy from a finite number of Fourier measurements. The accuracy is commensurate with the smoothness of the pieces and the discontinuity curve. We also provide a numerical implementation of the method and demonstrate its performance on synthetic data. |
| title | Algebraic Reconstruction of Piecewise-Smooth Functions of Two Variables from Fourier Data |
| topic | Numerical Analysis Primary 65T40, Secondary 65D15 |
| url | https://arxiv.org/abs/2503.02254 |