Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform
The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the in...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2023-06-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
| Subjects: | |
| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2023.1155484/full |
| _version_ | 1797793829739298816 |
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| author | Melanie Kircheis Daniel Potts |
| author_facet | Melanie Kircheis Daniel Potts |
| author_sort | Melanie Kircheis |
| collection | DOAJ |
| description | The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this article, we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT. |
| first_indexed | 2024-03-13T02:52:57Z |
| format | Article |
| id | doaj.art-de2d64bfc3094bba8d98af4e836adfce |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-03-13T02:52:57Z |
| publishDate | 2023-06-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-de2d64bfc3094bba8d98af4e836adfce2023-06-28T09:21:12ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872023-06-01910.3389/fams.2023.11554841155484Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transformMelanie KircheisDaniel PottsThe well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this article, we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT.https://www.frontiersin.org/articles/10.3389/fams.2023.1155484/fullinverse nonequispaced fast Fourier transformnonuniform fast Fourier transformdirect inversioniNFFTNFFT |
| spellingShingle | Melanie Kircheis Daniel Potts Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform Frontiers in Applied Mathematics and Statistics inverse nonequispaced fast Fourier transform nonuniform fast Fourier transform direct inversion iNFFT NFFT |
| title | Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform |
| title_full | Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform |
| title_fullStr | Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform |
| title_full_unstemmed | Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform |
| title_short | Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform |
| title_sort | fast and direct inversion methods for the multivariate nonequispaced fast fourier transform |
| topic | inverse nonequispaced fast Fourier transform nonuniform fast Fourier transform direct inversion iNFFT NFFT |
| url | https://www.frontiersin.org/articles/10.3389/fams.2023.1155484/full |
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