### Abstract

Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of functions with one jump discontinuity, and geometric convergence rates for functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for piecewise analytic functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.

Original language | English (US) |
---|---|

Pages (from-to) | 427-449 |

Number of pages | 23 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 2015 |

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### Keywords

- Chebyshev polynomials
- Edge detection
- Non-uniform Fourier data
- Resampling
- Spectral convergence

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied and Computational Harmonic Analysis*,

*39*(3), 427-449. https://doi.org/10.1016/j.acha.2014.10.002

**Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates.** / Platte, Rodrigo; Gutierrez, Alexander J.; Gelb, Anne.

Research output: Contribution to journal › Article

*Applied and Computational Harmonic Analysis*, vol. 39, no. 3, pp. 427-449. https://doi.org/10.1016/j.acha.2014.10.002

}

TY - JOUR

T1 - Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates

AU - Platte, Rodrigo

AU - Gutierrez, Alexander J.

AU - Gelb, Anne

PY - 2015/11/1

Y1 - 2015/11/1

N2 - Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of functions with one jump discontinuity, and geometric convergence rates for functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for piecewise analytic functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.

AB - Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of functions with one jump discontinuity, and geometric convergence rates for functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for piecewise analytic functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.

KW - Chebyshev polynomials

KW - Edge detection

KW - Non-uniform Fourier data

KW - Resampling

KW - Spectral convergence

UR - http://www.scopus.com/inward/record.url?scp=84942103768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84942103768&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2014.10.002

DO - 10.1016/j.acha.2014.10.002

M3 - Article

AN - SCOPUS:84942103768

VL - 39

SP - 427

EP - 449

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 3

ER -