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

Rodrigo Platte, Alexander J. Gutierrez, Anne Gelb

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)427-449
Number of pages23
JournalApplied and Computational Harmonic Analysis
Volume39
Issue number3
DOIs
StatePublished - Nov 1 2015

Fingerprint

Piecewise Smooth Functions
Exponential Convergence
Univariate
Convergence Rate
Fourier transform
Jump
Geometric Convergence
Magnetic Resonance Imaging
Compact Support
Iterative Process
Post-processing
Discontinuity
Analytic function
Fourier transforms
Sensing
Decay
Oscillation
Derivative
Polynomial
Integer

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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

In: Applied and Computational Harmonic Analysis, Vol. 39, No. 3, 01.11.2015, p. 427-449.

Research output: Contribution to journalArticle

@article{6483c9d273534236a22dec25442ba556,
title = "Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates",
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.",
keywords = "Chebyshev polynomials, Edge detection, Non-uniform Fourier data, Resampling, Spectral convergence",
author = "Rodrigo Platte and Gutierrez, {Alexander J.} and Anne Gelb",
year = "2015",
month = "11",
day = "1",
doi = "10.1016/j.acha.2014.10.002",
language = "English (US)",
volume = "39",
pages = "427--449",
journal = "Applied and Computational Harmonic Analysis",
issn = "1063-5203",
publisher = "Academic Press Inc.",
number = "3",

}

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 -