Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

Xinjuan Chen, Jae Hun Jung, Anne Gelb

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted L2 space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

Original languageEnglish (US)
Pages (from-to)1127-1147
Number of pages21
JournalJournal of Scientific Computing
Volume76
Issue number2
DOIs
StatePublished - Aug 1 2018
Externally publishedYes

Fingerprint

Polynomials
Polynomial
Approximation
Gibbs Phenomenon
Image reconstruction
Reconstruction Algorithm
Image Reconstruction
Weighted Spaces
Resampling
Numerical Approximation
Orthogonal Polynomials
Demonstrate
Spacing
Analytic function
Resolve
Projection
Transform
Numerical Results
Coefficient
Operator

Keywords

  • Fourier frame
  • Gibbs phenomenon
  • Inverse polynomial reconstruction method
  • Non-uniform Fourier data

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Engineering(all)
  • Computational Theory and Mathematics

Cite this

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method. / Chen, Xinjuan; Jung, Jae Hun; Gelb, Anne.

In: Journal of Scientific Computing, Vol. 76, No. 2, 01.08.2018, p. 1127-1147.

Research output: Contribution to journalArticle

Chen, Xinjuan ; Jung, Jae Hun ; Gelb, Anne. / Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method. In: Journal of Scientific Computing. 2018 ; Vol. 76, No. 2. pp. 1127-1147.
@article{3402beb3ca2e40718c0bde06e6f206d8,
title = "Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method",
abstract = "In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted L2 space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.",
keywords = "Fourier frame, Gibbs phenomenon, Inverse polynomial reconstruction method, Non-uniform Fourier data",
author = "Xinjuan Chen and Jung, {Jae Hun} and Anne Gelb",
year = "2018",
month = "8",
day = "1",
doi = "10.1007/s10915-018-0655-4",
language = "English (US)",
volume = "76",
pages = "1127--1147",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer New York",
number = "2",

}

TY - JOUR

T1 - Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

AU - Chen, Xinjuan

AU - Jung, Jae Hun

AU - Gelb, Anne

PY - 2018/8/1

Y1 - 2018/8/1

N2 - In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted L2 space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

AB - In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted L2 space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

KW - Fourier frame

KW - Gibbs phenomenon

KW - Inverse polynomial reconstruction method

KW - Non-uniform Fourier data

UR - http://www.scopus.com/inward/record.url?scp=85041538480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041538480&partnerID=8YFLogxK

U2 - 10.1007/s10915-018-0655-4

DO - 10.1007/s10915-018-0655-4

M3 - Article

AN - SCOPUS:85041538480

VL - 76

SP - 1127

EP - 1147

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 2

ER -