Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions

Mikhail Isaev; Roman G. Novikov; Grigory V. Sabinin

doi:10.1088/1361-6420/ac87cb

Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions

Mikhail Isaev
, Roman G. Novikov
, Grigory V. Sabinin

Research output: Contribution to journal › Article › peer-review

Abstract

We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318-33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform F [ v ] given within the ball B _r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.

Original language	English
Article number	105002
Journal	Inverse Problems
Volume	38
Issue number	10
DOIs	https://doi.org/10.1088/1361-6420/ac87cb
Publication status	Published - 1 Oct 2022

Keywords

Radon transform
band-limited Fourier transform
ill-posed inverse problems
prolate spheroidal wave functions
super-resolution

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10.1088/1361-6420/ac87cb

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title = "Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions",

abstract = "We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318-33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform F [ v ] given within the ball B r . For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.",

keywords = "Radon transform, band-limited Fourier transform, ill-posed inverse problems, prolate spheroidal wave functions, super-resolution",

author = "Mikhail Isaev and Novikov, \{Roman G.\} and Sabinin, \{Grigory V.\}",

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doi = "10.1088/1361-6420/ac87cb",

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AU - Isaev, Mikhail

AU - Novikov, Roman G.

AU - Sabinin, Grigory V.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318-33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform F [ v ] given within the ball B r . For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.

AB - We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318-33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform F [ v ] given within the ball B r . For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.

KW - Radon transform

KW - band-limited Fourier transform

KW - ill-posed inverse problems

KW - prolate spheroidal wave functions

KW - super-resolution

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DO - 10.1088/1361-6420/ac87cb

M3 - Article

AN - SCOPUS:85137128547

SN - 0266-5611

VL - 38

JO - Inverse Problems

JF - Inverse Problems

IS - 10

M1 - 105002

ER -

Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions

Abstract

Keywords

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