A sampling theorem for deconvolution in two dimensions

Joseph McDonald; Brett Bernstein; Carlos Fernandez-Granda

doi:10.1137/20M1329615

A sampling theorem for deconvolution in two dimensions

Joseph McDonald, Brett Bernstein, Carlos Fernandez-Granda

Mathematics

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

Abstract

This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the ℓ₁-norm exactly recovers the true spikes if they are sufficiently separated, and the samples are sufficiently dense. In addition, we provide numerical evidence that our results extend to non-Gaussian kernels relevant to microscopy and telescopy.

Original language	English (US)
Pages (from-to)	1754-1780
Number of pages	27
Journal	SIAM Journal on Imaging Sciences
Volume	13
Issue number	4
DOIs	https://doi.org/10.1137/20M1329615
State	Published - 2020

Keywords

Convex optimization
Deconvolution
Dual certificate
Gaussian convolution
Sampling theory
Sparsity
Super-resolution

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1137/20M1329615

Cite this

@article{1f38185fdc264c879afca37edca1028c,

title = "A sampling theorem for deconvolution in two dimensions",

abstract = "This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the ℓ1-norm exactly recovers the true spikes if they are sufficiently separated, and the samples are sufficiently dense. In addition, we provide numerical evidence that our results extend to non-Gaussian kernels relevant to microscopy and telescopy.",

keywords = "Convex optimization, Deconvolution, Dual certificate, Gaussian convolution, Sampling theory, Sparsity, Super-resolution",

author = "Joseph McDonald and Brett Bernstein and Carlos Fernandez-Granda",

note = "Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics.",

year = "2020",

doi = "10.1137/20M1329615",

language = "English (US)",

volume = "13",

pages = "1754--1780",

journal = "SIAM Journal on Imaging Sciences",

issn = "1936-4954",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - A sampling theorem for deconvolution in two dimensions

AU - McDonald, Joseph

AU - Bernstein, Brett

AU - Fernandez-Granda, Carlos

PY - 2020

Y1 - 2020

N2 - This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the ℓ1-norm exactly recovers the true spikes if they are sufficiently separated, and the samples are sufficiently dense. In addition, we provide numerical evidence that our results extend to non-Gaussian kernels relevant to microscopy and telescopy.

AB - This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the ℓ1-norm exactly recovers the true spikes if they are sufficiently separated, and the samples are sufficiently dense. In addition, we provide numerical evidence that our results extend to non-Gaussian kernels relevant to microscopy and telescopy.

KW - Convex optimization

KW - Deconvolution

KW - Dual certificate

KW - Gaussian convolution

KW - Sampling theory

KW - Sparsity

KW - Super-resolution

UR - http://www.scopus.com/inward/record.url?scp=85099016261&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85099016261&partnerID=8YFLogxK

U2 - 10.1137/20M1329615

DO - 10.1137/20M1329615

M3 - Article

AN - SCOPUS:85099016261

SN - 1936-4954

VL - 13

SP - 1754

EP - 1780

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

IS - 4

ER -

A sampling theorem for deconvolution in two dimensions

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this