Deconvolution of Point Sources

A Sampling Theorem and Robustness Guarantees

Brett Bernstein, Carlos Fernandez-Granda

Research output: Contribution to journalArticle

Abstract

In this work we analyze a convex-programming method for estimating superpositions of point sources or spikes from nonuniform samples of their convolution with a known kernel. We consider a one-dimensional model where the kernel is either a Gaussian function or a Ricker wavelet, inspired by applications in geophysics and imaging. Our analysis establishes that minimizing a continuous counterpart of the ℓ1-norm achieves exact recovery of the original spikes as long as (1) the signal support satisfies a minimum-separation condition and (2) there are at least two samples close to every spike. In addition, we derive theoretical guarantees on the robustness of the approach to both dense and sparse additive noise.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - Jan 1 2018

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Sampling Theorem
Geophysics
Additive noise
Convex optimization
Deconvolution
Point Source
Convolution
Spike
Sampling
Robustness
Imaging techniques
Recovery
kernel
Gaussian Function
Convex Programming
Additive Noise
One-dimensional Model
Superposition
Wavelets
Imaging

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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