Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

Brett Bernstein, Carlos Fernandez-Granda

Research output: Contribution to journalArticlepeer-review


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 ℓ 1 -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)
Pages (from-to)1152-1230
Number of pages79
JournalCommunications on Pure and Applied Mathematics
Issue number6
StatePublished - Jun 2019

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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