Blind inverse problems with isolated spikes. (16th June 2022)

• Record Type: Journal Article
• Authors: Debarnot, Valentin; Weiss, Pierre
• Is Part Of: Information and inference. Volume 12:Number 1(2023)
• Page Start: 26
• Page End: 71
• Publication Date: 2022-06-16
• Subjects: Blind inverse problem -- blind deconvolution -- sparse spike recovery -- off-the-grid -- chaos process -- 2010 Math Subject Classification: 15A29 -- 45Q05 -- 78A46 -- 62M15 -- 60G15 -- 62D05; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaac015
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators.