# Resolving scaling ambiguities with the $\ell_1/\ell_2$ norm in a blind deconvolution problem with feedback

 Title Resolving scaling ambiguities with the $\ell_1/\ell_2$ norm in a blind deconvolution problem with feedback Publication Type Conference Year of Publication 2015 Authors Ernie Esser, Rongrong Wang, Tim T.Y. Lin, Felix J. Herrmann Conference Name IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing Page 365-368 Month 12 Keywords $\ell_1/\ell_2$, blind deconvolution, CAMSAP, lifting, method of multipliers Abstract Compared to more mundane blind deconvolution problems, blind deconvolution in seismic applications involves a feedback mechanism related to the free surface. The presence of this feedback mechanism gives us an unique opportunity to remove ambiguities that have plagued blind deconvolution for a long time. While beneficial, this feedback by itself is insufficient to remove the ambiguities even with $\ell_1$ constraints. However, when paired with an $\ell_1/\ell_2$ constraint the feedback allows us to resolve the scaling ambiguity under relatively mild assumptions. Inspired by lifting approaches, we propose to split the sparse signal into positive and negative components and apply an $\ell_1/\ell_2$ constraint to the difference, thereby obtaining a constraint that is easy to implement. Numerical experiments demonstrate robustness to the initialization as well as to noise in the data. Notes (IEEE CAMSAP Workshop, Cancún, Mexico) URL https://www.slim.eos.ubc.ca/Publications/Public/Conferences/CAMSAP/2015/esser2015CAMSAPrsa/esser2015CAMSAPrsa.pdf DOI 10.1109/CAMSAP.2015.7383812 Presentation https://www.slim.eos.ubc.ca/Publications/Public/Conferences/CAMSAP/2015/... URL1 https://www.slim.eos.ubc.ca/Publications/Public/Conferences/CAMSAP/2015/... Citation Key esser2015CAMSAPrsa