Imaging with highly incomplete and corrupted data

dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Métodos Numéricos y Aplicacioneses
dc.affiliation.institutoUC3M. Instituto Universitario sobre Modelización y Simulación en Fluidodinámica, Nanociencia y Matemática Industrial Gregorio Millán Barbanyes
dc.contributor.authorMoscoso, Miguel
dc.contributor.authorNovikov, Alexei
dc.contributor.authorPapanicolau, George
dc.contributor.authorTsogka, Chrysoula
dc.contributor.funderMinisterio de Economía y Competitividad (España)es
dc.description.abstractWe consider the problem of imaging sparse scenes from a few noisy data using an L1-minimization approach. This problem can be cast as a linear system of the form Ap = b, where A is an N x K measurement matrix. We assume that the dimension of the unknown sparse vector p E Ck is much larger than the dimension of the data vector b E Cn, i.e. K >>N. We provide a theoretical framework that allows us to examine under what conditions the L1-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that L1-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of L1-minimization we propose to solve instead the augmented linear system [A|C]p = b, where the N = Σ matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension N, can be well approximated. Theoretically, the dimension Σ of the noise collector should be eN which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns Σ~10K.en
dc.description.sponsorshipPart of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, during the Fall 2017 semester. The work of M Moscoso was partially supported by Spanish MICINN grant FIS2016-77892-R. The work of A Novikov was partially supported by NSF grants DMS-1515187, DMS-1813943. The work of C Tsogka was partially supported by AFOSR FA9550-17-1-0238.en
dc.identifier.bibliographicCitationMoscoso, M., Novikov, A., Papanicolaou, G. & Tsogka, C. (2020). Imaging with highly incomplete and corrupted data. Inverse Problems, 36(3), 035010.en
dc.identifier.publicationtitleInverse Problemsen
dc.publisherIOP Publishingen
dc.relation.projectIDGobierno de España. FIS2016-77892-Res
dc.rights© 2020 IOP Publishing Ltd.en
dc.rights.accessRightsopen accessen
dc.subject.otherArray imagingen
dc.subject.otherL1-norm minimizationen
dc.subject.otherHighly corrupted dataen
dc.titleImaging with highly incomplete and corrupted dataen
dc.typeresearch article*
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