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Imaging with highly incomplete and corrupted data

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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.date.accessioned2021-04-27T08:08:43Z
dc.date.available2021-04-27T08:08:43Z
dc.date.issued2020-03
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.format.extent21
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.doihttps://doi.org/10.1088/1361-6420/ab5a21
dc.identifier.issn0266-5611
dc.identifier.publicationfirstpage1
dc.identifier.publicationissue3(035010)
dc.identifier.publicationlastpage21
dc.identifier.publicationtitleInverse Problemsen
dc.identifier.publicationvolume36
dc.identifier.urihttps://hdl.handle.net/10016/32486
dc.identifier.uxxiAR/0000026263
dc.language.isoeng
dc.publisherIOP Publishingen
dc.relation.projectIDGobierno de España. FIS2016-77892-Res
dc.rights© 2020 IOP Publishing Ltd.en
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMaterialeses
dc.subject.ecienciaQuímicaes
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*
dc.type.hasVersionAM*
dspace.entity.typePublication
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