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Phase retrieval for the Cauchy wavelet transform

hal.structure.identifierLaboratoire d'informatique de l'école normale supérieure [LIENS]
dc.contributor.authorMallat, Stéphane
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
hal.structure.identifierLaboratoire d'informatique de l'école normale supérieure [LIENS]
dc.contributor.authorWaldspurger, Irène
dc.date.accessioned2018-02-28T13:40:46Z
dc.date.available2018-02-28T13:40:46Z
dc.date.issued2015
dc.identifier.issn1069-5869
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/17488
dc.language.isoenen
dc.subjectPhase retrievalen
dc.subjectWavelet transformen
dc.subjectCauchy waveletsen
dc.subject.ddc515en
dc.titlePhase retrieval for the Cauchy wavelet transformen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in L 2-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.en
dc.relation.isversionofjnlnameJournal of Fourier Analysis and Applications
dc.relation.isversionofjnlvol21en
dc.relation.isversionofjnlissue6en
dc.relation.isversionofjnldate2015-12
dc.relation.isversionofjnlpages1251–1309en
dc.relation.isversionofdoi10.1007/s00041-015-9403-4en
dc.relation.isversionofjnlpublisherSpringeren
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2018-02-28T13:38:34Z
hal.author.functionaut
hal.author.functionaut


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