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SUMMARY:Stable Gabor Phase Retrieval and Spectral Clustering - Philipp Gro
 hs (University of Vienna)
DTSTART:20171005T140000Z
DTEND:20171005T150000Z
UID:TALK72520@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:We consider the problem of reconstructing a signal $f$ from it
 s spectrogram\, i.e.\, the magnitudes $|V_\\varphi f|$ of its Gabor transf
 orm\n	$$V_\\varphi f (x\,y):=\\int_{\\mathbb{R}}f(t)e^{-\\pi (t-x)^2}e^{-2
 \\pi \\i y t}dt\, \\quad x\,y\\in \\mathbb{R}.$$ Such problems occur in a 
 wide range of applications\, from optical imaging of nanoscale structures 
 to audio processing and classification.\n\nWhile it is well-known that the
  solution of the above Gabor phase retrieval problem is unique up to natur
 al identifications\, the stability of the reconstruction has remained wide
  open. The present paper discovers a deep and surprising connection betwee
 n phase retrieval\, spectral clustering and spectral geometry. We show tha
 t the stability of the Gabor phase reconstruction is bounded by the recipr
 ocal of the \\emph{Cheeger constant} of the flat metric on $\\mathbb{R}^2$
 \, conformally multiplied with $|V_\\varphi f|$.  The Cheeger constant\, i
 n turn\, plays a prominent role in the field of spectral clustering\, and 
 it precisely quantifies the `disconnectedness' of the measurements $V_\\va
 rphi f$.\n\nIt has long been known that a disconnected support of the meas
 urements results in an instability -- our result for the first time provid
 es a converse in the sense that there are no other sources of instabilitie
 s.\n\nDue to the fundamental importance of Gabor phase retrieval in cohere
 nt diffraction imaging\, we also provide a new understanding of the stabil
 ity properties of these imaging techniques: Contrary to most classical pro
 blems in imaging science whose regularization requires the promotion of sm
 oothness or sparsity\, the correct regularization of the phase retrieval p
 roblem promotes the `connectedness' of the measurements in terms of boundi
 ng the Cheeger constant from below. Our work thus\, for the first time\, o
 pens the door to the development of efficient regularization strategies.
LOCATION:MR 14\, CMS
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