BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Applied and Computational Analysis
SUMMARY:Stable Gabor Phase Retrieval and Spectral Clusteri
ng - Philipp Grohs (University of Vienna)
DTSTART;TZID=Europe/London:20171005T150000
DTEND;TZID=Europe/London:20171005T160000
UID:TALK72520AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/72520
DESCRIPTION:We consider the problem of reconstructing a signal
$f$ from its spectrogram\, i.e.\, the magnitudes
$|V_\\varphi f|$ of its Gabor transform\n $$V_\\va
rphi f (x\,y):=\\int_{\\mathbb{R}}f(t)e^{-\\pi (t-
x)^2}e^{-2\\pi \\i y t}dt\, \\quad x\,y\\in \\math
bb{R}.$$ Such problems occur in a wide range of ap
plications\, from optical imaging of nanoscale str
uctures to audio processing and classification.\n\
nWhile it is well-known that the solution of the a
bove Gabor phase retrieval problem is unique up to
natural identifications\, the stability of the re
construction has remained wide open. The present p
aper discovers a deep and surprising connection be
tween phase retrieval\, spectral clustering and sp
ectral geometry. We show that the stability of the
Gabor phase reconstruction is bounded by the reci
procal of the \\emph{Cheeger constant} of the flat
metric on $\\mathbb{R}^2$\, conformally multiplie
d with $|V_\\varphi f|$. The Cheeger constant\, i
n turn\, plays a prominent role in the field of sp
ectral clustering\, and it precisely quantifies th
e `disconnectedness' of the measurements $V_\\varp
hi f$.\n\nIt has long been known that a disconnect
ed support of the measurements results in an insta
bility -- our result for the first time provides a
converse in the sense that there are no other sou
rces of instabilities.\n\nDue to the fundamental i
mportance of Gabor phase retrieval in coherent dif
fraction imaging\, we also provide a new understan
ding of the stability properties of these imaging
techniques: Contrary to most classical problems in
imaging science whose regularization requires the
promotion of smoothness or sparsity\, the correct
regularization of the phase retrieval problem pro
motes the `connectedness' of the measurements in t
erms of bounding the Cheeger constant from below.
Our work thus\, for the first time\, opens the doo
r to the development of efficient regularization s
trategies.
LOCATION:MR 14\, CMS
CONTACT:Carola-Bibiane Schoenlieb
END:VEVENT
END:VCALENDAR