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CATEGORIES:Applied and Computational Analysis
SUMMARY:Histogram tomography - Bill Lionheart (University
of Manchester)
DTSTART;TZID=Europe/London:20181025T150000
DTEND;TZID=Europe/London:20181025T160000
UID:TALK113968AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/113968
DESCRIPTION:In many tomographic imaging problems the data cons
ist of integrals along lines or curves. Increasing
ly we encounter "rich tomography" problems where t
he quantity imaged is higher dimensional than a sc
alar per voxel\, including vectors tensors and fun
ctions. The data can also be higher dimensional an
d in many cases consists of a one or two dimension
al spectrum for each ray. In many such cases the d
ata contain not just integrals along rays but the
distribution of values along the ray. If this is d
iscretized into bins we can think of this as a his
togram. In this paper we introduce the concept of
"histogram tomography". For scalar problems with h
istogram data this holds the possibility of recons
truction with fewer rays. In vector and tensor pro
blems it holds the promise of reconstruction of im
ages that are in the null space of related integra
l transforms. For scalar histogram tomography prob
lems we show how bins in the histogram correspond
to reconstructing level sets of function\, while m
oments of the distribution are the x-ray transform
of powers of the unknown function. In the vector
case we give a reconstruction procedure for potent
ial components of the field. We demonstrate how th
e histogram longitudinal ray transform data can be
extracted from Bragg edge neutron spectral data a
nd hence\, using moments\, a non-linear system of
partial differential equations derived for the str
ain tensor. In x-ray diffraction tomography of str
ain the transverse ray transform can be deduced fr
om the diffraction pattern the full histogram tran
sverse ray transform cannot. We give an explicit e
xample of distributions of strain along a line tha
t produce the same diffraction pattern\, and chara
cterize the null space of the relevant transform.
LOCATION:MR 14
CONTACT:Carola-Bibiane Schoenlieb
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