BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Histogram tomography - Bill Lionheart (University of Manchester)
DTSTART:20181025T140000Z
DTEND:20181025T150000Z
UID:TALK113968@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:In many tomographic imaging problems the data consist of integ
 rals along lines or curves. Increasingly we encounter "rich tomography" pr
 oblems where the quantity imaged is higher dimensional than a scalar per v
 oxel\, including vectors tensors and functions. The data can also be highe
 r dimensional and in many cases consists of a one or two dimensional spect
 rum for each ray. In many such cases the data contain not just integrals a
 long rays but the distribution of values along the ray. If this is discret
 ized into bins we can think of this as a histogram. In this paper we intro
 duce the concept of "histogram tomography". For scalar problems with histo
 gram data this holds the possibility of reconstruction with fewer rays. In
  vector and tensor problems it holds the promise of reconstruction of imag
 es that are in the null space of related integral transforms. For scalar h
 istogram tomography problems we show how bins in the histogram correspond 
 to reconstructing level sets of function\, while moments of the distributi
 on are the x-ray transform of powers of the unknown function. In the vecto
 r case we give a reconstruction procedure for potential components of the 
 field. We demonstrate how the histogram longitudinal ray transform data ca
 n be extracted from Bragg edge neutron spectral data and hence\, using mom
 ents\, a non-linear system of partial differential equations derived for t
 he strain tensor. In x-ray diffraction tomography of strain the transverse
  ray transform can be deduced from the diffraction pattern the full histog
 ram transverse ray transform cannot. We give an explicit example of distri
 butions of strain along a line that produce the same diffraction pattern\,
  and characterize the null space of the relevant transform.
LOCATION:MR 14
END:VEVENT
END:VCALENDAR