Transcript case because

Can Phase Space Tomography be
Spun-off back into Medical Imaging?
S. Hancock
What is Tomography?
Here is a trivial density distribution and six of its projections. The aim of tomography in this
example is to estimate the two-dimensional distribution – which would normally be hidden
from us – using only the one-dimensional information. There are a variety of algorithms to
achieve this.
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ART Algorithm – Back Projection
Back projection is, naturally enough, a sort of inverse process to projection. However, it
admits no two-dimensional knowledge so the contents of one bin is shared along a “track”
over all pixels which could have contributed to that bin. Combining all back projections gives
a crude first estimate.
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ART Algorithm – Projection
A projection of the estimated distribution is different from the corresponding original profile
because of all the so-called “point spread”. So, subtracting bin by bin each (red) projection
from its (black) original gives a set of difference profiles in which some bins necessarily have
negative contents.
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ART Algorithm – Iteration
Back projection onto the approximate distribution of the bin-by-bin difference profiles yields
an improved approximation. The process can be iterated many times.
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What is the Relevance to Phase Space?
The application to accelerators becomes obvious once the imaginative leap is made between
the x-ray projections of a patient in a rotating body scanner and the turn-by-turn profiles of a
bunch of particles rotating in longitudinal phase space. On each turn around the machine, a
longitudinal pick-up provides a snapshot of the bunch projected at a slightly different angle.
However, the problem with conventional ART is that its strategies for estimating the
redistribution coefficients are based on straight-line back projection, which is not the
geometry of longitudinal phase space. Large-amplitude synchrotron motion is simply not
circular nor can all the projections be measured simultaneously. This issue has been resolved
in a hybrid algorithm which combines particle tracking with ART. Indeed, the tracking code
can be made arbitrarily complex while, afterwards, the tomography proceeds in exactly the
same way.
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Online Phase Space Tomography (1)
The idea is to “see” into longitudinal phase space at the click of a mouse. The Tomoscope
application serves to simplify the acquisition of a mountain range of bunch profile data and to
automate the reconstruction.
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Online Phase Space Tomography (2)
Tomography provides a phase space image that is not only instructive, but affords beam
measurements of unprecedented precision that are remarkably immune to both systematic
errors and noise. The resultant particle distribution is consistent with all the measured profiles
and the physics of synchrotron motion.
Cf., a gaussian fit.
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Speculation (1998!)
The method constitutes an extension of computerized tomography to cater for non-rigid bodies. A model is required for
the deformation over the duration of data-taking, but, because the algorithm is an iterative one, the parameters of the
model can be refined by their influence on convergence.
In phase space tomography, the distribution of particles under study effectively rotates and a single projection is
measured at successive times. The geometry of this non-linear rotation is analysed using particle tracking before the
reconstruction is made. Since the analysis of the motion is entirely decoupled from the tomography part of the code,
could applications outside the accelerator domain be treated by replacing the tracking code with a suitable model? In
medical imaging, it might allow a patient’s breathing to be taken into account, or perhaps a beating heart could be imaged
using a much lower dose of radioisotope than presently required.
Conceptually, it sounds feasible enough, but I see some potential stumbling blocks that I am not qualified to assess. As
far as I understand, all projections in a PET scanner are concurrently acquired throughout the course of data-taking and
the images are produced in a single pass of some highly sophisticated algorithm. In the accelerator case, each projection
is completely acquired as a single snapshot, but only one projection can be measured at a time. It’s a bit like having a
patient who, rather obligingly, is rotating in a stationary body scanner which has a very limited field of view, but which
does not need to integrate over time to build up each projection. A knowledge of how the particle distribution evolves as
it rotates allows the information in all the discrete time slices to be translated back to the same instant and combined in a
single image. And here is the potential to reduce radiation dose in the medical case because data taken, for example,
throughout a few cardiac cycles could all be usefully combined instead of binning together short data samples taken at
the same point in each of many heartbeats.
I guess an iterative algorithm versus a single-pass one is not the issue and the need to integrate detector signals over time
probably precludes the convenient decoupling of the model from the reconstruction which occurs in the accelerator case.
The question is: can the time evolution due to morphological changes be incorporated in an alreadly complex medical
tomography code? Maybe people have already tried.
(Contact [email protected])
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Why KT?
I still don’t have answers to these speculative
questions, but I have found someone prepared to
help investigate them:
http://people.bath.ac.uk/ms350/Engineering%20to
mography%20laboratory.htm
The idea would be for Dr. Soleimani to supervise a
PhD student to build an optical phantom with
flexible voids at the University of Bath’s
Engineering Tomography Laboratory in England.
The aim would be to make a proof of principle,
with only minor consultancy from me, that a
CERN-like hybrid algorithm can better reconstruct
the phantom when the varying air pressure in the
voids is included in the model.
The bottom line would be of the order of £25,000
over one year.
(The existing algorithm entered the public domain
in 1999 and the IP rights are CERN’s.)
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