1. Can Phase Space Tomography be Spun-off back into Medical Imaging?
S. Hancock

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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

10. Why KT?
I still don’t have answers to these speculative questions, but I have found someone prepared to help investigate them:
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.)