Inversion of the 3D exponential x-ray transform.

Rolf Clackdoyle
University of Utah

Co-Authors: J-M Wagner, F Noo

The 3D parallel-beam x-ray transform maps object functions
to collections of parallel-beam projections. The collection
of projection directions, a subset of the unit sphere, must satisfy Orlov's condition for stable reconstruction to be possible. A more general statement, also due to Orlov, is that any projection inside the convex hull (on the unit sphere) of the known projections can be synthesized from the available projections.

This talk concerns the exponential case. Wagner has shown that if exponential projections are known for a circle of directions on the unit sphere, then exponential projections can be synthesized for directions inside the circle. Furthermore, it is possible to alter the exponential constant, mu. The approach involves introducing a 3D exponential Radon transform, and uses a result of Kuchment and Shneiberg on the angular dependent 2D exponential transform.

The motivation for this work is the rotating slant-hole (RSH) SPECT scanner, whose measurement data can be modelled as exponential projections for directions lying on some circles on the unit sphere. Wagner's result admits an exact rebinning procedure for reconstruction for this RSH SPECT geometry.


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