Inversion of the 3D exponential x-ray transform for a half equatorial band and other semi-circular geometries.
This work presents new mathematical results on the inversion of the exponential x-ray transform. It is shown that a reconstruction formula can be obtained for any dataset whose projection directions consist of a union of half great circles on the unit sphere. A basic example of such a dataset is the semi-equatorial band. The discussion in the paper is mostly focused on this example. The reconstruction formula takes the form of a Neumann (geometric) series and is both exact and stable. The exponential x-ray transform has been mainly studied in SPECT imaging. In this context, our results demonstrate mathematically that fully 3D image reconstruction in SPECT with non-zero attenuation does not always require symmetric datasets (opposing views).
Noo
Frédéric
F
Department of Radiology, University of Utah, CAMT Building, 729 Arapeen Drive, Salt Lake City, UT 84112, USA. noo@doug.med.utah.edu
Clackdoyle
Rolf
R
Wagner
Jean-Marc
JM
D000465:Algorithms; D003198:Computer Simulation; D006321:Heart / Q000000981:diagnostic imaging*; D006801:Humans; D007089:Image Enhancement / Q000379:methods*; D021621:Imaging, Three-Dimensional / Q000379:methods*; D019047:Phantoms, Imaging; D012680:Sensitivity and Specificity; D015899:Tomography, Emission-Computed, Single-Photon / Q000295:instrumentation / Q000379:methods*; D014965:X-Rays