A constrained dual-energy reconstruction method for material-selective transmission tomography

A new reconstruction method has been developed for material-selective tomography that allows non-negativity and maximum density constraints to be enforced on each pixel. In addition, uncertainty in the projection data is appropriately accounted for . Results of applying the constrained reconstruction to simulated data demonstrate significant improvements over unconstrained reconstructions. Dual energy X-ray transmission tomography can be useful for estimating attenuation coefficients for PET and SPECT imaging [1] . In an effort to minimize the dose to the patient in the transmission scan, we are investigating new reconstruction approaches that allow reduced X-ray exposure for a given error in attenuation estimates . As in Ref. [2], we consider the object to be composed to two basis materials. However, instead of allowing all 1~ Full paper submitted to Nucl . Instr. and Meth . A. Research supported under NIH grants ROI CA54362 and R01 CA32846. * Tel . + 1 313 763 9244, fax + 1 313 764 0288, e-mail [email protected] .edu. NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SectionA linear combinations of the basis vectors, we constrain the feasible attenuation values to only convex combinations, which corresponds to non-negative densities of each material . Further performance enhancements are obtained by restricting the maximum allowable density of each material . Imposing these physically meaningful constraints stabilizes the reconstruction error as the X-ray exposure (and hence dose) is reduced. Conventional reconstruction techniques neither impose these constraints nor correctly weight uncertainty in the projection data . We solve the reconstruction problem by minimizing an uncertainty-weighted, penalized least-squares objective function parameterized by the unknown material densities . A grouped coordinate descent modification to the sequentially updating Gauss-Seidel algorithm used by Sauer and Fig . 1 . Reconstructions of soft-tissue (top row) and bone (bottom row) densities for simulated tomographic data . Left : unconstrained reconstruction . Middle: non-negativity constraints only Right: non-negativity and maximum density constraints . True soft-tissue densities: background, 1.0 g/cm3 . Regions clockwise starting from 12 o'clock, 0 g/cm3, 0 5 g/cm 3 , 0.2 g/cm3, 0.5 g/cm 3 . True bone densities: background, 0 g/cm 3 . Regions in same order as previous, 2.0 g/cm3, 0.5 g/cm3, 0 g/cm3 1.0 g/cm 3 . 0168-9002/94/$07.00 C 1994 Elsevier Science B.V. All nghts reserved SSDI0168-9002(94)00771-3 V. IMAGING 348 N.H. Chnthorne/Nucl. Instr. andMeth. in Phys. Res. A 353 (1994) 347-348 Bouman [3] is employed for the iterative solution . Each pair of pixels, corresponding to a single spatial location, is updated by choosing new values from the constraint set to minimize the objective. A single iteration is complete when all pixels have been updated in this manner. The method has been tested using simulated dual-energy tomographic data for soft-tissue and bone at 40 and 80 keV. Results are shown in Fig. 1 where unconstrained reconstructions, those employing only non-negativity constraints, and the reconstructions using both non-negativity and maximum density constraints are shown left-to-right respectively . The advantage of the constraints is clearly seen when the densities lie on the boundary of the constraint set (all regions except the one at 3 o'clock) . If the density is in the interior of the constraint set, the improvement is not as dramatic. Nevertheless, with the exception of lung tissue, most body tissues-being either soft-tissue or soft-tissue/bone mixtures-will be at or near the maximum feasible density, and it is precisely this situation that favors the fully constrained reconstruction over alternatives.