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Technical Note: Convergence analysis of a polyenergetic SART algorithm

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  • PURPOSE: We analyze a recently proposed polyenergetic version of the simultaneous algebraic reconstruction technique (SART). This algorithm, denoted pSART, replaces the monoenergetic forward projection operation used by SART with a post-log, polyenergetic forward projection, while leaving the rest of the algorithm unchanged. While the proposed algorithm provides good results empirically, convergence of the algorithm was not established mathematically in the original paper. METHODS: We analyze pSART as a nonlinear fixed point iteration by explicitly computing the Jacobian of the iteration. A necessary condition for convergence is that the spectral radius of the Jacobian, evaluated at the fixed point, is less than one. A short proof of convergence for SART is also provided as a basis for comparison.. RESULTS: We show that the pSART algorithm is not guaranteed to converge, in general. The Jacobian of the iteration depends on several factors, including the system matrix and how one models the energy dependence of the linear attenuation coefficient. The authors provide a simple numerical example that shows that the spectral radius of the Jacobian matrix is not guaranteed to be less than one. A second set of numerical experiments using realistic CT system matrices, however, indicates that conditions for convergence are likely to be satisfied in practice CONCLUSION: Although pSART is not mathematically guaranteed to converge, our numerical experiments indicate that it will tend to converge at roughly the same rate as SART for system matrices of the type encountered in CT imaging. Thus we conclude that the algorithm is still a useful method for reconstruction of polyenergetic CT data.
  • Keywords: ART, algebraic reconstruction technique, nonlinear fixed point iteration, beam hardening, computed tomography, polyenergetic CT, SART
  • This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by American Association of Physicists in Medicine and can be found at: http://www.medphys.org/
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  • Humphries, T. (2015). Technical Note: Convergence analysis of a polyenergetic SART algorithm. Medical Physics, 42(7), 4007-4014. doi:10.1118/1.4922204
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  • 42
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  • 7
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