A dual formulation of a penalized maximum likelihood x-ray CT reconstruction Problem

Research output: Chapter in Book/Report/Conference proceedingConference contribution


This work studies the dual formulation of a penalized maximum likelihood reconstruction problem in x-ray CT. The primal objective function is a Poisson log-likelihood combined with a weighted cross-entropy penalty term. The dual formulation of the primal optimization problem is then derived and the optimization procedure outlined. The dual formulation better exploits the structure of the problem, which translates to faster convergence of iterative reconstruction algorithms. A gradient descent algorithm is implemented for solving the dual problem and its performance is compared with the filtered back-projection algorithm, and with the primal formulation optimized by using surrogate functions. The 3D XCAT phantom and an analytical x-ray CT simulator are used to generate noise-free and noisy CT projection data set with monochromatic and polychromatic x-ray spectrums. The reconstructed images from the dual formulation delineate the internal structures at early iterations better than the primal formulation using surrogate functions. However the body contour is slower to converge in the dual than in the primal formulation. The dual formulation demonstrate better noise-resolution tradeoff near the internal organs than the primal formulation. Since the surrogate functions in general can provide a diagonal approximation of the Hessian matrix of the objective function, further convergence speed up may be achieved by deriving the surrogate function of the dual objective function.

Original languageEnglish (US)
Title of host publicationMedical Imaging 2009
Subtitle of host publicationPhysics of Medical Imaging
StatePublished - Jun 15 2009
EventMedical Imaging 2009: Physics of Medical Imaging - Lake Buena Vista, FL, United States
Duration: Feb 9 2009Feb 12 2009

Publication series

NameProgress in Biomedical Optics and Imaging - Proceedings of SPIE
ISSN (Print)1605-7422


OtherMedical Imaging 2009: Physics of Medical Imaging
Country/TerritoryUnited States
CityLake Buena Vista, FL


  • Computed tomography
  • Dual optimization
  • Emission computed tomography
  • Poisson likelihood
  • Primal optimization
  • Rreconstruction algorithm
  • X-ray CT

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Biomaterials
  • Atomic and Molecular Physics, and Optics
  • Radiology Nuclear Medicine and imaging


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