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

Jingyan Xu; Katsuyuki Taguchi; Grant T. Gullberg; Benjamin M. W. Tsui

doi:10.1117/12.813873

13 March 2009 A dual formulation of a penalized maximum likelihood x-ray CT reconstruction problem

Jingyan Xu, Katsuyuki Taguchi, Grant T. Gullberg, Benjamin M. W. Tsui

Proceedings Volume 7258, Medical Imaging 2009: Physics of Medical Imaging; 725828 (2009) https://doi.org/10.1117/12.813873
Event: SPIE Medical Imaging, 2009, Lake Buena Vista (Orlando Area), Florida, United States

Abstract

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.

Citation Download Citation

Jingyan Xu, Katsuyuki Taguchi, Grant T. Gullberg, and Benjamin M. W. Tsui "A dual formulation of a penalized maximum likelihood x-ray CT reconstruction problem", Proc. SPIE 7258, Medical Imaging 2009: Physics of Medical Imaging, 725828 (13 March 2009); https://doi.org/10.1117/12.813873

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KEYWORDS

X-rays

Reconstruction algorithms

CT reconstruction

Evolutionary algorithms

X-ray detectors

Computed tomography

Data modeling

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