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Faster maximum-likelihood reconstruction via explicit conjugation of search directions

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3 Author(s)
Pratx, G. ; Stanford University, Department of Radiology, and Molecular Imaging Program at Stanford, USA ; Reader, A.J. ; Levin, C.S.

Conjugate gradient (CG) is useful to perform maximum-likelihood (ML) reconstruction in positron emission tomography (PET). Although first derived to solve linear systems of equations, CG has been used for non-quadratic objectives. For the log-likelihood of inhomogeneous Poisson processes, the search directions generated by the generic Polak-Ribière formulation are not quite conjugate. We investigated a new CG formulation specific to the ML criterion in PET that preserves the conjugation. We first established a new relationship between the search direction and the image residual. We then derived a new method to generate a basis of search directions conjugate in the ML sense. Conjugation was enforced explicitly by forming the new search direction from a linear combination of the gradient and all the past search directions. The new formulation converges faster to the ML optimal solution. The equivalent of 50 Polak-Ribière iterations is reached in 39 iterations (1.3× faster) and the equivalent of 2000 Polak-Ribière iterations is reached in 451 iterations (4.4× faster). The truncation of the new formulation converges at the same rate as the Polak-Ribière method. The new formulation requires only a negligible amount of extra computation, but very large amounts of memory to store past search directions.

Published in:

Nuclear Science Symposium Conference Record, 2008. NSS '08. IEEE

Date of Conference:

19-25 Oct. 2008