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J Nucl Med. 2008; 49 (Supplement 1):61P
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Instrumentation & Data Analysis: Image Generation

PET - Reconstruction and Compensation

Fast maximum likelihood image reconstruction via PCG without a line search

Guillem Pratx1, Andrew Reader2 and Craig Levin1

1 Stanford University, Stanford, California; 2 University of Manchester, Manchester, United Kingdom

244

Objectives: Maximum Likelihood (ML) image reconstruction using the preconditioned conjugate gradient algorithm (ML-PCG) has been shown to converge to the ML estimate faster than expectation maximization (ML-EM). ML-PCG is however slower in the early iterations than ordered-subset (OS) EM, but since OS-EM can suffer from limit cycles, ML-PCG is still preferred for converged regularized image reconstruction. ML-PCG requires the computation of an optimal step size {alpha} for each iteration, which is conventionally computed by performing an iterative 1D line search, and often tens of iterations are needed (~17% of the reconstruction time). We have derived a new analytical approximation for {alpha} that is accurate and 100X faster to compute.

Methods: The new formulation for {alpha} was derived directly from the log-likelihood for an inhomogeneous Poisson process, and requires just two scalar products (using the ML gradient and the projection of the current search direction, which are readily available in PCG). The new method was evaluated for simulated 2D PET phantom with warm background.

Results: The accuracy of the new analytical {alpha}, compared to a line search, was assessed for two noise levels at different iterations. At the 1X noise level, the error in evaluating {alpha} was 0.8% RMSE (maximum error 2.0%). At the 4X noise level, the error was 1.7% RMSE (4.7% max error). Appealingly, the RMSE decreases to 0 at higher iterations. The analytical formulation was then used in place of the line search in ML-PCG. The resulting algorithm converges in as few iterations as regular PCG to the ML estimate, but in 17% less total computation time.

Conclusions: We have derived a new analytical expression for computing the optimal step size {alpha}, eliminating the need for an iterative line search. This formulation is accurate and fast, yielding a convergent ML-PCG algorithm.





This Article
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Right arrow Articles by Levin, C.
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Right arrow Articles by Pratx, G.
Right arrow Articles by Levin, C.