The Annals of Statistics

A Study of Least Squares and Maximum Likelihood for Image Reconstruction in Positron Emission Tomography

Finbarr O'Sullivan

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Positron emission tomography (PET) is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo. A number of biological and physical factors limit the resolution of PET so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study. To a first approximation, reconstruction involves the solution of a linear inverse problem with a line-integral Radon-type transform. Standard filtered back-projection reconstruction is based on the method of least squares. Although computationally efficient, the method does not enforce positivity constraints leading to undesirable negative artifacts in the results. Maximum likelihood based approaches to reconstruction do not suffer from this problem, but their computational complexity has limited the ability to determine quantitatively the improvements in image quality. In this paper, asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail. The studies are carried out for idealized representations of conventional and time-of-flight tomographs. The asymptotic analysis indicates that for a range of Sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order. This is borne out by numerical studies. However, in these studies maximum likelihood is found to be more efficient than least squares: on a conventional distance-angle tomograph, the root mean square error is on the order of 10-20% smaller for maximum likelihood reconstructions. The corresponding reduction in the root mean square error on a time-of-flight tomograph is in the 5% range. Similar results are found using more complex region of interest based measures of reconstruction accuracy. In addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts. Since the postprocessing scheme is computationally fast, this suggests that in PET it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the EM algorithm.

Article information

Ann. Statist., Volume 23, Number 4 (1995), 1267-1300.

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62P10: Applications to biology and medical sciences 41A35: Approximation by operators (in particular, by integral operators) 41A25: Rate of convergence, degree of approximation 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 45L10 45M05: Asymptotics

Asymptotic approximation rates of estimation filtered back-projection positron emission tomography time-of-flight maximum likelihood positivity regularization Sobolev norms


O'Sullivan, Finbarr. A Study of Least Squares and Maximum Likelihood for Image Reconstruction in Positron Emission Tomography. Ann. Statist. 23 (1995), no. 4, 1267--1300. doi:10.1214/aos/1176324709.

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