Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind |
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Authors: | P P B Eggermont |
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Institution: | (1) Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA, US |
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Abstract: | We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing
step of the form , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (à la positron emission tomography)
the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast
to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original
EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence
of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence
of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that
the new algorithm behaves roughly like the EMS algorithm.
Accepted 1 April 1997 |
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Keywords: | , EM, Maximum likelihood, Ill-posed problem, Regularization, AMS Classification, 45L10, 65R30, 65U05, 62G05, |
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