# Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind 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 ${\cal N} f = \exp(S^{*}\log f)$ , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

, Volume 39 (1) – Jun 1, 2007
17 pages

/lp/springer_journal/nonlinear-smoothing-and-the-em-algorithm-for-positive-integral-LgaQa6OnVT
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900099
Publisher site
See Article on Publisher Site

### 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 ${\cal N} f = \exp(S^{*}\log f)$ , 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2007

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