A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography

• Published in: Journal of the Royal Statistical Society. Series B, Methodological Methodological, 1990-01, Vol.52 (2), p.271-324
• Main Authors: Silverman, B. W., Jones, M. C., Wilson, J. D., Nychka, D. W.
• Format: Article
• Language: English
• Subjects: Algorithms; Data smoothing; Density estimation; Estimation methods; Exact sciences and technology; ill‐posed problems; Image reconstruction; indirect observations; intensity estimation; Mathematics; maximum likelihood; Maximum likelihood estimation; Nonparametric inference; penalized likelihood; Pixels; Positron emission tomography; Probability and statistics; Radius of a sphere; Sciences and techniques of general use; smoothing; Statistics; stereology; Tomography
• ISSN: 0035-9246; 1369-7412; 2517-6161; 1467-9868
• DOI: 10.1111/j.2517-6161.1990.tb01788.x

There are many practical problems where the observed data are not drawn directly from the density g of real interest, but rather from another distribution derived from g by the application of an integral operator. The estimation of g then entails both statistical and numerical difficulties. A natural statistical approach is by maximum likelihood, conveniently implemented using the EM algorithm, but this provides unsatisfactory reconstructions of g. In this paper, we modify the maximum likelihood-EM approach by introducing a simple smoothing step at each EM iteration. In our experience, this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration. Some theoretical background is given that relates this smoothed EM algorithm to a maximum penalized likelihood approach. Two applications are considered in detail. The first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium. The second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography; for this problem, we also suggest several technical improvements on existing methodology.