Speed of Estimation in Positron Emission Tomography
Abstract
Several algorithms for image reconstruction in positron emission tomography (PET) have been described in the medical and statistical literature. A continuous idealisation of the PET reconstruction problem is studied, considered as an example of bivariate density estimation based on indirect observations. Given a large sample of indirect observations, the size is considered of the equivalent sample of observations, whose original exact positions would allow equally accurate estimation of the image of interest. Both for indirect and for direct observations, the exact minimax rates are established for convergence of estimation, for all possible estimators, over suitable smoothness classes of functions. For indirect data and (in practice unobservable) direct data, these rates are n exp(-p/(p+2)) and (n/log n) exp(-p/ (p+1)) respectively, for densities in a class corresponding to bounded square- integrable pth derivatives. Numerical values are obtained for equivalent sample sizes for a particular class of orthogonal series estimators, based on the eigenfunctions of the Radon transform. Keywords: Density estimation; Fano's lemma; Image analysis; Information theory; Inverse problems; Minimax; Orthogonal series; Radon transform; Singular value decomposition; Tomography.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1987
- Accession Number
- ADA196948
Entities
People
- B. W. Silverman
- Iain M. Johnstone
Organizations
- Stanford University