Johnstone, Iain M.; Silverman, Bernard W. Speed of estimation in positron emission tomography and related inverse problems. (English) Zbl 0699.62043 Ann. Stat. 18, No. 1, 251-280 (1990). Summary: Several algorithms for image reconstruction in positron emission tomography (PET) have been described in the medical and statistical literature. We study a continuous idealization of the PET reconstruction problem, considered as an example of bivariate density estimation based on indirect observations. Given a large sample of indirect observations, we consider the size 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, we establish exact minimax rates of convergence of estimation, for all possible estimators, over suitable smoothness classes of functions. A key technical device is a modulus of continuity appropriate to global function estimation. For indirect data and (in practice unobservable) direct data, the rates for mean integrated square error are \(n^{-p/(p+2)}\) and (n/log n)\({}^{- p/(p+1)}\), respectively, for densities in a class corresponding to bounded square-integrable p th derivatives. We obtain numerical values for equivalent sample sizes for minimax linear estimators using a slightly modified error criterion. Modifications of the model to incorporate attenuation and the third dimension effect do not affect the minimax rates. The approach of the paper is applicable to a wide class of linear inverse problems. Cited in 1 ReviewCited in 59 Documents MSC: 62G05 Nonparametric estimation 62P99 Applications of statistics 62N99 Survival analysis and censored data 65R20 Numerical methods for integral equations 65C99 Probabilistic methods, stochastic differential equations Keywords:biased sampling; density estimation; Fano’s lemma; image analysis; information theory; metric entropy; orthogonal series; Radon; transform; rates of convergence; Sobolev norms; singular value; decomposition; algorithms; image reconstruction; positron emission tomography; continuous idealization of the PET reconstruction problem; bivariate density estimation; indirect observations; direct observations; exact minimax rates of convergence of estimation; modulus of continuity; global function estimation; mean integrated square error; bounded square- integrable p th derivatives; minimax linear estimators; attenuation; linear inverse problems × Cite Format Result Cite Review PDF Full Text: DOI