## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 1 (1990), 251-280.

### Speed of Estimation in Positron Emission Tomography and Related Inverse Problems

Iain M. Johnstone and Bernard W. Silverman

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 18, Number 1 (1990), 251-280.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347500

**Digital Object Identifier**

doi:10.1214/aos/1176347500

**Mathematical Reviews number (MathSciNet)**

MR1041393

**Zentralblatt MATH identifier**

0699.62043

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62C20: Minimax procedures 65R20: Integral equations 65U05

**Keywords**

Attenuation biased sampling density estimation Fano's lemma image analysis information theory linear inverse problems metric entropy modulus of continuity minimax orthogonal series Radon transform rates of convergence Sobolev norms singular value decomposition tomography

#### Citation

Johnstone, Iain M.; Silverman, Bernard W. Speed of Estimation in Positron Emission Tomography and Related Inverse Problems. Ann. Statist. 18 (1990), no. 1, 251--280. doi:10.1214/aos/1176347500. https://projecteuclid.org/euclid.aos/1176347500