Abstract: We study an estimation problem in PET when the time-of-fight information is available. The continuous idealization of the PET reconstruction problem, formulated by Johnstone and Silverman (1990) as a special case of bivariate density estimation based on indirect observations, is used. A Modified Deconvoluting Kernel Density Estimator (MDK) is proposed. For densities wit h mth derivatives satisfying α Lipschitz condition in L2 norm and in L∞ norm, the convergence rates of mean integrated square error and maximum mean square error are shown to bewhere n is the umber of counts. These rates are optimal. By comparing our results with those in the literature where no time-of-flight is considered, it is shown that although the time of flight does not yield better convergence rates in this model, it can yield better constants when the noise is small.
Key words and phrases: Density estimation, deconvoluting kernel estimator, minimax, Radon transform, tomography.