Efficient estimation of linear functionals in emission tomography

In emission tomography, the spatial distribution of a radioactive tracer is estimated from a finite sample of externally detected photons. We present an algorithm-independent theory of statistical accuracy attainable in emission tomography that makes minimal assumptions about the underlying image. Let f denote the tracer density as a function of position (i.e., f is the image being estimated). We consider the problem of estimating the linear functional Phi(f) = integral phi(x)f(x) dx, where phi is a smooth function, from n independent observations identically distributed according to the Radon transform of f. Assuming only that f is bounded above and below away from 0, we construct statistically efficient estimators for Phi(f). By definition, the variance of the efficient estimator is a best-possible lower bound (depending on phi and f) on the variance of unbiased estimators of Phi(f). Our results show that, in general, the efficient estimator will have a smaller variance than the standard estimator based on the filtered-backprojection reconstruction algorithm. The improvement in performance is obtained by exploiting the range properties of the Radon transform.