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 Φ(f) ≡ ∫ φ(x)f(x) dx, where φ 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 Φ(f). By definition, the variance of the efficient estimator is a best-possible lower bound (depending on φ and f) on the variance of unbiased estimators of Φ(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.