Estimation of integral functionals in emission tomography

The author presents an algorithm-independent theory of statistical accuracy attainable in emission tomography (ET) that makes minimal assumptions about the underlying image. They model the tracer density as a probability density, f, on a bounded domain, D (i.e., f is the underlying image) and consider the problem of estimating the integral functional Φ(f)≡∫φ(x)f(x) dx, where φ is a smooth function. Given n independent, identically-distributed observations distributed according to the Radon transform of f, Rf, the author constructs efficient, i.e., minimum-variance unbiased, estimators for Φ(f). Let L denote the set of lines through D and l₁ ,...,l_n∈L denote the observations in ET. The author shows there are many unbiased linear estimators of the form n ^-1Σ_i=1ⁿψ(l_i), where ψ is a function on L. A necessary and sufficient for ψ to generate an unbiased linear estimator is that ψ backproject to φ on D. The efficient estimator is generated by the ψ that minimizes ∫Lψ²(θ,s)Rf(θ,s)dsdθ while satisfying this unbiasedness constraint. The author represents the standard estimator based on filtered backprojection as the linear estimator generated by a function Fφ on L and show that the efficient estimator is generated by the projection of Fφ onto the orthogonal complement of the nullspace of the backprojection operator when these functions are viewed in a certain weighted Hilbert space. Numerical examples for functionals generated by Gaussian functions are presented. The results quantify the potential improvement attainable by incorporation of information on the domain of the image and the statistical uncertainty of the observations into the estimation process