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Efficient Estimation of Linear Functionals in Emission Tomography

Alvin Kuruc
SIAM Journal on Applied Mathematics
Vol. 57, No. 2 (Apr., 1997), pp. 426-452
Stable URL: http://www.jstor.org/stable/2951870
Page Count: 27
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Efficient Estimation of Linear Functionals in Emission Tomography
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Abstract

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 $\Phi(f) \equiv \int\phi(x)f(x)dx$, 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.

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