Continuous and discrete filtering of photon noise in nuclear medicine
Description
Let $(X,\chi)$ be a measurable space, let $M = M(X,\chi)$ be the space of nonnegative, bounded measures on $(X,\chi),$ let N be the subset of M with integer values, let ${\cal M}$ be the smallest $\sigma$-algebra on M with respect to which the coordinate functions$$\{f\sb A :M \to \Re\vert f\sb A (\mu) = \mu(A),\forall A \in \chi, \forall\mu \in M\}$$are measurable, and let ${\cal N}$ be the $\sigma$-algebra on N induced by its inclusion in $(M,{\cal M})$. A measurable map $\nu$ from a complete probability space ($\Omega,{\cal E},P)$ to $(M,{\cal M})$ is a random measure of $(X,\chi)$; such a map with range in $(N,{\cal N})$ is a point process. Let $\mu$ be a finite positive measure on $(X,\chi)$, random measure $\nu$ is a Poisson point process on $(X,\chi)$ with mean or directing measure $\mu$ if for all $A \in \chi\ \nu(A;\cdot)$ is a Poisson random variable with mean $\mu(A)$, $$P(\{\omega\vert\nu(A;\omega) = k\}) = e\sp{-\mu(A)}{\mu(A)\sp k\over k!},\ k = 0,1,2,\... ,$$and for any finite collection of pairwise disjoint sets $\{A\sb1,A\sb2,\...,A\sb p\} \subset \chi$ the random variables $\{\nu(A\sb1;\cdot),\nu(A\sb2;\cdot),\...,\nu(A\sb p;\cdot)\}$ are independent,$$P(\cap\sbsp{i = 1}{p}\{\omega\vert\nu(A\sb i;\omega) = k\sb i\}) = {\prod\limits\sbsp{i = 1}{p}}\ P(\{\omega\vert\nu(A\sb i;\omega) = k\sb i\}).$$ Following Peskin, Tranchina, and Hull, we use a Poisson point process to model the occurrence of $\gamma$-photons produced by the decay of a radioactive marker in the tissue of a given patient for a diagnostic procedure in nuclear medicine. The directing measure of the process represents the distribution of the marker during the scan; this directing measure can be estimated through a linear filtering technique. We prove the existence of a continuous, linear, least-squares filter and the approximability of this filter by a sequence of discrete filters of the kind developed by Peskin, Tranchina, and Hull. We also demonstrate techniques for filtering and filter error estimation at high resolution