Image Science with Particle Processing Detectors

In recent years, emission imaging systems have begun to incorporate a new detector design based on on-the-fly event attribute estimation. These new detectors - called particle processing detectors - lead to interesting new mathematical models for imaging systems. The goal is to quantify the performance of such detectors using the theory of inverse problems and statistical decision theory.

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\( \newcommand{\r}{\boldsymbol{r}} \newcommand{\e}{\mathcal{E}} \newcommand{\s}{\hat{\boldsymbol{s}}} \newcommand{\K}{\mathcal{K}} \) To be more precise, I work with imaging system models that are decomposed into two operators, a propagation operator \(\mathcal{P}\) and a detection operator \(\mathcal{D}\). The propogation operator models the physics of the energy used, so in my case (I work mostly in emission imaging) it’s the solution operator to a Radiative Transfer Equation (RTE):

$$\left\{\begin{array}{ll} c_m\s\cdot\nabla w + c_m\mu w - \K w = f & (\r,\s)\in \Gamma,\; \e\in [0,\infty) \\ w(\r,\s,\e) = g(\r,\s,\e) & (\r,\s)\in \partial_-\Gamma,\;\e\in [0,\infty) \end{array}\right.$$

(A figure describing optical phase space)

This equation models the propagation of high-frequency light from visible up to X-Ray, and even particles like \(\alpha\)s and \(\beta\)s, in biological tissue. Solving this equation results in the \(\mathcal{P}\) operator:

$$(\mathcal{P} f)(\r,\s,\e) = \frac{1}{c_m}\int_0^{\tau_-}f(\r-t\s)\exp\left(-\int_0^{\tau_-}\mu(\r-t^\prime\s)dt^\prime\right)dt$$

This

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Detector Modeling

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