\documentclass[a4paper,10pt]{article}
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%opening
\title{Resolution along LOR majority for limited angle tomography}
\author{Andrej Studen}

\begin{document}

\maketitle

\begin{abstract}
Resolution in limited angle PET is not homogeneous in all directions due to angular undersampling. Elongated artefacts oriented perpendicular to the missing LORs are common. The resolution along artefact elongation axis can derived from likelihood of collected events for a given underlying emission image. The treaty derives expressions for expected resolution depending on the scanner TOF resolution, spatial resolution and amount of collected angles.
\end{abstract}

\section{Expression derivation}

\textbf{Spatial resolution contribution}

\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.9\textwidth]{/home/studen/Downloads/limitedAngleSpatialResolution.png}
\end{minipage}
\begin{minipage}{0.5\textwidth}
 The probability of a LOR set given image $\lambda$ is determined by the likelihood function, a product of probability densities for each of the LOR in the set:
 \begin{equation}
   p(\{LOR\}\vert \lambda)=\prod_{LOR_i} p(LOR_i \vert \lambda)
 \end{equation}
Assuming a single point source at the origin, $\lambda=\delta(0)$, the likelihood of that source being at a distance away from the origin is given by:
\begin{equation}
   p(\{LOR\}\vert z)=\prod_{LOR_i} p(LOR_i \vert z)
\end{equation}
The probability density for given LOR depends only on projected distance z on direction perpendicular to LOR. For an azimuth angle of $\theta$ relative to axis z, this projection is:
\begin{equation}
 r=z\sin\theta
\end{equation}
\end{minipage}


Taking into account the central limit theorem and a sufficiently substantial non-linearity contribution, the probability density can be approximated by a Gaussian distribution with a resolution parameter $\sigma$, where $\sigma$ can be from example determined by the block equation:
\begin{equation}
 \sigma^2=1.25(R_{det}^2+R_{180^\circ}^2+R_{e^+}^2)
\end{equation}

The LOR probability density is then:
\begin{equation}
 p\left(LOR\;\vert\; z\right)=\exp\left(-\frac{r^2}{\sigma^2}\right)=\exp\left(-\frac{z^2\sin^2\theta}{\sigma^2}\right)
\end{equation}
For a $\theta$ of zero, pdf is independent of z and constant, which nicely reflects absence of positional information along z axis for straight LOR.

Asssuming uniform sensitivity (circular detectors along z axis) within a range of $[0,\theta_0]$, the  LOR can be assumed to be spread according to $dw/d\cos\theta$=const. Assuming N LOR at equal probability intervals the set of angles will be
\begin{equation}
\cos\theta_i=1-\frac{i}{N-1}(1-\cos\theta_0)\qquad i\in [0,N-1]
\end{equation}
The combined likelihood will be
\begin{equation}
 p\left(\{LOR\}\vert z\right)=\prod_{i} \exp\left(-\frac{z^2\sin^2\theta_i}{\sigma^2}\right)
 =\exp\left(-\frac{z^2}{\sigma^2}\sum_i (1-\cos\theta_i^2)\right)
\end{equation}
Or in summary:
\begin{equation}
 p\left(\{LOR\}\vert z\right)=\exp\left(-\frac{z^2}{\sigma_r^2}\right)
\end{equation}
with
\begin{equation}
 \frac{1}{\sigma_r^2}=\frac{1}{\sigma^2}\sum_i(1-\cos^2\theta_i)
\end{equation}
The angular acceptance is best parametrized by limiting tangens of largest acceptance angle $\theta_0$, which is equal to half detector size divided by the distance to the source:
\begin{equation}
 \tan\theta_0=\frac{D/2}{R}
\end{equation}
A 50~cm detector at 50~cm gives you a limiting tangens 0.5 and a multiplication factor of 6.73.

%\begin{minipage}{0.5\textwidth}
%\includegraphics[width=0.9\textwidth]{/home/studen/software/src/simulPET/zResolutionCTheta.png}
%\end{minipage}
%\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.9\textwidth]{/home/studen/software/src/simulPET/zResolutionTTheta.png}
%\end{minipage}

\textbf{TOF resolution}\\
\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.9\textwidth]{/home/studen/Downloads/limitedAngleTOF.png}
\end{minipage}
\begin{minipage}{0.5\textwidth}
The procedure is identical to the above, except that the limiting distance is now along the LOR, with the distance s from the expected source position, and that a different parameter, $\sigma_t$ allows for TOF resolution. The z pdf is:
\begin{equation}
  p\left(LOR\;\vert\; z\right)=\exp\left(-\frac{s^2}{\sigma_t^2}\right)=\exp\left(-\frac{z^2\cos^2\theta}{\sigma_t^2}\right)
\end{equation}

The combined contribution from all LORs is then:
\begin{equation}
 p\left(\{LOR\}\vert z\right)=\prod_{i} \exp\left(-\frac{z^2\cos^2\theta_i}{\sigma_t^2}\right)
 \end{equation}
Or:
\begin{equation}
 p\left(\{LOR\}\vert z\right)=\exp\left(-\frac{z^2}{\sigma_T^2}\right)
\end{equation}
with
\begin{equation}
 \frac{1}{\sigma_T^2}=\frac{1}{\sigma_t^2}\sum_i\cos^2\theta_i
\end{equation}
\end{minipage}

\includegraphics[width=0.9\textwidth]{/home/studen/software/src/simulPET/zResolutionTOFTTheta.png}

\textbf{Combined effect}\\

\begin{equation}
 \frac{1}{\sigma^2}=\frac{1}{\sigma_T^2}+\frac{1}{\sigma_r^2}
\end{equation}

\includegraphics[width=0.9\textwidth]{/home/studen/software/src/simulPET/zResolutionCombinedTTheta.png}


\end{document}
