KAHIBARO
28.4. Lines of Response
Table of Contents
Detector positions
In PET, each coincidence event corresponds to two detector elements that have both detected a 511 keV photon from the same annihilation. To build a line of response, or LOR, you must know where these detector elements are located in the scanner coordinate system.
In a typical GATE PET simulation, detector geometry is built from repeated modules and crystals arranged in rings. Each crystal has a unique set of identifiers, such as ring ID, module ID, and crystal ID, and a well-defined center position. During simulation, when a gamma interaction produces a valid single event after digitization, GATE records both the detector identifiers and the reconstructed interaction position, usually the crystal center.
There are two common ways to obtain detector positions for LOR building. The first is to use the position stored in the singles output, for example the $x$, $y$, $z$ coordinates of each detected event. These positions already include all geometry translations and rotations and are directly in world coordinates. The second is to reconstruct positions from detector IDs using your own mapping between identifiers and positions, based on how the scanner was built in the geometry script. This second approach is useful if you want to ignore fine interaction details inside the crystal and work only with idealized crystal centers.
It is helpful to remember that different PET geometries lead to different position patterns. A simple cylindrical ring scanner yields detector positions approximately on a circle in the transverse plane, with axial repetition along the scanner axis. More complex clinical scanners may have multiple rings, block detectors, or partial rings. Regardless of complexity, for each valid single the essential information is the detector center position in 3D space, $ \vec{r} = (x, y, z) $, expressed in a consistent global coordinate system.
When you read coincidence data, each coincidence entry contains two singles, often labeled as detector 1 and detector 2, with positions $\vec{r}_1 = (x_1, y_1, z_1)$ and $\vec{r}_2 = (x_2, y_2, z_2)$. These two positions are the endpoints from which you construct the LOR. If you later use time-of-flight information, you will also need the detection times for each single, but for basic LOR construction, positions alone are sufficient.
LOR construction
A line of response represents the possible locations of a positron annihilation event, given that two annihilation photons were detected in coincidence. In conventional non time-of-flight PET, you assume that the annihilation occurred somewhere along the straight line that joins the two detector elements that fired.
Mathematically, if $\vec{r}_1$ and $\vec{r}_2$ are the positions of the two detectors that recorded the coincidence, you can define a direction vector
$$
\vec{u} = \vec{r}_2 - \vec{r}_1 .
$$
You can then represent the LOR in parametric form as
$$
\vec{r}(t) = \vec{r}_1 + t \, \vec{u},
$$
where $t$ is a real parameter. For many reconstruction algorithms you normalize the direction vector. Define the distance between detectors as
$$
L = \|\vec{u}\| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2},
$$
and the unit direction vector as
$$
\hat{u} = \frac{\vec{u}}{L}.
$$
You can also define the midpoint of the LOR segment connecting the two detectors:
$$
\vec{m} = \frac{\vec{r}_1 + \vec{r}_2}{2}.
$$
Key LOR formulas:
- Direction vector:
$$\vec{u} = \vec{r}_2 - \vec{r}_1.$$ - Distance between detectors:
$$L = \|\vec{u}\|.$$ - Unit direction:
$$\hat{u} = \frac{\vec{u}}{L}.$$ - Midpoint:
$$\vec{m} = \frac{\vec{r}_1 + \vec{r}_2}{2}.$$ - Parametric LOR:
$$\vec{r}(t) = \vec{m} + t \, \hat{u}.$$
In practice, you usually do not work with infinite lines, but with line segments that either join the two detector centers or intersect a finite image volume. For a reconstruction grid, you only consider the part of the LOR that passes through the image. You can compute the intersection of the parametric line with the image box, then use that segment for projection and backprojection.
From a data analysis perspective, each coincidence in GATE can be converted to a compact LOR representation. A practical representation in a table or ROOT tree is to store either the two endpoints $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ or equivalently the midpoint and direction, $(m_x, m_y, m_z)$ and $(u_x, u_y, u_z)$. The table below summarizes the two common formats.
| Representation type | Stored values | Use case |
|---|---|---|
| Endpoint | $x_1, y_1, z_1, x_2, y_2, z_2$ | Direct from simulation, easy to build from data |
| Midpoint–direction | $m_x, m_y, m_z, u_x, u_y, u_z$ | Convenient for analytic operations and TOF PET |
To construct LORs from GATE output, you typically follow a fixed processing chain. First, apply energy and timing selection to the coincidence data so that only valid coincidences remain. Second, read the detector positions for the two singles in each coincidence. Third, compute the LOR representation using the formulas above and save the result for later reconstruction. If you are interested in sinograms, you can also convert each LOR into sinogram coordinates, such as radial coordinate and projection angle, by projecting the endpoints into the transverse plane and binning them into angular and radial bins.
It is important to maintain unit consistency when constructing LORs. If your simulation geometry is defined in millimeters, then the positions $x$, $y$, and $z$ in the output will also be in millimeters. All LOR computations must use the same length unit. If you change length units in any part of your pipeline, convert positions explicitly before computing direction vectors and distances, otherwise LOR geometry will not match the actual scanner.
In advanced analyses, you can enrich the LOR with additional attributes from GATE output, such as whether the coincidence is true, scattered, or random, or whether it is within a chosen energy window. These labels do not change the mathematical construction of the LOR itself, but they are crucial when you want to study how different event types contribute to image formation.
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