KAHIBARO
17.5. Energy Blurring
Table of Contents
Detector energy resolution
When a particle deposits energy in a detector crystal, the detector does not measure this energy perfectly. Even if every event deposits exactly the same true energy $E_{\text{true}}$, the measured energies will form a spread around that value. This spread is described by the detector energy resolution.
Energy resolution is usually specified as a full width at half maximum (FWHM) at a given reference energy, for example “12 % at 511 keV” for a PET detector or “9 % at 140 keV” for a NaI(Tl) gamma camera. The FWHM is the width of the energy peak at half of its maximum height.
In GATE digitizers, energy resolution is applied on the “true” summed energy of a hit or a crystal signal to produce a “measured” energy. This is critical whenever you use energy windows, for example PET photopeak windows around 511 keV, SPECT photopeak windows around 140 keV, or thresholds for rejecting low energy background. Without blurring, the simulated spectrum would contain unrealistically sharp lines and the fraction of events inside a given energy window would not match real detectors.
For many scintillation detectors, the energy resolution scales approximately as the inverse square root of the energy. A common parametrization is
$$
\frac{\text{FWHM}(E)}{E} = \sqrt{\left(\frac{a}{\sqrt{E}}\right)^2 + b^2}
$$
where $E$ is expressed in a reference unit such as keV, $a$ represents the stochastic term from photon statistics, and $b$ represents a constant term accounting for effects such as nonuniform light collection.
In most GATE configurations you do not have to implement this formula manually, but you need to understand what it represents so that you can choose parameters that reproduce your detector’s measured energy resolution.
Below is a summary of typical energy resolution behavior.
| Detector type | Typical FWHM at reference energy | Comment |
|---|---|---|
| NaI(Tl) SPECT crystal | 8 % to 10 % at 140 keV | Strong energy dependence |
| LYSO PET crystal | 10 % to 15 % at 511 keV | Near-linear in $1/\sqrt{E}}$ in keV units |
| Semiconductor (HPGe) | < 1 % at 1 MeV | Very good resolution, small constant term |
In a GATE digitizer, you normally pass this information in one of two forms. Either you provide a fixed relative FWHM at the energy of interest, if you simulate a narrow range of energies, or you provide parameters of an energy-dependent model that the digitizer uses to compute the width of the Gaussian at each energy.
Setting these parameters correctly is essential for realistic simulations of energy spectra and for any study that depends on scatter rejection, scatter fractions, or photopeak efficiency.
Key rule: Always configure detector energy resolution in the digitizer so that simulated peaks match measured FWHM values at the reference energies of your system. Use experimentally measured spectra from your scanner to calibrate these parameters.
Gaussian blurring
Energy blurring in GATE is implemented by applying a random fluctuation to the true deposited energy using a probability distribution. For most medical imaging detectors, the fluctuations around the mean energy are well approximated by a Gaussian distribution.
Mathematically, if $E_{\text{true}}$ is the true deposited energy, the digitizer produces a measured energy $E_{\text{meas}}$ according to
$$
E_{\text{meas}} \sim \mathcal{N}\left(E_{\text{true}},\, \sigma^2(E_{\text{true}})\right)
$$
where $\mathcal{N}$ denotes a normal distribution with mean $E_{\text{true}}$ and standard deviation $\sigma(E_{\text{true}})$.
The standard deviation $\sigma$ is related to the FWHM of the energy peak by
$$
\text{FWHM} = 2.355\, \sigma
$$
so for a given relative energy resolution $R(E)$ expressed as FWHM over energy,
$$
R(E) = \frac{\text{FWHM}(E)}{E},
$$
you can compute
$$
\sigma(E) = \frac{R(E)\,E}{2.355}.
$$
The digitizer uses this $\sigma(E)$ to draw a random value of $E_{\text{meas}}$ for each event.
In practice, the Gaussian blurring step in GATE comes after energies have been summed over all deposits that belong to the same detector channel or crystal. For example, if a gamma interacts multiple times in a single crystal, the energy deposits are combined first to give a single true energy, and then Gaussian noise is applied once to that total. This reflects the fact that the photodetector and electronics see the sum of all light produced in that crystal, not each microscopic deposit separately.
Gaussian blurring creates the characteristic bell-shaped photopeaks and Compton continua seen in measured spectra. The width of the Gaussian controls how much overlap there is between the photopeak and scattered events. A narrow Gaussian means better energy discrimination and more efficient scatter rejection. A broader Gaussian increases the fraction of scattered events that fall inside the photopeak window.
Important formulas for Gaussian energy blurring:
- Relation between FWHM and $\sigma$:
$$
\text{FWHM} = 2.355\,\sigma.
$$ - Relative energy resolution:
$$
R(E) = \frac{\text{FWHM}(E)}{E}.
$$ - Standard deviation used for blurring:
$$
\sigma(E) = \frac{R(E)\,E}{2.355}.
$$
By carefully choosing the Gaussian width as a function of energy, you can reproduce the measured spectra of your detector and ensure that downstream analysis, such as energy windowing, scatter estimation, and sensitivity evaluation, behaves in a realistic way.
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