Time-of-flight scatter rejection in x-ray radiography

Objective. Time-of-flight (TOF) scatter rejection allows for identifying and discarding scattered photons without the use of an anti-scatter grid (ASG). Although TOF scatter rejection was initially presented for cone-beam computed tomography, we propose, herein, to extend this approach to x-ray radiography. This work aims to evaluate with simulations if TOF scatter rejection can outperform ASGs for radiography. Approach. GATE was used to simulate the radiography of a head and a torso and a water cylinder with bone inserts in a system with total timing jitters from 0 ps up to 500 ps full-width-at-half-maximum. The transmission factor of TOF scatter rejection for primary and scattered photons was evaluated as if it were a virtual ASG. Main results. With a total timing jitter of 50 ps, TOF scatter rejection can reach a selectivity of 4.93 with a primary photons transmission of 99%. Reducing the timing jitter close to 0 ps increases the selectivity up to 15.85 for a head and torso radiography, outperforming typical ASGs which usually have a selectivity from 2.5 to 10 with a primary photons transmission from 50% to 70%. Significance. This suggests that TOF scatter rejection may be suitable to replace ASGs in applications requiring lower radiation exposure if sufficiently low timing jitter is achieved.


Introduction
x-ray radiography is the most commonly used two-dimensional (2D) medical imaging modality (Smith-Bindman et al 2019, Mahesh et al 2022).It offers an affordable solution to measure the attenuation of x-rays in biological tissues, and for this reason it is extensively used in both medical and dental practices.While both absorption and scattering contribute to the attenuation of x-ray beams, scattered photons cause an unwanted contribution to the attenuation measurement when they are redirected toward the detector (Niklason et al 1981).In abdominal imaging, the scattered photons contribution across the thickest part of the body can be more important than the primary beam (Porubszky 2012).In radiography, a scatter-to-primary ratio (SPR) δ leads to a (1 + δ)-fold reduction in the contrast-to-noise ratio (CNR).Statistical noise also increases, while image sharpness and background homogeneity decrease (Mazurov and Potrakhov 2015).The SPR represents the ratio of scattered photons to primary photons at the detector, expressed as a percentage.The SPR is equivalent to the error percentage between the measured photons count and the primary photons count.
Scattering also causes degradation of image quality in x-ray computed tomography (CT) (Siewerdsen and Jaffray 2001).However, in most conventional CT scanners, a small volume is simultaneously irradiated by x-rays, hence keeping the scattering contribution low.The development of cone-beam CT, where the irradiated volume is similar to that of radiography, has greatly increased the interest of the medical imaging community in finding a solution to the detrimental effects of the scatter contribution.
In both radiography and CT, anti-scatter grids (ASGs) remove part of the scatter contribution by intercepting photons not following a direct path from the source to the detector.These grids, usually made of lead or tungsten, are composed of multiple lamellae oriented toward the focal spot of the source.However, because of the narrow width of the grid lamellae, some scattered photons can pass through the grid and reach the detectors, while the same grid also stops part of the primary beam, thus reducing the detection efficiency and requiring a longer radiation exposure to measure the same amount of x-rays (Schafer et al 2011).Increasing the width of the lamella to further stop scattered photons also increases the proportion of primary photons stopped.Therefore, ASGs must be designed to balance the trade-off between stopping scattered photons and increasing radiation exposure.This is particularly a problem in CT where high radiation dose is a major concern.
In recent years, gridless approaches (Rührnschopf and Klingenbeck 2011a, 2011b, Mentrup et al 2016), based on software scatter correction, showed potential to replace ASGs in certain conditions where radiation dose is critical (Onodera et al 2020) or where ASGs are seldom used because of risk of misalignment (Lisson et al 2019).However, these methods have not been studied in a wide variety of scenarios (Sayed et al 2023) and fail to offers better image quality than ASGs (Precht et al 2019).
A recent proposal evaluated the use of measurements of the time-of-flight (TOF) of X photons to remove the scatter contribution in CT without an ASG (Rossignol et al 2020) even in highly scattering condition.This approach uses the photon TOF from the source to the detector for each photon and a threshold to discriminate scattered and primary photons.Since all X photons travel at the speed of light, scattered photons have a longer TOF than primary photons that travel through a straight path.To measure the TOF, a pulsed source emits photons toward a time-resolved detector.A trigger synchronizes the source and the detector.Since the differences in TOF between scattered photons and primary photons are usually below a few hundred picoseconds, both ultrashort x-ray pulses (<100 ps) and a precise time-resolved detector (< 300 ps) are required to discriminate both types of photons and observe a significant improvement in image quality in CT (Gaudreault et al 2020).
This work aims to evaluate through simulations if this approach of TOF scatter rejection can also significantly improve image quality in x-ray radiography and offer an alternative to ASGs.This paper is structured as follows: ASG performance is presented first, followed by a methodology to estimate the performance of TOF scatter rejection using the same metrics as for ASGs.Finally, the simulation results are presented and discussed.

Anti-scatter grid performance
ASGs can be evaluated by their ability to block scattered photons while allowing primary photons to pass through (Neitzel 1992).Such ability is assessed through the scattered photons transmission factor t s and the primary photons transmission factor t p given by: where x refers to either primary (p) or scattered (s) photons, N xg is the number of scattered or primary photons that reach the detector with a grid, and N xwog is the number of photons that reach the detector without a grid.
An ideal ASG has a low t s and a high t p .The ratio Σ = t p /t s is known as the selectivity, which is also equal to the ratio between the SPR without and with the grid.With an ASG, the SPR δ improves and becomes δ/Σ.Since the scatter contribution decreases the contrast by a factor of 1 + δ, the contrast improvement factor (CIF) for a given SPR is given by (Mazurov and Potrakhov 2015): However, removing both primary and scattered photons increases the statistical noise.The signal-to-noise ratio (SNR) improvement factor (SIF), which is also equal to the gain of the CNR (note that contrast should not be confused with CNR, see image quality metrics discussed later in the paper), is given by (Bor et al 2016): This represents the capacity of the grid to improve image visual quality.Although defined for ASGs, these parameters and performance metrics can be used as a basis to compare TOF scatter rejection to ASGs, as the scattered and primary photons transmission factors can be determined for TOF scatter rejection in simulation.Therefore, this work aims to find these values and offer an evaluation of the performance of TOF scatter rejection as a replacement for ASGs in radiography.

Simulations
A 72 × 72 cm 2 flat panel detector was modeled in GATE (Jan et al 2004) and divided into 57 600 pixels (figure 1).Each pixel is represented as a 3 × 3 × 2.5 mm 3 block with the density of LYSO, a prospective scintillator for TOF measurement in x-ray imaging (Lemaire et al 2023).Both photoelectric and scattering interactions inside each pixel were combined to output a single event per X photon.The scintillation process was not simulated.A source to image distance (SID) of 180 cm is used as a comparison with previous work on TOF x-ray imaging (Gaudreault et al 2020, Rossignol et al 2020) and on the effect of scattering (Siewerdsen and Jaffray 2001).The phantom to image distance is critical for TOF imaging as it increases the TOF difference between primary and scattered photons.Reducing the system size to that of a conventional radiography system would, however, have limited effects (Rossignol et al 2020).The time of the first interaction per pixel was kept as the timing measurement on which timing jitter is added by randomly sampling a Gaussian distribution.Herein, timing jitter is defined as a random variation of the TOF measurement regardless of its cause, including the limited time precision of the detector, the width of the source pulse and the uncertainty of the source's trigger period.The energy deposited by each interaction is summed together to obtain the energy measurement, which is only used to discard events that deposit less than 10 keV in a pixel.
A 100 kVp pulsed polychromatic x-ray source is placed 180 cm in front of the detector.The x-ray source is triggered at a rate of 100 MHz.The photons are emitted in a 1 ps window after the source trigger.The TOF of each photon is the difference between the source trigger time and the time measurement at the detector.Each X photon generated by the source is independently simulated, thus detector dead time and pile-up are not included.Each simulation tracked 10 billion photons and at least 4 billion photons were detected and included in the analysis for each simulation.Therefore, the figures are presented without error bars as they would be too small to be shown.
Two phantoms were used: part of a full-body phantom from head to torso (Figure 2 For each phantom, six simulations were ran with a Gaussian timing jitter of 0 ps, 50 ps, 100 ps, 200 ps, 300 ps, and 500 ps FWHM for a total of 12 simulations.This jitter represents the total timing jitter of the system, including the timing uncertainty of the source and detector timing jitter.

TOF scatter rejection and image generation
The TOF scatter rejection algorithm presented in (Rossignol et al 2020) is applied to the output of each simulation.
The maximum expected TOF (eTOF max ) for each pixel is computed by taking the maximum distance between the source and the apices of the detector and dividing it by the speed of light.TOF deviation (TOF d ), given by = -TOF TOF eTOF d m a x , offers a pixel-independent evaluation of the lateness of a photon.Thus, before applying the effect of the system jitter, photons having a positive TOF d have arrived later than expected for a primary photon, and photons with a negative TOF d have arrived on time regardless of the position of the pixel in the flat panel.
For each detector pixel, events are binned based on their TOF d in two histograms containing 500 bins of 10 ps.Primary and scattered photons, as identified by GATE, are binned in separated histograms to allow independent evaluation of both contributions.The same range, from −1 to 4 ns, is used for all histograms.Since simulations are performed here, it is possible to obtain separately, for a given detector, a histogram for primary photons and a histogram for scattered photons (which is not possible in real measurements), and then to add these histograms together to emulate a histogram acquired in a real acquisition.To apply TOF scatter rejection, the histograms' bins below the threshold are summed (those bins corresponding to short TOFs).An image is generated by taking this number for each detector pixel.
For each simulation, an image is generated with three different thresholds.These thresholds are chosen according to how many primary photons are kept, which is given by t p .An early time threshold discards more primary photons than a later threshold would.The first image is generated with a threshold yielding a t p of 99 %.This represents a conservative virtual ASG with little impact on the primary photons beam, thus with a low dose penalty.The second image is generated with a threshold yielding a t p of 66 % to compare with typical ASGs that usually have such primary photons transmission.The third image is generated using the threshold that maximizes the selectivity, which is always 0 ps, at the cost of an important dose penalty.This is used as a reference to evaluate the absolute limit of TOF scatter rejection.
The generated images are normalized to the number of emitted photons and the sensitivity of the detectors.For comparison, an image using only the primary photons (ground truth) and an image using all detected photons are also generated.

TOF scatter rejection performances as a virtual ASG
TOF scatter rejection acts as a virtual ASG.Thus, the primary and scattered photons transmission can be evaluated, which yields its selectivity.These parameters are a function of the chosen time threshold for a given timing jitter.
To evaluate the primary photons transmission for all thresholds, the primary photon histogram of each pixel is normalized in order to obtain a probability distribution function (PDF) of the TOF d .From this PDF the cumulative distribution function (CDF) is computed by summing up the bin values up to each time and this for all times.Because TOF scatter rejection sums all photons until the threshold is reached, this CDF corresponds to t p as a function of the time threshold.This procedure is carried out as well for the scattered photons to obtain t s .The selectivity per threshold is obtained by dividing t p by t s .Since t s is dependent on the travel of scattered photons in the phantom, t s can also be computed independently for each pixel and an image of t s can be produced.CIF and SIF are computed from equations (2) and (3) using the average SPR from the four central pixels of each image.

Image quality
Two regions of interest (ROIs) are selected in each phantom as shown in figure 4.
The contrast and CNR are evaluated for each image using TOF scatter rejection.The equations used for the contrast and CNR are: where μ roix is the mean value of region x and σ roi2 is the standard deviation of the second region.
Both contrast and CNR recovery values are then calculated for each image using the following equation: where RV is the recovery value, V represents contrast or CNR, V TOF is the value obtained when applying TOF scatter rejection, V NoTOF is the value obtained with all photons and V GT is the value obtained with only primary photons.Higher contrast recovery values lead in principle to better images.

TOF scatter rejection and image generation
Figures 5 and 6 show eight simulated radiography images for each phantom.In both cases, a degradation of the CNR is visible from the top-left image, in which only primary photons have been retained, to the bottom-right image, in which all photons have been retained.As the simulated timing jitter increases, more scattered photons are retained, which in turn decreases the CNR of the image.This is in line with previous study on TOF x-ray imaging (Gaudreault et al 2020, Rossignol et al 2020) and on scattering (Siewerdsen and Jaffray 2001).The same behavior is observed when the threshold is set to retain only 66% of primary photons or to provide the best selectivity.This fits what is expected from the TOF scatter rejection algorithm.The line profiles in figure 7 showcase how the contrast improves with lower timing jitters.

TOF scatter rejection as a virtual ASG
The primary photons transmission for both phantoms is shown in figure 8.As expected, there is no difference between the results from the two phantoms, as primary photons are, by definition, unaffected by the phantom.However, t p differs significantly depending on the amount of timing jitter in the simulations.In an ideal case, the primary contribution to the measured temporal signal can be considered to be a Dirac delta function.Once convolved with a Gaussian timing jitter, the simulated primary contribution yields therefore the same Gaussian distribution.Applying a threshold to this distribution yields the same CDF as integrating the Gaussian distribution from negative infinity to the threshold.The scattered photons transmission factor showcases a different behavior as shown in figure 9 (left).For scattered photons, there is a significant difference between the phantoms.For instance, the water cylinder phantom yields a shorter average TOF difference for scattered photons.This makes them harder to remove with TOF scatter rejection; thus, higher transmission of scattered photons is obtained.
Figure 9 (right) also highlights that the timing jitter affects t s less than t p .This is due to the TOF distribution of scattered photons which decreases quickly from its maximum at 0 ps but showcases a long slowly decaying tail (Rossignol et al 2020).Adding Gaussian timing jitter to the TOF distribution of the primary photons (figure 10 (left)) completely changes the distribution, explaining the difference in t p with different timing jitters.On the contrary, adding Gaussian timing jitter to the scattered distribution does not significantly affect the tail as seen in figure 10 (right).However, this is not true at the beginning of the distribution.This explains the difference in t s observable when the threshold is close to 0 ps.The combination of t p and t s yields the selectivity as shown in figure 11.In each case, the maximum selectivity is achieved with the lowest threshold.Figure 12 showcases that the selectivity decreases quickly as the timing jitter increases.While the selectivity is always higher with a lower threshold, t p is also at its lowest.This means that while the reduction of the scattered contribution is maximum, using such a threshold would cause a high dose penalty.Figure 13 shows selectivity with a dose penalty similar to commercial ASGs (t p = 66%) and with a minimal dose penalty (t p = 99%).They follow the same trend as in figure 12.Most of the gain in selectivity is achieved with a total timing jitter below 100 ps.figure 14 show the CIF and SIF achieved with t p = 99% for both phantoms.Figure 9. Scattered photons transmission factor (t s ) for phantom 1 and 2 for a 300 ps timing jitter (left) and phantom 1 for timing jitters ranging from 0 to 500 ps (right).
Figure 15 shows an example of t s calculated for each pixel with a 100 ps timing jitter.One can appreciate the dependence of t s on the phantom's geometry since part of the structure of the head and torso can be seen for phantom 1 and the outline of the water cylinder can be seen for phantom 2. This means that the selectivity, CIF, and SIF are all also dependent on the phantom geometry and vary from pixel to pixel.Therefore, since the contrast, CNR, and SNR are computed using ROIs, the real improvements will differ from the theoretical CIF and SIF.  Figure 12.Highest selectivity achievable as a function of timing jitter for each phantom.A fit using a power function (ax b + c) is shown to highlight an overall tendency.However, the underlying model is more complex as the selectivity is a function of t s which depends on the scattering interactions in the phantoms.Figures 16 and 17, respectively, show the contrast and CNR recovery value for all images with TOF scatter rejection.In phantom 1, the images with a t p of 99% outperform the more aggressive rejections in some instances, whereas one normally expects lower t p to perform better.This is most probably due to the spatial variation of the t s that may sometimes artificially increase the contrast between two regions by being less effective   on the lower attenuation region.In phantom 2, where t s varies less near the ROIs, the more aggressive rejections always outperform the less aggressive ones.The CNR recovery values do not follow the same trend as the contrast recovery value.With the noise being taken into account, the most aggressive rejections yield better recovery in phantom 1.However, in phantom 2, they only perform better at low-timing jitter.At high-timing jitter, the increased noise caused by the significant removal of primary photons outweights the gain in contrast.This even leads to the most aggressive rejection to worsen image quality with timing jitters over 200 ps.

Discussion
Most ASGs have reported selectivity from 2.5 to 10 (Keevilt et al 1987, Mizuta et al 2012, Mazurov andPotrakhov 2015), but values as high as 69 were obtained under particular conditions (Bor et al 2016).These outlier values must be considered with caution since they were measured with very low photon counts.Values of t p from 50% to 70% are reported (Schafer et al 2011, Mazurov andPotrakhov 2015).In comparison, TOF scatter rejection allows choosing the desired t p , theoretically up to 100%.Any reduction of t p must be compensated with a comparable increment in radiation exposure.Therefore TOF scatter rejection allows for a significant reduction of the exposure in comparison with ASGs with equal selectivity.At a t p of 99%, a selectivity competing with the less aggressive ASGs can be reached with a timing jitter of 200 ps.With a timing jitter below 50 ps, TOF scatter rejection can equal or even outperform ASGs with no significant loss of primary photons.It is however still challenging to design such a system.The simulation conditions are also ideal for TOF scatter rejection, which means that reaching the same selectivity in a real system with the same total timing jitter may be challenging.However, the selectivities are most likely underestimated in comparison with those available in the literature for ASGs.The selectivity of ASGs is generally measured using beam-blockers, which yield a lower SPR and t s than in simulation.With simulation, all scattered photons are considered both for the SPR and for t s , even those that do not change direction when scattering.With beam-blockers, only scattered photons that scattered sufficiently to circumvent the beam-blocker are measured, thus small angle scattering is not included.Even with a perfect timing resolution, it is nearly impossible to reach SPRs below 4% when considering all scattered photons.Using equations (2) and (3) can give an idea of the impact of ASGs on image quality based on their selectivity.However, in our case, these equations overestimate the impact of scatter noise on the contrast and on the CNR.This is most likely due to our artificially higher SPR.Using the values of t p , t s , and the selectivity seems to be the most robust way to evaluate between scatter rejection approaches.Moreover, the spatial variation of the selectivity makes the use of equations (2) and (3) inherently inaccurate.
The trend of CNR recovery values found in x-ray radiography matches those previously published for CT (Gaudreault et al 2020).A significant difference in the magnitude of CNR recovery values is observed between phantoms 1 and 2. As previous studies in CT also used the water cylinder phantom (Gaudreault et al 2020, Rossignol et al 2020), this may have led to an underestimation of the true achievable improvements for chest imaging.
This work also highlights how dependent on the imaged object scatter rejection is.This makes it particularly challenging to design an optimal general-purpose algorithm.Although t p can be chosen to be as good (close to 1) or as bad as desired based on the chosen threshold, a more aggressive threshold (i.e. with lower value of t p ) does not necessarily lead to a better CNR, as shown in figure 17 (right).While not shown in the results, it could technically be possible to be even more aggressive.Since a Gaussian jitter is assumed and its mean is equal to the expected TOF for primary photons, some photons have a negative TOF deviation.However, placing a threshold before the average of the primary distribution would have an important impact on t p and would not offer a gain over ASGs.
t s is also dependent on the geometry of the phantom.Interestingly, contrary to what previous studies on TOF CT would suggest (Rossignol et al 2020), figure 9 highlights that the full-body phantom, which generates a smaller scatter contribution, yields a better t s .One hypothesis to explain this phenomenon is survivorship bias since only the TOF of photons reaching the detector is measured.This phenomenon was accidentally highlighted in other simulations done with GATE to test some functionalities of the tool.We detected a larger number of Rayleigh scattered photons versus Compton scattered photons, while at medical x-ray energy, Compton scattering is significantly more likely than Rayleigh scattering.However, Compton-scattered photons usually scatter with a higher angle thus, fewer are detected.As a phantom gets larger, it increases the likelihood that photons scatter, but it also reduces the chance for a photon that lost a large portion of its energy to scattering to pass through and be detected.A similar bias is used as part of the air gap method (Neitzel 1992) which is also used to reduce the impact of the scatter contribution.Increasing the air gap between the object and the detector reduces the chance that scattered photons be detected, but this does not mean that the amount of scattered photons in the object has changed.In addition, phantom 2 is thicker than phantom 1; therefore each photon beam traveling from the source to a detector pixel goes through more material.Since the irradiated volume surrounding each beam is larger in the water cylinder phantom it therefore generates a lot of low TOF deviation photons.This could also explain part of the difference between the t s of both phantoms.
Contrary to most chest imaging schemes, our simulation setup has a significantly larger air gap.This is necessary to increase the TOF deviation between scattered and primary photons and improves the effectiveness of the method.However, increasing the air gap also reduces the amount of scattered radiation reaching the detector.This effect was not evaluated in this study as the selectivity and transmission factors were measured with the same air gap.The effect of an air gap on t p and t s are presented in details in (Neitzel 1992) and previous work on TOF x-ray imaging have shown marginal changes in TOF for SID from 800 to 1600 mm (Rossignol et al 2020).Adding such an air gap will require a larger detector than in conventional radiography with accordingly larger pixels to avoid increasing the statistical noise.Increasing the SID may be required to avoid increasing too much the required detector solid angle.
As stated before, a total timing jitter of 50 ps FWHM or better is required to outperform ASGs with no loss of primary photons.This is still unreachable with state-of-the-art system, which have been able to reach a timing jitter of 155 ps FWHM on a single pixel (Lemaire et al 2023).According to figure 11 (left) this level of jitter performs similarly to some ASGs in term of selectivity and primary transmission.
For the emission, a 100MHz pulsed sourced triggered by a 60 ps wide laser pulse is reported in (Lemaire et al 2023).While 1 ps pulse width was used in our study, increasing the pulse width is equivalent to increasing the time jitter of the system.In a clinical application, since the x-ray source does not emit continuously, a longer acquisition may be required to reach the same exposure.This effect has yet to be quantified.
Current semiconductor detectors used for photon counting imaging can only reach 1 ns resolution for TOF x-ray measurements (Sundberg et al 2023), making scintillators the only option.Counting individual photons using scintillators is a challenge because the large decay time of common materials makes separating events difficult, but it was recently shown that sufficient count rate was achievable using LuAP: Ce and LaBr 3 (van der Sar et al 2021).
In terms of timing, comparable scintillator detector requirements are needed for TOF positron emission tomography (PET) where it was deemed feasible to reach 10 ps timing jitter (Lecoq 2017).While PET scintillators, such as LYSO, may be used for this application, the lower energy of the photons in x-ray imaging degrades the timing precision since less scintillation photons are generated.On the other hand, this also lowers the stopping power requirement, making possibe the use of scintillators that are unsuitable in PET.

Conclusion
TOF scatter rejection is an interesting candidate to replace ASGs for x-ray radiography if a total timing jitter below 50 ps is achieved.It can offer similar or better selectivity to ASGs, up to 10 or 16 depending on the phantom geometry.Contrary to what can be achieved with ASGs, the transmission of primary photons t p can be selected to be as good as desired which would result in a significant reduction of the radiation.In an chest imaging scheme, 71% of the degradation of the CNR caused by scattered photons is mitigated with a timing jitter of 50 ps with a t p of 99%.In an ideal case with no timing jitter, 94% of the degradation is mitigated.
The complexity of measuring TOF in comparison with the simplicity of an ASG must, however, be underlined.TOF scatter rejection is therefore most interesting for applications in which the radiation exposure must be limited as much as possible.
(a)), henceforth called phantom 1, and a water and bone inserts phantom (figure 2(b)), called phantom 2, is used for comparison with previous studies on TOF x-ray imaging; the dimensions of phantom 2 are depicted in figure 3.No ASG is used.

Figure 2 .
Figure 2. 3D rendering of (a) part of a full-body phantom comprising a head and a torso (phantom 1) and (b) a water cylinder with bone inserts phantom (phantom 2).

Figure 3 .
Figure 3. Dimensions of the water cylinder with bone insert phantom.Its height (not shown) is 240 mm.

Figure 4 .
Figure 4. Simulation of only primary photons (ground truth) for (a) phantom 1 and (b) phantom 2. The regions of interest (ROIs) for CNR computation are shown on each image.

Figure 5 .
Figure5.Simulated radiography of phantom 1 with (a) only primary photons, with the threshold set so that 99% of primary photons are retained with a timing jitter of (b) 0 ps, (c) 50 ps, (d) 100 ps, (e) 200 ps, (f) 300 ps, (g) 500 ps, and (h) all primary and scattered photons.The average SPR of the four central pixels of each image is shown.

Figure 6 .
Figure6.Simulated radiography of phantom 2 with (a) only primary photons, with the threshold set so that 99% of primary photons are retained with a timing jitter of (b) 0 ps, (c) 50 ps, (d) 100 ps, (e) 200 ps, (f) 300 ps, (g) 500 ps, and (h) all primary and scattered photons.The average SPR of the four central pixels of each image is shown.

Figure 7 .
Figure 7. Horizontal line profile at the center of images (a), (b), (d), (f) and (h) for phantom 1 (left) and phantom 2 (right).For clarity, some values of timing jitter are not shown, but they follow the same trend.

Figure 8 .
Figure8.Primary photons transmission factor (t p ) for phantom 1 (left) and phantom 2 (right) for timing jitters ranging from 0 to 500 ps.The 66% and 99% limits used to set the threshold for TOF scatter rejection are shown respectively with a dashed and a dotted line.

Figure 10 .
Figure 10.TOF deviation of primary photons (left) and scattered photons (right) for different timing jitters.

Figure 13 .
Figure13.Selectivity achieved as a function of timing jitter for each phantom while keeping 66% of primary photons (left) and 99% of primary photons (right).

Figure 14 .
Figure 14.CIF (left) and SIF (right) achieved as a function of timing jitter for each phantom while keeping 99% of primary photons.

Figure 15 .
Figure15.t s at 100 ps timing jitter with a threshold selected for 99% t p for phantom 1 (left) and phantom 2 (right).