Trajectory patterns illuminate PT method variationsTo understand PT performance at a visual level, we plotted a small window of trajectories for the bimodal and unimodal simulations for \(PSDR\le 1.7\) (Fig. 4). Some selected trajectories for \(PSDR\ge 3.0\) are also given in Supplementary Fig 1. For the bimodal simulation at \(PSDR=3\), trajectories from all PT codes reasonably mirror the simulations. However, this congruence quickly diminishes at \(PSDR=1.7\) (Fig. 4b), with marked deviations underscoring the nuances of each linking algorithm. It’s noteworthy that for \(PSDR\le 1.7\) (Fig. 4a and b), trajectories close to the cylinders (i.e., slow trajectories) are detected more accurately compared to the faster trajectories in the pore throat. In addition to increased speed, particles in the pore throat exhibit spatial convergence, which results in a large local decrease in PSDR. This observation carries over to the unimodal simulations (Fig. 4c), which exhibit consistent patterns across PT codes. A pervasive trend emerges: PT codes tend to underestimate the likelihood of particles moving at high velocities for a variety of simulated speed distributions, especially when spatial convergence further reduces PSDR.Fig. 4Sample trajectories for all PT codes for the low-PSDR bimodal and unimodal simulations. Each plot shows a 400×400 pixel section of the whole domain. Within a plot, each line corresponds to a unique trajectory (with random colors used to show the contrast between individual trajectories). (a) Bimodal simulations for \(PSDR=1.1\). (b) Bimodal simulations for \(PSDR=1.7\). (c) Unimodal simulations for \(PSDR=1.6\). All algorithms suffer from trajectory splitting and erroneous linking for these low-PSDR simulations. TM-Kalman, TM-LAP and V-TrackMat clearly outperform TP in all scenarios. Although TM-Kalman and V-TrackMat have more clearly false links that stretch across the pore space (jump from one group of streamlines to another), TM-LAP has a much larger amount of zig-zagging trajectories caused by erroneous links between close particles. The low-PSDR unimodal simulations generally show the same trends as the bimodal simulations; however, the differences between TM-Kalman, TM-LAP, and V-TrackMat are less significan.Probing deeper into individual PT code performances for the bimodal simulations, especially at lower simulated PSDRs, TM-Kalman stands out with superior accuracy, although it’s not exempt from erroneous links at elevated speeds. In particular, TM-Kalman shows a significant amount of erroneous long links across pore spaces (streamlines don’t cross the pore space in our bimodal simulations, so any link across the pore space is a false link). TP shows the greatest amount of false links and split trajectories (further explained in next section) at \(PSDR\le 1.7\) (Fig. 4a and b). TM-Lap similarly exhibits pronounced difficulties in linking high-speed particles, though not as significant as TP. At \(PSDR=1.7\), V-TrackMat trajectories generally resemble those of TM-LAP and TM-Kalman in terms of accuracy, although there are a smaller number of V-TrackMat trajectories. At \(PSDR=1.1\) (Fig. 4a), TM-Kalman and V-TrackMat appear to outperform TM-LAP. Although TM-Kalman and V-TrackMat may have more erroneous links across the pore space, TM-LAP has a much greater number of zig-zagging trajectories (links going back and forth between two or more different particles), and less true trajectories that last a significant distance. Thus, although V-TrackMat’s linking algorithm is less aggressive than either TM algorithm, V-TrackMat captures a substantial portion of accurate trajectories.The unimodal simulations at \(PSDR=4\) (Supplementary Fig. 1) further show that all algorithms besides TP have robust performance regardless of geometry. At \(PSDR=1.6\) (Fig. 4c), all algorithms show problems with false links and split trajectories. Similar to the bimodal results, V-TrackMat and TM-Kalman seem to have a larger amount of accurate trajectories than TM-LAP. Thus, general algorithm performance is largely independent of the geometry in which the particles are tracked. However, it should be noted here that the range of possible particle speeds in our simulations, which is largely impacted by geometry and flow conditions, only spans 3-4 magnitudes (Fig. 2a and b). High fidelity simulations of Lagrangian particles in porous media show speed distributions that range up to 8 orders of magnitude53, so we can’t be confident that our findings (relative rankings of PT performance) would remain accurate for transport in any geometry or flow condition. Furthermore, to focus on linking, we didn’t include any background. However, in real experiments that image bacteria in microfluidic devices, the geometry has a significant impact on tracking performance due to the presence of light scattering around grains21.To further understand differences in our PT codes, we plot both the simulated (ground truth) and PT-generated normalized speed distributions for our lowest PSDR bimodal (Fig. 5a) and unimodal (Fig. 5b) simulations. To quantify these differences, we calculate the 1-Wasserstein distance (\(W_1\)) between each ground-truth and tracked speed PDF (Table 2). Visual inspection of the PDFs, as well as the trends in (\(W_1\)), indicate that TM-Kalman is able to reproduce the simulated speed distributions the best, followed by V-TrackMat, then TM-LAP, then TP. Interestingly, each PT code besides TP overpredicts the fastest speeds for the bimodal simulation, but underpredicts the fastest speeds for the unimodal simulations. During tracking, an effort was made to use the highest possible linking distance that did not result in a significant number of mislinks. Because the range of speeds for the unimodal simulations is greater than that of the bimodal simulations, we were unable to capture the fastest speeds in the unimodal simulation without causing significant false links.Fig. 5This figure shows the a comparison between the speed distributions for each PT code for the lowest-PSDR bimodal (a) and lowest-PSDR unimodal (b) simulations. The speed distributions in (a) are normalized by the mean speed of the lowest-PSDR bimodal simulation (19.6 pixels/frame). The speed distributions in (b) are normalized by the mean speed of the lowest-PSDR unimodal simulation (11.3 pixels/frame). This figure shows the ability of each PT code to handle significantly different distributions of particle speeds.Table 2 \(W_1\) between ground truth speed distributions and the speed distributions from each PT method for the lowest PSDR bimodal and unimodal simulations.Relationship between classical statistics and PSDRTo develop a more large-scale understanding of the performance of each PT code, we use a variety of classical and experimental statistics (Fig. 3). Each of these classical statistics target different potential sources of linking error. Because the imagery had a high signal to noise ratio, there were not many errors in the detection stage of PT for each simulation (only occurring due to overlapping particles). Therefore, the false link rate (FLR) primarily shows the potential for a particle to be unlinked, meaning there were no probable candidates for linking in nearby frames (Fig. 3a). The mean path length (MPL) shows the propensity for trajectories to be fractured due to lack of linking (Fig. 3b), and the Euclidean distance (ED) indicates the likelihood for links to move back and forth between particles, sampling a large number of particles for a single trajectory (Fig. 3c). A realistic diagram of each of these potential errors is shown in Fig. 3d.Plotting these statistics over a range of PSDRs reveals that TM-Kalman and TM-LAP consistently eclipse the performance of other PT methods (Fig. 6). In particular, the bimodal simulations reveal several task-relevant patterns. The mean path lengths (Fig. 6a) illuminate the tendency for V-TrackMat and TP to generally have shorter trajectories compared to either TM method. This shortening in TP’s trajectories is significantly accentuated, especially at low PSDR levels. We attribute this phenomenon to ’trajectory splitting’, where a particle is tracked for only a fragment of its presence in the field of view. Intricacies of TP’s linking algorithm, which narrows the search space when inundated with potential particles for the ensuing frame, underpin this observation. While effective for slower-moving particles, especially in terms of memory requirements and algorithm speed, this linking strategy is less adept at tracking high-velocity particles in a directed flow. For V-TrackMat, the trajectory splitting seems to be a result of its more stringent linking algorithm. Although all PT codes try to match all trajectories during linking, V-TrackMat seems to have more extreme criteria that prevent incorrect links, as shown from the sample trajectories (Fig. 4). Thus, many trajectories are lost by V-TrackMat due to the algorithm’s necessity for high-probability links.Fig. 6This figure shows the results of the classical comparative statistics for both the bimodal and unimodal simulations. For all plots, the size of the scatter points represent the particle density of the simulation (larger points means greater particle density). a-c correspond to the bimodal simulations, and d-f correspond to the unimodal simulations. (a) Mean false link rate (error due to detection and temporally local missed links). (b) Mean path length of all PT-obtained and simulated trajectories. The ground truth is shown as a black X. This statistic describes how often full trajectories are split (linking error over time). (c) Mean Euclidean distance between true and predicted trajectories (error due to localization and linking error). (d–f) Repeat of a-c but for the unimodal simulations. These figures generally indicate that V-TrackMat and TP have the worst “classical” performance. Furthermore, classical statistics tend to follow a power law trend as a function of PSDR. Power law fit equations and goodness of fit are given in Table 3.Table 3 Power law fit equations and goodness of fit for ED, MPL and FLR.Furthermore, the FLRs (Fig. 6b) point towards V-TrackMat’s propensity to either miss or inaccurately record a particle in a frame. However, because this error is likely a result of careful linking, the classical statistics may exaggerate the experimental errors for tracking algorithms such as V-TrackMat’s. The EDs (Fig. 6c) further highlight that TP and V-TrackMat often record the most substantial discrepancies between the actual and tracked positions. This observation, particularly for V-TrackMat, implies that a rigorous linking algorithm doesn’t invariably lead to precise trajectory reconstructions. Although untested, it is theoretically plausible that during velocity-based linking or gluing, particles are incorrectly linked because they have similar velocities.For unimodal simulations, classical statistics (Fig. 6d–f) generally perform better than their bimodal counterparts. The bimodal simulations have higher mean speeds than the unimodal solutions (Table 1). Furthermore, the bimodal simulations (Fig. 1a) have a larger number of particles at high speeds, which causes more difficulty in particle tracking. In addition, the unimodal simulations show a greater range of speeds and are generally more reminiscent of speed distributions of particles in porous media54. Thus, the unimodal simulations likely offer a more comprehensive representation of generic PT code efficacy in porous media. While the general trends mirror those in the bimodal findings, V-TrackMat performs comparatively better in the FLR metric (Fig. 6d) and worse in the ED metric (Fig. 6f), and TP performs better in the ED metric (Fig. 6f). TP’s aforementioned challenges with fast-moving particles mean its performance slightly improves in unimodal settings, which aren’t dominated by high speed trajectories. Still, TP’s mean path lengths (Fig. 6e) depict a sharp decline as PSDR decreases, implying the persistent issue of trajectory splitting in both bimodal and unimodal settings.The results of the classical statistics imply that TM-Kalman and TM-LAP outperform V-TrackMat in all cases, but from the sample trajectories (Fig. 4), we have shown this to not be true. Also, the trajectories show TP performs much worse than the other algorithms at low PSDR, but this is not reflected by the FLR and ED metrics. We posit that the primary reason for the disconnect between the classical statistics and the sample trajectories is that the FLR and ED metrics underpenalize aggressive linking. The FLR will always be lower when more links are forced, since the probability of false positive detection is very low. The ED metric will always be higher when more links occur between different trajectories, but if the trajectories are nearest neighbors, then the error will be relatively small. Thus, long trajectories and links across the pore space (such as those of V-TrackMat and TM-Kalman) will result in more ED error than zig-zagging trajectories between close neighbors (such as TM-LAP) will. Ultimately, the FLR and ED underpredict PT error for nearest-neighbor based algorithms with little constraint for linking. As a result, these statistics fail to grasp the nuanced differences between PT codes.Beyond comparing PT codes’ performances, we also demonstrate that all classical statistics have a power law relationship with PSDR, although some relationships are more significant than others (Table 3). As PSDR is reduced, all PT codes generally exhibit increased ED and FDR, and decreased MPL. V-TrackMat and TP show a steeper relationship between FDR and PSDR than either TM algorithm, which generally indicates that the TM algorithms are more robust with respect to FLR performance over a range of PSDRs (Fig. 6a and d). V-TrackMat and TP also show steeper relationships between MPL and PSDR, further demonstrating the resilience of the TM algorithms when considering classical linking failures. V-TrackMat and TP also generally show more significant (lower RMSE) power-law relationships than the TM algorithms, indicating that classical PT error for V-TrackMat and TP is more predictable. Furthermore, classical statistics from unimodal simulations (Fig. 6d–f) present slightly different power law relationships compared to those from bimodal simulations (Fig. 6a–c). Thus, the choice of PT algorithm, and variations in ground truth particle speed distributions, can influence the specifics of these power law relationships.Experimental statistics highlight task-specific PT performanceThe classical statistics from bimodal simulations (Fig. 6) echo many patterns observed in the sample trajectories (Fig. 4). However, there are notable deviations. The sample trajectories, for instance, present V-TrackMat as clearly superior to TP and comparable or superior to TM-LAP. To discern which mode of analysis — comparative statistics or visual trajectory inspection — offers a more accurate picture of PT performance, we used a variety of experimental statistics.In the bimodal simulations, the normalized speed-angle joint probability density difference heatmaps rank TM-Kalman as the top performer, with V-TrackMat and TM-LAP occupying intermediate positions and TP trailing (Fig. 7). All codes demonstrate strong tracking performance at \(PSDR\ge 2.5\), but V-TrackMat and TP’s limitations become evident at \(PSDR\le 2.3\). TM-LAP and TM-Kalman significantly outperform V-TrackMat for \(PSDR\ge 1.7\). However, at \(PSDR=1.1\), V-TrackMat performs better than TM-LAP, as shown by the large amount of overprediction for the probability of low speed and high turn angle particles (Fig. 7). This disparity is likely rooted in the LAP algorithm’s propensity for aggressive linking that doesn’t take particle velocities into account, in contrast to V-TrackMat’s more conservative velocity-based approach. Consequently, at \(PSDR=1.1\), while LAP is prone to errant predictions for high speed particles and forces links with large turn angles, V-TrackMat is more likely to keep particles unlinked, and only significantly overpredicts low turn angles. In other words, V-TrackMat often refrains from making connections altogether, and when V-TrackMat does have false links, its reliance on expected particle velocities, akin to TM-Kalman, ensures that the errors are relatively benign (with respect to velocity and angle distributions) compared to TM-LAP.Fig. 7Speed-angle joint probability density difference heatmaps for the bimodal simulation. Speeds determined from particle tracking (\(S_p\)) are normalized by the mean speed of the respective simulation (\(S_{sim}\)). Red corresponds to an underprediction of probability density, blue corresponds to an overprediction of probability density, and white corresponds to an accurate probability density prediction within the speed-angle feature space. These results generally show the same trends as the sample trajectories (Fig. 1). At \(PSDR=2.5\), all algorithms show strong performance as indicated by the lack of strong color. All PT methods besides Trackpy and V-TrackMat show good replication of the simulation for \(PSDR\ge 1.7\). At \(PSDR=1.1\), TM-Kalman still performs best and TP performs worst, but V-TrackMat surprisingly outperforms TM-LAP. Thus, at very low PSDR, velocity-based algorithms result in significant improvements to PT performance.In the context of unimodal simulations (Supplementary Fig. 2), both the V-TrackMat and TM algorithms predict speed and angle statistics with near perfection. V-TrackMat and TM-Kalman perform slightly better than TM-LAP, which can be seen from the slightly greater underprediction of high speed and low turn angle particles for TM-LAP at \(PSDR\le 2.6\). TP shows relatively poor performance for all \(PSDR\le 3.1\). These observations further reinforce the general trends seen in the sample trajectories (Fig. 4). They confirm the case presented by the classical statistics that TM-Kalman has superior performance, but they significantly contrast the relative classical results of V-TrackMat and TM-LAP. Specifically, the speed-angle distributions (both bimodal and unimodal) show that TM-LAP may be favorable for \(PSDR\ge 1.7\), but that V-TrackMat is superior for \(PSDR\le 1.6\).Velocity autocorrelation function (\(C_v\)) and mean squared displacement (MSD) analysis (Fig. 8) further corroborates the trends evident in the speed-angle heatmaps. It should be noted here that we only present the first 20 frames of the lowest and highest-PSDR simulations in the main text of this paper, although the full \(C_v\) and MSDs for all simulations can be observed in Supplementary Figs. 3 and 3. Because our simulations don’t use reinjection to keep the number of particles in the field of view relatively constant, the \(C_v\) and MSDs for our simulated particles are unrealistic past 20-30 frames. Since the focus of our analysis is on the relatively accurate simulation of dispersing particles in porous media, we chose to focus on the subset of our results that are the most realistic.Fig. 8MSD ratios and VACFs for unimodal and bimodal simulations for high (a and c) and low (b and d) PSDRs. The MSD ratio is caluculated as the MSD obtained from particle tracking divided by the simulated MSD. An MSD ratio of 1 implies perfect accuracy. The bimodal MSD ratios are shown by dashed lines, and the unimodal MSD ratios are shown by solid lines. The simulation, or ground truth, is black, and the results from each PT method are different colors. For the unimodal simulations, \(PSDR=34.3\) (a and c) or \(PSDR=1.6\) (b and d). For the bimodal simulations, \(PSDR=42.8\) (a and c) or \(PSDR=1.1\) (b and d). These figures generally confirm trends present in the other experimental results. Furthermore, the MSDs and VACFs generally show the same trends, implying that a good prediction of MSD allows for a good prediction of \(C_v\). However, unlike the other experimental statistics, the \(C_v\) is not a reliable proxy for general PT performance.At \(PSDR=34.3 – 42.8\), all PT methods align closely with the simulated autocorrelations and MSD ratios. There is some slight deviation for the MSD ratio at late times for the bimodal simulation for TP and V-TrackMat (Fig. 8c), but generally, all results are highly accurate. However, at \(PSDR=1.1 – 1.6\), all PT methods show large deviations in autocorrelation and MSD ratio. The autocorrelation for the low PSDR bimodal simulation (Fig. 8b) shows decent performance for TM-Kalman, but poor performance for all other PT methods. The repetitive motion of the \(C_v\) is indicative of the wave-like periodic movement of the particles dispersing through the lattice-like geometry of the bimodal simulations. TM-Kalman is slightly able to capture this feature of the autocorrelation, but the other PT codes are not. The most likely explanation for this lies in the false links and splitting of fast trajectories. As previously discussed, as particles travel through the pore throat, they get closer together and speed up, which causes a decrease in the local PSDR. Thus, the \(C_v\) reveals that TM-Kalman is more likely to capture these fast/dense particles in the pore throats than the other PT codes are. The unimodal results for the \(C_v\) at low PSDR (Fig. 8b) surprisingly show that V-TrackMat outperforms TM-Kalman, and TP outperforms TM-LAP. However, the full \(C_v\) (Supplementary Fig. 4) indicates the TM-LAP outperforms TP at \(t\ge 30\). Likely, the \(C_v\) for TP is relatively accurate at early times because TP can only track very slow particles, so there are no significant false links that would cause velocity decorrelation between successive timesteps. TM-LAP, on the other hand, can track much faster particles, but may also erroneously link these fast particles, meaning a greater amount of velocity decorrelation. Thus, although it is important to know how accurate the \(C_v\) is for general analysis of particle transport, the \(C_v\) accuracy can’t be used as a general proxy for total particle tracking accuracy.The MSD ratios for low PSDR (Fig. 8d) show significant deviations from the simulated MSD for each PT code. Both the bimodal and unimodal results show TM-Kalman is able to most closely follow the true MSD (i.e., have an MSD ratio of 1), then V-TrackMat, then TM-LAP, and finally TP shows a complete disconnection from the true MSD. Interestingly, the unimodal simulations show an improvement in the MSD ratio over time, which indicates that for each PT code, the history of previous particle positions and links can improve the accuracy of tracking. For the bimodal simulations, we see a decrease in the accuracy of the MSD ratio over time (Fig. 8d). However, the full time-series for the lowest PSDR bimodal simulation (Supplementary Fig. 3) shows a significant improvement in the MSD accuracy over time for both V-TrackMat and TM-Kalman. Thus, velocity-based algorithms show a clear advantage in late time prediction of MSDs for low-PSDR scenarios, regardless of geometry.Generally, our experimental statistics reveal that while rudimentary comparative statistics can offer broad insights into PT code competencies across various tracking scenarios, they might fall short in pinpointing optimal codes for specific particle motions with particular analytical objectives. In our bacterial dispersion simulation within porous media, these statistics fail to elucidate speed, angle, autocorrelation or displacement distribution accuracies – all crucial for comprehending bacterial transport. Furthermore, these comparative statistics tend to underpenalize aggressive linking. Thus, basic comparative statistics might not capture the full spectrum of PT code capabilities. A more complete analysis, which can be done through a variety of statistical and visual methods, is indispensable for discerning the optimal PT code tailored to specific conditions. PT performance for simulations with noisy trajectoriesWhile the primary analysis in this paper revolves around simulations where trajectories only vary in speed and particle density, we have also provided an analysis of PT performance for simulations that contain more noise depicting experimental errors in video capture and processing. Specifically, we analyzed PT performance for simulations in which the particles had enhanced random displacement (Gaussian distribution with \(\mu =0\) and \(\sigma =2\) pixels) added to the purely advective tracks, and 2% of the particles were dropped in any given frame to account for particle intermittency. The random displacement is a simple representation of a variety of experimental phenomena/positioning errors such as diffusion, camera jitter, and/or oscillations in particle brightness. The intermittency represents particles moving in and out of the focal plane, which can also be impacted by diffusion, particle-particle interactions, particle-wall interactions, and camera exposure time. Both of these changes can be generalized as increasing the noise of the trajectories in the simulations. The sample trajectories of the lowest PSDR simulations with the intermittent and random-motion particles (Fig. 9) show the same general trends as those of the simulations with minimal noise (Fig. 4), but for each PT code the errors are slightly higher in the case of the noisy trajectories. The speed-angle distributions for the noisy unimodal simulations (Fig. 10) show that TP clearly has the worst performance for \(PSDR\le 2.3\) . For the highest PSDR noisy unimodal simulation, the performance of all PT codes are comparable. At the lowest PSDR, TM-Kalman once again shows the best performance. Similar to the unimodal results with minimal trajectory noise (Supplementary Fig. 2), V-TrackMat performs better than TM-LAP in all cases besides the highest-PSDR simulation.Fig. 9Sample trajectories for all PT codes for the low-PSDR bimodal and unimodal simulations with random motion and 2% particle intermittency. Each plot shows a 400×400 pixel section of the whole domain. Within a plot, each line corresponds to a unique trajectory (with random colors used to show the contrast between individual trajectories). (a–e) Bimodal simulations for \(PSDR=1.5\). (f–j) Unimodal simulations for \(PSDR=1.5\). (k–o) Unimodal simulations for \(PSDR=8.1\). Compared with Fig. 4, this figure (specifically the top left of f–j) shows a slight decrease in tracking performance for similar PSDR due to the addition of trajectory noise.Fig. 10Speed-angle joint probability density difference heatmaps for the unimodal simulations with random motion and 2% particle intermittency. Speeds determined from particle tracking (\(S_p\)) are normalized by the mean speed of the respective simulation (\(S_{sim}\)). Red corresponds to an underprediction of probability density, blue corresponds to an overprediction of probability density, and white corresponds to an accurate probability density prediction within the speed-angle feature space. Although these noisy simulations are slightly harder to track, the general trends in PT performance remain the same.Ultimately, these results indicate that trajectory noise such as large random motions and particle intermittency make the tracking process more error-prone, although the rankings of the linking algorithms are not impacted by these potential experimental issues. However, a more robust analysis of potential experimental errors would deal with a number of other factors such as signal to noise ratio and particle shape/size. This would also require a rigorous comparison of detection methods, which was beyond the scope of our work, but we recommend that future researchers compare PT codes in the context of more diverse simulations.Consequences of particle tracking errors on transport analysisBuilding on our comparative exploration of PT codes, the findings from the bivariate speed-angle heatmaps, MSDs, and \(C_v\) (Figs. 7 and 8, and Supplementary Figs. 2, 3, and 4) shed light on the dispersion dynamics of tracer particles within porous media. Specifically, they underscore how inaccuracies introduced by PT errors can skew transport analysis. A predominant manifestation of PT error arises from false links (Fig. 4d), leading to a systematic underestimation of high-speed particles (Figs. 4, 5 and 7). This, in turn, results in a conservative estimation of particle speeds (Figs. 5 and 7) and MSDs (Fig. 8). TP, which shows the most significant error due to trajectory splitting, underestimates the particle speeds and MSDs to an extreme degree for low PSDR.Other consequences of PT error, which can primarily be observed in the TP results, are inflated turn angles (Fig. 7) and diminished or enhanced \(C_v\) (Fig. 8a and b), attributable mainly to trajectory splitting and erroneous linking. Furthermore, we find that in the case of the bimodal geometry, where there is a periodic nature to the velocity of particles over time, only TM-Kalman is able to slightly capture the periodicity of this autocorrelation.Our analysis also emphasizes the paramount importance of experimental conditions (related to particle speed and density) in achieving reliable PT outcomes. In the case of minimal-noise simulations, a PSDR exceeding 3 ensures nearly flawless tracking, regardless of PT algorithm. Conversely, a PSDR near or below 1 presents challenges for all PT codes. In scenarios characterized by low PSDR coupled with directed particle movements, algorithms that harness velocity-based linking emerge as the more prudent choice.In addition, the results for noisy simulations are worse than for simulations with minimal noise (Figs. 4 and 9), which highlights the need for tight experimental controls to improve the visual quality of the particles. Although some amount of noise is unavoidable, these results show the importance of trying to ensure that all particles remain in the focal plane of the acquisition device.PT algorithm speed comparisonIn addition to performance analysis, we also report how long each PT code takes to link trajectories (Fig. 11). Generally, TM – LAP is the fastest linking algorithm, then TP, then TM – Kalman, and V-TrackMat is the slowest. Thus, we observe a significant trade-off between performance and computation time – the best PT methods at low PSDR also take the longest. However, we must also note that each PT code is developed in a different programming language (Python, Matalb and Javascript), so we are unable to fairly assess the speed of the underlying algorithms.Fig. 11Amount of time each PT code takes during the linking stage for selected unimodal simulations. Simulation speed is represented by scatter point size (large = 9.9 px/frame, medium = 2.6 px/frame, small = 0.9 px/frame). V-TrackMat consistently has the longest linking times, and TM-LAP has the shortest linking times. All algorithms show a power law relationship between linking time and particle density. High speed simulations generally take longer to link than low speed ones, and this difference increases at higher particle density.
