Track-to-Flow (T2F): Mapping Tracks to Flows

• Track-to-Flow (T2F) is a family of methods that convert discrete observations and trajectories into continuous flow representations using domain-specific mapping rules.
• It is applied in various fields such as optical flow in satellite imaging, fluid field reconstruction with Navier–Stokes constraints, traffic monitoring, perturbative QCD analysis, and psychological flow decoding.
• Despite diverse implementations, T2F techniques share common challenges like occlusion, sparse observations, and calibration complexities, driving continuous methodological refinements.

Track-to-Flow (T2F) is a label used in several research literatures for methods that convert tracked observations, trajectories, or associations into a flow representation, or that realize tracking by composing flow estimates. In the cited work, T2F spans satellite tracking of ship-emitted aerosols, dense long-term pixel tracking, inversion of Eulerian velocity–pressure fields from sparse Lagrangian tracks, normalizing-flow-based multi-object association, track-function matching in perturbative QCD, conversion of multi-target traffic tracks into microscopic flow variables, and decoding psychological flow from motor performance (Larson et al., 2021, Neoral et al., 2023, Wan et al., 13 Jul 2025, Wu et al., 21 Sep 2025, Mancusi et al., 2023, Li et al., 2021, Teknomo et al., 2016, Tian et al., 2023).

1. Terminological scope and domain-specific meanings

In the cited literature, T2F does not denote a single standardized algorithm. The same term is applied to different objects called “tracks” and to different meanings of “flow,” ranging from optical flow and fluid flow to traffic flow, energy flow, probabilistic normalizing flows, and psychological flow states.

Usage	Track input	Flow output or operation
Optical-flow tracking	Ship-track ROIs or image pixels	Trajectories, pathlines, persistence
Sparse-trajectory inversion	Particle tracks or one agent trajectory	Dense velocity, pressure, temperature fields
Multi-object association	Track–detection cue vector	Conditional normalizing-flow likelihood
Traffic measurement	Multi-target centroids	Speed, acceleration, headways
Perturbative QCD	Partonic observables with track functions	Restricted-hadron energy flow
Behavioral decoding	Performance metrics	Continuous flow intensity

The table shows that the common element is not a shared implementation but a shared mapping logic: some tracked entity is converted into a field, a transport description, or a flow-valued score. This suggests that T2F is best understood as a family of track-conditioned inference procedures rather than as a single method class.

2. Optical-flow formulations for tracking visible structures

In atmospheric remote sensing, T2F appears as the use of optical flow to follow the motion and persistence of ship-emitted aerosol signatures that manifest as bright, quasi-linear cloud features in GOES-17 ABI imagery (Larson et al., 2021). The method uses ABI Level-1b radiances from C06 at 2.24 μm and C07 at 3.90 μm, forms the day–night visualization $C06 - C07$, applies histogram equalization, and operates on 2 km imagery at 5-minute cadence over 24 manually selected ship-track intersections in the North Pacific Ocean off the U.S. West Coast. Motion is estimated with Shi–Tomasi “Good Features to Track” and pyramidal Lucas–Kanade sparse optical flow, with the local brightness-constancy constraint

$I_x u + I_y v + I_t = 0.$

Tracking boxes are updated by averaging feature displacements, and generalized propagation follows

$$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$

The principal confound is motion from high, optically thick clouds; the paper addresses this by manual ROI placement on the low cloud deck, feature selection within the ROI, omission of frames with more than 2% corrupted pixels, and a solar-zenith-angle criterion with $c=83^\circ$ and $d=97^\circ$ to bridge sunrise and sunset using a constant-velocity predictor. Compared with NOAA HYSPLIT forward trajectories initialized from GDAS and ERA-5, the optical-flow tracker maintains higher SSIM than HYSPLIT and random-box baselines across the 24 cases. The reported outcome is that ship tracks persist beyond 9 hours and sometimes longer than 24 hours (Larson et al., 2021).

A different optical-flow-based T2F formulation is the Multi-Flow dense Tracker (MFT), which performs dense, pixel-level, long-term tracking by composing optical flows across logarithmically spaced temporal intervals (Neoral et al., 2023). For frames $i,j,k$, flow composition is

$$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$

MFT uses the default interval set $D=\{\infty,1,2,4,8,16,32\}$, where $\Delta=\infty$ denotes direct flow from the reference frame to the current frame. It augments RAFT with two CNN heads that output per-pixel occlusion probability and uncertainty, then selects for each pixel the candidate chain with minimum accumulated uncertainty subject to an occlusion gate $\theta_o=0.02$. The reference-frame accumulators $I_x u + I_y v + I_t = 0.$0 reduce drift and allow recovery from medium-length occlusions by longer temporal hops. On TAP-Vid DAVIS, MFT reports $I_x u + I_y v + I_t = 0.$1, $I_x u + I_y v + I_t = 0.$2, and $I_x u + I_y v + I_t = 0.$3 for first/strided evaluation; on TAP-Vid Kinetics first, it reports $I_x u + I_y v + I_t = 0.$4, $I_x u + I_y v + I_t = 0.$5, and $I_x u + I_y v + I_t = 0.$6. Runtime is about $I_x u + I_y v + I_t = 0.$7 FPS densely at $I_x u + I_y v + I_t = 0.$8, and exceeds 100 FPS with precomputed segments (Neoral et al., 2023).

These two optical-flow usages differ in scale and objective. The ship-track system isolates a meteorological structure within a carefully chosen ROI and quantifies persistence, whereas MFT tracks every pixel in parallel and uses learned uncertainty and occlusion modeling to select among competing flow chains. Both, however, treat tracking as a composition of local motion estimates rather than as repeated detection.

3. From sparse trajectories to dense Eulerian fields

In fluid mechanics, T2F denotes the reconstruction of dense, physically consistent flow fields from sparse Lagrangian observations. TrajectoryFlowNet reconstructs a continuous velocity field $I_x u + I_y v + I_t = 0.$9, pressure field $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$0, and particle trajectories $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$1 from sparse particle tracks by coupling a Lagrangian trajectory network with an Eulerian flow-field network under Navier–Stokes constraints (Wan et al., 13 Jul 2025). The Lagrangian description uses the material derivative

$$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$2

and imposes the advection constraint

$$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$3

The Eulerian module enforces incompressible Navier–Stokes without external forcing, while both modules are parameterized as fully connected networks with Fourier feature mappings and tanh activations. Training minimizes a data loss for observed particle positions and velocities and a physics residual loss enforcing both Lagrangian kinematics and the Navier–Stokes residuals at Latin-hypercube collocation points. In the lid-driven cavity case, the trajectory block uses 4 hidden layers with 40 neurons, the flow-field block uses 6 hidden layers with 60 neurons, and 160,400 collocation points are sampled. Optimization uses L-BFGS with learning rate 1.0 and up to 50,000 iterations. The reported cavity performance includes trajectory mean error $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$4, standard deviation $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$5, velocity RMSE and MAE $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$6, and Pearson correlations $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$7 for $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$8, $$x_{t+\Delta t} = x_t + u(x_t,t)\Delta t,\qquad y_{t+\Delta t} = y_t + v(y_t,t)\Delta t.$$9 for $c=83^\circ$0, $c=83^\circ$1 for $c=83^\circ$2, $c=83^\circ$3 for $c=83^\circ$4, and $c=83^\circ$5 for $c=83^\circ$6. In the complex-cylinder case, velocity and pressure correlations are $c=83^\circ$7, $c=83^\circ$8, and $c=83^\circ$9, and in the aortic and left-ventricle experimental cases correlations exceed 0.9 for trajectories and velocities despite moving boundaries and sparse measurements (Wan et al., 13 Jul 2025).

A more extreme sparse-observation variant is the 2025 T2F model for super-resolution reconstruction from a single actively migrating Lagrangian trajectory (Wu et al., 21 Sep 2025). The agent is not passively advected: it self-propels at fixed speed $d=97^\circ$0 and chooses a swimming direction $d=97^\circ$1 with a Soft Actor-Critic controller from the state

$d=97^\circ$2

with kinematics

$d=97^\circ$3

From a trajectory segment of length $d=97^\circ$4, T2F extracts small patches around the agent, tokenizes them, encodes the sequence with a Vision Transformer using $d=97^\circ$5 layers and $d=97^\circ$6 attention heads, and decodes a $d=97^\circ$7 Eulerian window with a three-layer transposed-convolution CNN. The data-only variant minimizes an MSE loss, whereas T2F+PINN adds PDE residuals for continuity, momentum, and temperature transport. In a cylinder wake at $d=97^\circ$8, T2F reports normalized $d=97^\circ$9 errors for velocity of about $i,j,k$0 for $i,j,k$1 and about $i,j,k$2 for $i,j,k$3, while T2F+PINN reduces vorticity error by about $i,j,k$4. In turbulent Rayleigh–Bénard convection at $i,j,k$5 and $i,j,k$6, both models achieve similar primitive-variable errors, but T2F+PINN reduces normalized $i,j,k$7 error by $i,j,k$8 for vorticity, $i,j,k$9 for $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$0, and $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$1 for $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$2 relative to T2F (Wu et al., 21 Sep 2025).

Taken together, these studies define a strong fluid-mechanical meaning of T2F: sparse trajectory data become dense field reconstructions through learned Lagrangian–Eulerian couplings, and physics-informed regularization mainly improves gradient-sensitive quantities rather than primitive variables.

4. Tracking, association, and traffic-flow measurement

In traffic engineering, T2F has a direct operational meaning: multi-target trajectories extracted from video are converted into microscopic traffic variables such as individual speeds, accelerations, and headways (Teknomo et al., 2016). The underlying tracker is intentionally simple. A fixed-focus camera is mounted above the road or walkway, a background image is captured, and each frame at about 30 fps is differenced against the background, filtered, thresholded, and analyzed by connected components to obtain centroid points $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$3. Association between consecutive slices uses a binary distance matrix

$$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$4

with $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$5. The method assigns IDs by nearest-neighbor matching under a distance gate $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$6, manages entries and exits through zero columns and zero rows, and does not perform re-identification. After planar calibration, trajectories are converted into

$$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$7

time headway $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$8, and distance headway $$F_{i\to k}(x)=F_{i\to j}(x)+F_{j\to k}(x+F_{i\to j}(x)).$$9. The reported validation is qualitative but explicit: the automatic method works “without error” against manual collection as long as there is no occlusion (Teknomo et al., 2016).

In multi-object tracking, the paper “TrackFlow” uses the term in a different sense: it converts track-to-detection association scoring into conditional density estimation with normalizing flows (Mancusi et al., 2023). For a candidate pair $D=\{\infty,1,2,4,8,16,32\}$0, the feature vector $D=\{\infty,1,2,4,8,16,32\}$1 consists of $D=\{\infty,1,2,4,8,16,32\}$2, $D=\{\infty,1,2,4,8,16,32\}$3, and $D=\{\infty,1,2,4,8,16,32\}$4, while the conditioning variable $D=\{\infty,1,2,4,8,16,32\}$5 includes the track temporal context $D=\{\infty,1,2,4,8,16,32\}$6 and a scene cluster $D=\{\infty,1,2,4,8,16,32\}$7 obtained from CLIP-based clustering. The association cost is

$D=\{\infty,1,2,4,8,16,32\}$8

with $D=\{\infty,1,2,4,8,16,32\}$9 defined by a conditional normalizing flow and trained by maximum likelihood on ground-truth positive associations. The model uses $\Delta=\infty$0 flow blocks with 64 hidden neurons per block, a context window $\Delta=\infty$1, and $\Delta=\infty$2 scene clusters. It replaces heuristic weighted sums of IoU, appearance, motion, and coarse 3D cues with a learned joint conditional model of cue dependencies. On MOTSynth, MOT17, and MOT20, the reported effect is consistent but moderate for estimated distances and much larger when ground-truth distances are available. For example, on MOT17 validation, TrackFlow improves SORT by $\Delta=\infty$3 HOTA and $\Delta=\infty$4 IDF1, ByteTrack by $\Delta=\infty$5 HOTA and $\Delta=\infty$6 IDF1, and OC-SORT by $\Delta=\infty$7 HOTA and $\Delta=\infty$8 IDF1 (Mancusi et al., 2023).

These two uses share a reliance on track-level state evolution, but they differ in what “flow” denotes. In traffic analysis, flow is the measurable stream of moving agents derived from trajectories. In TrackFlow, flow is the normalizing-flow machinery that turns heterogeneous association cues into a likelihood.

5. Track functions and restricted energy flow in perturbative QCD

In perturbative QCD, T2F has a formal theoretical meaning: it is the mapping from partonic predictions for energy-flow observables to charged-particle or otherwise restricted-hadron energy flow through nonperturbative track functions (Li et al., 2021). A track function $\Delta=\infty$9 gives the energy fraction $\theta_o=0.02$0 of a parton $\theta_o=0.02$1 converted into a subset $\theta_o=0.02$2 of hadrons defined by quantum numbers, such as electrically charged hadrons. Unlike fragmentation functions, track functions sum over all hadrons in the subset and therefore encode multi-hadron correlations. Their $\theta_o=0.02$3-th moments are

$\theta_o=0.02$4

with the sum rule $\theta_o=0.02$5.

The paper identifies a shift symmetry in track-function evolution and uses it to organize nonlinear renormalization-group equations in terms of central moments such as

$\theta_o=0.02$6

It explicitly computes the evolution of the first three moments at next-to-leading order and shows that, for charged hadrons in QCD where $\theta_o=0.02$7 is small, the evolution of the first three central moments is governed by standard timelike twist-2 anomalous dimensions up to suppressed nonlinearities. For energy correlators, the core factorization is

$\theta_o=0.02$8

The paper then computes the two-point energy correlator on charged particles at $\theta_o=0.02$9 and demonstrates that infrared singularities in perturbation theory are absorbed by moments of the track functions. It also states that the first three moments suffice for up to three-point energy correlations (Li et al., 2021).

Here the word “track” refers not to a trajectory in physical space but to the subset of hadrons seen by a tracking detector, and “flow” refers to energy flow. This usage is mathematically distant from optical-flow or trajectory-inversion methods, yet it fits the same broader template: a restricted observable built from tracks is related to a more complete flow observable through an explicit transformation law.

6. Decoding psychological flow from tracked performance

In cognitive and motor-control research, T2F denotes the mapping from tracked task performance to a continuous estimate of psychological flow intensity (Tian et al., 2023). The cited study uses a Fine Fingertip Force Control task in VR, in which participants press a force transducer with the right index finger to keep the force $I_x u + I_y v + I_t = 0.$00 inside a target band around $I_x u + I_y v + I_t = 0.$01 N for at least 500 ms within a 3 s trial. Force is sampled at 1 kHz, each trial is followed by 2 s rest, and three main sessions of 100 trials each are preceded by a 50-trial skill-measurement session. Difficulty is calibrated by an adaptive staircase to a success probability of about 0.5, with mean measured skill $I_x u + I_y v + I_t = 0.$02 N across participants. Flow labels come from 12 sparse self-report probes using a three-item subset of the Flow Short Scale, and 24 of 32 recruited participants are retained after exclusions.

The T2F decoder uses eight performance metrics computed from the force trajectory: reaction time, arriving time, completing time, in-range time, force overshooting, average deviation, average adjusting rate, and success rate. For probe $I_x u + I_y v + I_t = 0.$03, metrics are averaged across the five trials immediately preceding the probe to form $I_x u + I_y v + I_t = 0.$04, and a participant-specific linear model predicts flow intensity through

$I_x u + I_y v + I_t = 0.$05

where $I_x u + I_y v + I_t = 0.$06 contains at most four selected metrics. Training uses leave-one-out cross-validation over the 12 probes, with exhaustive feature-subset search under the constraint that labels outnumber features by at least a factor of three. The reported group-level decoding accuracy is $I_x u + I_y v + I_t = 0.$07 with $I_x u + I_y v + I_t = 0.$08, and mean NRMSE is $I_x u + I_y v + I_t = 0.$09 for overall flow intensity, $I_x u + I_y v + I_t = 0.$10 for fluency, and $I_x u + I_y v + I_t = 0.$11 for absorption. All participants pass both random-label and permuted-timestamp tests at $I_x u + I_y v + I_t = 0.$12. The decoded series has one estimate per trial, corresponding to a cadence of $I_x u + I_y v + I_t = 0.$13 Hz, and power spectral density analysis indicates that more than 70% of power lies at timescales of $I_x u + I_y v + I_t = 0.$14 s and faster (Tian et al., 2023).

The study also reports systematic metric differences between in-flow and out-flow states. Reaction time and completing time are shorter in-flow, in-range time and success rate are higher, and overshooting, average deviation, and average adjusting rate are smaller; arriving time is not significant in the paired $I_x u + I_y v + I_t = 0.$15-test. This T2F usage is conceptually the farthest from fluid or optical flow, but it preserves the same inferential structure: a tracked time series is mapped into a latent flow-valued quantity that varies continuously between sparse labels.

7. Recurrent design patterns, limitations, and methodological implications

Taken together, these usages suggest three recurrent T2F patterns. The first is flow-for-tracking, in which motion fields are estimated and composed to maintain trajectories, as in ship-track optical flow and MFT (Larson et al., 2021, Neoral et al., 2023). The second is track-to-field inversion, in which sparse trajectories are used to reconstruct dense state variables or transport fields, as in TrajectoryFlowNet and the single-trajectory T2F/T2F+PINN models (Wan et al., 13 Jul 2025, Wu et al., 21 Sep 2025). The third is track-conditioned flow scoring, in which tracks are turned into probabilistic, energetic, or psychological flow quantities, as in TrackFlow, track functions in QCD, and performance-based flow decoding (Mancusi et al., 2023, Li et al., 2021, Tian et al., 2023).

The limitations reported across these domains are also structurally similar. Optical-flow trackers are vulnerable to occlusion, diurnal radiance changes, thin structures, textureless regions, and drift (Larson et al., 2021, Neoral et al., 2023). Fluid-mechanical inversions depend on trajectory coverage, may struggle in regions with few or stagnant particles, and in the single-trajectory PINN variant use temporal derivatives from reference fields during training rather than pure autodiff (Wan et al., 13 Jul 2025, Wu et al., 21 Sep 2025). Normalizing-flow association depends on domain transfer and the reliability of coarse 3D distance estimates (Mancusi et al., 2023). The centroid-based traffic tracker assumes minimal occlusion and small inter-frame displacement relative to inter-object spacing (Teknomo et al., 2016). The psychological decoder depends on sparse self-reports, individualized calibration, and stability of the task-performance relationship under fatigue, learning, and distraction (Tian et al., 2023). In perturbative QCD, the main assumptions are perturbative factorization, universality of collinear limits, and controlled treatment of nonlinear moment evolution (Li et al., 2021).

A plausible implication is that T2F has become a reusable research pattern rather than a single discipline-specific term. Across fields, it typically combines sparse or partial observations, an inductive bias about transport or correspondence, and an explicit rule for propagating, aggregating, or regularizing tracks into a flow-like object. What changes from domain to domain is the ontology of both sides of the mapping: pixels, aerosols, particles, vehicles, hadrons, or human performance can all serve as “tracks,” while the target “flow” can be optical motion, a Navier–Stokes field, traffic variables, a likelihood under a normalizing flow, restricted energy flow, or a latent cognitive state.