Trajectory Extractor (TE) Overview

• Trajectory Extractor (TE) is a system that reconstructs spatiotemporal paths from sensor data using both hardware-based methods and algorithmic frameworks.
• TE methods employ electromagnetic probe techniques, inversion strategies, and machine learning to analyze dynamic phenomena in various scientific disciplines.
• Applications of TE span accelerator physics, astrophysics, and computer vision, enhancing diagnostic accuracy and automating trajectory tracking.

A Trajectory Extractor (TE) is a class of measurement apparatus, algorithm, or signal processing workflow designed to reconstruct, infer, or extract spatiotemporal paths (“trajectories”) from a set of observations. TE architectures are fundamental in disciplines such as accelerator physics, particle physics, computer vision, astrophysics, and fluid dynamics. Core implementations can rely on electromagnetic probe methods (as in electron cloud diagnostics), geometric or algebraic inversion (as with detector cluster data), neural forecasting, or spatiotemporal grouping in large multidimensional sensor outputs. The TE concept encompasses both physical instrumentation (such as the TE wave method for electron clouds in accelerators) and algorithmic engines (as in tracklet extraction in astronomical surveys or regression-based trajectory disentanglement in VLBI jet studies).

1. Measurement Principles and Physical Realizations

Trajectory Extractors can be either direct hardware-based systems or algorithmic frameworks built on top of sensor data:

• TE Wave Method: In accelerator diagnostics (Sikora et al., 2013), the TE wave approach couples microwaves into a beam-pipe using Beam Position Monitor (BPM) electrodes, enabling the extraction of local electron cloud (EC) densities and profiles along the beam path. The electron cloud perturbs the phase or frequency of the transmitted microwave, which is measured to infer the EC distribution—the “trajectory” here is the spatial profile of the cloud. The method evolved from a transmission analysis (treating the beam pipe as a waveguide) to a resonant cavity analysis, since real-world geometries induce strong standing wave patterns and discrete resonances.
• Bench and Simulation-Enhanced TE: Physical TE methods may be augmented by bench measurements (such as bead-pull techniques to map spatial variation in the resonant field) and fully electromagnetic simulations (e.g., VORPAL) to predict and analyze resonance shifts under varying EC conditions or applied magnetic fields.

2. Mathematical Formulations and Inversion Methods

TE systems often rely on perturbation theory or geometric transformations to connect observations to the underlying trajectory:

• Transmission Phase Shift (Accelerator Physics):

$$\Delta\phi = \frac{L e^2 n_e}{2 m_e \varepsilon_0 c \sqrt{\omega^2 - \omega_c^2}}$$

This formula links phase shift in a uniform waveguide to local EC density.

• Resonant Frequency Shift:

$$\frac{\Delta\omega}{\omega} = \frac{e^2}{2\varepsilon_0 m_e \omega^2} \frac{\int n_e E_0^2 dV}{\int E_0^2 dV}$$

Allows for inference of average or spatially-varying EC density from measured frequency shifts.

• Signal Extraction via Transform Methods: In pattern recognition, e.g., the Legendre Transform can convert the problem of finding the tangent to a family of ellipses (sensor hits) into the identification of an intersection in dual (Legendre) space (Alexopoulos et al., 2016). In this context, intersections in $(r, \theta)$ parameter space correspond to candidate tangent (track) solutions in $(x, y)$. This enables robust identification of charged particle tracks in detectors.

3. Algorithms for Automated Trajectory Extraction

Modern TE workflows include highly automated and scalable algorithms for track identification, object association, and trajectory hypothesis sampling:

• Tracklet Extraction Engines: Algorithms such as tracee (Ohsawa, 2021) handle vast point clouds (e.g., detections in astronomical imaging or video) by constructing approximate $k$-Nearest Neighbor graphs and then grouping co-linear elemental line segments in $(x, y, t)$ (space-time) into candidate trajectories. Robustness is gained by angular, positional, and temporal consistency checks, and the approach is resistant to noise, distractors, and intersecting trajectories.
• Regression Strip Algorithms: In VLBI imaging pipelines (Zhang et al., 2017), jet component trajectories are disentangled by iterative pattern extraction and orthogonal distance regression, enabling automated proper motion measurement without subjective operator intervention. Statistical significance is quantified via a “degree of separation” metric (DS): $\mathrm{DS} = \Delta\mu/\sigma_\mathrm{max}$, informing the algorithm's ability to resolve nearby patterns.
• Neural and Probabilistic Forecasting: In computer vision, modules such as TrajE deploy recurrent Mixture Density Networks to estimate future position distributions and propagate trajectory hypotheses forward with beam search, improving robustness to occlusion and identity switches (Girbau et al., 2021).

4. Applications in Experimental and Computational Domains

Trajectory Extractors play critical roles in diverse domains:

Domain / Instrument	Workflow / Output	Typical TE Type
Accelerator EC Diagnostics	Microwave resonance phase/freq analysis	Hardware (RF), physicist-driven
Particle/Beam Telescopes	Cluster, hit alignment, track fit	Algorithmic (GBL, regression)
Astrophysical Imaging	Component strip, regression, matching	Statistical pattern extraction
Computer Vision (Tracking)	Hypothesis propagation, feature memory	ML module, aggregator
Autonomous Sensing	Tracklet grouping, kNN-temporal links	Graph/grouping algorithm
Oceanography	Rotation-averaged path metrics, field	Single-trajectory diagnostics

• In accelerator physics, TE wave methods support detailed EC mapping, enabling improved mitigation strategies and contributing to reliable high-current operations (e.g., CESRTA, DAΦNE) (Sikora et al., 2013).
• In imaging arrays and particle detection, TE methods (such as in EUTelescope (Bisanz et al., 2020)) offer modular pipelines—clustering, noise rejection, alignment, advanced track fitting (GBL)—adaptable to a wide range of sensor geometries and experimental requirements.
• In radio astronomy and VLBI, regression strip algorithms automate the extraction of proper motions from multi-epoch jet component data, providing a statistical approach to disentangling overlapping trajectories (Zhang et al., 2017).
• In fluid dynamics and geoscience, quasi-objective metrics such as the trajectory rotation average (TRA) provide objective diagnostics of coherent structures (e.g., oceanic eddies) from sparse trajectory data (Bartos et al., 2021).
• Algorithms such as tracee are applied in large-scale astronomical surveys or video streams for extraction of moving objects (e.g., asteroids, satellites), with scalability to thousands of concurrent trajectories (Ohsawa, 2021).

5. Advancements, Challenges, and Future Directions

Current and future TE systems address limitations and enable new capabilities:

• Resonant Analysis Superiority: Transition from transmission to resonant cavity models in TE wave methods was motivated by measurement inconsistencies and strong presence of resonances in real beam-pipe geometries. This allows for greater fidelity and spatial selectivity in EC profile extraction (Sikora et al., 2013).
• Numerical and Sensor Constraints: Challenge areas include inversion of nonlinear models (e.g., iterative solution for drift time vs. depth in CdTe detectors), limited timing or energy resolution in ASICs, and the need for statistical robustness under high background.
• Integration with Simulation: Electromagnetic and beam dynamics simulations (e.g., VORPAL for microwave cavities) under external fields or in nontrivial geometries provide predictive support for TE design and benchmarking.
• Probabilistic/ML-based Enhancement: The integration of trajectory distribution modeling (e.g., MDN-based) into broader detection and association pipelines is improving real-world robustness to missing data and occlusion. Beam search and explicit confidence handling is essential in coping with environmental uncertainty.
• Cross-disciplinary Reusability: Frameworks such as TrajPy (Moreira-Soares et al., 2023) are expanding the reach of TE concepts by providing cross-domain feature extraction, simulation, and classification utilities, crucial for applications from molecular dynamics to clinical trajectory analysis.

6. Impact and Significance

The development and refinement of Trajectory Extractors have had a profound effect in both foundational and applied research:

• In accelerator science, TEs underpin the ability to diagnose and mitigate electron cloud effects, directly impacting luminosity and beam stability.
• In experimental physics and engineering, TEs support high-precision reconstruction and alignment, critical for detector calibration and fundamental measurements.
• In astrophysics and geoscience, they enable automated, objective extraction of trends and proper motions from massive datasets, improving statistical confidence and reproducibility.
• In sensing and computer vision, TEs—augmented by machine learning—are an enabling architecture for robust multi-target tracking, object re-identification, and prediction under uncertainty.

Ongoing research focuses on enhancing measurement selectivity, improving algorithmic scalability, integrating richer prior information (semantic, physical, or linguistic), and expanding the applicability of TE frameworks across additional domains and measurement modalities.