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CoHelix: Event-Based Multi-Propeller Rotation-Rate Tracking with Shared Projective Geometry

Radim Spetlik, Tobiáš Vinklárek, Jan Klivan, Tomáš Kestřánek, Matouš Vrba, Jiří Matas

arXiv:2610.31384v1cs.RO

TL;DR

Moving-camera propeller-rate tracking must separate rotation from changing image geometry and background events. CoHelix estimates one shared homography and four separate rotation states, achieving higher lock than the evaluated methods on both datasets.

  • Problem

    Moving-camera rate tracking must recover propeller rotations while camera motion changes their image geometry and background events mix with blade events.

    Source: §1 · p. 1

  • Method

    CoHelix recursively estimates a shared homography and four independent rotation states, updating the assigned state and shared geometry from admitted events.

    Source: §1 · p. 2

  • Results

    62.9% lock on moving-camera data exceeds HelixTrack’s 39.8% and EventPro’s 22.2%; 95.9% lock on target–follower exceeds the corresponding methods.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    Neither CoHelix nor HelixTrack runs in real time in the reported tests, with CoHelix faster on TQE and HelixTrack faster on target–follower.

    Source: §4.7 · p. 8

    The method assumes localized rings and rigid, coplanar propellers; evaluation covers two tracked quadcopters and one sensor type.

    Source: §5 · p. 8

Abstract

from arXiv · show

Estimating propeller rates from a moving event camera requires separating blade rotation from the changing image geometry. Background activity mixes with blade events, while repeated blade appearances admit half- and double-rate ambiguities. We propose CoHelix, a recursive filter that estimates one shared homography and separate rotation states for four rigidly mounted, coplanar propellers. Each admitted event updates its assigned rotation state, and its accepted geometry measurements correct the shared homography used to interpret later events from every propeller. We report lock, the fraction of evaluated time for which an estimate remains within 2% of synchronized ground truth, because mean error alone does not show how long a tracker follows the true rate. With fixed-rate initialization, leave-one-recording-out evaluation on moving-camera data gives CoHelix 62.9% lock, compared with 39.8% for HelixTrack and 22.2% for EventPro. With other settings fixed, allowing four geometry contributors instead of one raises lock from 3.4% to 62.9%. On a private target-follower dataset with a different target quadcopter, CoHelix retains the measurement settings selected on moving-camera data and uses a fixed initial rate. It reaches 95.9% lock, compared with 56.2% for HelixTrack and 61.1% for EventPro.

1. Introduction

Moving event cameras must separate propeller rotation from changing image geometry and background activity. CoHelix couples one shared homography with four independent rotation states, achieving the highest reported lock among the evaluated methods on two datasets.

1. Introduction

  • A moving camera adds background events to blade events, while four nearly identical propellers make event assignment ambiguous.Propeller projections change with relative camera–airframe motion even as the blades rotate.Source: §1 · p. 1
  • One homography maps the fixed layout of rigidly mounted, coplanar propellers into the image, while each propeller retains its own phase, rate, and acceleration.A two-blade propeller looks unchanged after a half turn, creating a rate ambiguity.Source: §1 · p. 2
  • CoHelix recursively estimates one shared homography and four independent rotation states, updating the assigned state and shared geometry from individual events.An accepted geometry measurement changes how later events from every propeller are interpreted.Source: §1 · p. 2
  • Lock percentage measures the fraction of evaluated time that rates stay within 2% of synchronized ground truth; half- and double-rate estimates fail this test.Missing output also counts as not locked.Source: §1 · p. 2
  • 62.9% lock on moving-camera data, versus 39.8% for HelixTrack and 22.2% for EventPro; on target–follower, CoHelix reaches 95.9%.The moving-camera result uses leave-one-recording-out evaluation with a fixed initial rate; target–follower uses the selected measurement settings unchanged.Source: §1 · p. 2

2. Related Work

Prior event-based methods detect or track objects, estimate periodic motion, or estimate propeller rates, but the reviewed rate methods do not update shared geometry from individual events while keeping propeller rates separate.

2. Related Work

  • Appearance-based trackers follow blobs, corners, or learned templates, while multi-object methods use learned association and sequence models.These methods output positions or feature tracks rather than rotation rates; identical blades also look the same after a half turn.Source: §2 · p. 2
  • Periodic-motion methods correlate templates, register stream slices, estimate pixel frequencies [22], or apply fixed kernels.EventPro estimates centres from clustered events and rate from an adaptive event batch, but its experiments use a stationary camera and it maintains no state for camera translation or perspective changes.Source: §2 · p. 2
  • HelixTrack updates one propeller’s phase per event but refines its pose from accumulated events and radial counts after each half revolution.It processes propellers independently and does not update shared geometry from individual events.Source: §2 · p. 2
  • Event-contrast methods warp events by candidate motion and optimize accumulation sharpness, but repeated window warps are costly and an incorrect motion model can also produce sharp accumulations.Processing cascades reduce cost by rejecting events before expensive operations.Source: §2 · p. 2
  • None of the reviewed propeller-rate methods updates shared geometry from individual events while keeping propeller rates separate.CoHelix’s shared-map, separate-rate design addresses this gap.Source: §2 · p. 2

3. Method

CoHelix represents four coplanar propellers with a shared homography and separate rotation states, then processes events through assignment, measurement, and state updates. Geometry measurements from individual events update the shared map used for later events from every propeller.

3. Method

  • CoHelix uses one shared homography for the propeller layout and separate phase, rate, and acceleration states for each propeller.Both CoHelix and HelixTrack update phase per event; CoHelix additionally uses accumulated centre measurements and an ellipse-based drift observer.Source: §3 · pp. 2–3, Table 1 · p. 3

3.1. State and dynamics

  • Each propeller’s state contains phase, rate, and acceleration, with constant-acceleration prediction, white-jerk noise, and an additional rate-scaled random walk.The phase is the B-fold blade-passage phase; the variance cap keeps its Gaussian approximation within one equivalence interval.Source: §3.1 · p. 3

3.2. Event admission

  • Events outside padded projected-ring regions are rejected, and remaining events are inverse-mapped and assigned to the nearest fixed rotor-plane centre.A propeller-specific radial gate then accepts events within normalized inner and outer bounds, with the outer bound clipped to avoid overlap.Source: §3.2 · p. 3

3.3. Measurements

  • Individual-event radial residuals constrain scale and perspective by measuring normalized distance from the assigned centre relative to the unit ring.Other geometry measurements use boundary and polarity information, signed crossings, and a rotating silhouette; centre measurements are formed after 2,000 admitted events and omitted for one-sided arcs.Source: §3.3 · p. 4

3.4. Update and validity

  • The filter applies individual-event residuals on arrival and centre residuals after each 2,000 admitted events, using Kalman updates for both phase and geometry.Geometry residuals pass a normalized-innovation gate because a geometry update changes the back-warp used for later events from all propellers; phase residuals are not gated.Source: §3.4 · p. 4

4. Experiments

The experiments evaluate rate tracking on moving-camera recordings and test how shared geometry, update choices, and calibration affect performance. CoHelix achieves higher lock than the compared methods on both datasets, but calibration sensitivity and sub-real-time throughput remain practical constraints.

4.1. Datasets / 4.2. Measures and protocol

  • Evaluation uses synchronized rate references, leave-one-recording-out tuning on TQE, and lock within 2% of ground truth alongside mean errors and coverage.TQE contains 12 scored recordings; target–follower evaluation uses three recordings and settings selected on TQE.Source: §4 · p. 4, §4.1 · p. 5, §4.2 · pp. 5–6

4.3. Comparison with prior methods

  • 62.9% TQE lock exceeds HelixTrack’s 39.8% and EventPro’s 22.2% in leave-one-recording-out evaluation.CoHelix also reports 1383 RPM shaft-rate MAE and 99.9% coverage on the 12 TQE recordings.Source: Table 2 · p. 6, §4.3 · p. 6
  • On target–follower recordings, CoHelix reaches 95.9% lock, versus 61.1% for EventPro and 56.2% for HelixTrack.CoHelix has higher lock on each of the three recordings; its measurement settings were selected on TQE and applied unchanged.Source: Table 3 · p. 6, §4.3 · p. 6

4.4. Shared geometry ablation

  • Allowing one, two, three, or four geometry contributors yields 3.4%, 17.7%, 50.2%, and 62.9% lock, respectively.More contributors provide observations at more image locations and more geometry measurements. Under equal geometry-event caps with the ellipse observer disabled, both one- and four-contributor conditions reach at most 2.9% lock versus 62.3% unbudgeted; differing realized counts mean this diagnostic does not isolate spatial support.Source: §4.4 · p. 7

4.5. Geometry updates and stored uncertainty

  • Per-event geometry updates beat tested windowed Gauss–Newton by 23.1 points, while correlation retention (−2.1 points) and applying all same-event measurements (1.9 points) show no established benefit.The update comparison also changes solver and uncertainty treatment; internal uncertainty is not interpreted as calibrated confidence.Source: §4.5 · pp. 7–8

4.6. Calibration sensitivity

  • A 5% increase in supplied ring radii reduces target–follower lock from 95.9% to 65.7%, with the loss concentrated in one recording.Offsets of 5% of mean annotated radius yield 61.0% TQE and 88.4% target–follower lock; both diagnostics alter the fixed layout.Source: §4.6 · p. 8

4.7. Runtime and diagnostic tests

  • On one CPU thread, four-rate real-time factors are 0.0306 versus 0.0189 on TQE and 0.0478 versus 0.0593 on target–follower; neither method runs in real time.CoHelix is faster on TQE, while HelixTrack is faster on target–follower.Source: §4.7 · p. 8

4.8. Scope of the evaluation

  • The evaluation assumes localized rings and rigid, coplanar propellers; flex, non-coplanarity, or relative hub motion require richer models.Whole-airframe motion preserves the shared-plane assumption.Source: §4.8 · p. 8

5. Conclusion

CoHelix tracks four propellers through a shared homography and separate rotation states, outperforming the reported baselines on moving-camera and target–follower data. Its scope includes assumptions about localization and rigid, coplanar propellers.

5. Conclusion

  • 62.9% lock on moving-camera data and 95.9% on target–follower data; both exceed the corresponding HelixTrack and EventPro lock percentages.CoHelix separates shared image motion from independent propeller rotations through one homography and four phase states.Source: §5 · p. 8
  • Additional geometry contributors improve tracking, while per-event Kalman geometry updates exceed the tested windowed Gauss–Newton formulation by 23.1 percentage points.The update comparison also differs in solver, uncertainty treatment, and update schedule.Source: §5 · p. 8
  • The method assumes localized rings and rigid, coplanar propellers; a 5% increase in supplied ring radii reduces target–follower lock from 95.9% to 65.7%.Evaluation covers two tracked quadcopters and one sensor type, not other cameras, airframes, or scenes.Source: §5 · p. 8

S1. Measurement model

The measurement model uses individual events to update propeller phase and shared geometry, with selected radial, annulus, polarity, centre, and blade-edge measurements. It also tests several geometry and measurement alternatives that are not retained.

S1. Measurement model

  • Scalar residual measurements update either a propeller’s phase state or the shared geometry state, with geometry measurements discarded for low weight or excessive normalized innovation.The geometry update also requires the candidate mean to pass geometry checks.Source: S1. Measurement model · p. 11
  • Each admitted event updates phase without an innovation gate; the filter bounds acceleration and signed rate, resetting acceleration if either limit is reached.The reported configuration disables the optional phase gate.Source: S1. Measurement model · p. 11
  • The reported configuration omits phase-based geometry updates; radial, annulus, and polarity weights are selected together, while rotating silhouette and signed crossing have separate weights.The radial, annulus, polarity, silhouette, and signed-crossing measurements are used; polarity chord and same-edge transport are not.Source: S1. Measurement model · pp. 11–12, Table S1 · p. 13
  • Radial alignment uses the widest symmetric interval around each ring and omits the measurement when no such interval exists, avoiding scale bias from one-sided events.The radial residual measures distance from the normalized ring radius.Source: S1. Measurement model · p. 11
  • Every 2,000 admitted events per propeller, a weighted centre estimate is used only when effective sample size is at least 50 and antipodal blade segments are sufficiently balanced.The centre update omits measurements when Neff < 50 or the angular resultant exceeds 0.5.Source: S1. Measurement model · p. 12
  • Blade-edge tests include rotating silhouettes and signed crossings, while polarity chords and same-edge transport are disabled in the reported configuration.The signed-crossing measurement requires event polarity to agree with the direction of contour crossing.Source: S1. Measurement model · p. 12
  • A four-centre similarity fit and a separate similarity constraint were tested as geometry alternatives but are not used in the reported configuration.The four-centre fit requires all four centre estimates to be less than 50 ms old.Source: S1. Measurement model · pp. 12–13
  • The tested windowed Gauss–Newton geometry update is not retained because the tested windows reduce lock percentage.The comparison uses zero forgetting and damping values of 100, 100, and 10 for 50, 100, and 200 µs windows, respectively.Source: S1. Measurement model · p. 13

S2. Configuration used in the experiments

The experimental configuration represents rates internally as blade-passage frequency and converts reported quantities to shaft RPM. It also specifies process noise, covariance handling, and an auxiliary observer for slow centre drift.

S2. Configuration used in the experiments

  • The filter uses blade-passage frequency internally; figures, tables, and errors convert it to shaft RPM by dividing by the known two-blade count.No accuracy correction is fitted.Source: S2. Configuration used in the… · p. 13
  • Rate-process-noise spectral density is set to (f|ω|)^2 with f = 0.0025 s^-1/2; phase covariance is limited at the uniform-circular threshold while preserving positive semi-definiteness.The covariance row and column are scaled to set phase variance to its limit.Source: S2. Configuration used in the… · p. 13
  • An auxiliary observer handles slow centre drift by fitting event ellipses and applying bounded translation and scale corrections to shared geometry.Its 0.2 s time constant prevents short-lived ellipse fits from moving geometry abruptly.Source: S2. Configuration used in the… · p. 13

S3. Geometry validity checks

CoHelix rejects geometry updates that violate numerical and projective validity checks, while noting that passing those checks does not guarantee accurate pose or prevent drift.

S3. Geometry validity checks

  • Passing the homography checks does not establish pose accuracy or prevent drift within the allowed bounds.The initial homography is subject to the same checks.Source: S3. Geometry validity checks · p. 13
  • Candidate homographies must have finite entries, satisfy the applicable orientation and scale constraints, and pass translation, perspective, determinant, and condition-number bounds.The full eight-parameter model also has an orientation test.Source: S3. Geometry validity checks · p. 13
  • Four corner checks ensure every modelled ring point remains in front of the projective horizon, with its denominator constrained to at least 0.05.The denominator is affine in the body point, so its minimum over the bounding box occurs at a corner.Source: S3. Geometry validity checks · p. 13
  • A cached homography anchor avoids repeated singular-value decompositions for sufficiently small perturbations while maintaining the condition-number bound.Candidates within the Frobenius-distance radius reuse the last checked decomposition; larger changes trigger a Jacobi SVD.Source: S3. Geometry validity checks · pp. 14–15

S4. Evaluation details

Evaluation uses a shared 5 ms scoring grid and a 2% lock criterion, with missing outputs counted as unlocked. The section also reports a damaged-stream stress test, recording-level uncertainty, and sensitivity to the lock threshold.

S4. Evaluation details

  • The scorer tests lock every 5 ms against a 2% relative-error threshold; missing estimates count as unlocked, and half- or double-rate estimates fail.TQE rates and blade-pass estimates are converted to shaft RPM using the known blade count; target–follower supplies a direct shaft-rate timeline.Source: S4. Evaluation details · p. 15
  • The target–follower reference checks support timing and internal consistency, but do not independently calibrate its absolute rate.After clock alignment, the 95th-percentile timing residuals are 3.0, 3.0, and 5.3 ms across its three recordings.Source: S4. Evaluation details · p. 15

S4.1. Missing events and their treatment

  • Only the 2 m, Mego = 4 recording meets the input-only damage rule; it is excluded from selection and the primary result, then scored as a separate stress test.The rule is more than 0.5 seconds of complete sensor silence within any 5-second interval.Source: S4.1. Missing events and their treatment · p. 15
  • 21.68% lock on the damaged recording leaves CoHelix ahead of HelixTrack at 15.30% and EventPro at 9.11%.Including that recording in the 13-recording aggregate gives 59.69%, 37.83%, and 21.19%, respectively, with the ordering unchanged.Source: S4.1. Missing events and their treatment · p. 15

S4.2. Recordings shown in the figures

  • Figure 2 uses six timestamps fixed before tracker output inspection, beginning about 0.5 seconds after each recording’s evaluation cut and advancing every 0.4 seconds.The displayed rings use the final selected CoHelix configuration; the figures do not define an evaluation subset.Source: S4.2. Recordings shown in the figures · p. 15

S4.3. Recording-level uncertainty

  • CoHelix exceeds HelixTrack by 23.18 percentage points and EventPro by 40.71 points on moving-camera data, and by 39.66 and 34.78 points on target–follower.Recording-bootstrap 95% intervals are [9.64, 38.77] and [27.36, 54.44] points for moving-camera, and [18.01, 79.64] and [25.78, 38.96] for target–follower.Source: S4.3. Recording-level uncertainty · p. 15

S4.4. Lock-threshold sensitivity

S5. Selection of measurements and weights

The moving-camera benchmark selects CoHelix measurement weights inside recording-level folds, then transfers the selected settings unchanged to target–follower. A separate propeller-wise protocol tests transfer across propeller identity within known recordings.

S5. Selection of measurements and weights

  • Configuration selection uses nested leave-one-recording-out folds, with both CoHelix selection stages repeated using the other 11 recordings and a fixed 4,750 shaft-RPM start.The damaged stream is excluded from selection; the 4 m, motion-level-six recording remains an ordinary fold.Source: S5. Selection of measurements and… · pp. 15–16
  • HelixTrack’s nested search gives 39.75% lock, versus 40.31% with its public ring-bound pair applied to all 12 recordings.The nested result is 0.56 points lower, with a 95% recording-bootstrap interval of [−1.60, 0.02] points.Source: S5. Selection of measurements and… · p. 16
  • Each selection stage maximizes the smallest total gain remaining after removing one training recording, so one unusually favourable recording cannot decide the choice.Ties are resolved by total gain and then configuration name.Source: S5. Selection of measurements and… · p. 16

S5.1. Propeller-wise four-fold protocol

  • The propeller-wise protocol holds out propeller identity rather than recordings, retaining the same camera motions and shared geometry state across folds.It measures transfer across propellers within known recordings, unlike the primary protocol’s transfer to an omitted recording.Source: S5.1. Propeller-wise four-fold protocol · p. 16
  • 69.98% lock is reported for the propeller-wise protocol, with 100% coverage, 679.5 shaft-RPM MAE, and 15.49% MArE.This protocol includes the damaged recording to match the 13-recording benchmark.Source: S5.1. Propeller-wise four-fold protocol · p. 16
  • Fixed rather than reference-rate initialization changes lock by −0.29 points on moving-camera data, with a 95% interval of [−7.96, 7.98] points.On target–follower, the difference is −0.005 points, with an interval of [−0.099, 0.083].Source: S5.1. Propeller-wise four-fold protocol · p. 16

S6. Comparison protocols

The comparison evaluates baseline reimplementations, initialization, tracking ablations, runtime, calibration sensitivity, and geometry-contributor budgets. The reported evidence includes strong gains from multiple geometry contributors, alongside scope and measurement caveats.

S6.1. Baseline methods

  • 22.22% lock on moving-camera data transfers to 61.09% on target–follower for the EventPro reimplementation, which reports no value when its reliability checks fail.The EventPro paper provides neither public code nor evaluation data, so these are replay results rather than author-certified outputs.Source: S6.1. Baseline methods · p. 17
  • With missing HelixTrack suffixes counted as locked, its upper bounds remain below CoHelix: 40.54% versus 62.94% on moving-camera and 58.22% versus 95.87% on target–follower.The ordering does not depend on how the two shortened HelixTrack outputs are treated.Source: S6.1. Baseline methods · p. 18
  • Four geometry contributors yield 62.9% lock versus 3.4% with one, a 59.6-point increase with higher lock on all 12 recordings.This comparison changes both spatial support and the amount of geometry evidence; it is not a separately optimized single-propeller tracker.Source: S6.1. Baseline methods · p. 18
  • Fixed-rate initialization recovers 50 of 52 moving-camera tracks at 4,750 shaft RPM and all 12 target–follower tracks at 9,500 shaft RPM.The corresponding median acquisition times are 0.397 s and 0.007 s.Source: S6.1. Baseline methods · p. 19
  • Applying all active scalar measurements from an event changes lock by 1.88 points over the one-measurement control, with a 95% interval of [−3.33, 6.38].Normalized innovations change substantially, so the stored covariance is not treated as calibrated.Source: S6.1. Baseline methods · p. 20
  • CoHelix is 1.62× faster than HelixTrack on moving-camera recordings, while HelixTrack is 1.24× faster on target–follower; neither reaches real time.The four-rate geometric-mean real-time factors are 0.0306 versus 0.0189 and 0.0478 versus 0.0593, respectively.Source: S6.1. Baseline methods · p. 20, Table S10 · p. 20

S6.2. Sensitivity to ring calibration

  • A 5% radius increase reduces one target–follower recording’s lock from 96.5% to 0.4% despite 100% coverage, so these responses do not establish a monotonic tolerance bound.Sensitivity is concentrated in one of the three recordings, and centre-offset responses are also non-monotonic.Source: S6.2. Sensitivity to ring calibration · p. 21

S6.3. Contributor comparison with fixed budgets

  • Under fixed geometry-event opportunity caps, every one- and four-contributor condition remains at or below 2.9% aggregate lock, versus 62.3% without a cap.The experiment establishes neither a practical contributor advantage under these caps nor a benefit caused by spatial support alone.Source: S6.3. Contributor comparison with… · p. 21

S7. Additional negative results

The additional results preserve cases where EventPro has lower shaft-rate MAE but CoHelix has higher lock, and report unchanged rate traces under repeated radial relinearization. Separate tables examine ring-calibration sensitivity and geometry-contributor conditions.

S7. Additional negative results

  • Two, three, or five radial relinearizations produce bit-identical rate traces, so the method retains one radial linearization per event.This test does not compare iterations of the windowed solver.Source: S7. Additional negative results · p. 21
  • EventPro has lower shaft-rate MAE than CoHelix in five listed recordings, while CoHelix has higher lock in each.The cases are at 2 m, Mego = 6 and 7, and at 4 m, Mego = 1, 2 and 3.Source: S7. Additional negative results · p. 21
  • Table S11 examines supplied ring-calibration sensitivity on 12 TQE and three target–follower recordings with selected measurement settings fixed.Its lock score covers all four rates at 2% relative tolerance on the complete 5 ms grid; missing estimates count as unlocked.Source: Table S11 · p. 22
  • Table S12 compares geometry contributors under four equal geometry-event caps across all 12 TQE recordings, with the ellipse observer disabled.It scores all four rate states at 2% lock tolerance; equal caps do not imply equal realized measurements.Source: Table S12 · p. 22
  • Table S13 reports selected-event and accepted-geometry-row totals for the Table S12 conditions, while phase updates use every admitted event.Lock includes all four rates, even those supplying no geometry input; measurement rejection remains active.Source: Table S13 · p. 23

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