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Navigation by Skylight Polarization via Rayleigh Model Optimization on a Road Vehicle

Ben Potter, Eslam Mounier, Aboelmagd Noureldin, Yahia Antar, Muhammad Alam

arXiv:2610.20797v1cs.ROcs.SYeess.SY

TL;DR

Reliable vehicle heading is needed when conventional references are degraded, but skylight-polarization methods face occlusion and camera-tilt challenges. This paper estimates heading by optimizing a camera-projected Rayleigh sky model against polarimetric images. Across three urban trajectories, it reports sub-degree RMS yaw error and lower heading error than representative methods.

  • Problem

    Absolute heading references can be impaired by inertial drift, magnetic interference, and GNSS obstruction or attack, while polarization landmark methods can degrade under occlusion and camera tilt can introduce transformation errors.

    Source: §I · pp. 1–2

  • Method

    The method optimizes candidate vehicle states by comparing measured AoP images with full-image predictions from a pinhole camera, Rayleigh sky, and solar-position model.

    Source: §IV · p. 4, A. Objective Function · p. 5, §VII · pp. 9–10

  • Results

    Across three urban road-vehicle trajectories, RMS yaw errors ranged from 0.718° to 0.889°, and the method reduced heading error versus representative feature-based and model-based methods.

    Source: §VII · p. 10

  • Takeaways & Limitations

    The results suggest skylight polarization can provide a useful passive, infrastructure-independent absolute-heading constraint for ground-vehicle navigation, particularly as a complement to inertial navigation systems.

    Source: §VII · p. 10

Abstract

from arXiv · show

Reliable absolute heading estimation is critical for road-vehicle navigation, particularly when magnetic, inertial, or GNSS-based references are degraded. This paper presents a skylight polarization navigation method that estimates vehicle heading by globally optimizing a Rayleigh sky model against measurements from a division-of-focal-plane polarimetric camera. The proposed framework combines a pinhole camera model, a single-scattering Rayleigh polarization model, and a solar position algorithm to synthesize the expected full image angle of polarization distribution for candidate vehicle states. A degree of polarization weighted root mean square error objective function compares measured and simulated polarization patterns, allowing the estimator to use all available sky information while attenuating low confidence pixels caused by clouds, buildings, foliage, and other depolarizing occlusions. Experimental evaluation is performed using three urban road vehicle trajectories collected with a roof mounted polarimetric camera and a high precision dual-antenna GNSS/INS reference system. Results show sub-degree RMS yaw error across all trajectories, ranging from $0.718^{\circ}$ to $0.889^{\circ}$. Compared with representative feature-based and model-based polarization navigation methods, the proposed approach reduces RMS heading error by up to 87% and demonstrates improved robustness to partial sky occlusion.

I. INTRODUCTION

The paper addresses the need for a passive absolute-heading reference when conventional sensors are impaired and local skylight features may be occluded. It proposes full-image Rayleigh-model optimization, while prior polarization methods rely on landmarks, analytic models, or preprocessing.

I. INTRODUCTION

  • Conventional inertial, magnetic, and GNSS heading references face drift, interference, vehicle distortion, multipath, shadowing, or jamming in relevant operating conditions.The paper motivates a passive, drift-free, infrastructure-independent absolute orientation reference for sensor fusion.Source: §I · p. 1
  • Landmark-based methods can degrade when sky features are occluded, while non-horizontal vehicle camera tilt can introduce systematic AoP transformation errors.The introduction notes that tilt-related errors are rarely modeled or compensated in the literature.Source: §I · p. 2
  • The proposed framework optimizes a full-image AoP model combining a pinhole camera model, single-scattering Rayleigh theory, and solar position, with DoP weighting across visible sky pixels.The weighting attenuates low-polarization noise from clouds and solid occlusions.Source: §I · p. 2
  • Up to a 73 % decrease in heading-estimate mean squared error (RMSE) is reported compared with other model optimization methods.The introduction also reports a decrease compared with traditional HT-based methods, but gives no value in the supplied passage.Source: §I · p. 2

II. RELATED WORK

  • Prior analytic sky-model methods estimate heading using different polarization properties, while model-based AoP restoration is typically followed by a separate heading-extraction method.The proposed approach combines sky-model handling of occlusion with heading estimation directly, reducing the importance of preprocessing schemes.Source: §II · pp. 2–3

III. THEORY OF SKYLIGHT POLARIZATION FOR NAVIGATION

The theory defines horizontal, camera-body, and ray coordinate systems to express camera orientation and ray-specific polarization geometry. Heading is the camera’s rotation about the upward axis.

1) Horizontal Coordinate System:

2) Camera Body Coordinate System:

  • The camera body b-frame shares the observer origin with the n-frame; their rotation defines camera orientation, and its angle about U is yaw or heading.The method seeks to constrain this component of camera rotation.Source: 2) Camera Body Coordinate System · p. 3

3) Ray Coordinate System:

  • Each ray’s r-frame depends on its propagation direction; its Zr axis points from sky point P toward observer O.The Y r axis is orthogonal to Zr in the EN plane, and Xr completes the right-handed system.Source: 3) Ray Coordinate System · p. 3

4) Superpixel Coordinate System:

The superpixel frame is fixed to the camera hardware, while the paper represents each linearly polarized ray by its angle and degree of polarization. The analysis restricts polarization states to linear.

4) Superpixel Coordinate System:

  • Each superpixel s-frame shares the camera body axes but has an origin determined by its image-sensor position.The frame is defined relative to the b-frame because each superpixel is physically embedded in the camera hardware.Source: 4) Superpixel Coordinate System · p. 3

B. Representations of Linearly Polarized Light

  • AoP represents e-vector orientation in the plane transverse to propagation, while DoP is the fraction of total intensity in polarized components.The paper considers only linear polarization because prior work establishes that elliptical atmospheric-scattering states are negligible.Source: B. Representations of Linearly… · p. 3
  • The ray-frame AoP is measured from Xr in the right-handed rotation direction and is represented on (−π/2, π/2].Negating the e-vector leaves the polarization state unchanged.Source: B. Representations of Linearly… · p. 3

C. Rayleigh Model

The Rayleigh model predicts skylight polarization from the geometry of single scattering relative to the sun. The method uses that predicted AoP pattern, with camera and solar-position modeling, for comparison with measurements.

C. Rayleigh Model

  • Single-scattering models assume a ray interacts once before reaching the observer; the paper chooses Rayleigh because neutral-point modeling is unnecessary for its method.The paper also cites its common use in prior work and reported agreement with measurements; the Berry model can represent neutral points, while radiative-transfer solutions are computationally prohibitive for navigation.Source: C. Rayleigh Model · p. 3
  • The Rayleigh model predicts a ring-like distribution of e-vector orientations around the solar point, with orientations orthogonal to the direction toward the sun.The e-vector after scattering is orthogonal to the plane defined by observer O, scattering point P, and solar point S.Source: C. Rayleigh Model · p. 3, Figure 1 · p. 4
  • The model derives AoP from the e-vector expressed in the ray frame and derives DoP from the scattering angle.The solar-position algorithm supplies the solar vector from observer location and time.Source: C. Rayleigh Model · p. 4
  • In the illustrated example, measured ray-frame AoP and Rayleigh-model AoP generated from the reference state match closely, forming the optimization constraint.The figure also presents measured superpixel-frame AoP, transformed ray-frame AoP, and measured DoP.Source: C. Rayleigh Model · p. 4, Figure 2 · p. 4

IV. MODEL OPTIMIZATION

The framework estimates camera state by simulating a skylight AoP image from a candidate state and comparing it with a measured polarimetric image. Its objective combines pixelwise measured-versus-expected AoP errors using degree-of-polarization weights.

IV. MODEL OPTIMIZATION

  • Candidate position, time, and orientation drive the simulated polarization image, which is compared with the measurement to evaluate each state.The camera model maps pixels to sky points, and the sky model predicts each pixel’s polarization state.Source: §IV · p. 4, Figure 3 · p. 5

A. Objective Function

  • Measured DoP weights pixelwise error between measured and expected AoP in the WRMSE objective.The objective combines the weighted expectation error for each pixel in the measured image.Source: A. Objective Function · pp. 4–5
  • Camera geometry maps each pixel location to a sky point, where the Rayleigh model computes its expected AoP.The expected AoP is generated by combining the camera model with the Rayleigh model.Source: A. Objective Function · p. 5

B. Pinhole Camera Model

The pinhole camera model traces sensor pixels to sky-ray directions and maps those rays into the sky using the estimated camera orientation. Its focal length sets the image-plane geometry and field of view.

B. Pinhole Camera Model

  • The pinhole geometry traces a light ray from a sensor pixel through the lens into a sky-ray direction.The image-plane location and optical properties determine the corresponding sky point.Source: B. Pinhole Camera Model · p. 5, Figure 4 · p. 5
  • Focal length f constrains image-plane position and field of view; 8 mm matched the lens hardware and can be tuned.The camera model sets the image plane’s position using the focal length.Source: B. Pinhole Camera Model · p. 5
  • Azimuth and polar angles parameterize each ray, and the inverse projection uses atan2 to select its correct quadrant.The polar angle ranges from 90° to 180°; atan2 selects the correct quadrant unlike arctan’s restricted interval.Source: B. Pinhole Camera Model · p. 5
  • The estimated camera orientation transforms a body-frame ray to the navigation frame to identify the viewed sky point.The frame transformation is determined by the orientation of the state estimate.Source: B. Pinhole Camera Model · p. 5

C. Measurement with Polarization Camera

A division-of-focal-plane camera reconstructs polarization measurements for each superpixel, whose sensor-frame angle of polarization must be transformed to the ray-frame reference used by the sky model. The zenith-based transform is exact for a level camera but approximates the geometry when the camera is tilted.

C. Measurement with Polarization Camera

  • Four micropolarizers in each superpixel measure orthogonal polarization intensities used to reconstruct the Stokes vector, AoP, and DoP.The sensor records the polarization state for each superpixel across the image.Source: C. Measurement with Polarization Camera · pp. 5–6, Figure 5 · p. 6
  • Measured AoP is referenced to the superpixel frame, while the sky model defines it in the ray frame, requiring a per-ray reference transformation.The ray-frame reference direction varies with each ray’s azimuthal orientation.Source: C. Measurement with Polarization Camera · p. 6
  • The zenith-path approximation is exact for a level sensor but incurs error when tilted, with error increasing as tilt magnitude grows.It reduces computation compared with simulating optical paths for every pixel.Source: C. Measurement with Polarization Camera · p. 6
  • For typical road-vehicle pitch and roll below 5°, the mean absolute approximation error is less than a degree; domains with more tilt should consider alternative transforms.The paper reports that the approximation’s error remains relatively small even for large pitch and roll angles.Source: C. Measurement with Polarization Camera · pp. 6–7

A. Instrumentation

A roof-mounted polarimetric camera records skylight measurements, while a dual-antenna GNSS/INS system supplies the reference navigation solution. Images are paired with nearby reference measurements by timestamp.

A. Instrumentation

  • A NovAtel PwrPak7 with dual GNSS antennas and a NovAtel IMU-µIMU-IC provides 50 Hz position, velocity, and orientation ground truth.TerraStar Pro corrections support centimeter-level precise point positioning.Source: A. Instrumentation · p. 7
  • The roof-mounted LUCID PHX050S-P/Q camera points its optical axis toward the zenith and captures 2048 by 2448 images every 200 ms.It uses a 3.5 mm focal-length lens with a 120° field of view.Source: A. Instrumentation · p. 7
  • Each image is labeled with the nearest reference-system measurement, with a maximum time difference of 50 ms.The reference system polls faster than the camera, so each image has a measurement within that interval.Source: A. Instrumentation · p. 7

B. Data Collection

The dataset comprises three urban Kingston trajectories with varying sky visibility and intermittent building and foliage occlusion. The reported results table summarizes yaw-error metrics against two comparison methods, while the selected grid search and camera model use fixed settings.

B. Data Collection

  • Three Kingston trajectories recorded sequentially on November 24, 2025, lasted 10.10, 18.92, and 13.93 minutes, respectively.The routes passed through sunny conditions with few clouds, intermittent building and foliage occlusion, and solar altitude angles of about 23.27°–25.11°.Source: B. Data Collection · p. 7
  • Trajectory 2 had the lowest percentage of sky pixels because of buildings along the roadway, while trajectory 1 had periods of tree cover not fully captured by the classification.Trajectory 3 mostly passed through residential areas where buildings covered only a small portion of the sky.Source: B. Data Collection · p. 8, Figure 9 · p. 8

A. Hyperparameter Selection

  • Yaw is selected by grid search within ±5° of the reference yaw at 0.1° resolution.The bounds are fixed across estimates in this evaluation, though a fused solution could adjust them to current state uncertainty.Source: A. Hyperparameter Selection · p. 8
  • The camera model uses an 8.0 mm focal length and superpixel-matched sensor dimensions, without correcting real optical-system intrinsics or tuning camera parameters.The authors suggest this may support generalization to other measurement methodologies.Source: A. Hyperparameter Selection · pp. 8–9

VI. RESULTS

  • Table I summarizes RMS, P95, and maximum absolute yaw error for the proposed method and Lu’s and Wang’s methods.The passage identifies the metrics and comparison methods but does not provide their values.Source: §VI · p. 8, Table I · p. 9

B. Merits of Proposed Method

The method’s yaw error remains drift-free over time and stable as sky visibility varies. Degree-of-polarization weighting lets the estimator use available polarization information while down-weighting uncertain pixels.

B. Merits of Proposed Method

  • The yaw error exhibits a drift-free property because each frame supplies a new sky image with minimal coupling between adjacent state estimates.The paper identifies this as useful for applying heading constraints in fused systems with IMUs.Source: B. Merits of Proposed Method · p. 9
  • The proposed method maintains low, stable absolute yaw error across environmental settings, while Lu’s and Wang’s errors generally depend on available sky pixels.Their error and variance increase when fewer sky pixels are available; the authors attribute the proposed method’s occlusion handling to its DoP-weighted objective.Source: B. Merits of Proposed Method · p. 9
  • Strategic pixel weighting uses polarization information that binary keep-or-ignore methods may discard.Pixels with large weights can guide estimation while lower-weight pixels remain considered rather than being excluded by a binary threshold.Source: B. Merits of Proposed Method · p. 9

C. Remaining Challenges with Proposed Method

The remaining challenges include a high-noise local minimum near the antisolar meridian and computationally expensive grid search. The conclusion reports sub-degree RMS yaw error on three urban trajectories and identifies broader evaluation and more efficient optimization as future work.

C. Remaining Challenges with Proposed Method

  • A local minimum remains at the antisolar meridian and may cause challenges in high-noise environments.The method avoids reliance on any single AoP feature, making it more robust to the 180° ambiguity associated with symmetric features.Source: C. Remaining Challenges with Proposed… · p. 9

VII. CONCLUSION

  • Grid search is computationally expensive because it repeatedly evaluates full-image simulations across many candidate states.Gradient-based or manifold search could reduce evaluations; pixel-level parallelism makes GPU or embedded parallel hardware a possible path toward online operation.Source: C. Remaining Challenges with Proposed… · p. 9, §VII · p. 10
  • 0.718°–0.889° RMS yaw error across three urban road-vehicle trajectories, with drift-free estimates under realistic sky occlusions.Figure 10 compares yaw-error time series and distributions for the proposed method, Lu’s method, and Wang’s method across the trajectories.Source: Figure 10 · p. 10, §VII · p. 10
  • The conclusion identifies broader atmospheric, seasonal, and geographic evaluation, full-attitude estimation, and tighter inertial or vehicle-motion integration as future work.It also notes that more efficient optimization strategies could reduce objective-function evaluations per estimate.Source: §VII · p. 10
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