Full-DoF Egomotion Estimation for Event Cameras Using Geometric Solvers

Conference: CVPR 2025
arXiv: 2503.03307
Code: https://github.com/jizhaox/relpose-event
Area: Others
Keywords: Event camera, egomotion estimation, geometric solver, 6-DoF pose, coplanarity constraint

TL;DR

Proposes the first geometric solver method to estimate full 6-DoF egomotion (angular and linear velocities) solely using event streams. By establishing line-segment geometric constraints on the eventail manifold—specifically incidence and novel coplanarity relations—a sparse solver requiring as few as 8 events is designed, enabling decoupled rotation and translation estimation without requiring an IMU.

Background & Motivation

Background

Background: Event cameras are widely utilized in robotic navigation due to their high temporal resolution and high dynamic range. Most existing event-camera motion estimation methods assume that rotational displacement is known (provided by an IMU) and only estimate translation, or they estimate rotation only.

Limitations of Prior Work: (1) Relying on IMU-provided rotational priors limits the independence and lightweight nature of the system; (2) estimating translation only or rotation only fails to meet the demand for full 6-DoF motion estimation; (3) existing methods lack a theoretical foundation to recover full motion solely from event streams.

Key Challenge: Event streams (asynchronous pixel-level brightness changes) differ from traditional frames, making it impossible to directly apply classical epipolar geometry methods. How to extract 6-DoF motion signals from the spatiotemporal structure of event streams remains an open problem.

Goal: To recover full 6-DoF egomotion (3 angular velocity and 3 linear velocity components) solely from event streams, without requiring an IMU or other external sensors.

Key Insight: Leverage line-segment geometry on the eventail manifold: each event forms a ray in spatiotemporal space, and a set of collinear events forms a line segment. Motion constraint equations are established using the incidence relations (shared points) and coplanarity relations (shared plane normal vectors) among these line segments.

Core Idea: Model the event stream as a collection of spatiotemporal line segments, construct systems of equations using incidence and coplanarity relations among the segments, and recover full 6-DoF egomotion using a geometric solver requiring as few as 8 events.

Method

Overall Architecture

Given an input event stream \(\{(x_i, y_i, t_i, p_i)\}\), line segments on the eventail manifold are constructed in spatiotemporal space. Systems of equations regarding the motion parameters \((\omega_x, \omega_y, \omega_z, v_x, v_y, v_z)\) are constructed via two types of geometric constraints: (1) linear equations from incidence relations (point-on-line constraints); (2) bilinear equations from coplanarity relations (normal vector constraints). An Adam optimizer and a first-order rotation approximation are used for efficient solving. Pure rotational degeneracy is handled as a special case.

Key Designs

1. Geometric Modeling of the Eventail Manifold:

   • Function: Convert the event stream into usable geometric structures.
   • Mechanism: Each event \((x, y, t)\) defines a ray in 3D spatiotemporal space, and continuous events on the same edge form a line segment. The directions of these line segments are proportional to pixel velocities, which are determined by scene depth and egomotion. Reliable line-segment direction estimation is achieved by aggregating multiple events.
   • Design Motivation: A single event contains too little information (only a single brightness change trigger), but the structure of line segments in spatiotemporal space encodes motion information.

2. Incidence Relation Solver:

   • Function: Establish linear equations through point-on-line constraints.
   • Mechanism: If an event point lies on a line segment, its coordinates must satisfy the line equation. Substituting the motion parameters into the line-segment direction expression yields linear constraints on \((\omega, v)\). A minimum of 8 constraints (8 event-line pairs) are required to form the system of linear equations \(Ax=0\).
   • Design Motivation: Solving linear equations is fast and stable, making it the preferred choice for a minimal solver.

3. Coplanarity Relation Solver:

   • Function: Provide additional constraints through normal vector constraints.
   • Mechanism: If two line segments are coplanar, their direction vectors and connecting vector must satisfy a zero triple product constraint. This yields a bilinear equation \(n_1^T \cdot d_{12} = 0\), where \(n_1\) is the normal vector and \(d_{12}\) is the connection between the two line segments. This constraint does not require shared points, making it applicable to disjoint line segments in space.
   • Design Motivation: The incidence relation requires a clear association between line segments and points, whereas the coplanarity relation is more flexible—any two line segments can generate a constraint.

Loss & Training

As a non-learning method, the Adam optimizer is utilized to minimize geometric residuals. A first-order rotation approximation \(R \approx I + [\omega]_\times \Delta t\) simplifies the non-linear equations. Dedicated theoretical analysis and handling are provided for pure rotational degeneracy.

Key Experimental Results

Main Results

Evaluated on real event camera sequences from the VECtor dataset, split into 0.3-second non-overlapping intervals:

Sequence	IncBat \(\varepsilon_{ang}\)	IncBat \(\varepsilon_{lin}\)(°)	CopBat \(\varepsilon_{ang}\)	CopBat \(\varepsilon_{lin}\)(°)
desk-normal	0.232	23.0	0.236	25.1
mountain-normal	0.195	17.5	0.216	18.7
sofa-normal	0.229	21.1	0.221	20.6

Where \(\varepsilon_{ang} \in [0,1]\) (lower is better), and \(\varepsilon_{lin}\) is the angular error (°).

Ablation Study

Runtime and numerical stability (\(M=5\) line segments, \(N=100\) events/segment, noise-free synthetic data, 1000 scenes):

Solver	Rotation Parameterization	SR1 (threshold 0.01)	SR2 (threshold 0.05)	Median Runtime
IncBat	+cascad	97.3%	—	17.1 ms
CopBat	+cascad	—	—	16.7~48.7 ms
IncBat	+exact	Lower	—	Slower
IncBat	+approx	Medium	—	Faster

Key variable analysis (\(M=10\), \(N=8\sim1000\), pixel noise \(0.5\) pix, timestamp jitter \(0.5\) ms): - IncBat outperforms CopBat when the number of events is \(<100\); as the number of events increases, both converge towards similar performance. - The error drops significantly as the number of line segments increases from 1 to 50; all methods fail at \(M=1\) due to rotational-translational ambiguity. - The error increases monotonically with noise; in the absence of noise, the solver achieves nearly 100% success rate.

Key Findings

• Cascade rotation parameterization performs best: utilizing a first-order approximation for rapid initialization, followed by fine-tuning with exact parameterization to balance efficiency and accuracy.
• Coplanarity relations provide key complementary constraints in scenarios with non-intersecting line segments, presenting a major theoretical innovation.
• Pure rotation is a degenerate case (translation is unestimable); practical deployment requires motion classification and detection.
• The actual error levels (\(\varepsilon_{ang} \approx 0.2\), \(\varepsilon_{lin} \approx 20°\)) are sufficient for integration into VIO/SLAM pipelines.

Highlights & Insights

• Outstanding Theoretical Contribution: It is proven for the first time that full 6-DoF motion can be recovered solely from event streams, and a theoretical lower bound for the minimum number of events (8) is established.
• Novelty of Coplanarity Relations: Traditional line-segment geometry mostly utilizes incidence relations; coplanarity relations do not require explicit intersections of line segments, significantly expanding the number of usable constraints.
• No External Sensors Required: Removing the reliance on IMUs enables motion estimation using only event cameras.

Limitations & Future Work

• The first-order rotation approximation is only valid under small motions; fast-rotation scenarios require higher-order expansions.
• Noise in practical event streams affects the accuracy of line-segment direction estimation.
• There is a lack of quantitative comparison against learning-based event-camera motion estimation methods.
• The analysis of computational efficiency is not detailed enough.

Related Work & Insights

• vs. CMax methods: The CMax series only estimates rotation, whereas ours simultaneously estimates both rotation and translation.
• vs. Traditional frame-based methods (e.g., 5-point algorithm): Frame-based methods require feature matching; ours extracts motion directly from the spatiotemporal structure of events.
• vs. Event + IMU fusion: Removing IMU dependency makes the system more lightweight.

Rating

• Novelty: ⭐⭐⭐⭐⭐ The first pure event-based 6-DoF solver, with a major theoretical contribution from coplanarity relations.
• Experimental Thoroughness: ⭐⭐⭐ Theoretically rigorous, but the experimental scale is relatively small, lacking large-scale quantitative comparisons.
• Writing Quality: ⭐⭐⭐⭐ Clear geometric derivation and a complete theoretical framework.
• Value: ⭐⭐⭐⭐ Provides fundamental contributions to event-camera SLAM and autonomous navigation.

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