An N-Point Linear Solver for Line and Motion Estimation with Event Cameras (2404.00842v1)

Abstract: Event cameras respond primarily to edges--formed by strong gradients--and are thus particularly well-suited for line-based motion estimation. Recent work has shown that events generated by a single line each satisfy a polynomial constraint which describes a manifold in the space-time volume. Multiple such constraints can be solved simultaneously to recover the partial linear velocity and line parameters. In this work, we show that, with a suitable line parametrization, this system of constraints is actually linear in the unknowns, which allows us to design a novel linear solver. Unlike existing solvers, our linear solver (i) is fast and numerically stable since it does not rely on expensive root finding, (ii) can solve both minimal and overdetermined systems with more than 5 events, and (iii) admits the characterization of all degenerate cases and multiple solutions. The found line parameters are singularity-free and have a fixed scale, which eliminates the need for auxiliary constraints typically encountered in previous work. To recover the full linear camera velocity we fuse observations from multiple lines with a novel velocity averaging scheme that relies on a geometrically-motivated residual, and thus solves the problem more efficiently than previous schemes which minimize an algebraic residual. Extensive experiments in synthetic and real-world settings demonstrate that our method surpasses the previous work in numerical stability, and operates over 600 times faster.

• The paper introduces a linear solver that linearizes incidence relations into a minimal form for direct estimation of motion parameters.
• The approach employs an angle-axis line representation and a geometrically motivated velocity averaging scheme to mitigate numerical instability and handle degenerate cases.
• The paper demonstrates superior performance with over 600 times faster processing in both synthetic and real-world tests compared to polynomial solvers.

An Efficient Linear Approach for Line and Motion Estimation Using Event Cameras

Introduction

Event cameras have emerged as a promising alternative to traditional frame-based cameras, especially in scenarios requiring high dynamic range and temporal resolution. These cameras, which capture pixel-level intensity changes, are particularly adept at handling tasks like line-based motion estimation due to their ability to sharply capture edges. Recent developments have introduced a novel incident relation that facilitates the extraction of motion parameters from events generated by lines. However, existing solutions leveraging this relation are not without their limitations, chiefly rooted in their reliance on polynomial solvers and non-minimal representations. Addressing these challenges, this paper presents a new linear solver that simplifies the estimation process by linearizing the system of constraints obtained from event camera data, resulting in a highly efficient and stable approach for motion estimation.

Related Work

Ego-motion estimation has been a focal point in mobile vision systems, with a plethora of methodologies spanning single, stereo, and multi-camera systems, often supplemented by inertial measurement units. Among these, event cameras present a unique proposition, offering solutions unconstrained by the issues plaguing traditional camera systems, such as motion blur and low temporal resolution. Existing literature features a variety of approaches including optimization, filtering, and learning-based methods, with a notable focus on leveraging line features for motion estimation. This body of work lays the groundwork for exploring geometric incidence relationships for motion calculation, a field where the current paper makes significant contributions.

Methodology

The proposed method introduces a linear solver to estimate motion parameters and line characteristics from event data. It begins by redefining the line representation using an angle-axis formulation, which inherently handles the scale ambiguity present in monocular setups. This representation simplifies the system of equations derived from the incidence relations of events to a line, facilitating a straightforward linear solution.

• Minimal Form Transition: A pivotal innovation is the transition of the incidence relation into a minimal form, allowing for the direct recovery of motion parameters from the events without the need for complex polynomial solving methods. This transition hinges on a three-degree-of-freedom (DoF) representation of lines and a simplified camera velocity representation in the line-coordinate frame.
• Solver Implementation: The linear solver efficiently addresses minimal and overdetermined cases, circumventing the numerical instability issues of previous solvers. It also introduces a mechanism to deal with degenerate cases and solution multiplicity, underpinning its robustness.
• Velocity Averaging Scheme: To reconcile partial motion estimates from multiple lines, the paper proposes a geometrically motivated averaging scheme. This approach not only enhances efficiency but also retains the theoretical underpinnings necessary for understanding the solution's optimality.

Experimental Results

Extensive testing in both synthetic and real-world scenarios illustrates the solver's superior performance over existing methods. Notably, it achieves a substantial speed increase, operating over 600 times faster than the previously leading polynomial solver, while maintaining comparable or improved accuracy. These results are a testament to the effectiveness of the linear approach, especially in handling the complex dynamics captured by event cameras.

Implications and Future Directions

The successful development of a linear solver for motion and line estimation using event cameras marks a significant advance in the field. Theoretically, it offers insights into the geometric conditions underpinning the interaction between event data and motion estimation, paving the way for further exploration of geometric models in event-based vision systems. Practically, the efficiency and stability improvements open up new possibilities for real-time applications in robotics and augmented/virtual reality systems, where rapid and accurate motion estimation is crucial.

Looking forward, the conversation around incorporating uncertainties in the velocity measurements and extending the method to work asynchronously over time suggests a roadmap for continued innovation. The potential for integrating this approach with traditional frame-based systems and IMU data also points toward a hybrid future, where the strengths of different sensing modalities are combined for unprecedented motion estimation capabilities in challenging environments.