Reliability-based G1 Continuous Arc Spline Approximation

TL;DR

The paper addresses robust arc-spline fitting under data uncertainty by introducing covariance-weighted, $G^{1}$ continuous arc approximations. A single-arc framework combines an anchor model and an arc-measurement model with an equality constraint on the middle node, solved as a CNLS problem, and is extended to a two-phase multi-arc framework with initialization, data association, $G^{1}$ continuity, and Chi-squared validation. Core contributions include the concrete CNLS formulations, phase-wise initialization and validation procedures, and covariance-aware lane-map parameterization validated on real vehicle data, demonstrating improved compactness and robustness over prior methods such as Song2009. The approach yields reliable lane representations by leveraging per-point covariances, enabling efficient and robust data fitting suitable for traffic-lane mapping and related applications; limitations include the need for well-ordered data and accurate covariance estimates, with potential enhancements via 0-norm/MOO techniques.

Abstract

In this paper, we present an algorithm to approximate a set of data points with G1 continuous arcs, using points' covariance data. To the best of our knowledge, previous arc spline approximation approaches assumed that all data points contribute equally (i.e. have the same weights) during the approximation process. However, this assumption may cause serious instability in the algorithm, if the collected data contains outliers. To resolve this issue, a robust method for arc spline approximation is suggested in this work, assuming that the 2D covariance for each data point is given. Starting with the definition of models and parameters for single arc approximation, the framework is extended to multiple-arc approximation for general usage. Then the proposed algorithm is verified using generated noisy data and real-world collected data via vehicle experiment in Sejong City, South Korea.

Figures (19)

• Figure 1: 3 Points are set as parameters(optimization variables) to define a single arc
• Figure 2: Anchor Model for Single Arc Approximation: Arc Nodes ($\mathbf{A}_{1}, \mathbf{A}_{2}$) are matched with first and last data points ($\mathbf{P}_{1}, \mathbf{P}_{n}$) respectively. Middle node is not included in the anchor model.
• Figure 3: Anchor Model for Cases: (a)$\Sigma_{\mathrm{AC}_1}=\left[1001\right]$ (b)$\Sigma_{\mathrm{AC}_2}=\left[10000100\right]$
• Figure 4: Arc Measurement Model for Single Arc Approximation: (1) Arc center $\mathbf{X}_{C}$ is computed geometrically, by using two Arc Nodes and the Middle Node. (2) A virtual point(marked green) $\mathbf{P}_i^{v}$ is defined by finding the intersection of line $\mathbf{P}_{i}\mathbf{X}_{C}$ and arc $\widearc{A_{1}N_{1}A_{2}}$. (3) Matching error (model residual) $\mathbf{r}^{i}_{\mathrm{ME}}$ is computed by subtracting data point vector $\mathbf{P}_{i}$ from virtual point vector $\mathbf{P}^{v}_{i}$. (4) Residual is weighted using the pre-obtained covariance matrix $\Sigma^{i}_{\mathrm{ME}}$ of point $\mathbf{P}_{i}$. (5) Arc measurement model cost is accumulated as $i$ iterates from 1 to $n$, the data size.
• Figure 5: Single Arc Approximation Example 1 (Generated Data Points)