Generating Synthetic Ground Truth Distributions for Multi-step Trajectory Prediction using Probabilistic Composite Bézier Curves

TL;DR

The paper tackles the lack of ground-truth distribution data for multi-step trajectory prediction by introducing composite probabilistic Bézier curves, or N-Curves, which generate full-trajectory distributions and permit posterior conditioning via Gaussian-process equivalence. It defines a multi-path dataset as a mixture of N-Curves, derives the prior as a Gaussian mixture with a structured mean and covariance, and computes posteriors by conditioning on observed trajectory segments. An exemplary evaluation demonstrates training a multi-modal predictor (RED) on synthetic data and assessing performance with the Wasserstein distance in addition to NLL, arguing for the greater interpretability and variance-awareness of distributional metrics. The work enables more expressive benchmarking and posterior analysis for probabilistic trajectory predictors, at the cost of additional computational complexity in evaluating distributional distances.

Abstract

An appropriate data basis grants one of the most important aspects for training and evaluating probabilistic trajectory prediction models based on neural networks. In this regard, a common shortcoming of current benchmark datasets is their limitation to sets of sample trajectories and a lack of actual ground truth distributions, which prevents the use of more expressive error metrics, such as the Wasserstein distance for model evaluation. Towards this end, this paper proposes a novel approach to synthetic dataset generation based on composite probabilistic Bézier curves, which is capable of generating ground truth data in terms of probability distributions over full trajectories. This allows the calculation of arbitrary posterior distributions. The paper showcases an exemplary trajectory prediction model evaluation using generated ground truth distribution data.

Figures (6)

• Figure 1: Exemplary composite $\mathcal{N}$-Curve consisting of $N_{seg} = 3$ segments with control point sets $\mathcal{P} = \{\mathcal{P}_1, \mathcal{P}_2, \mathcal{P}_3\}$, where $\mathcal{P}_1 = \{P^1_0, P^1_1, P^1_2\}$, $\mathcal{P}_2 = \{P^2_0, P^2_1, P^2_2, P^2_3\}$ and $\mathcal{P}_3 = \{P^3_0, P^3_1, P^3_2\}$, with $L_1 = L_3 = 2$ and $L_2 = 3$. Each control point $P^j_l \sim \mathcal{N}(\cdot|\cdot)$ follows a Gaussian distribution with respective mean $\mu^j_l$ and covariance matrix $\Sigma^j_l$. Left: The resulting mean curve with control point locations. Covariance ellipses are omitted for clarity. Right: Gaussian curve points along the composite $\mathcal{N}$-Curve at curve positions $t_1 = 0.15$, $t_2 = 0.55$ and $t_3 = 0.8$. The influence of control points on each curve point is indicated by solid lines.
• Figure 2: Example for a $C^0$ (left) and a $C^1$ (right) continuous curve.
• Figure 3: Example of a mean vector and covariance matrix derived from an $\mathcal{N}$-Curve covering $N=4$ Gaussian curve points.
• Figure 4: Posterior distributions given by conditioning on different subsets of the same trajectory. Fig. \ref{['fig:dataset']} depicts the underlying prior distribution.
• Figure 5: A dataset's prior distribution in terms of a Gaussian mixture covering full trajectories (left) and samples drawn from the prior (right).