A Geometric Gaussian Mixture Representation of Plane Curves

2026-06-08 · Source: cs.CV updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

This paper introduces a Geometric Gaussian Mixture Model (GMM) representation for plane curves, derived from a user-defined probabilistic polygonal approximation. The method selects vertices on a curve, connects them with line segments, and assigns a user-defined uncertainty (standard deviation) in the normal direction to each segment. This forms probabilistic geometric primitives that preserve local tangent, normal, and arc length. Each segment is then associated with a Random Variable, uniformly distributed tangentially and Gaussian distributed normally. Moment matching induces a Gaussian component with its mean at the segment midpoint and covariance encoding both tangential and normal uncertainty. These components are combined with weights proportional to segment length to form the GMM. The framework is analytically tractable, preserving local position, orientation, length scale, and normal uncertainty. It applies to diverse curve types, supporting adaptive discretization and varying uncertainty, with experiments confirming accurate shape capture for applications in uncertainty-aware CAD, digital twins, and probabilistic robotics.

Key takeaway

For Robotics Engineers developing autonomous systems, this GMM representation offers a robust method to incorporate tunable uncertainty into geometric models. You can now represent obstacle boundaries with explicit safety margins or sensor noise, enabling more reliable collision probability queries and risk-aware trajectory optimization. This analytically tractable approach is efficient enough for embedded platforms, enhancing real-time decision-making.

Key insights

A GMM representation of plane curves is derived from probabilistic polygonal approximations, encoding local uncertainty.

Principles

• Polygonal segments can encode normal direction uncertainty.
• Gaussian component parameters are derived in closed form.
• GMMs preserve local geometric properties and uncertainty.

Method

Discretize a plane curve into polygonal segments, each assigned a normal uncertainty τ₋. Define a Random Variable (uniform tangentially, Gaussian normally) per segment. Moment matching yields Gaussian components, combined with length-proportional weights into a GMM.

In practice

• Encode manufacturing tolerances in CAD models.
• Model probabilistic obstacle boundaries in robotics.
• Furnish compact priors for Bayesian filtering tasks.

Topics

• Geometric Gaussian Mixture Model
• Plane Curves
• Probabilistic Polygonal Representation
• Uncertainty Quantification
• Robotics
• Computer-Aided Design

Best for: AI Scientist, Research Scientist, Robotics Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.CV updates on arXiv.org.