Probabilistic Point-to-Curve Proximity Under Geometric Uncertainty
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Solution Overview
Problem
Current proximity detection methods fail to accurately measure the distance between uncertain geometric entities such as points and lines or curves in autonomous applications, as they do not effectively consider noise statistics, leading to inaccurate results.
Innovation Solution
The implementation of probabilistic computing techniques to determine probabilistic proximity by identifying a second point on a polynomial curve, transforming uncertainty associated with the curve into uncertainty at that point, and calculating the proximity using statistical distances like Mahalanobis or Bhattacharyya distances.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional proximity detection methods are used to measure distance between geometric entities, then the computational process is simple, but the measurement precision is poor because noise statistics are not considered
Solution Approach 1:
The patent transforms the proximity measurement problem by changing the parameter representation from deterministic distance to probabilistic distance. Instead of computing a single distance value, the system computes statistical parameters (mean and covariance) that characterize the uncertainty in proximity measurements. This allows the system to account for noise statistics while maintaining computational tractability through analytical solutions rather than exhaustive simulations.
Solution Approach 2:
The patent replaces traditional geometric computation methods with probabilistic computing techniques. Instead of using deterministic geometric algorithms to measure distance between points, lines, and curves, the system employs statistical methods that model measurement uncertainties and noise characteristics. This substitution enables more accurate proximity detection by incorporating noise statistics into the computational framework.
2Measurement precision
If probabilistic computing techniques are implemented to consider noise statistics, then the measurement precision improves, but the computational complexity increases
Solution Approach 1:
The patent segments the complex probabilistic computation into distinct computational stages: (1) computing the mean proximity value, (2) computing the covariance matrix representing uncertainty, and (3) combining these to obtain the final probabilistic distance. This segmentation allows each stage to be handled with appropriate mathematical tools and simplifies the overall computational process while maintaining accuracy.
Solution Approach 2:
The patent performs preliminary computations of statistical parameters (mean and covariance of geometric entities) before executing the main proximity detection algorithm. By pre-computing these statistical characteristics of points, lines, and curves, the system avoids redundant calculations during the actual proximity measurement, thereby reducing overall computational complexity while preserving measurement precision.
3Measurement precision
If uncertainty transformation is performed from polynomial curve to point, then the probabilistic proximity can be calculated, but additional computational steps are required
Solution Approach 1:
The patent introduces an intermediary computational step that transforms the uncertainty representation from the polynomial curve domain to the point domain. This intermediary transformation uses the Jacobian matrix to map covariance information, enabling the system to leverage standard point-to-point distance calculations while preserving the uncertainty information from the curve representation. This approach avoids the need for complex direct curve-to-point probabilistic calculations.
Data Source
AI summary
A method includes identifying, using at least one processor, a first point associated with an uncertain location of an object in a space and a polynomial curve associated with an uncertain location of a feature in the space. The method also includes determining, using the at least one processor, a probabilistic proximity of the object and the feature. The probabilistic proximity is determined by identifying a second point on the polynomial curve, transforming an uncertainty associated with the polynomial curve into an uncertainty associated with the second point, and identifying the probabilistic proximity of the object and the feature using the first and second points and the uncertainty associated with the second point.