3D Voronoi Diagram Computation for Complex Generators
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Solution Overview
Problem
Existing methods struggle to automatically construct accurate Voronoi diagrams for complex generators and higher dimensional spaces, particularly in 3D environments, due to relaxed definitions and issues with handling intersections of generators, which invalidate the diagrams.
Innovation Solution
Two novel methods are introduced for deriving Voronoi diagrams: one involving a mathematical game to map input data to generators and another for computing equations defining the diagram, allowing for accurate partitioning and handling of complex generators and intersections.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If traditional Voronoi diagram construction methods are used for complex generators in 3D, then the construction process becomes intractable, but the definition must be relaxed to accommodate curved faces and complex generators
Solution Approach 1:
The patent segments complex 3D generators into simpler components (line segments, curves, surfaces) and constructs Voronoi diagrams hierarchically. By dividing the complex construction problem into manageable sub-problems for each generator type, the method maintains diagram validity while achieving adaptability to complex generators.
Solution Approach 2:
The patent extends Voronoi diagram construction from 2D to 3D space, introducing a new dimension to handle complex generators. This dimensional extension allows the method to accommodate curved faces and three-dimensional generator types while preserving the fundamental Voronoi property that each cell contains points closer to its generator than to any other generator.
2Ease of manufacture
If previous methods approximate Voronoi diagrams for line segments, then construction is simplified, but the generated cells do not meet the requirements for valid Voronoi cells
Solution Approach 1:
The patent performs preliminary classification and identification of generator types before constructing Voronoi cells. By pre-categorizing generators into specific types (point, line segment, curve, surface) and selecting appropriate construction methods for each type, the method ensures both ease of construction and accuracy of the resulting Voronoi cells.
Solution Approach 2:
The patent changes the parameters and properties of Voronoi cell construction based on the type of generator being processed. Different distance metrics, cell topology rules, and boundary conditions are applied according to the generator type, ensuring that each cell accurately represents the region of points closer to its generator while maintaining construction simplicity.
3Adaptability or versatility
If generators are allowed to be associated with multiple Voronoi cells, then construction flexibility increases, but the fundamental definition of Voronoi diagrams is violated
Solution Approach 1:
The patent applies different rules for generator-cell associations based on the local context and generator type. Each generator type (point, line, curve, surface) has specific local quality requirements for its Voronoi cell association. This localized approach maintains construction flexibility while ensuring that the fundamental Voronoi property of unique closest generator association is preserved in each local region.
Data Source
AI summary
Two methods are provided for the automated derivation of Voronoi diagrams in 3D. The invention, implementable via various means such as a processing system, method, or data structure in a recording medium such as memory or as a self-contained electronic circuit, has wide ranging applicability to numerous fields such as big data analysis, computer graphics and animation, route planning, collision avoidance, computer vision, robotic vision, and etc. The first method of the invention details steps necessary to segment data according to a set of generators so as to produce a Voronoi partitioning of the data. The second method of the invention includes steps associated with the derivation of the mathematical specification of a 3D Voronoi diagram.