Pseudo-random Interval Bisection for Graphics Error Bounds
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Solution Overview
Problem
Current interval analysis methods in computer graphics face challenges such as constant bounded error due to midpoint bisection and the 'curse of dimensionality,' which lead to undesirable results and increased computation time, especially when dealing with high-dimensional problems.
Innovation Solution
The introduction of pseudo-random interval bisection and undersampling techniques, where intervals are subdivided irregularly and selectively sampled, respectively, to vary the error bound and reduce computation time while maintaining rigorous interval computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If midpoint bisection is used in interval analysis, then the interval domain is systematically divided, but constant bounded error is introduced leading to undesirable results
Solution Approach 1:
The patent applies asymmetry by replacing the symmetric midpoint bisection with asymmetric pseudo-random bisection. Instead of always dividing at the midpoint, the method uses pseudo-random points to divide intervals, creating asymmetric subdivisions that vary with each iteration. This transforms the constant bounded error from symmetric midpoint bisection into bounded noise through asymmetric division, eliminating the harmful constant error while maintaining rigorous interval computations.
2Reliability
If traditional interval analysis methods are used, then rigorous error bounds are computed, but computation time increases significantly for high-dimensional problems
Solution Approach 1:
The patent applies dynamics by transforming the static, deterministic midpoint bisection process into a dynamic pseudo-random bisection process. The division points are no longer fixed but vary dynamically based on pseudo-random numbers generated at each iteration. This dynamic approach allows the algorithm to adaptively sample the interval domain, reducing the number of iterations needed and thereby decreasing computation time for high-dimensional problems while maintaining rigorous error bounds through the deterministic pseudo-random number generation.
Solution Approach 2:
The patent applies parameter changes by modifying the bisection parameter from a fixed midpoint value to a variable pseudo-random value. Instead of using a constant division point, the method changes the division parameter to a pseudo-random number within the interval, which varies with each iteration. This parameter change enables more efficient exploration of the interval domain, reducing computation time while preserving the rigorous error bound properties through controlled randomness.
3Stability of the object's composition
If regular sampling patterns are used in interval analysis, then systematic coverage is achieved, but aliasing issues occur in the rendered image
Solution Approach 1:
The patent applies asymmetry to eliminate aliasing by replacing symmetric regular sampling patterns with asymmetric pseudo-random sampling. Instead of systematically dividing intervals at midpoints creating regular patterns, the method uses asymmetric pseudo-random division points that create irregular sampling patterns. This asymmetric approach breaks the periodicity that causes aliasing, while the pseudo-random nature maintains systematic coverage through deterministic reproduction, effectively eliminating aliasing artifacts in rendered images.
Data Source
AI summary
An improved branch-and-bound process of interval arithmetic subdivision in furtherance of computation of rigorous error bounds on integrated digital scene information for two dimensional display is provided. More particularly, a first aspect of the subject process includes pseudo-randomly subdividing an interval domain comprising a set of interval variables in furtherance of ascertaining a characteristic contribution of the interval variables of said set of interval variables to an image space comprising at least a sub-pixel area. A further aspect, either alone or in combination with the first aspect contemplates pseudo-randomly discarding a select partitioning of interval variables of a set of interval variables of a geometric function from a computed solution of an interval arithmetic branch-and-bound process.