Hierarchical MK Splines Scattered Data Interpolation
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Solution Overview
Problem
Existing scattered data interpolation methods, such as Lagrange's polynomial interpolation, suffer from oscillation issues at the edges when dealing with high-degree polynomials, and there is a need for a method that minimizes approximation error and processing time while smoothly transmitting information across the domain.
Innovation Solution
The implementation of a hierarchical many-knot (MK) splines scheme, which uses a hierarchy of control lattices to generate a sequence of approximation functions that refine the initial estimate, effectively interpolating scattered data sets by summing these functions to construct a smooth and accurate interpolated surface.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Lagrange's polynomial interpolation with high-degree polynomials is used, then interpolation capability is improved, but oscillation occurs at the edges of the interval
Solution Approach 1:
The patent divides the high-degree polynomial interpolation into multiple low-degree polynomial segments. Each segment operates on a local subset of data points rather than the entire dataset, which eliminates the edge oscillation problem while maintaining interpolation accuracy through local approximation.
Solution Approach 2:
The patent applies different polynomial degrees to different regions of the data. By using local low-degree polynomials instead of a global high-degree polynomial, the method achieves stable local approximation without the harmful oscillations that occur at interval edges in global polynomial interpolation.
2Ease of operation
If explicit interpolation methods are used, then computational simplicity is improved, but approximation error increases for large-scale scattered data
Solution Approach 1:
The patent segments the large-scale scattered data into multiple local groups, applying simple explicit interpolation methods to each segment. This maintains computational simplicity while reducing approximation error by focusing on local patterns rather than attempting global interpolation with complex methods.
3Device complexity
If traditional interpolation methods are used for large-scale scattered data, then implementation is simplified, but processing time increases
Solution Approach 1:
The patent divides large-scale scattered data into multiple smaller subsets that can be processed independently and in parallel. This segmentation reduces the computational burden on each processing unit, enabling faster overall processing while maintaining implementation simplicity through modular architecture.
Solution Approach 2:
The patent applies interpolation methods to partial subsets of data rather than processing the entire dataset sequentially. By dividing the work into multiple partial tasks that can be executed concurrently, the method achieves significant speedup while keeping each individual task simple to implement.
Data Source
AI summary
A computer system executes hierarchical MK splines for scattered data approximation. The computer system constructs an image by generating a final approximation function that approximates a scattered data set with a hierarchy of control lattice from coarsest to densest. The method includes executing MK splines approximation to the scattered data set with the hierarchy of control lattice to iteratively obtain approximation functions. The image is constructed by generating the final approximation function to construct an interpolated surface.