N-Dimensional Hedron Interpolation for Color Conversion
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Solution Overview
Problem
Existing methods for color conversion between different color spaces, such as RGB to CMYK, are inefficient due to slow computation times, especially in multi-dimensional color tables, as they require calculating multiple terms using linear interpolation.
Innovation Solution
A method to quickly calculate the hedron that encloses the input point in an n-dimensional color lookup table by determining the addresses of its vertices, allowing for faster interpolation without conditional branching, which is particularly effective for large n dimensions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If linear interpolation method is used with all 2^n adjacent lattice points, then the output value calculation is accurate, but the computation time is excessively long
Solution Approach 1:
The n-dimensional cube is segmented into n! subvolumes called hedra, each with n+1 vertices. This segmentation allows the interpolation to be performed on a smaller subset of vertices (n+1 instead of 2n), significantly reducing computation time while maintaining accuracy within each hedral region.
Solution Approach 2:
The method uses local linear approximation within each hedral subvolume. By determining which hedral contains the input point and performing interpolation only on the n+1 vertices of that specific hedral, the solution achieves accurate local approximation without the computational burden of global interpolation across all 2n lattice points.
2Productivity
If hedral methods are used to reduce computation to n+1 vertices, then the computation speed improves, but the complexity of determining the enclosing hedral increases
Solution Approach 1:
The patent transforms the hedral determination problem from a geometric complexity issue into a systematic coordinate-based calculation. By using the formula that computes vertex addresses directly from the input coordinates and sorted channel values, the method avoids complex geometric queries and conditional branching, reducing the determination process to straightforward arithmetic operations.
Solution Approach 2:
The method pre-sorts the input channel values to determine the permutation that defines the enclosing hedral. This preliminary sorting action establishes the vertex addresses in a systematic order, allowing the interpolation to proceed efficiently without repeated conditional checks during the actual interpolation calculation.
3Measurement precision
If conventional linear interpolation is used, then all 2^n lattice points are utilized for accurate interpolation, but the method becomes impractical for high-dimensional color spaces
Solution Approach 1:
The n-dimensional space is segmented into manageable hedral subvolumes, each requiring interpolation from only n+1 vertices. This segmentation makes high-dimensional interpolation practical by reducing the exponential complexity (2^n) to linear complexity (n+1) per hedral, enabling application to color spaces with 4 or more dimensions.
Solution Approach 2:
The method changes the interpolation parameter from using all 2^n lattice points to using only the n+1 vertices of the enclosing hedral. This parameter change maintains interpolation accuracy within each hedral while making the method scalable to high dimensions where conventional methods become computationally infeasible.
Data Source
AI summary
A method, apparatus, and computer program product implementing techniques to calculate the hedron that contains an input point. An input color value representing a color having n color components is received. A color lookup table organized as an n-dimensional lattice is received, where n is the number of input channels in the color lookup table, each entry in the color lookup table being a vertex in the lattice, the color lookup table having 2n adjacent lattice points that form an n-dimensional cube around the input value, and the color lookup table defining an output color value at each lattice point. n+1 vertices from the 2n vertices of the n-dimensional cube are calculated, the n+1 vertices defining an enclosing hedron that encloses the input value, where the instructions to calculate n+1 vertices perform no branching operations. The enclosing hedron is used to calculate an output value from the input value.