Barycentric Projection for Color Space Interpolation
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Solution Overview
Problem
Existing color conversion systems face inefficiencies in handling geometrically complex Look-Up Tables (LUTs) due to limitations in interpolation methods, such as Sequential Linear Interpolation (SLI), which fail to preserve symmetries, are not compatible with all geometrical configurations, and struggle with 'gap interpolation' between incompatible grids.
Innovation Solution
The implementation of a 'barycentric projection' method within a unit hypercube, allowing for Sequential Tetrahedral Interpolation (STI), which imposes boundary conditions on facets and vertices to efficiently interpolate values, even in complex geometrical situations, by performing a series of barycentric projections and back substitutions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If Sequential Linear Interpolation (SLI) is used for color conversion, then the interpolation process is simple to implement, but it fails to preserve symmetries and is not compatible with complex geometrical configurations
Solution Approach 1:
The patent segments the color space into multiple simplices (tetrahedra in 3D) that form a triangulated mesh. Each simplex is processed independently using linear interpolation, while the overall structure handles complex geometries. This segmentation allows the method to maintain simplicity within each cell while achieving versatility across the entire color space, resolving the contradiction between ease of implementation and geometrical compatibility.
2Quantity of substance
If traditional interpolation methods are used on adaptive rectangular grids, then storage efficiency is improved by reducing sampling rate, but interpolation accuracy deteriorates in regions with incompatible grids
Solution Approach 1:
The patent transitions from processing data in the original dimensional space to embedding it in a barycentric coordinate system. By representing points as convex combinations of simplex vertices and using barycentric coordinates for interpolation, the method accurately handles points in regions between incompatible grids. This dimensional transformation maintains storage efficiency while recovering interpolation accuracy in previously problematic regions.
3Device complexity
If SLI is applied to reduce dimensionality step-by-step, then computational complexity is reduced, but the method cannot handle gap interpolation between incompatible grids
Solution Approach 1:
The patent introduces barycentric coordinates as an intermediary representation between the input color value and the interpolation result. By expressing the input point in barycentric coordinates relative to the containing simplex and using these coordinates to weight the vertex values, the method seamlessly handles gap regions between incompatible grids. This intermediary approach maintains computational efficiency while enabling versatile handling of all geometrical configurations including gap interpolation.
Data Source
AI summary
A method and apparatus for interpolating values for a color space from an input color value. A unit hypercube enclosing the input value is generated based on values from a look up table. A set of boundary conditions are then imposed on the unit hypercube. To perform the actual interpolation, an initial barycentric projection is performed from a selected vertex of the unit hypercube through the input value onto a boundary of the unit hypercube. If the projection satisfies one of the boundary conditions, an interpolated value is calculated using the projection by back substitution. If the initial projection does not satisfy a boundary condition, an intermediate value is generated from the previous projection and successive barycentric projections are performed using respectively different vertices of the unit hypercube through intermediate values onto a boundary of the unit hypercube until a projection satisfies one of the boundary conditions.