# Performing color space conversions with three-dimensional linear interpolation

## Abstract

Three-dimensional interpolation is suitable for many kinds of color space transformations. We examine and analyze several linear interpolation schemes - some standard, some known, and one novel. An interpolation algorithm design is divided into three parts: packing (filling the space of the input variable with sample points), extraction (selecting from the constellation of sample points those appropriate to the interpolation of a specific input point), and calculation (using the extracted values and the input point to determine the interpolated approximation to the output point). We focus on regular (periodic) packing schemes. Seven principles govern the design of linear interpolation algorithms: (1) Each sample point should be used as a vertex of as many polyhedra as possible; (2) the polyhedra should completely fill the space; (3) polyhedra that share any part of a face must share the entire face; (4) the polyhedra used should have the fewest vertices possible; (5) polyhedra should be small; (6) in the absence of information about curvature anisotropy, polyhedra should be close to regular in shape; and (7) polyhedra should be of similar size. A test for interpolation algorithm performance in performing actual color space conversion is described, and results are given for an example color space conversion using several linear interpolation methods. The extractions from cubic, body-centered-cubic, and face-centered-cubic lattices are described and analyzed. The results confirm Kanamori's claims for the accuracy of PRISM interpolation; it comes close to the accuracy of trilinear interpolation with roughly three-quarters the computations. The results show that tetrahedral interpolation, with close to half the computational cost of trilinear interpolation, is capable of providing better accuracy. Of the tetrahedral interpolation techniques, one diagonal extraction from cubic packing is useful as a general-purpose color space interpolator. The disphenoid extraction from body-centered cubic packing provides the most accurate approximations of all the methods tested.