Efficient Computation of Geodesics in Color Space

Although scientists agree that a perceptual color space is not Euclidean and color difference measures, such as CIELAB's <inline-formula><tex-math notation="LaTeX">$\Delta E_{2000}$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mn>2000</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="bujack-ieq1-3346673.gif"/></alternatives></inline-formula>, model these aspects of color perception, colormaps are still mostly evaluated through piecewise linear interpolation in a Euclidean color space. In a non-Euclidean setting, the piecewise linear interpolation of a colormap through control points translates to finding shortest paths. Alternatively, a smooth interpolation can be generalized to finding the straightest path. Both approaches are difficult to solve and are compute intensive. We compare the 11 most promising optimization algorithms for the computation of a geodesic either as the shortest or as the straightest path to find the most efficient one to use for colormap interpolation in real-world applications. For two control points, the zero curvature algorithms excelled, especially the 2D relaxation method. For multiple control points, only the mimimal curvature algorithms can produce smooth curves, amongst which the 1D relaxation method performed best.