Mathematical Discontinuities in CIEDE2000 Color Difference Computations

We examine mathematical properties of the CIEDE2000 color difference formula and illustrate that the CIEDE2000 color difference is not a continuous function of the CIELAB color pairs between which the difference is calculated. Particularly, we illustrate that ambiguities in the computation of mean hue angle and hue angle differences of color pairs contribute to a discontinuity in the formula for color pairs chosen 180-degrees apart in CIELAB hue. We analyze and characterize the discontinuity and give a visual presentation that allows better appreciation of its nature. For color pairs under 5 CIELAB delta E units apart, the maximum discontinuity magnitude is bounded by 0.274. For color pairs further apart in CIELAB, the magnitude of the discontinuity rises sharply. The results indicate that care should be exercised in using the formula in Taylor series approximations and that its use should be restricted to small color differences as recommended by the CIEDE2000 standard.