Making the calculation of Logvinenko's coordinates easy

In 2009 Logvinenko introduced a new object colour space that utilises representation of non-luminous surfaces as reflectance spectra of a rectangular shape i.e. a reflectance spectrum takes two values 1+α/2 or 1−α/2 for 0 ≤ α ≤ 1 with two transitions at λ1 and λ2. The calculation of Logvinenko colour coordinates for a large set of reflectance spectra (or tristimulous values) is time consuming, even despite a more recent more efficient algorithm of Godau and Funt. In this paper, we propose two approximate, but fast solutions to finding the Logvinenko coordinates (ADL) that exploit the combinatorial properties of rectangular spectra. The proposed algorithm takes around 0.02s to calculate the ADL coordinates for 1600 surface database as opposed to earlier implementation that reported 90s. Both algorithms are approximate, but the precision should be acceptable for most of the applications as the calculated mean, median, 95 percentile and max ΔE Lab errors were respectively 0.3, 0.2, 0.7 and 1.6.