Calculating cone response curves (MAGAZINE)

May 13, 2012
In this article, George Kelly explains how to calculate cone response curves.
Calculating cone response curves is a step that helps understanding the concept of color matching. The first equation below shows how to calculate luminous flux for instance by integrating an SPD over the visible spectrum (380-780 nm) weighted by the V(λ) curve. Intensity, luminance or illuminance can be calculated in the same manner provided SPD(λ) is in the proper units, (cd, nits, lux respectively).

Similarly cone receptor response, for the L, M, and S cones, can be calculated using the cone response curves in place of the V() curve.

In color analysis, we typically measure the SPD of a light source with a spectroradiometer that samples the power of the source at discrete narrow wavelength bands on the order of a few nanometers. The integrals of above trio of equations are then replaced with the equivalent summation in the equations below.
The i subscript denotes the i th sample of both the cone responses and the SPD, while Δλ is the sampling period. We can think of this discrete sampling of the SPD, as an N dimensional vector where N is the number of wavelength samples and the response of each type of cone receptors is simply the inner product of the SPD with the cone response curves.