Grassmann's laws (color science)

Grassmann's laws describe empirical results about how the perception of mixtures of colored lights (i.e., lights that co-stimulate the same area on the retina) composed of different spectral power distributions can be algebraically related to one another in a color matching context. Discovered by Hermann Grassmann these "laws" are actually principles used to predict color match responses to a good approximation under photopic and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions.

Modern interpretation


The four laws are described in modern texts with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources ):

These laws entail an algebraic representation of colored light. Assuming beam 1 and 2 each have a color, and the observer chooses $$(R_1,G_1,B_1)$$ as the strengths of the primaries that match beam 1 and $$(R_2,G_2,B_2)$$ as the strengths of the primaries that match beam 2, then if the two beams were combined, the matching values will be the sums of the components. Precisely, they will be $$(R,G,B)$$, where:


 * $$R= R_1+R_2\,$$
 * $$G= G_1+G_2\,$$
 * $$B= B_1+B_2\,$$

Grassmann's laws can be expressed in general form by stating that for a given color with a spectral power distribution $$I(\lambda)$$ the RGB coordinates are given by:


 * $$R= \int_0^\infty I(\lambda)\,\bar r(\lambda)\,d\lambda$$
 * $$G= \int_0^\infty I(\lambda)\,\bar g(\lambda)\,d\lambda$$
 * $$B= \int_0^\infty I(\lambda)\,\bar b(\lambda)\,d\lambda$$

Observe that these are linear in $$I$$; the functions $$\bar r(\lambda), \bar g(\lambda), \bar b(\lambda)$$ are the color matching functions with respect to the chosen primaries.