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^[1] 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.^[2][3]

Modern interpretation

Grassmann expressed his first law with respect to a circular arrangement of spectral colors in this 1853 illustration.^[4]

The four laws are described in modern texts^[5] with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources^[6]):

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First law:	Two colored lights appear different if they differ in either dominant wavelength, luminance or purity. Corollary: For every colored light there exists a light with a complementary color such that a mixture of both lights either desaturates the more intense component or gives uncolored (grey/white) light.
Second law:	The appearance of a mixture of light made from two components changes if either component changes. Corollary: A mixture of two colored lights that are non-complementary result in a mixture that varies in hue with relative intensities of each light and in saturation according to the distance between the hues of each light.
Third law:	There are lights with different spectral power distributions but appear identical. First corollary: such identical appearing lights must have identical effects when added to a mixture of light. Second corollary: such identical appearing lights must have identical effects when subtracted (i.e., filtered) from a mixture of light.
Fourth law:	The intensity of a mixture of lights is the sum of the intensities of the components. This is also known as Abney's law.

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These laws entail an algebraic representation of colored light.^[7] Assuming beam 1 and 2 each have a color, and the observer chooses ${\displaystyle (R_{1},G_{1},B_{1})}$ as the strengths of the primaries that match beam 1 and ${\displaystyle (R_{2},G_{2},B_{2})}$ 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 ${\displaystyle (R,G,B)}$, where:

${\displaystyle R=R_{1}+R_{2}\,}$

${\displaystyle G=G_{1}+G_{2}\,}$

${\displaystyle B=B_{1}+B_{2}\,}$

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

${\displaystyle R=\int _{0}^{\infty }I(\lambda )\,{\bar {r}}(\lambda )\,d\lambda }$

${\displaystyle G=\int _{0}^{\infty }I(\lambda )\,{\bar {g}}(\lambda )\,d\lambda }$

${\displaystyle B=\int _{0}^{\infty }I(\lambda )\,{\bar {b}}(\lambda )\,d\lambda }$

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

See also

• Color space
• CIE 1931 color space

References

1. Grassmann, H. (1853). "Zur Theorie der Farbenmischung". Annalen der Physik und Chemie. 165 (5): 69–84. Bibcode:1853AnP...165...69G. doi:10.1002/andp.18531650505.
2. Pokorny, Joel; Smith, Vivianne C.; Xu, Jun (1 February 2012). "Quantal and non-quantal color matches: failure of Grassmann's laws at short wavelengths". Journal of the Optical Society of America A. 29 (2): A324-36. Bibcode:2012JOSAA..29A.324P. doi:10.1364/JOSAA.29.00A324. PMID 22330396.
3. Brill, Michael H.; Robertson, Alan R. (2007). "Open Problems on the Validity of Grassmann's Laws". Colorimetry: Understanding the CIE System. John Wiley & Sons, Inc. pp. 245–259. doi:10.1002/9780470175637.ch10. ISBN 978-0-470-17563-7.
4. Hermann Grassmann; Gert Schubring (1996). Hermann Günther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar: papers from a sesquicentennial conference. Springer. p. 78. ISBN 978-0-7923-4261-8.
5. Stevenson, Scott. "University of Houston Vision OPTO 5320 Vision Science 1 Lecture Notes" (PDF). University of Houston Vision OPTO 5320 Vision Science 1 Course Materials. Archived from the original (PDF) on 5 January 2018. Retrieved 4 January 2018.
6. Judd, Deane Brewster; Technology, Center for Building (1979). Contributions to Color Science. NBS. p. 457. Retrieved 6 January 2018.
7. Reinhard, Erik; Khan, Erum Arif; Akyuz, Ahmet Oguz; Johnson, Garrett (2008). Color Imaging: Fundamentals and Applications. CRC Press. p. 364. ISBN 978-1-4398-6520-0.