View source for Biology:Grassmann's laws (color science)

{{Short description|Perception of color mixtures}}
'''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 [[Physics:Spectral power distribution|spectral power distribution]]s can be algebraically related to one another in a color matching context. Discovered by [[Biography:Hermann Grassmann|Hermann Grassmann]]<ref>{{cite journal|last1=Grassmann|first1=H.|title=Zur Theorie der Farbenmischung|journal=Annalen der Physik und Chemie|date=1853|volume=165|issue=5|pages=69–84|doi=10.1002/andp.18531650505|bibcode=1853AnP...165...69G|url=https://zenodo.org/record/1423628}}</ref> these "laws" are actually principles used to predict color match responses to a good approximation under [[Biology:Photopic vision|photopic]] and [[Biology:Mesopic vision|mesopic]] vision.  A number of studies have examined how and why they provide poor predictions under specific conditions.<ref>{{cite journal|last1=Pokorny|first1=Joel|last2=Smith|first2=Vivianne C.|last3=Xu|first3=Jun|title=Quantal and non-quantal color matches: failure of Grassmann's laws at short wavelengths|journal=Journal of the Optical Society of America A|date=1 February 2012|volume=29|issue=2|pages=A324-36|doi=10.1364/JOSAA.29.00A324|pmid=22330396|bibcode=2012JOSAA..29A.324P}}</ref><ref>{{cite book|last1=Brill|first1=Michael H.|last2=Robertson|first2=Alan R.|chapter=Open Problems on the Validity of Grassmann's Laws|title=Colorimetry: Understanding the CIE System|date=2007|pages=245–259|doi=10.1002/9780470175637.ch10|publisher=John Wiley & Sons, Inc.|language=en|isbn=978-0-470-17563-7}}</ref>

==Modern interpretation==

[[File:Grassman color circle 1853.png|right|thumb|Grassmann expressed his first law with respect to a circular arrangement of spectral colors in this 1853 illustration.<ref>{{cite book | title = Hermann Günther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar: papers from a sesquicentennial conference
 |author=Hermann Grassmann |author2=Gert Schubring
 | publisher = Springer
 | year = 1996
 | isbn = 978-0-7923-4261-8
 | page = 78
 | url = https://books.google.com/books?id=PUifpzKWhLkC&q=grassman%27s+second+law+color+linear
 }}</ref>]]

The four laws are described in modern texts<ref>{{cite web|last1=Stevenson|first1=Scott|title=University of Houston Vision OPTO 5320 Vision Science 1 Lecture Notes|url=https://www.opt.uh.edu/onlinecoursematerials/stevenson-5320/vis_sci_02.pdf|website=University of Houston Vision OPTO 5320 Vision Science 1 Course Materials|access-date=4 January 2018|archive-date=5 January 2018|archive-url=https://web.archive.org/web/20180105180347/https://www.opt.uh.edu/onlinecoursematerials/stevenson-5320/vis_sci_02.pdf}}</ref> with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources<ref>{{cite book|last1=Judd|first1=Deane Brewster|last2=Technology|first2=Center for Building|title=Contributions to Color Science|date=1979|publisher=NBS|page=457|url=https://books.google.com/books?id=jgz_iI8NAzYC&q=%22Grassmann%27s+laws%22&pg=PA457|access-date=6 January 2018|language=en}}</ref>):
----
{|
|valign="top"|{{nowrap|'''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.
|-
|valign="top"|{{nowrap|'''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.
|-
|valign="top"|{{nowrap|'''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.
|-
|valign="top"|{{nowrap|'''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.
|}
----

These laws entail an algebraic representation of colored light.<ref>{{cite book|last1=Reinhard|first1=Erik|last2=Khan|first2=Erum Arif|last3=Akyuz|first3=Ahmet Oguz|last4=Johnson|first4=Garrett|title=Color Imaging: Fundamentals and Applications|date=2008|publisher=CRC Press|isbn=978-1-4398-6520-0|page=364|url=https://books.google.com/books?id=suLqBgAAQBAJ&q=grassman%27s+laws+additive&pg=PA364|language=en}}</ref> Assuming beam 1 and 2 each have a color, and the observer chooses <math>(R_1,G_1,B_1)</math> as the strengths of the primaries that match beam 1 and <math>(R_2,G_2,B_2)</math> 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 <math>(R,G,B)</math>, where:

:<math>R= R_1+R_2\,</math>
:<math>G= G_1+G_2\,</math>
:<math>B= B_1+B_2\,</math>

Grassmann's laws can be expressed in general form by stating that for a given color with a [[Physics:Spectral power distribution|spectral power distribution]] <math>I(\lambda)</math> the RGB coordinates are given by:

:<math>R= \int_0^\infty I(\lambda)\,\bar r(\lambda)\,d\lambda</math>
:<math>G= \int_0^\infty I(\lambda)\,\bar g(\lambda)\,d\lambda</math>
:<math>B= \int_0^\infty I(\lambda)\,\bar b(\lambda)\,d\lambda</math>

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


== See also ==

* [[Color space]]
* [[CIE 1931 color space]]

== References ==

{{reflist}}

[[Category:Visual system]]

{{Sourceattribution|Grassmann's laws (color science)}}