Biology: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
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]):
| 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. |
These laws entail an algebraic representation of colored light.[7] Assuming beam 1 and 2 each have a color, and the observer chooses [math]\displaystyle{ (R_1,G_1,B_1) }[/math] as the strengths of the primaries that match beam 1 and [math]\displaystyle{ (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]\displaystyle{ (R,G,B) }[/math], where:
- [math]\displaystyle{ R= R_1+R_2\, }[/math]
- [math]\displaystyle{ G= G_1+G_2\, }[/math]
- [math]\displaystyle{ B= B_1+B_2\, }[/math]
Grassmann's laws can be expressed in general form by stating that for a given color with a spectral power distribution [math]\displaystyle{ I(\lambda) }[/math] the RGB coordinates are given by:
- [math]\displaystyle{ R= \int_0^\infty I(\lambda)\,\bar r(\lambda)\,d\lambda }[/math]
- [math]\displaystyle{ G= \int_0^\infty I(\lambda)\,\bar g(\lambda)\,d\lambda }[/math]
- [math]\displaystyle{ B= \int_0^\infty I(\lambda)\,\bar b(\lambda)\,d\lambda }[/math]
Observe that these are linear in [math]\displaystyle{ I }[/math]; the functions [math]\displaystyle{ \bar r(\lambda), \bar g(\lambda), \bar b(\lambda) }[/math] are the color matching functions with respect to the chosen primaries.
See also
References
- ↑ Grassmann, H. (1853). "Zur Theorie der Farbenmischung". Annalen der Physik und Chemie 165 (5): 69–84. doi:10.1002/andp.18531650505. Bibcode: 1853AnP...165...69G. https://zenodo.org/record/1423628.
- ↑ 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. doi:10.1364/JOSAA.29.00A324. PMID 22330396. Bibcode: 2012JOSAA..29A.324P.
- ↑ Brill, Michael H.; Robertson, Alan R. (2007). "Open Problems on the Validity of Grassmann's Laws" (in en). Colorimetry: Understanding the CIE System. John Wiley & Sons, Inc.. pp. 245–259. doi:10.1002/9780470175637.ch10. ISBN 978-0-470-17563-7.
- ↑ 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. https://books.google.com/books?id=PUifpzKWhLkC&q=grassman%27s+second+law+color+linear.
- ↑ Stevenson, Scott. "University of Houston Vision OPTO 5320 Vision Science 1 Lecture Notes". https://www.opt.uh.edu/onlinecoursematerials/stevenson-5320/vis_sci_02.pdf.
- ↑ Judd, Deane Brewster; Technology, Center for Building (1979) (in en). Contributions to Color Science. NBS. p. 457. https://books.google.com/books?id=jgz_iI8NAzYC&q=%22Grassmann%27s+laws%22&pg=PA457. Retrieved 6 January 2018.
- ↑ Reinhard, Erik; Khan, Erum Arif; Akyuz, Ahmet Oguz; Johnson, Garrett (2008) (in en). Color Imaging: Fundamentals and Applications. CRC Press. p. 364. ISBN 978-1-4398-6520-0. https://books.google.com/books?id=suLqBgAAQBAJ&q=grassman%27s+laws+additive&pg=PA364.
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