Abstract
In Chap. 14, we saw that one can characterize a light source in terms of its spectral intensity with respect to the wavelength I(λ), which is an objective function. We now turn to the question as to the relationship between the objective characteristics of a light source and the subjective perception of the light. We have already identified three attributes of one’s visual perception − hue, saturation, and brightness. The first two are the attributes that, together, are referred to as the color, or alternatively, the chromaticity. The technical term for the third attribute, brightness, is referred to as the luminance, which has the symbol Y. All three are numerical parameters and are therefore objective parameters.
We will learn how to calculate these three parameters from the spectral intensity with respect to the wavelength I(λ). Then we will discuss how this relationship can be understood in terms of the physiological behavior of the visual apparatus, the rods of the retina.
In addition to the elementary book, Light and Color by Overheim and Wagner (op, cit. in Chapter 13), the reader is referred to the advanced texts: T. N. Cornsweet’s Visual Perception (Academic Press, N.Y., 1970) and Y. Le Grand’s Light, Color, and Vision (Dover, N.Y., 1957).
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- 1.
A few decades ago I was privy to a conversation between my violin teacher’s wife and her close friend—the first a pianist, the second a painter. They were discussing the differences between their personal artistic experiences. My pianist friend pointed out that a performance takes place in time, having a beginning and an end in time. An audio recording cannot fully capture the depth of the experience she had. She expressed her envy of the painter. While the pianist will have only with great difficulty a deep pleasure contemplating her past experience, the painter produces a painting in space that is confined only by its spatial boundary, so that she has all time in the world to contemplate and treasure her piece of art.
- 2.
Dealing with variations of spectral intensity on a surface leads to complexity of the excitation of light sensors on an area of the retina; this interesting and very important interaction will not be dealt with in this book. Of course, quite important variations are associated with paintings.
- 3.
In the late 1800s, the mathematician Georg Cantor introduced the concept of different orders of infinity and provided a clearly defined method of comparing these levels. The lowest order of infinity is the number of integers, given the symbol ℵ 0 . The next order of infinities is the set of real numbers, given the symbol \(\mathcal {C}\) . It can be shown that \(\mathcal {C}=10^{\aleph _0}\). As surprising as it may seem, Cantor was able to show using his method of comparing infinities that \(\mathcal {C}\) is also the number of points along a line, a finite line segment, as well as the number of points in an area, as well as in a volume! This infinity is the infinite number of different chromaticities. The number of distinct spectral intensities is the number of ways you can draw a continuous graph along an axis. This number is an even higher infinity than \(\mathcal {C}\) and can be shown to be equal to \(\aleph _0^{\mathcal {C}}\). See the Wikipedia article (1-8-2011): http://en.wikipedia.org/wiki/Georg_Cantor.
- 4.
It should be recognized that any particular individual is limited in their ability to discriminate one color from another. There are Just Noticeable Differences in Color in analogy with Just Noticeable Differences in Frequency Therefore, a particular individual can discriminate among only a finite number of distinct colors.
- 5.
See the interesting article on this complex subject on the website https://lizerbramlaw.com/2012/10/30/color-as-a-trademark/.
- 6.
We will discuss the modern developments in the field of color. The history of the science of color vision started with Isaac Newton, who in the 1600s proposed that there are seven primaries that can be mixed in appropriate proportions to produce any color sensation. The basis of the proposal was Newton’s studies of the decomposition of white light by a prism into its rainbow of colors. He identified the color of an object as an attribute of the response of the eye to various wavelengths of light that are reflected off the object as opposed to the idea that the color “resides” in the object itself. The proposal that there are three primaries is due to Thomas Young (1807). Many other scientists helped develop the basic principles of color mixing—in particular, James C. Maxwell, who provided a theoretical basis for electromagnetic waves, as discussed in Chap. 5. In 1860, Maxwell produced the first, albeit crude, set of color matching functions , which will be discussed in detail in this chapter. For an excellent history of studies of color vision, see Deane Judd in the publication: NATIONAL BUREAU OF STANDARDS: VOL. 55, p. 1313, (1966).
- 7.
The figure was produced using the applet on the following wonderful website: http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/spectrum/metamers_guide.html. The applet enables you to play around with pairs of spectral intensities, each independently and see their respective color patches. You can then produce metamers galore.
- 8.
Frank Preucil, Color Hue and Ink Transfer Their Relation to Perfect Reproduction, TAGA Proceedings, p 102–110 (1953).
- 9.
Wright, William David (1928). “A re-determination of the trichromatic coefficients of the spectral colours.” Transactions of the Optical Society 30: 141–164. Guild, John (1931). “The colorimetric properties of the spectrum.” Philosophical Transactions of the Royal Society of London (Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 230) A230: 149–187.
- 10.
If a given spectral intensity is increased uniformly for all wavelengths by the same factor, it has been found from studies of human subjects that the color coördinates do not change. This observation amounts to saying that color is independent of brightness or, analogously, that pitch is independent of loudness.
- 11.
See Sect. 14.2.3.
- 12.
For those who know the calculus: The sums should actually be integrals. E.g., R=\(\int _{\mbox{400-nm}}^{\mbox{700-nm}} \overline {r}(\lambda ) \text{I}\lambda ) d\lambda \) ≈ Sum [\(\overline {r}(\lambda ) \text{I}(\lambda ) \Delta \lambda \)]. If the spacing is halved, the number of entries will be doubled, but Δλ is reduced to 5-nm to compensate.
- 13.
In the table, I have cut off the range of wavelengths below 400-nm in order to coincide with Chap. 14.
- 14.
S. J. Williamson and H. Z. Cummins, Light and Color, (John Wiley and Sons, New York, 1983)
- 15.
This situation is comparable to a see-saw, wherein one can balance the see-saw with unequal lengths ℓ 1 and ℓ 2 opposite the pivot point, as long as the weights satisfy the condition ℓ 1 w 1 = ℓ 2 w 2.
- 16.
See also Sect. 10.4.
- 17.
The values in this table were obtained by interpolation, using Table 15.2.
- 18.
See Stiles, Walter Stanley & Birch, Jennifer M. (1958), N.P.L. colour matching investigation: final report. Optica Acta 6: 1–26. See also the website: http://cvrl.ioo.ucl.ac.uk/database/text/cmfs/sbrgb2.htm.
- 19.
Source: https://en.wikipedia.org/wiki/Luminous_eflca cy#/media/File:CIE_1931_Luminosity.png. The reader should note that the luminous efficiency is symbolized by either V (λ) or \(\overline y_{\lambda }\). In more extensive studies, the two parameters are different but barely distinguishable.
- 20.
In the case of light, if we double I(λ) we can maintain equal brightness with a source of wavelength λ 0=555-nm merely by doubling Iλ 0). Let us consider sound: Suppose that we have a sound of frequency f and sound level SL. If we double the sound intensity, the SL will increase by 3-dB. To match the change in loudness, we cannot generally double the intensity of the 1000-Hz sound, which would correspond to an increase of the phon level by 3-phons. Loudness appears to be more complex than brightness.
- 21.
We have introduced a number of photopic parameters that characterize a light beam: luminance, relative illuminance, luminous flux, and luminous efficacy. There are many more important parameters that are used by experts who need to characterize lighting very precisely. The reader is forewarned that there are often differences of the terminology for the parameters.
- 22.
For a source of many tables of color matching functions as well as comments about their reliability, see the website of the Color Vision Research Laboratory with the link http://www.cvrl.org.
- 23.
The word is pronounced á-lichnee, with an accent on the “a” and with “ch” pronounced as in the Scottish word for lake, “loch.” The Greek word λυχνo, pronounced “lichno,” means “light.” The term is based upon an ancient Greek word meaning “no” light, since the prefix “a” means “without.” It is understood to have been coined by the great twentieth century theoretical physicist, Erwin Schrödinger.
- 24.
The subject of gamma and its related gamma correction is very complex. As a result it is extremely difficult to find resources that are reliable. Articles abound with contradictory information. For what I consider a very reliable reference I highly recommend Charles Poynton, Video and HDTV [Morgan Kaufmann Publishers and
Elsevier Science, San Francisco, 2003].
- 25.
Tests of some monitors have revealed that their three RGB values do not have the same value of gamma. In this case, the chromaticity will change.
- 26.
The table was produced by using transformation matrices between the CIE table of color matching functions and the RGB coördinates for the sRGB primaries.
- 27.
For example, in the website (1-12-2011): http://www.gizmag.com/sharp-4-primary-color-tvs-enables-trillion-colors/13823/ we read: “By adding yellow to the colors red, green and blue, the televisions are capable of rendering nearly all the colors a human eye can discern.”
- 28.
Each element, e.g. a microscopic particle with two possible orientations, can store one bit of information. Two particles can store 22=4 possible bits of information. Eight particles can store 28 = 256 bits of information; and so on.
- 29.
Adding a fourth color with 8-bits will increase this number by a factor of 256, so that we would have over 4-billion different combinations! In a recent (May, 2010) website of the SHARP Corporation, it was claimed that their monitor would produce trillions of colors. It is incomprehensible to understand how they can arrive at such a number. See SHARP website: http://www.sharpusa.com/AboutSharp/NewsAndEvents/PressReleases/2010/January/2010_01_06_Booth_Overview.aspx.
- 30.
Note that the ellipses are largest in the green region, indicating that the eye does not discriminate changes in chromaticity well in the green. On the other hand, the ellipses are much smaller towards the blue region. We can see this variation in discrimination in the CIE chromaticity diagram of Fig. 15.15.
- 31.
See D. B. Judd and G. Wyszecki (1975), Color in Business, Science and Industry, Wiley Series in Pure and Applied Optics (3rd ed.). New York: Wiley-Interscience. p. 388.
- 32.
J. M. Linhares, et. al, J Optical Society of America, volume 25, p. 2918 (2008).
- 33.
This number is just under twenty times the number of distinguishable pitches of pure tones, which has been found to be about 1400. See Wikipedia (1-7-2011): http://en.wikipedia.org/wiki/Pitch_(music).
- 34.
In Appendix M I show how this set of primaries is close to producing the largest possible gamut of colors.
- 35.
I am grateful to Raymond Soneira for communication on this subject. You are invited to see his extremely informative website (1-27-2011): http://www.displaymate.com/eval.html.
- 36.
For more details, see http://en.wikipedia.org/wiki/Retina and http://webvision.med.utah.edu/sretina.html.
- 37.
See the article by Jeremy Nathans, who first identified the genes: Scientific American, volume 260, pp. 42–49 (1989).
- 38.
The figure is based upon Bowmaker J.K. and Dartnall H.J.A., “Visual pigments of rods and cones in a human retina.” J. Physiol. 298: pp501–511 (1980).
- 39.
If we plot the absorption spectrum as a function of the frequency, we would obtain a peak for the L-cone that has a width in frequency that is about 34% of the frequency at the peak.
- 40.
See a full discussion of this subject in the section “Complex Scenarios of Absorption and Emission” in Chap. 6.
- 41.
In mathematics, we say that r is a monotonically increasing function of n R, and so on.
- 42.
Omitted is the absorption of the macula, which contains the fovea. See the Wikipedia site (1-26-2011): http://en.wikipedia.org/wiki/Macular_degeneration, wherein it is pointed out that while “the macula comprises only 2.1% of the area of the retina …almost half of the visual cortex [in the brain] is devoted to processing macular information.”
- 43.
See the following website for a wonderful resource on color-blindness https://en.wikipedia.org/wiki/Color_blindness. It includes a fascinating set of figures that displays how the rainbow of colors appears for various types of colorblindness. It also discusses anomalous dichromacy, wherein there are three cones, but one of them is defective. Most interesting is the article’s claim that a remote ancestor of the primates was a tetranope . In addition, the article points out that mothers of male dichromats tend to have a fourth color receptor in the green region.
- 44.
For studies of unilateral dichromats, see Martin Bodian’s article “What do the Color Blind See?” in the American Journal of Ophthalmology, volume 35, p. 1471 (1952) and the article by Kurt Feig and Hans-Hilger Ropers in the journal Human Genetics, volume 41, p. 313 (1978).
- 45.
One approach, used by EnChroma , is to enhance the saturation of both red and green hues. See https://www.technologyreview.com/s/601782/how-enchromas-glasses-correct-color-blindness/. Another approach is the ColorCorrection System TM of Dr. Thomas Azman that uses a system of color filters.
- 46.
See the websites http://www.neitzvision.com/content/home.html and http://www.handprint.com/HP/WCL/color1.html#dichromat for details.
- 47.
- 48.
See http://prometheus.med.utah.edu/~bwjones/wp-content/uploads/2010/06/iPhone-4-Display_.jpg].
- 49.
http://sites.tufts.edu/pmclg/. The table is to be found as a Link. You may need the password: pmc2012. The labels of the color matching functions are primed r, g, and b, as opposed to overbarred letters.
- 50.
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Gunther, L. (2019). Theory of Color Vision. In: The Physics of Music and Color. Springer, Cham. https://doi.org/10.1007/978-3-030-19219-8_15
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