Abstract

The prior statistics of object colors is of much interest because extensive statistical investigations of reflectance spectra reveal highly non-uniform structure in color space common to several very different databases. This common structure is due to the visual system rather than to the statistics of environmental structure. Analysis involves an investigation of the proper sample space of spectral reflectance factors and of the statistical consequences of the projection of spectral reflectances on the color solid. Even in the case of reflectance statistics that are translationally invariant with respect to the wavelength dimension, the statistics of object colors is highly non-uniform. The qualitative nature of this non-uniformity is due to trichromacy.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  75. E. H. Land, “The retinex theory of colour vision,” Proc. R. Inst. Great Britain 47, 23-58 (1974).

2006 (6)

A. Lewis and L. Zhaoping, “Are cone sensitivities determined by natural color statistics?” J. Vision 6, 285-302 (2006).
[CrossRef]

F. Long, Z. Yang, and D. Purves, “Spectral statistics in natural scenes predict hue, saturation, and brightness,” Proc. Natl. Acad. Sci. U.S.A. 103, 6013-6018 (2006).
[CrossRef] [PubMed]

D. L. Philipona and J. K. O'Regan, “Color naming, unique hues, and hue cancellation predicted from singularities in reflection properties,” Visual Neurosci. 23, 331-339 (2006).
[CrossRef]

D. Sen, M. N. S. Swamy, and M. O. Ahmad, “Unbiased homomorphic system and its application in reducing multiplicative noise,” IEEE Proc. Vis. Image Signal Process. 153, 521-537 (2006).
[CrossRef]

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-388 (2006).
[CrossRef]

D. Davis, “Generic affine differential geometry of curves in Rn,” Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 136, 1195-1205 (2006).
[CrossRef]

2005 (1)

R. G. Kuehni, “Focal color variability and unique hue stimulus variability,” J. Cogn. Culture 5, 409-426 (2005).
[CrossRef]

2004 (4)

R. G. Kuehni, “Development of the idea of simple colors in the 16th and 17th centuries,” Color Res. Appl. 32, 158-162 (2004).

D. T. Lindsey and A. M. Brown, “Sunlight and 'Blue',” Psychol. Sci. 15, 291-294 (2004).
[CrossRef] [PubMed]

H. S. Fairman and M. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104-110 (2004).
[CrossRef]

J. A. Worthey and M. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261-266 (2004).
[CrossRef]

2003 (1)

F. Long and D. Purves, “Natural scene statistics as the universal basis of color context effects,” Proc. Natl. Acad. Sci. U.S.A. 100, 15190-15193 (2003).
[CrossRef] [PubMed]

2002 (1)

T.-W. Lee, T. T. Wachtler, and T. J. Sejnowski, “Color opponency is an efficient representation of spectral properties in natural scenes,” Vision Res. 42, 2095-2103 (2002).
[CrossRef] [PubMed]

1997 (1)

R. A. Crone, “Schopenhauer on vision and the colors,” Doc. Ophthalmol. 93, 61-71 (1997).
[CrossRef] [PubMed]

1991 (1)

1986 (1)

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. A 3, 1973-1683 (1986).
[CrossRef]

1982 (1)

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszeckis metameric blacks,” Am. J. Psychol. 95, 537-564 (1982).
[CrossRef] [PubMed]

1974 (1)

E. H. Land, “The retinex theory of colour vision,” Proc. R. Inst. Great Britain 47, 23-58 (1974).

1968 (1)

A. V. Oppenheim, R. W. Shafer, and T. G. Stockham, Jr., “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264-1291 (1968).
[CrossRef]

1965 (1)

1957 (1)

L. M. Hurvich and D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384-404 (1957).
[CrossRef] [PubMed]

1953 (1)

G. Wyszecki, “Valenzmetrische Undersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie,” Farbe 2, 39-52 (1953).

1950 (1)

M. Richter, “Untersuchungen zur Aufstellung eines empfindungsgemäß gleichabständigen Farbsystems,” Z. Wiss. Photographie 45, 139-162(1950).

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 92-99 (1931).

1928 (1)

S. Rösch, “Die Kennzeichnung der Farben,” Phys. Z. 29, 83-91 (1928).

1925 (1)

E. Schrödinger, “Über das Verhältnis der Vierfarben-zur Driefarbentheorie,” Sitzungsber. Akad. Wiss. Wien Math. Naturwiss. Kl. Abt 2a 134, 471-490 (1925).

1920 (1)

E. Schrödinger, “Theorie der Pigmente von größter Leuchtkraft,” Ann. Phys. 62, 603-622 (1920).
[CrossRef]

1917 (1)

A. Kirschmann, “Das Umgekehrte Spektrum Seine Komplementarverhaltnisse,” Phys. Z. 18, 195-205 (1917).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1-22 (1905).
[CrossRef]

1872 (1)

E. Hering, “Zur Lehre vom Lichtsinn,” Sitzungsber. K. Preuss. Akad. Wiss. Math. Naturwiss. Kl. 5-24 (1872).

1860 (1)

J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57-84 (1860).
[CrossRef]

Ahmad, M. O.

D. Sen, M. N. S. Swamy, and M. O. Ahmad, “Unbiased homomorphic system and its application in reducing multiplicative noise,” IEEE Proc. Vis. Image Signal Process. 153, 521-537 (2006).
[CrossRef]

Axler, S.

S. Axler, Linear Algebra Done Right (Springer, 1997).

Berns, R. S.

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology (Wiley, 2000).

Birkhoff, G.

G. Birkhoff, Lattice Theory, 3rd ed., Colloquium Publications 25 (American Mathematical Society, 1967).

Blaschke, W.

W. Blaschke, Vorlesungen der Differentialgeometrie und geometrische Grundlagen von Einsteins Relativittstheorie. II. Affine Differentialgeometrie, bearbeitet von K. Reidemeister. Erste und zweite Auflage (Julius Springer, 1923).

W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativiteitstheorie. Volume 1, Elementare Differentialgeometrie (Julius Springer, 1921).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ. Press, 1999).
[PubMed]

Bouguer, P.

P. Bouguer, L'Essai d'Optique sur la Gradation de la Lumière (Gauthier-Villars et Cie, 1729).

Bouma, P. J.

P. J. Bouma, Physical Aspect of Colour (Macmillan, 1971).

Brill, M.

H. S. Fairman and M. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104-110 (2004).
[CrossRef]

J. A. Worthey and M. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261-266 (2004).
[CrossRef]

Brown, A. M.

D. T. Lindsey and A. M. Brown, “Sunlight and 'Blue',” Psychol. Sci. 15, 291-294 (2004).
[CrossRef] [PubMed]

Cohen, J. B.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszeckis metameric blacks,” Am. J. Psychol. 95, 537-564 (1982).
[CrossRef] [PubMed]

Coxeter, H. S. M.

H. S. M. Coxeter, Introduction to Geometry, 2nd ed. (Wiley, 1989).

Crone, R. A.

R. A. Crone, “Schopenhauer on vision and the colors,” Doc. Ophthalmol. 93, 61-71 (1997).
[CrossRef] [PubMed]

Davis, D.

D. Davis, “Generic affine differential geometry of curves in Rn,” Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 136, 1195-1205 (2006).
[CrossRef]

Fairman, H. S.

H. S. Fairman and M. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104-110 (2004).
[CrossRef]

Feiner, S. K.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice in C (Addison-Wesley Professional, 1995).

Foley, J. D.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice in C (Addison-Wesley Professional, 1995).

Hering, E.

E. Hering, “Zur Lehre vom Lichtsinn,” Sitzungsber. K. Preuss. Akad. Wiss. Math. Naturwiss. Kl. 5-24 (1872).

Hughes, J. F.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice in C (Addison-Wesley Professional, 1995).

Hurvich, L. M.

L. M. Hurvich and D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384-404 (1957).
[CrossRef] [PubMed]

Jääskeläinen, T.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-388 (2006).
[CrossRef]

Jameson, D.

L. M. Hurvich and D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384-404 (1957).
[CrossRef] [PubMed]

Jaynes, E. T.

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge Univ. Press, 1931).

Jeffreys, H.

H. Jeffreys, Scientific Inference (Cambridge Univ. Press, 2003).

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (Wiley, 1975).

Kappauf, W. E.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszeckis metameric blacks,” Am. J. Psychol. 95, 537-564 (1982).
[CrossRef] [PubMed]

Katz, D.

D. Katz, Die Erscheinungsweisen der Farben und ihre Beeinflussung durch die individuelle Erfahrung (Barth, 1911).

Kirschmann, A.

A. Kirschmann, “Das Umgekehrte Spektrum Seine Komplementarverhaltnisse,” Phys. Z. 18, 195-205 (1917).

Klee, P.

P. Klee, The Thinking Eye (Lund Humpries, 1961).

Kohonen, O.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-388 (2006).
[CrossRef]

Kortüm, G.

G. Kortüm, Reflexionsspektroskopie (Springer, 1969).

Krinov, E. L.

E. L. Krinov, “Spectral'naye otrzhatel'naya sposobnost' prirodnykh obrazovanii,” Izd. Akad. Nauk USSR, emphSpectral reflectance properties of natural formations, G. Belkov, transl., TT-439 (National Research Council of Canada, 1953).

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 92-99 (1931).

Kuehni, R. G.

R. G. Kuehni, “Focal color variability and unique hue stimulus variability,” J. Cogn. Culture 5, 409-426 (2005).
[CrossRef]

R. G. Kuehni, “Development of the idea of simple colors in the 16th and 17th centuries,” Color Res. Appl. 32, 158-162 (2004).

Lambert, J. H.

J. H. Lambert, Photometria sive de mensure de gratibus luminis, colorum umbræ (Eberhard Klett, 1760).

Land, E. H.

E. H. Land, “The retinex theory of colour vision,” Proc. R. Inst. Great Britain 47, 23-58 (1974).

Lee, T.-W.

T.-W. Lee, T. T. Wachtler, and T. J. Sejnowski, “Color opponency is an efficient representation of spectral properties in natural scenes,” Vision Res. 42, 2095-2103 (2002).
[CrossRef] [PubMed]

Lewis, A.

A. Lewis and L. Zhaoping, “Are cone sensitivities determined by natural color statistics?” J. Vision 6, 285-302 (2006).
[CrossRef]

Lindsey, D. T.

D. T. Lindsey and A. M. Brown, “Sunlight and 'Blue',” Psychol. Sci. 15, 291-294 (2004).
[CrossRef] [PubMed]

Long, F.

F. Long, Z. Yang, and D. Purves, “Spectral statistics in natural scenes predict hue, saturation, and brightness,” Proc. Natl. Acad. Sci. U.S.A. 103, 6013-6018 (2006).
[CrossRef] [PubMed]

F. Long and D. Purves, “Natural scene statistics as the universal basis of color context effects,” Proc. Natl. Acad. Sci. U.S.A. 100, 15190-15193 (2003).
[CrossRef] [PubMed]

Maloney, L. T.

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. A 3, 1973-1683 (1986).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, “On the theory of compound colours, and the relations of the colours of the spectrum,” Philos. Trans. R. Soc. London 150, 57-84 (1860).
[CrossRef]

Minnaert, M. G. J.

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, 1993).
[CrossRef]

Moebius, A. F.

A. F. Moebius, Der baryzentrische Calcul (orig. 1827) (Georg Olms, 1976).

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 92-99 (1931).

Munsell, A. H.

A. H. Munsell, A Color Notation (G. H. Ellis Co., 1905).

Newton, I.

I. Newton, Opticks or a Treatise on the Reflexions, Refractions, Inflexions and Colours of Light, 4th ed. (William Innys, 1730).

Nicodemus, F.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shafer, and T. G. Stockham, Jr., “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264-1291 (1968).
[CrossRef]

O'Regan, J. K.

D. L. Philipona and J. K. O'Regan, “Color naming, unique hues, and hue cancellation predicted from singularities in reflection properties,” Visual Neurosci. 23, 331-339 (2006).
[CrossRef]

Ostwald, W.

W. Ostwald, Farbkunde (S. Hirzel, 1923).

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Other (44)

Ph. O. Runge, Farbenkugel oder Construction des Verhältnisses aller Mischungen der Farben zu einander und ihrer vollständigen Affinität; mit angehängtem Versuch einer Ableitung der Harmonie in den Zusammenstellungen der Farben (Friedrich Perthes, 1810).

P. Klee, The Thinking Eye (Lund Humpries, 1961).

S. Quiller, Color Choices (Watson-Guptill Publications, 1989).

H. S. M. Coxeter, Introduction to Geometry, 2nd ed. (Wiley, 1989).

The volume of the color solid is 1.43 times the volume of the maximum volume crate. The cube root of 1.43 is 1.13, which yields a more intuitive (because linear instead of volumetric) measure of the difference.

A. H. Munsell, A Color Notation (G. H. Ellis Co., 1905).

W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativiteitstheorie. Volume 1, Elementare Differentialgeometrie (Julius Springer, 1921).

Cohen implicitly applies the usual Cartesian inner product on radiant power spectra on wavelength basis. This turns the space of spectra into a Hilbert space. The visual projection then can be turned into an orthogonal projection, thus enabling the definition of “visual space.” Accepting an orthonormal basis for visual space then leads to a representation very similar to the one constructed here. However, this seems more of a lucky strike. Moreover, the metric adopted by Cohen is rather ad hoc and hard to defend. It is certainly preferable to avoid the choice of metric altogether.

A. Schopenhauer, Über das Sehn und die Farben (Johann Friedrich Hartknoch, 1816).

J. W. von Goethe, Zur Farbenlehre (Cotta, 1808-1810).

I. Newton, Opticks or a Treatise on the Reflexions, Refractions, Inflexions and Colours of Light, 4th ed. (William Innys, 1730).

The volume of the color solid is only 1.43 times that of the inscribed parallelopiped of maximum volume, whereas the circumscribed sphere of a cube has 2.72 (namely, π3/2) times the volume of that cube. (Such numbers are very relevant yet cannot be found in textbooks on colorimetry.) Thus the color solid is rather close to the linear transform of a unit cube.

W. Blaschke, Vorlesungen der Differentialgeometrie und geometrische Grundlagen von Einsteins Relativittstheorie. II. Affine Differentialgeometrie, bearbeitet von K. Reidemeister. Erste und zweite Auflage (Julius Springer, 1923).

The dominant wavelengths of the secondary colors are 483 nm for cyan and 567 nm for yellow. (Magenta is complementary to green and might--formally--be said to have dominant wavelength −527 nm.)

S. Skiena, “Hasse diagrams,” in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (Addison-Wesley, 1990), pp. 163, 169-170, and 20-208.

G. Birkhoff, Lattice Theory, 3rd ed., Colloquium Publications 25 (American Mathematical Society, 1967).

B. S. W. Schröder, Ordered Sets: An Introduction (Birkhäuser, 2003).
[CrossRef]

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice in C (Addison-Wesley Professional, 1995).

A. F. Moebius, Der baryzentrische Calcul (orig. 1827) (Georg Olms, 1976).

Ratios of areas are not invariant in the chromaticity diagram. Almost all representations in the textbooks give an extremely lopsided view of the gamut of object colors. The boundary of the gamut (that is the spectrum locus) is the image of the black point and is irrelevant for most applications, despite the fact that it is usually suggested that it is the locus of the most vivid colors (apparently because it is monochromatic). But the spectral reflectances of monochromatic object colors are blip functions of vanishing support, thus these colors are blacks!

Of the NCSU objects data base only 3 out of the 170 samples yield more than 1% over- or underflow, the worst case being a 3% underflow in the green channel, which is perceptually irrelevant.

W. Ostwald, Farbkunde (S. Hirzel, 1923).

W. Ostwald, Die Farbenfibel (Unesma, 1926).

W. Ostwald, Er und Ich (Theodor Martins Textilverlag, 1936).

W. Ostwald, Der Farbatlas (Unesma, 1917).

H. Jeffreys, Scientific Inference (Cambridge Univ. Press, 2003).

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge Univ. Press, 1931).

E. L. Krinov, “Spectral'naye otrzhatel'naya sposobnost' prirodnykh obrazovanii,” Izd. Akad. Nauk USSR, emphSpectral reflectance properties of natural formations, G. Belkov, transl., TT-439 (National Research Council of Canada, 1953).

Here the arc length of the short-wavelength-pass edge colors is used to define the quartiles and median of the wavelength domain. The value of the relative bandwidth quoted here is the interquartile range divided by the median. The visual band is thus extremely narrow by physical standards, and one has no reason to expect significant variations due to the physics.

CIE Proceedings Vienna Session 1963, Vol. B, 209-220(Committee Report E-1.4.1), (Bureau Central de la CIE, Paris, 1964).

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology (Wiley, 2000).

P. J. Bouma, Physical Aspect of Colour (Macmillan, 1971).

Consider the locus of points {x1,x2,...,xn}∊R2 under the constraint 0⩽xi⩽ai. For n=1, one has the segment [0,a1], for n=2 the rectangle with vertices {0,0}, {a1,0}, {a1,a2} and {0,a2}, i.e., a rectangle, and so forth. In general, one obtains a “cuboid” (for a1,...,an=1 a unit hypercube) which is--in the absence of a metric--more appropriately called a “hyperparallelepiped.”

S. Axler, Linear Algebra Done Right (Springer, 1997).

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (Wiley, 1975).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, 1967).

The problem is that the experimental uncertainly is usually fixed in absolute terms (for typical databases of the order of 0.01), whereas for reflectance factors near zero or one it is the difference from zero or one that is relevant, and this difference easily drowns in the experimental uncertainty.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge Univ. Press, 1999).
[PubMed]

G. Kortüm, Reflexionsspektroskopie (Springer, 1969).

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, 1993).
[CrossRef]

D. Katz, Die Erscheinungsweisen der Farben und ihre Beeinflussung durch die individuelle Erfahrung (Barth, 1911).

P. Bouguer, L'Essai d'Optique sur la Gradation de la Lumière (Gauthier-Villars et Cie, 1729).

J. H. Lambert, Photometria sive de mensure de gratibus luminis, colorum umbræ (Eberhard Klett, 1760).

A “reflectance factor” is the ratio of the radiance scattered to the eye by some surface and the radiance that would have been scattered to the eye by an ideal Lambertian surface. The advantage of the “reflectance factor” is that this avoids complications with the viewing and illumination geometries.

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Figures (13)

Fig. 1
Fig. 1

Left: histogram of reflectance factors (all samples, all wavelengths pooled) from the NCSU objects database. The histograms are normalized to a maximum value of unity. This is the histogram in the “phenomenological domain” (range of the reflection factor between zero and one). Notice that the histogram is almost exponential. Right: the same data in the “physical domain” (range minus to plus infinity). Though the histogram is still a little skew, it is much more centrally symmetric than the one at left, as well as defined on the full axis. A simple normal distribution captures much of the structure of the available data bases. In all cases the flanks of the distribution are ill defined due to observational errors (actual reflectance factors too close to either zero or one).

Fig. 2
Fig. 2

Fourier spectral power of the spectral envelope of the radiant power falls by (about) the inverse second power of the frequency. The frequency is given in terms of the number of cycles per 100 nm on a logarithmic scale. Notice that the “visual range” measures about 200 400 nm . This is a double logarithmic plot: on the abscissa the actual values are indicated; on the ordinate the values of log spectral power (base 10). (Notice that 1 decade on the abscissa has the length of 4 decades on the ordinate; thus the slope is about 4 ).

Fig. 3
Fig. 3

Six samples from randomly generated spectral reflectances. The parameters of the random generator have been selected so as to simulate the statistics of typical databases of “natural reflectance spectra.” This random generator has stationary statistics; there is no wavelength preference whatsoever.

Fig. 4
Fig. 4

The Schrödinger color solid looks like a “slightly inflated cube” when represented in the canonical coordinates introduced later in the text. (In the CIE x y z system it looks rather less symmetrical.) It is composed of two smooth patches (the spectral and the non-spectral optimal colors) that are mutually congruent by central symmetry and hang together via two sharp creases (the edge color loci) and two conical points (the white and black points).

Fig. 5
Fig. 5

RGB cube, Schrödinger color solid, and edge color loci in the canonical projection, shown as viewed from three mutually orthogonal viewing directions (at far right the gray axis, at far left a direction in the yellow–blue plane, at center a direction in the red–cyan plane). This structure is the very backbone of object color space. Notice that the edge color loci closely follow the KRYW and WCBK edge progressions of the cube, whereas the closed YGCBMR edge progression approximates the “color circle,” that is, the locus of colors maximally removed from the gray (KW) axis.

Fig. 6
Fig. 6

Hasse diagram of the partial order of cardinal colors by set inclusion. Notice how it looks like a skeleton projection (“wireframe rendering”) of the RGB cube.

Fig. 7
Fig. 7

Equilateral RGB triangle is the preferred (most symmetrical) chromaticity diagram in the RGB representation. Notice that the spectrum locus runs far outside the triangle. This might easily lead to erroneous impressions, however; the volume of the Schrödinger color solid that is mapped outside the triangle is only about 8% of the total volume. It is very rare (probability less than 1%) for a color from one of the databases to fall outside the RGB triangle. The wavelengths 483 nm and 567 nm are the cut loci. Notice that the spectrum locus represents the chromaticity of the black point. The fat curves are the edge color loci.

Fig. 8
Fig. 8

Spectrum of the color x = { 0.8 , 0.6 , 0.3 } can be conceived of as addition of a uniform spectrum (white), nothing (black), yellow (extending over the red and green regions), and red (extending over the red region). Thus any color is the convex combination of white, black, and a “full color,” that is, the interpolation between a primary color and one of its nearest secondary colors.

Fig. 9
Fig. 9

Result of a simulation of the simple model. The number of transitions was 0, 1, 2, … with probability 1 2 , 1 4 , 1 8 . Of 10,000 samples about 5000 ended up at either the white or the black point; these are not represented here. The spheres represent samples of volume density over volumes of about one thousandth of the volume of the RGB cube; the radius is taken proportional to the logarithm of the density. Notice that almost all samples end up on either the long-wavelength-pass edge color locus (KRYW edge sequence) or the short-wavelength edge color locus (WCBK edge sequence). Some density accumulates on the MW edge (near W) and on the GK edge (near K). The remaining density concentrates on the surface of the RGB cube with only a sprinkle of samples in the interior. Thus the most frequent object colors are R and B (dark), C, and Y (light), with some representation of dark G and light M. Thus the simple model produces very pronounced non-uniformities.

Fig. 10
Fig. 10

Nexus of parameter lines for the passband center and passband width of the ideal colors (“slit location and width in spectroscopic terms”). The area of the meshes indicates the function J. Notice that J becomes zero at the edge color curves (the area collapses to a curve). Here the boundary of the color solid has been plotted in cylinder coordinates: vertically the “height” in the achromatic dimension (bottom horizontal edge represents the black point, top horizontal edge the white point), and horizontally the azimuth, which has been marked with the cardinal colors of the color circle. This is the type of plot conventionally shown in discussions of color preferences. The plot implicates a preponderance of dark reds and blues, light yellows, and cyans as well as (though less pronounced) dark greens and light purples.

Fig. 11
Fig. 11

Result of simulations 10,000 samples) with a random spectral reflectance generator with stationary statistics. The generator simulates generic databases of natural colors. In the contour plot one sees that the density is concentrated on the edge color loci.

Fig. 12
Fig. 12

Density of chromaticities for a random reflectance generator with stationary statistics. The parameters are such that the samples are similar to what one might obtain from a typical database of “natural reflectance spectra.” The density is based on 10,000 samples. Notice that the density concentrates upon the edge color loci.

Fig. 13
Fig. 13

Density of hue angles for the simulation. At left the density for dark colors, at right the density for light colors. Notice that dark colors are predominantly red or blue and light colors predominantly yellow or cyan.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

F ( x ) = 1 2 ( 1 + tanh x ) ,
F ( 1 ) ( r ) = arctanh ( 2 r 1 ) .
r = α r 1 β r 2 = F ( α F ( 1 ) ( r 1 ) + β F ( 1 ) ( r 2 ) ) ,
c ( r λ ) = 0 r λ I λ A λ T d λ ,
s ( λ ) = 0 λ I λ A λ T d λ ;
i ( λ 1 , λ 2 ) = λ 1 λ 2 I λ A λ T d λ = s ( λ 2 ) s ( λ 1 ) ,
i ( λ 1 , λ 2 ) λ 1 = | d s d λ | λ 1 = m λ 1 ,
i ( λ 1 , λ 2 ) λ 2 = + | d s d λ | λ 2 = + m λ 2 ,
d s 2 = E d λ 1 2 + 2 F d λ 1 d λ 2 + G d λ 2 2 ,
E = m λ 1 m λ 1 ,
F = m λ 1 m λ 2 ,
G = m λ 2 m λ 2 ,
J ( λ 1 , λ 2 ) = det | E F F G | = m λ 1 × m λ 2 .

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