Abstract

Spectre-photometric curves uniquely determined by every set of color specifications can be based on a Fourier-type analysis, using normal and orthogonal color-mixture functions. Normal and orthogonal color-mixture functions are convenient for the investigation of the propagation of errors in spectrophotometric colorimetry. The space based on normal and orthogonal color-mixture functions serves as a suggestive model for the as yet unavailable visually homogeneous color space, in which visual thresholds of color shall be represented by equal-size spheres, at least for some fixed conditions of adaptation. Recent puzzling results of investigations of color discrimination and heterochromatic photometry can be explained in a straightforward manner in terms of general properties of visually homogeneous color space. Those properties are independent of the detailed structure of that space and are exemplified, though probably in a simplified manner, by the color space based on normal and orthogonal color-mixture functions.

© 1954 Optical Society of America

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References

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  1. D. L. MacAdam, J. Opt. Soc. Am. 43, 533–538 (1953).
    [Crossref] [PubMed]
  2. G. Wyszecki, Die Farbe 2, 39–52 (1953).
  3. W. L. Brewer and F. R. Holly, J. Opt. Soc. Am. 38, 858–874 (1948).
    [Crossref] [PubMed]
  4. I. Nimeroff, J. Opt. Soc. Am. 43, 531–533 (1953).
    [Crossref] [PubMed]
  5. See L. P. Eisenhart, Differential Geometry (Princeton University Press, 1947), pp. 282–293. This analysis is based on suggestions made by Herman von Schelling, U. S. Naval Medical Research Laboratory, Submarine Base, New London, Connecticut.
  6. D. L. MacAdam, J. Opt. Soc. Am. 32, 2–6 (1942).
    [Crossref]
  7. The name apeiron is based on the Greek word for infinity απ∊ℓρos. Apeiron (pronounced a˙ · pī’rŏn) is defined in the Merriam-Webster “New International Dictionary” (unabridged) as “the indeterminate and indefinite ground, matter, or first principle of all being.” In the present specialized application to colorimetry, apeiron may be defined as “The line in any chromaticity diagram which is indeterminate in a second chromaticity diagram, but which determines the plane of projection of the latter.”For example, the alychne [lightless line, as defined by E. Schrödinger, Sitzber. Akad. Wiss. Wien, Abt. IIa,  134, 476 (1925), andD. B. Judd, J. Research, Natl. Bur. Standards 4, 545 (1929)] is the apeiron of all constant luminosity chromaticity diagrams, such as that represented by the (V= 1)-plane in Fig. 3, and shown in Fig. 6 of reference 1. By definition, the apeiron of a chromaticity diagram cannot be represented in it. In all other chromaticity diagrams in which the apeiron of a particular diagram can be represented, it represents the same series of chromaticities. Like the alychne, the apeiron of every chromaticity diagram used in the past represents chromaticities all exceeding spectral purity.
  8. D. B. Judd, J. Opt. Soc. Am. 25, 24–35 (1935).
    [Crossref]
  9. OSA Committee on Colorimetry, The Science of Color (T. Y. Crowell, New York, 1953), p. 302.
  10. W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808–835 (1949).
    [Crossref] [PubMed]
  11. A. Dresler, Trans. Illum. Eng. Soc. (London) 18, 141–165 (1953).
  12. H. E. Ives, Phil. Mag. 24, 353, 744, 845, and 853 (1912).
  13. M. Luckiesh, Elec. World 61, 620, 735, and 835 (1913);M. Luckiesh, Phys. Rev. 4, 1 (1914).
    [Crossref]
  14. E. Schrödinger, Ann. Physik 63, 397–456 (1920).
    [Crossref]
  15. On the basis of Schrödinger’s criterion, the surfaces of constant brightness can be plane only if all the loci of constant chromaticness are parallel straight lines. Such would be the case if there were no Bézold-Brücke effect, and if the Weber-Fechner law applied to the incremental thresholds of the separate stimuli. The latter assumption has been tested, modified, and found wanting by H. von Helmholtz, [Wiss. Abhandl.3, 407–459, 460–475 (Leipzig, 1895)], Schrödinger (see reference 14),R. H. Sinden [J. Opt. Soc. Am. 27, 124–131 (1937);J. Opt. Soc. Am. 28, 339–347 (1938)],L. Silberstein, [J. Opt. Soc. Am. 28, 63–85 (1938)], andW. S. Stiles, [Proc. Phys. Soc. (London) 18, 41–65 (1945)]. It seems most unlikely that the Bézold-Brücke effect would compensate for the discrepancies between the Weber-Fechner law and visually homogeneous color space, so as to yield straight, parallel loci of constant chromaticness and planes of constant brightness.
    [Crossref]
  16. L. Silberstein, J. Opt. Soc. Am. 33, 1–9 (1943).
    [Crossref]
  17. D. L. MacAdam, J. Franklin Inst. 238, 195–209 (1944);D. L. MacAdam, Rev. optique 28, 161–173 (1949); andD. L. MacAdam, Ophthalmologica 3, 214–239 (1949).
    [Crossref]
  18. D. L. MacAdam, J. Opt. Soc. Am. 40, 589–595 (1950).
    [Crossref]
  19. M. Tessier and F. Blottiau, Rev. optique 30, 309–322 (1951).
  20. W. S. Stiles, Ilium. Eng. 49, 77 (1954).

1954 (1)

W. S. Stiles, Ilium. Eng. 49, 77 (1954).

1953 (4)

D. L. MacAdam, J. Opt. Soc. Am. 43, 533–538 (1953).
[Crossref] [PubMed]

G. Wyszecki, Die Farbe 2, 39–52 (1953).

I. Nimeroff, J. Opt. Soc. Am. 43, 531–533 (1953).
[Crossref] [PubMed]

A. Dresler, Trans. Illum. Eng. Soc. (London) 18, 141–165 (1953).

1951 (1)

M. Tessier and F. Blottiau, Rev. optique 30, 309–322 (1951).

1950 (1)

1949 (1)

1948 (1)

1944 (1)

D. L. MacAdam, J. Franklin Inst. 238, 195–209 (1944);D. L. MacAdam, Rev. optique 28, 161–173 (1949); andD. L. MacAdam, Ophthalmologica 3, 214–239 (1949).
[Crossref]

1943 (1)

1942 (1)

1935 (1)

1925 (1)

The name apeiron is based on the Greek word for infinity απ∊ℓρos. Apeiron (pronounced a˙ · pī’rŏn) is defined in the Merriam-Webster “New International Dictionary” (unabridged) as “the indeterminate and indefinite ground, matter, or first principle of all being.” In the present specialized application to colorimetry, apeiron may be defined as “The line in any chromaticity diagram which is indeterminate in a second chromaticity diagram, but which determines the plane of projection of the latter.”For example, the alychne [lightless line, as defined by E. Schrödinger, Sitzber. Akad. Wiss. Wien, Abt. IIa,  134, 476 (1925), andD. B. Judd, J. Research, Natl. Bur. Standards 4, 545 (1929)] is the apeiron of all constant luminosity chromaticity diagrams, such as that represented by the (V= 1)-plane in Fig. 3, and shown in Fig. 6 of reference 1. By definition, the apeiron of a chromaticity diagram cannot be represented in it. In all other chromaticity diagrams in which the apeiron of a particular diagram can be represented, it represents the same series of chromaticities. Like the alychne, the apeiron of every chromaticity diagram used in the past represents chromaticities all exceeding spectral purity.

1920 (1)

E. Schrödinger, Ann. Physik 63, 397–456 (1920).
[Crossref]

1913 (1)

M. Luckiesh, Elec. World 61, 620, 735, and 835 (1913);M. Luckiesh, Phys. Rev. 4, 1 (1914).
[Crossref]

1912 (1)

H. E. Ives, Phil. Mag. 24, 353, 744, 845, and 853 (1912).

Blottiau, F.

M. Tessier and F. Blottiau, Rev. optique 30, 309–322 (1951).

Brewer, W. L.

Brown, W. R. J.

Dresler, A.

A. Dresler, Trans. Illum. Eng. Soc. (London) 18, 141–165 (1953).

Eisenhart, L. P.

See L. P. Eisenhart, Differential Geometry (Princeton University Press, 1947), pp. 282–293. This analysis is based on suggestions made by Herman von Schelling, U. S. Naval Medical Research Laboratory, Submarine Base, New London, Connecticut.

Holly, F. R.

Ives, H. E.

H. E. Ives, Phil. Mag. 24, 353, 744, 845, and 853 (1912).

Judd, D. B.

Luckiesh, M.

M. Luckiesh, Elec. World 61, 620, 735, and 835 (1913);M. Luckiesh, Phys. Rev. 4, 1 (1914).
[Crossref]

MacAdam, D. L.

Nimeroff, I.

Schrödinger,

On the basis of Schrödinger’s criterion, the surfaces of constant brightness can be plane only if all the loci of constant chromaticness are parallel straight lines. Such would be the case if there were no Bézold-Brücke effect, and if the Weber-Fechner law applied to the incremental thresholds of the separate stimuli. The latter assumption has been tested, modified, and found wanting by H. von Helmholtz, [Wiss. Abhandl.3, 407–459, 460–475 (Leipzig, 1895)], Schrödinger (see reference 14),R. H. Sinden [J. Opt. Soc. Am. 27, 124–131 (1937);J. Opt. Soc. Am. 28, 339–347 (1938)],L. Silberstein, [J. Opt. Soc. Am. 28, 63–85 (1938)], andW. S. Stiles, [Proc. Phys. Soc. (London) 18, 41–65 (1945)]. It seems most unlikely that the Bézold-Brücke effect would compensate for the discrepancies between the Weber-Fechner law and visually homogeneous color space, so as to yield straight, parallel loci of constant chromaticness and planes of constant brightness.
[Crossref]

Schrödinger, E.

The name apeiron is based on the Greek word for infinity απ∊ℓρos. Apeiron (pronounced a˙ · pī’rŏn) is defined in the Merriam-Webster “New International Dictionary” (unabridged) as “the indeterminate and indefinite ground, matter, or first principle of all being.” In the present specialized application to colorimetry, apeiron may be defined as “The line in any chromaticity diagram which is indeterminate in a second chromaticity diagram, but which determines the plane of projection of the latter.”For example, the alychne [lightless line, as defined by E. Schrödinger, Sitzber. Akad. Wiss. Wien, Abt. IIa,  134, 476 (1925), andD. B. Judd, J. Research, Natl. Bur. Standards 4, 545 (1929)] is the apeiron of all constant luminosity chromaticity diagrams, such as that represented by the (V= 1)-plane in Fig. 3, and shown in Fig. 6 of reference 1. By definition, the apeiron of a chromaticity diagram cannot be represented in it. In all other chromaticity diagrams in which the apeiron of a particular diagram can be represented, it represents the same series of chromaticities. Like the alychne, the apeiron of every chromaticity diagram used in the past represents chromaticities all exceeding spectral purity.

E. Schrödinger, Ann. Physik 63, 397–456 (1920).
[Crossref]

Silberstein, L.

Stiles, W. S.

W. S. Stiles, Ilium. Eng. 49, 77 (1954).

Tessier, M.

M. Tessier and F. Blottiau, Rev. optique 30, 309–322 (1951).

von Helmholtz, H.

On the basis of Schrödinger’s criterion, the surfaces of constant brightness can be plane only if all the loci of constant chromaticness are parallel straight lines. Such would be the case if there were no Bézold-Brücke effect, and if the Weber-Fechner law applied to the incremental thresholds of the separate stimuli. The latter assumption has been tested, modified, and found wanting by H. von Helmholtz, [Wiss. Abhandl.3, 407–459, 460–475 (Leipzig, 1895)], Schrödinger (see reference 14),R. H. Sinden [J. Opt. Soc. Am. 27, 124–131 (1937);J. Opt. Soc. Am. 28, 339–347 (1938)],L. Silberstein, [J. Opt. Soc. Am. 28, 63–85 (1938)], andW. S. Stiles, [Proc. Phys. Soc. (London) 18, 41–65 (1945)]. It seems most unlikely that the Bézold-Brücke effect would compensate for the discrepancies between the Weber-Fechner law and visually homogeneous color space, so as to yield straight, parallel loci of constant chromaticness and planes of constant brightness.
[Crossref]

von Schelling, Herman

See L. P. Eisenhart, Differential Geometry (Princeton University Press, 1947), pp. 282–293. This analysis is based on suggestions made by Herman von Schelling, U. S. Naval Medical Research Laboratory, Submarine Base, New London, Connecticut.

Wyszecki, G.

G. Wyszecki, Die Farbe 2, 39–52 (1953).

Ann. Physik (1)

E. Schrödinger, Ann. Physik 63, 397–456 (1920).
[Crossref]

Die Farbe (1)

G. Wyszecki, Die Farbe 2, 39–52 (1953).

Elec. World (1)

M. Luckiesh, Elec. World 61, 620, 735, and 835 (1913);M. Luckiesh, Phys. Rev. 4, 1 (1914).
[Crossref]

Ilium. Eng. (1)

W. S. Stiles, Ilium. Eng. 49, 77 (1954).

J. Franklin Inst. (1)

D. L. MacAdam, J. Franklin Inst. 238, 195–209 (1944);D. L. MacAdam, Rev. optique 28, 161–173 (1949); andD. L. MacAdam, Ophthalmologica 3, 214–239 (1949).
[Crossref]

J. Opt. Soc. Am. (8)

Phil. Mag. (1)

H. E. Ives, Phil. Mag. 24, 353, 744, 845, and 853 (1912).

Rev. optique (1)

M. Tessier and F. Blottiau, Rev. optique 30, 309–322 (1951).

Sitzber. Akad. Wiss. Wien, Abt. IIa (1)

The name apeiron is based on the Greek word for infinity απ∊ℓρos. Apeiron (pronounced a˙ · pī’rŏn) is defined in the Merriam-Webster “New International Dictionary” (unabridged) as “the indeterminate and indefinite ground, matter, or first principle of all being.” In the present specialized application to colorimetry, apeiron may be defined as “The line in any chromaticity diagram which is indeterminate in a second chromaticity diagram, but which determines the plane of projection of the latter.”For example, the alychne [lightless line, as defined by E. Schrödinger, Sitzber. Akad. Wiss. Wien, Abt. IIa,  134, 476 (1925), andD. B. Judd, J. Research, Natl. Bur. Standards 4, 545 (1929)] is the apeiron of all constant luminosity chromaticity diagrams, such as that represented by the (V= 1)-plane in Fig. 3, and shown in Fig. 6 of reference 1. By definition, the apeiron of a chromaticity diagram cannot be represented in it. In all other chromaticity diagrams in which the apeiron of a particular diagram can be represented, it represents the same series of chromaticities. Like the alychne, the apeiron of every chromaticity diagram used in the past represents chromaticities all exceeding spectral purity.

Trans. Illum. Eng. Soc. (London) (1)

A. Dresler, Trans. Illum. Eng. Soc. (London) 18, 141–165 (1953).

Other (3)

On the basis of Schrödinger’s criterion, the surfaces of constant brightness can be plane only if all the loci of constant chromaticness are parallel straight lines. Such would be the case if there were no Bézold-Brücke effect, and if the Weber-Fechner law applied to the incremental thresholds of the separate stimuli. The latter assumption has been tested, modified, and found wanting by H. von Helmholtz, [Wiss. Abhandl.3, 407–459, 460–475 (Leipzig, 1895)], Schrödinger (see reference 14),R. H. Sinden [J. Opt. Soc. Am. 27, 124–131 (1937);J. Opt. Soc. Am. 28, 339–347 (1938)],L. Silberstein, [J. Opt. Soc. Am. 28, 63–85 (1938)], andW. S. Stiles, [Proc. Phys. Soc. (London) 18, 41–65 (1945)]. It seems most unlikely that the Bézold-Brücke effect would compensate for the discrepancies between the Weber-Fechner law and visually homogeneous color space, so as to yield straight, parallel loci of constant chromaticness and planes of constant brightness.
[Crossref]

OSA Committee on Colorimetry, The Science of Color (T. Y. Crowell, New York, 1953), p. 302.

See L. P. Eisenhart, Differential Geometry (Princeton University Press, 1947), pp. 282–293. This analysis is based on suggestions made by Herman von Schelling, U. S. Naval Medical Research Laboratory, Submarine Base, New London, Connecticut.

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Figures (10)

F. 1
F. 1

Top: Plane perpendicular to constant luminance planes in normal and orthogonal color space, showing relation between constant luminance cross sections of spheres of uncertainties and zones of uncertainty of chromaticity determinations. Below: Constant luminance plane, showing cross sections of spheres of uncertainties (dotted) and ellipses of uncertainty of chromaticities.

F. 2
F. 2

Top: Same as Fig. 1, showing trace of paraboloid for which chromaticity uncertainties can be represented by equal circles in plane. Below: Curvilinear coordinate system, spectrum locus, and circular zones of uncertainties of chromaticities in plane corresponding to paraboloidal surface shown above.

F. 3
F. 3

Diagram of (U,V,W)-space. Plane (u″,w″) is inclined at angle ϕ to (V = 1)-plane, and intersects (V = 1)-plane in line: U sinθW cosθ=0.

F. 4
F. 4

Diagram showing correspondences of W′, in (V = 1)-plane, with w″, in (u″,w″)-plane. This is a vertical cross section through (U,V,W)-space as shown in Fig. 3, through the U(W = 0)-axis. Corresponding values of W and w″ (for identical chromaticities) are determined by points connected by straight lines through the origin, 0. In particular, w″ = 0 corresponds to W′ = 0, and w″ = ∞ corresponds to W′ = 1/tanϕ. Luminance (0.26V) is not constant in (u″,w″)-plane, but V = 1+w″ sinϕ. The alychne, which is at W′ = ∞ in (V = 1)-plane, is at w″ =− 1/sinϕ in (u″,w″)-plane.

F. 5
F. 5

Comparison of (u′, υ′)-diagram with UCS (u, υ)-diagram. Axis of υ′ is perpendicular to u′ and has same scale units as u′. Line which corresponds to υ-axis in (u′, υ′)-diagram is at angle of 90° 38′ with u′-axis, which corresponds exactly with u-axis. When 10-mµ weighted ordinate method of colorimetric computation is used, spectrophotometric uncertainties (σR = 0.01), equal at all wavelengths, produce uncertainties indicated by spheres in (U,V,W)-space which intersect (u′, υ′)-diagram in equal-size circles (shown dotted, 10 times enlarged) for samples having luminous reflectances equal to 0.33υ′. Corresponding uncertainties of chromaticities are indicated by ellipses all having equal minor axes, with major axes pointing toward u′=0.3677, υ′ = 0.737.

F. 6
F. 6

Dotted circles: Chromaticities corresponding to intersections of planes parallel to (u′,υ′)-plane, through centers of spheres of uncertainty (enlarged 10 times) arising from uniform spectrophotometric uncertainties (σR = 0.01) and 10-mµ weighted ordinate calculations, for samples having 10 percent luminous reflectance. Radii are proportional to υ′. Solid ellipses: Corresponding uncertainties of determination of chromaticities.

F. 7
F. 7

Curvilinear coordinate system in plane, in which uncertainties of chromaticities are represented by circles. For samples represented, in (U,V,W)-space, on the paraboloid of revolution specified in text, circles of uncertainty are all of equal size.

F. 8
F. 8

Representation of plane corresponding to 1931 CIE chromaticity diagram in (U,V,W)-space.

F. 9
F. 9

Intersections (dotted circles) of plane corresponding to 1931 CIE chromaticity diagram with spheres of uncertainty-caused by uniform spectrophotometric uncertainties. The CIE coordinate system (x,y) is shown. Corresponding uncertainties of chromaticities for samples having luminous reflectances proportional to y are represented by solid ellipses, whose major axes all point toward u″ = 0, w″= − 0.6651.

F. 10
F. 10

Dotted ellipses: Intersections with spheres of uncertainty (shown as dotted circles in Fig. 9) transformed to standard 1931 CIE chromaticity diagram. These intersections are ellipses, all of the same shape, orientation, and size (for samples having luminous reflectance equal to 0.48y). Solid ellipses: Corresponding uncertainties of determination of chromaticities. Diameters shown correspond to major axes of uncertainty ellipses in Fig. 9 and all point toward x = 0.3275, y = 0.3025.

Tables (1)

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Table I

Equations (35)

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U V = V W = U W = 0 .
U 2 = V 2 = W 2 = 1 .
U 2 = V 2 = W 2 = 1 ,
g 11 Δ r 2 + 2 g 12 Δ r Δ ψ + g 22 Δ ψ r 2 = σ 2 .
K = ( δ A / δ r + δ B / δ ψ ) / 2 g ,
g = g 11 g 22 g 12 2 , A = g 1 2 [ ( g 12 / g 11 ) ( δ g 11 / δ ψ ) δ g 22 / δ r ] ,
B = g 1 2 [ 2 δ g 12 / δ r δ g 11 / δ ψ ( g 12 / g 11 ) ( δ g 11 / δ r ) ] .
( 1 ) d s 2 = d r 2 / ( 1 + r 2 ) + r 2 d ψ 2 .
( 2 ) r = sinh u , ψ = c υ .
( 3 ) d s 2 = d u 2 + c 2 sinh 2 u d υ 2 .
u = ( a 1 U + a 2 W ) / ( a 5 U + a 6 W + 1 ) ,
w = ( a 3 U + a 4 W ) / ( a 5 U + a 6 W + 1 ) .
a 4 = a 6 ( 1 / sin ϕ ) = 1 / cos ϕ .
u = U cos ϕ / ( W sin ϕ + cos ϕ ) ,
w = W / ( W sin ϕ + cos ϕ ) .
u = ( 0.4661 x + 0.1593 y ) / ( y 0.15735 x + 0.2424 ) . υ = 0.6581 y / ( y 0.15735 x + 0.2424 ) .
U = ( 0.7348 x 0.2274 y 0.1691 ) S . V = 0.3598 y S . W = ( 0.0207 x 0.3977 y + 0.2239 ) S .
U = ( 0.7166 x 0.3008 y 0.1220 ) / 0.3598 y . W = ( 0.1640 x + 0.3455 y 0.2527 ) / 0.3598 y .
u = ( 2.652 x 1.113 y 0.451 ) / ( y 0.15735 x + 0.2424 )
w = ( 0.627 x + 1.321 y 0.966 ) / ( y 0.15735 x + 0.2424 ) .
cot 1 ( 0.0443 / 3.9851 ) = cot 1 ( 0.0111 ) = 90 ° 3 8
u = ( u + 1.8606 ) / 5.062
υ = ( w + 3.9851 ) / 5.062 .
u = ( 0.4661 x + 0.1477 y ) / ( y 0.15735 x + 0.2424 )
υ = 1.0482 y / ( y 0.15735 x + 0.2424 ) .
σ ( 1 + r 2 / 0.968 2 ) = σ ( 1 + 1.063 r 2 ) ,
σ ( 1 + 5.062 2 / 0.968 2 ) = σ ( 1 + 27.25 r 2 ) .
u = 3.7676 x 1.1077 y 0.8987 . w = 2.7725 y 1.5035 .
U sin 1 2 ( β 1 + β 2 ) + V cos 1 2 ( β 1 + β 2 ) = ( U 2 + V 2 + W 2 ) 1 2 cos 1 2 ( β 2 β 1 ) .
V = 1 ( U 2 + W 2 ) / 4 .
U sin 1 2 ( β 1 + β 2 ) + 2 V cos 1 2 β 1 cos 1 2 β 2 = 2 cos 1 2 ( β 2 β 1 ) .
[ U ( tan 1 2 β 1 + tan 1 2 β 2 ) 2 ] + W 2 = ( tan 1 2 β 2 tan 1 2 β 1 ) 2 .
( U + 2 m ) 2 + W 2 = 4 ( 1 + m ) 2 = R 2 .
R = 2 ( 1 + m 2 ) 1 2 = 2 ( 1 + U 1 2 ) 1 2 / U 1 .
U m = 2 [ ( 1 + m 2 ) 1 2 m ] = 2 [ ( 1 + U 1 2 ) 1 2 1 ] / U 1 .