Abstract

This paper is a supplementary article on counting metamers to one presented earlier in this journal. Whereas in the previous study the basic collection of object-color stimuli contained “jagged” spectral reflectance curves, including a great many that exhibit extreme reflectance variations within small spectral ranges, the present paper makes use of basic collections which contain “smooth” spectral reflectance curves generated by frequency-limited functions. The results obtained in the present study are similar to those obtained previously, particularly when the limiting frequency of the spectral reflectance functions is set at ω = 1/50. Such a limiting frequency leads to spectral reflectance curves of typically four oscillations within the visible spectrum, making them resemble practical spectral reflectance curves.

© 1977 Optical Society of America

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References

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  1. W. S. Stiles and G. W. Wyszecki, J. Opt. Soc. Am. 52, 313 (1962).
    [Crossref]
  2. See, for example, S. Goldman, Information Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1953).
  3. Commission Internationale de l’Eclairage (CIE), Colorimetry: Official Recommendations of the International Commission on Illumination, Publ. CIE No. 15 (E–1. 3. 1), 1971 (Bureau Central de la CIE, 4 Av. du Recteur Poincare, 75 Paris 16e, France).
  4. See, for example, R. K. Eisenschitz, Matrix Algebra for Physicists (Heinemann, London, 1966),p. 7.

1962 (1)

Eisenschitz, R. K.

See, for example, R. K. Eisenschitz, Matrix Algebra for Physicists (Heinemann, London, 1966),p. 7.

Goldman, S.

See, for example, S. Goldman, Information Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1953).

Stiles, W. S.

Wyszecki, G. W.

J. Opt. Soc. Am. (1)

Other (3)

See, for example, S. Goldman, Information Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1953).

Commission Internationale de l’Eclairage (CIE), Colorimetry: Official Recommendations of the International Commission on Illumination, Publ. CIE No. 15 (E–1. 3. 1), 1971 (Bureau Central de la CIE, 4 Av. du Recteur Poincare, 75 Paris 16e, France).

See, for example, R. K. Eisenschitz, Matrix Algebra for Physicists (Heinemann, London, 1966),p. 7.

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Figures (4)

FIG. 1
FIG. 1

Two examples of frequency-limited functions representing spectral reflectance curves. The solid curve has a limiting frequency of ω = 1 50; the dashed curves one of ω = 1 100.

FIG. 2
FIG. 2

Portion of CIE 1931 (x, y) chromaticity diagram showing 7 cross sections of color-mismatch ellipsoid containing 95% of all possible (D65)-gray metamers of Y = 50 viewed by the CIE 1931 standard observer under CIE standard illuminant A. The cross sections are located at different Y values as indicated in the graph. The calculations are based on spectral reflectance curves with a limiting frequency of ω = 1 50. The point marked A is the chromaticity point of CIE standard illuminant A.

FIG. 3
FIG. 3

Portion of CIE 1931 (x, y) chromaticity diagram showing three cross sections of color-mismatch ellipsoid containing 95% of all possible (D65)-gray metamers of Y = 50 viewed by the CIE 1931 standard observer under CIE standard illuminant A. The cross sections are located at different Y values as indicated in the graph. The calculations are based on spectral reflectance curves with a limiting frequency of ω = 1 100. The point marked A is the chromaticity point of CIE standard illuminant A,

FIG. 4
FIG. 4

Portion of CIE 1931 (x, y) chromaticity diagram showing the envelopes of 3 color-mismatch ellipsoids. The two solid lines represent the ellipsoids derived on the basis of basic collections contracted by means of frequency-limited functions (one with ω = 1 50, the other with ω = 1 100). The dashed line represents the ellipsoid derived on the basis of a basic collection constructed by means of jagged spectral reflectance functions. The point marked A is the chromaticity point of CIE standard illuminant A.

Equations (39)

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ρ ( λ ) = 1 2 + 1 2 ϕ ( λ ) ,
ϕ ( λ ) = i = + Ψ i 2 sin 2 [ π ( λ λ 0 ) ω i π / 2 ] [ π ( λ λ 0 ) ω i π / 2 ] 2 .
1 Ψ i + 1 ( i = 0 , ± 1 , ± 2 , etc . ) ;
i = + may be replaced by i = N + N ,
0 ρ ( λ ) 1
F ( Ψ N , Ψ N + 1 , , Ψ 0 , , Ψ N 1 , Ψ N ) × d Ψ N d Ψ N + 1 d Ψ 0 d Ψ N 1 d Ψ N .
F ( Ψ N , , Ψ 0 , , Ψ N ) = F N ( Ψ N ) F 0 ( Ψ 0 ) F N ( Ψ N ) .
F i ( Ψ i ) d Ψ i ,
1 + 1 F i ( Ψ i ) d Ψ i = 1 .
F i ( Ψ i ) = 1 2 .
X = λ ρ ( λ ) E ( λ ) x ¯ ( λ ) d λ , Y = λ ρ ( λ ) E ( λ ) y ¯ ( λ ) d λ , Z = λ ρ ( λ ) E ( λ ) z ¯ ( λ ) d λ .
X k = λ ρ ( λ ) x ¯ k ( λ ) d λ ,
X 1 = Y x ¯ 1 ( λ ) = E ( λ ) y ¯ ( λ ) , X 2 = X x ¯ 2 ( λ ) = E ( λ ) x ¯ ( λ ) , X 3 = Z x ¯ 3 ( λ ) = E ( λ ) z ¯ ( λ ) .
X k = 1 2 λ x ¯ k ( λ ) d λ + 1 2 λ ϕ ( λ ) x ¯ k ( λ ) d λ = X k c + i = N + N p k i Ψ i i ,
X k c = 1 2 λ x ¯ k ( λ ) d λ ,
p k i = 1 4 λ x ¯ k ( λ ) sin 2 [ π ( λ λ 0 ) ω i π / 2 ] [ π ( λ λ 0 ) ω i π / 2 ] 2 d λ .
X 1 = Y x ¯ 1 ( λ ) = E ( λ ) y ¯ ( λ ) , X 2 = X x ¯ 2 ( λ ) = E ( λ ) x ¯ ( λ ) , X 3 = Z x ¯ 3 ( λ ) = E ( λ ) z ¯ ( λ ) , X 4 = Y x ¯ 4 ( λ ) = F ( λ ) y ¯ ( λ ) , X 5 = X x ¯ 5 ( λ ) = F ( λ ) x ¯ ( λ ) , X 6 = Z x ¯ 6 ( λ ) = F ( λ ) z ¯ ( λ ) .
ū k ( λ ) = l = 1 6 t k l x ¯ l ( λ ) ,
U k = λ ρ ( λ ) ū k ( λ ) d λ = λ ρ ( λ ) l = 1 6 t k l x ¯ l ( λ ) d λ = l = 1 6 t k l λ ρ ( λ ) x ¯ l ( λ ) d λ = l = 1 6 t k l X l .
ν k = U k l = 1 6 t k l X l c .
ν k = i = N + N Ψ i l = 1 6 t k l p l i .
Ω ( ν 1 , ν 2 , , ν 6 ) d ν 1 d ν 2 d ν 6 = d ν 1 d ν 6 ( 2 π ) 3 exp 1 2 [ ( ν 1 ν 1 m ) 2 + + ( ν 6 ν 6 m ) 2 ] ,
ν k m = i = N + N l = 1 6 t k l p l i 1 + 1 Ψ i F i ( Ψ i ) d Ψ i
i = N + N ( l = 1 6 t k l p l i ) 2 θ i = 1
i = N + N ( l = 1 6 t k l p l i ) ( l = 1 6 t j l p l i ) θ i = 0
θ i = 1 + 1 Ψ 2 i F i ( Ψ i ) d Ψ i ( 1 + 1 Ψ i F i ( Ψ i ) d Ψ i ) 2 ,
θ i = const = 1 3
t 11 0 0 0 0 0 t 21 t 22 0 0 0 0 t 31 t 32 t 33 0 0 0 t 41 t 42 t 43 t 44 0 0 t 51 t 52 t 53 t 54 t 55 0 t 61 t 62 t 63 t 64 t 65 t 66 .
Ω ( ν 4 , ν 5 , ν 6 ) d ν 4 d ν 5 d ν 6 = d ν 4 d ν 5 d ν 6 ( 2 π ) 3 / 2 exp 1 2 [ ( ν 4 ν 4 m ) 2 + ( ν 5 ν 5 m ) 2 + ( ν 6 ν 6 m ) 2 ] .
X 1 = X 10 , X 2 = X 20 , X 3 = X 30 .
( ν 4 ν 4 m ) 2 + ( ν 5 ν 5 m ) 2 + ( ν 6 ν 6 m ) 2 = χ 2 ,
k = F ( χ 2 ) ,
τ 11 ( X 4 X 40 ) 2 + τ 22 ( X 5 X 50 ) 2 + τ 33 ( X 6 X 60 ) 2 + 2 τ 12 ( X 4 X 40 ) ( X 5 X 50 ) + 2 τ 23 ( X 5 X 50 ) ( X 6 X 60 ) + 2 τ 31 ( X 6 X 60 ) ( X 4 X 40 ) = χ 2 ,
τ 11 = t 44 2 + t 54 2 + t 64 2 , τ 12 = t 54 t 55 + t 64 t 65 , τ 22 = t 55 2 + t 65 2 , τ 23 = t 65 t 66 , τ 33 = t 66 2 , τ 31 = t 64 t 66 ,
X 40 = X 4 c 1 t 44 l = 1 3 t 4 l ( X l X l c ) , X 50 = X 5 c + t 54 t 44 t 55 l = 1 3 t 4 l ( X l X l c ) 1 t 55 l = 1 3 t 5 l ( X l X l c ) , X 60 = X 6 c + t 64 t 66 t 44 t 65 t 54 t 44 t 55 t 66 l = 1 3 t 4 l ( X l X l c ) + t 65 t 55 t 66 l = 1 3 t 5 l ( X l X l c ) 1 t 66 l = 1 3 t 6 l ( X l X l c ) ,
X l c = 1 2 λ x ¯ l ( λ ) d λ
λ 0 = 550 nm ,
sin 2 [ π ( λ λ 0 ) ω i π / 2 ] [ π ( λ λ 0 ) ω i π / 2 ] 2
i = 30 + 30 p 2 i / i = 55 + 55 p 2 i = 0.997 .