Abstract

Isometric embedding of non-Euclidean color spaces into Euclidean color spaces is investigated. Owing to regions of nonzero Gaussian curvature within common non-Euclidean color spaces, we focus on the determination of transformations into Euclidean spaces with minimal isometric disagreement. A computational method is presented for deriving such a color space transformation by means of a multigrid optimization, resulting in a simple color look-up table. The multigrid optimization is applied on the CIELAB space with the CMC, CIE94, and CIEDE2000 formulas. The mean disagreement between distances calculated by these formulas and Euclidean distances within the new spaces is far below 3% for all investigated color difference formulas. Color space transformations containing the inverse transformations are provided as MATLAB scripts at the first author’s website.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Committee of the Society of Dyers and Colorists, "BS 6923: method for calculation of small color differences," (British Standards Institution, 1988).
  2. CIE Publication No. 116, "Industrial color-difference evaluation," (CIE Central Bureau, 1995).
  3. CIE Publication No. 142, "Improvement to industrial color difference evaluation," (CIE Central Bureau, 2001).
  4. M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
    [CrossRef]
  5. D. B. Judd, "Ideal color space: curvature of color space and its implications for industrial color tolerances," Palette 29, 25-31 (1968).
  6. D. B. Judd, "Ideal color space: the super-importance of hue differences and its bearing on the geometry of color space," Palette 30, 21-28 (1968).
  7. D. B. Judd, "Ideal color space: ideal color space redefined," Palette 31, 23-29 (1969).
  8. R. G. Kuehni, Color Space and Its Divisions, 1st ed. (Wiley, 2003).
    [CrossRef]
  9. G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 2000).
  10. CIE Publication No. 101, "Parametric effects in color difference evaluation" (CIE Central Bureau, 1993).
  11. D. L. MacAdam, "Nonlinear relations of psychometric scale values to chromaticity differences," J. Opt. Soc. Am. 53, 754-757 (1963).
    [CrossRef]
  12. H. G. Völz, "Die Berechnung grosser Farbabstände in nichteuklidischen Farbräumen," Farbe 44, 1-45 (1998).
  13. H. G. Völz, "Transformation der CIE94-Formel in einen euklidischen Farbraum," Farbe 44, 97-105 (1998).
  14. H. G. Völz, "Die Euklidisierung des CMC-Raumes zur Berechnung grosser Farbabstände," Farbe 45, 1-23 (1998).
  15. H. G. Völz, "Euclidization of the first quadrant of the CIEDE2000 color difference system for the calculation of large color differences," Color Res. Appl. 31, 5-12 (2006).
    [CrossRef]
  16. K. Thomsen, "A Euclidean color space in high agreement with the CIE94 color difference formula," Color Res. Appl. 64-65, 404-411 (2000).
  17. E. Rohner and D. C. Rich, "Eine angenährt gleichförmige Farbabstandsformel für industrielle Farbtoleranzen," Farbe 42, 207-220 (1996).
  18. D. C. Rich, "Euclidean color spaces with logarithmic compression: a comment on Knud Thomsen's note," Color Res. Appl. 25, 293 (2000).
    [CrossRef]
  19. DIN6176, "Farbmetrische Bestimmung von Farbabständen bei Körperfarben nach der DIN99-Formel" (DIN Deutsches Institut für Normung e.V, 2000).
  20. G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
    [CrossRef]
  21. J. J. Stocker, Differential Geometry (Wiley, 1969).
  22. H. W. Guggenheimer, Differential Geometry (McGraw-Hill, 1963).
  23. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).
  24. R. Fletcher, Practical Methods of Optimizations, 2nd ed. (Wiley, 1997).
  25. S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
    [CrossRef]
  26. G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
    [CrossRef]

2006

H. G. Völz, "Euclidization of the first quadrant of the CIEDE2000 color difference system for the calculation of large color differences," Color Res. Appl. 31, 5-12 (2006).
[CrossRef]

2005

G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
[CrossRef]

2001

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

2000

D. C. Rich, "Euclidean color spaces with logarithmic compression: a comment on Knud Thomsen's note," Color Res. Appl. 25, 293 (2000).
[CrossRef]

K. Thomsen, "A Euclidean color space in high agreement with the CIE94 color difference formula," Color Res. Appl. 64-65, 404-411 (2000).

1999

S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
[CrossRef]

1998

H. G. Völz, "Die Berechnung grosser Farbabstände in nichteuklidischen Farbräumen," Farbe 44, 1-45 (1998).

H. G. Völz, "Transformation der CIE94-Formel in einen euklidischen Farbraum," Farbe 44, 97-105 (1998).

H. G. Völz, "Die Euklidisierung des CMC-Raumes zur Berechnung grosser Farbabstände," Farbe 45, 1-23 (1998).

1996

E. Rohner and D. C. Rich, "Eine angenährt gleichförmige Farbabstandsformel für industrielle Farbtoleranzen," Farbe 42, 207-220 (1996).

1969

D. B. Judd, "Ideal color space: ideal color space redefined," Palette 31, 23-29 (1969).

1968

D. B. Judd, "Ideal color space: curvature of color space and its implications for industrial color tolerances," Palette 29, 25-31 (1968).

D. B. Judd, "Ideal color space: the super-importance of hue differences and its bearing on the geometry of color space," Palette 30, 21-28 (1968).

1963

Cui, G.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
[CrossRef]

Dalal, E. N.

G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimizations, 2nd ed. (Wiley, 1997).

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Guan, S. S.

S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
[CrossRef]

Guggenheimer, H. W.

H. W. Guggenheimer, Differential Geometry (McGraw-Hill, 1963).

Judd, D. B.

D. B. Judd, "Ideal color space: ideal color space redefined," Palette 31, 23-29 (1969).

D. B. Judd, "Ideal color space: the super-importance of hue differences and its bearing on the geometry of color space," Palette 30, 21-28 (1968).

D. B. Judd, "Ideal color space: curvature of color space and its implications for industrial color tolerances," Palette 29, 25-31 (1968).

Kuehni, R. G.

R. G. Kuehni, Color Space and Its Divisions, 1st ed. (Wiley, 2003).
[CrossRef]

Luo, M. R.

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
[CrossRef]

MacAdam, D. L.

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Rich, D. C.

D. C. Rich, "Euclidean color spaces with logarithmic compression: a comment on Knud Thomsen's note," Color Res. Appl. 25, 293 (2000).
[CrossRef]

E. Rohner and D. C. Rich, "Eine angenährt gleichförmige Farbabstandsformel für industrielle Farbtoleranzen," Farbe 42, 207-220 (1996).

Rigg, B.

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

Roesler, G.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

Rohner, E.

E. Rohner and D. C. Rich, "Eine angenährt gleichförmige Farbabstandsformel für industrielle Farbtoleranzen," Farbe 42, 207-220 (1996).

Sharma, G.

G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
[CrossRef]

Stiles, W.

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 2000).

Stocker, J. J.

J. J. Stocker, Differential Geometry (Wiley, 1969).

Thomsen, K.

K. Thomsen, "A Euclidean color space in high agreement with the CIE94 color difference formula," Color Res. Appl. 64-65, 404-411 (2000).

Völz, H. G.

H. G. Völz, "Euclidization of the first quadrant of the CIEDE2000 color difference system for the calculation of large color differences," Color Res. Appl. 31, 5-12 (2006).
[CrossRef]

H. G. Völz, "Transformation der CIE94-Formel in einen euklidischen Farbraum," Farbe 44, 97-105 (1998).

H. G. Völz, "Die Euklidisierung des CMC-Raumes zur Berechnung grosser Farbabstände," Farbe 45, 1-23 (1998).

H. G. Völz, "Die Berechnung grosser Farbabstände in nichteuklidischen Farbräumen," Farbe 44, 1-45 (1998).

Witt, K.

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Wu, W.

G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
[CrossRef]

Wyszecki, G.

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 2000).

Color Res. Appl.

M. R. Luo, G. Cui, and B. Rigg, "The development of the CIE 2000 color-difference formula: CIEDE2000," Color Res. Appl. 26, 340-350 (2001).
[CrossRef]

H. G. Völz, "Euclidization of the first quadrant of the CIEDE2000 color difference system for the calculation of large color differences," Color Res. Appl. 31, 5-12 (2006).
[CrossRef]

K. Thomsen, "A Euclidean color space in high agreement with the CIE94 color difference formula," Color Res. Appl. 64-65, 404-411 (2000).

D. C. Rich, "Euclidean color spaces with logarithmic compression: a comment on Knud Thomsen's note," Color Res. Appl. 25, 293 (2000).
[CrossRef]

G. Cui, M. R. Luo, B. Rigg, G. Roesler, and K. Witt, "Uniform color spaces based on the DIN99 colour-difference formula," Color Res. Appl. 27, 282-290 (2001).
[CrossRef]

S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
[CrossRef]

G. Sharma, W. Wu, and E. N. Dalal, "The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations," Color Res. Appl. 30, 21-30 (2005).
[CrossRef]

Farbe

E. Rohner and D. C. Rich, "Eine angenährt gleichförmige Farbabstandsformel für industrielle Farbtoleranzen," Farbe 42, 207-220 (1996).

H. G. Völz, "Die Berechnung grosser Farbabstände in nichteuklidischen Farbräumen," Farbe 44, 1-45 (1998).

H. G. Völz, "Transformation der CIE94-Formel in einen euklidischen Farbraum," Farbe 44, 97-105 (1998).

H. G. Völz, "Die Euklidisierung des CMC-Raumes zur Berechnung grosser Farbabstände," Farbe 45, 1-23 (1998).

J. Opt. Soc. Am.

Palette

D. B. Judd, "Ideal color space: curvature of color space and its implications for industrial color tolerances," Palette 29, 25-31 (1968).

D. B. Judd, "Ideal color space: the super-importance of hue differences and its bearing on the geometry of color space," Palette 30, 21-28 (1968).

D. B. Judd, "Ideal color space: ideal color space redefined," Palette 31, 23-29 (1969).

Other

R. G. Kuehni, Color Space and Its Divisions, 1st ed. (Wiley, 2003).
[CrossRef]

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 2000).

CIE Publication No. 101, "Parametric effects in color difference evaluation" (CIE Central Bureau, 1993).

Committee of the Society of Dyers and Colorists, "BS 6923: method for calculation of small color differences," (British Standards Institution, 1988).

CIE Publication No. 116, "Industrial color-difference evaluation," (CIE Central Bureau, 1995).

CIE Publication No. 142, "Improvement to industrial color difference evaluation," (CIE Central Bureau, 2001).

DIN6176, "Farbmetrische Bestimmung von Farbabständen bei Körperfarben nach der DIN99-Formel" (DIN Deutsches Institut für Normung e.V, 2000).

J. J. Stocker, Differential Geometry (Wiley, 1969).

H. W. Guggenheimer, Differential Geometry (McGraw-Hill, 1963).

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

R. Fletcher, Practical Methods of Optimizations, 2nd ed. (Wiley, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Gaussian curvature of the a * × b * plane for the CMC formula is negative (anticlastic) as well as positive (synclastic).

Fig. 2
Fig. 2

Gaussian curvature of the a * × b * plane for the CIE94 metric is in each point positive (synclastic) and, except in the region around the gray axes, very small.

Fig. 3
Fig. 3

Gaussian curvature of the a * × b * plane for the CIEDE2000 metric. The effect of the rotation term at a hue of about 275 ° is clearly noticeable. In this region the curvature is negative (anticlastic).

Fig. 4
Fig. 4

Two starting grids (solid and dashed) covering the a * × b * plane of the CIELAB space.

Fig. 5
Fig. 5

Preliminary step: Calculation of the Δ E distances d i , k 0 , d i , k 1 , d i , k 2 , d i , k 3 between the mesh vertices y k 0 , y k 1 , y k 2 , y k 3 of one starting grid (solid lines) and the enclosed vertex x i of the other starting grid (dashed lines).

Fig. 6
Fig. 6

Step of the gradient-descent method for the vertex x i and the mesh with vertices y k 0 , y k 1 , y k 2 , y k 3 M j . Since the step length s optim would destroy the grid topology, we have to choose a smaller step length s topo to ensure that x i does not move outside the area enclosed by the mesh polygon (thick solid lines).

Fig. 7
Fig. 7

Triangulation (dashed lines) of a grid resulting from the multigrid optimization (solid gray lines). To transform a point z T Δ E (CIELAB) into CIELAB, the enclosing triangle has to be found, i.e., the triangle defined by z i 1 , z i 2 , z i 3 . The transformation can be performed by triangular interpolation using the areas A ( z , z i 2 , z i 3 ) , A ( z i 1 , z , z i 3 ) , A ( z i 1 , z i 2 , z ) .

Fig. 8
Fig. 8

Lightness L C M C * according to the integral in Eq. (27) calculated for l = 1 .

Fig. 9
Fig. 9

(a) Result of the multigrid optimization for the CMC color difference formula. The calculation has been performed for l , c = 1 . (b) Result of the transformation of the rectangular grid H from the new color space into the CIELAB color space using the look-up table inverse transformation T C M C 1 as described in Subsection 2H2.

Fig. 10
Fig. 10

(a) Unity color-tolerance ellipsoids for the CMC metric, shown here as ellipses projected onto the a * × b * plane. (b) The same ellipsoids transformed into the new color space by T C M C and shown here as ellipses projected onto the a C M C * × b C M C * plane.

Fig. 11
Fig. 11

(a) Euclidean distances in CIELAB (i.e., Δ E a b * ) plotted against CMC distances. (b) DETCMC (i.e., Euclidean metric in the new color space) distances plotted against CMC distances.

Fig. 12
Fig. 12

(a) Result of the multigrid optimization for the CIE94 color difference formula. The calculation has been performed for k C , k H = 1 . (b) Result of the transformation of a rectangular grid from the new color space into the CIELAB color space using the look-up table inverse transformation as described in Subsection 2H2.

Fig. 13
Fig. 13

(a) Unity color-tolerance ellipsoids for the CIE94 metric, shown here as ellipses projected onto the a * × b * plane. (b) The same ellipsoids transformed into the new color space by T 94 and shown here as ellipses projected onto the a 94 * × b 94 * plane.

Fig. 14
Fig. 14

(a) Euclidean distances in CIELAB (i.e., Δ E a b * ) plotted against CIE94 distances. (b) DET94 (i.e., Euclidean metric in the new color space) distances plotted against CIE94 distances.

Fig. 15
Fig. 15

Lightness L 00 * according to the integral in Eq. (30) calculated for k L = 1 .

Fig. 16
Fig. 16

Result of the multigrid optimization for the CIEDE2000 color difference formula. The calculation has been performed for k C , k H = 1 . (b) Result of the transformation of a rectangular grid from the new color space into the CIELAB color space using the look-up table inverse transformation T 00 1 as described in Subsection 2H2.

Fig. 17
Fig. 17

(a) Unity color-tolerance ellipsoids for the CIEDE2000 metric, shown here as ellipses projected onto the a * × b * plane. (b) The same ellipsoids transformed into the new color space by T 00 and shown here as ellipses projected onto the a 00 * × b 00 * plane.

Fig. 18
Fig. 18

(a) Euclidean distances in CIELAB (i.e., Δ E a b * ) plotted against CIEDE2000 distances. (b) DET00 (i.e., Euclidean metric in the new color space) distances plotted against CIEDE2000 distances.

Tables (3)

Tables Icon

Table 1 Disagreement with the CMC Color Difference Formula

Tables Icon

Table 2 Disagreement with the CIE94 Color Difference Formula

Tables Icon

Table 3 Disagreement with the CIEDE2000 Color Difference Formula

Equations (31)

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

CIELAB ( K ( y , r ) ( Δ E ( x , y ) T Δ E ( x ) T Δ E ( y ) 2 ) 2 d x ) 1 2 d y = ! min .
G : { I R 2 i x i .
G ( i + 1 , j ) G ( i , j ) = ( c 1 , 0 ) T , c 1 > 0 , i = 1 , , M 1 ,
G ( i , j + 1 ) G ( i , j ) = ( 0 , c 2 ) T , c 2 > 0 , j = 1 , , N 1 .
I j { j 1 , j 1 + 1 } × { j 2 , j 2 + 1 } .
M j G ( I j ) .
G ¯ = { G 1 G = G 2 G 2 G = G 1 , I ¯ = { I 1 I = I 2 I 2 I = I 1 .
d i , k Δ E ( x i , y k ) .
x i y k 2 = x i y l 2 .
F j ( x i ) = k I j ( d i , k x i y k 2 ) 2 = min .
x i = x i + s x i j d x i j ,
d x i j F j ( x i ) F j ( x i ) 2 ,
s x i j min { s optim , s topo } ,
0 < μ 1 s optim F j ( x i ) T d x i j F j ( x i ) F j ( x i + s optim d x i j ) μ 2 s optim F j ( x i ) T d x i j ,
F j ( x i + s optim d x i j ) F j ( x i ) + s optim ρ F j ( x i ) T d x i j ,
F j ( x i + s optim d x i j ) T d x i j μ F j ( x i ) T d x i j ,
D max ( x i , M j ) = max { max { d i , k , x i y k 2 } min { d i , k , x i y k 2 } k I j } .
M j G ( I ) x i enclosed by M j D max ( x i , M j ) D max ( x i + s x i j d x i j , M j ) 2 2 < ϵ .
f j ( x i ) = k I j ( d i , k x i y k 2 ) x i y k x i y k 2 ,
f j ( x i ) = 0.5 F j ( x i ) .
T Δ E : { G ( I ) G ( I ) x i G ( G 1 ( x i ) ) .
T Δ E : { CIELAB R 3 ( L * , a * , b * ) ( L Δ E * ( L * ) , a Δ E * ( a * , b * ) , b Δ E * ( a * , b * ) ) ,
( a Δ E * 1 ( z ) , b Δ E * 1 ( z ) ) A ( z , z i 2 , z i 3 ) A ( z i 1 , z i 2 , z i 3 ) x i 1 + A ( z i 1 , z , z i 3 ) A ( z i 1 , z i 2 , z i 3 ) x i 2 + A ( z i 1 1 , z i 2 , z ) A ( z i 1 , z i 2 , z i 3 ) x i 3 ,
T Δ E 1 : { T Δ E ( CIELAB ) CIELAB ( L Δ E * , a Δ E * , b Δ E * ) ( L Δ E * 1 ( L Δ E * ) , a Δ E * 1 ( a Δ E * , b Δ E * ) , b Δ E * 1 ( a Δ E * , b Δ E * ) ) .
DET X ( u 1 , u 2 ) = T X ( u 1 ) T X ( u 2 ) 2 ,
Disagreement ( CMC , DETCMC ) = [ max { CMC , DETCMC } min { CMC , DETCMC } 1 ] * 100 % .
L C M C * ( L * ) = 0 L * d t l S L ( t ) ,
S L ( t ) = { 0.511 t < 16 0.04097 t 1 + 0.01765 t otherwise .
L 94 * ( L * ) = L * .
L 00 * ( L * ) = 0 L * d t k L S L ( t ) ,
S L ( t ) = 1 + 0.015 ( t 50 ) 2 20 + ( t 50 ) 2 .

Metrics