OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics


  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 7 — Jul. 16, 2007

Embedding non-Euclidean color spaces into Euclidean color spaces with minimal isometric disagreement

Philipp Urban, Mitchell R. Rosen, Roy S. Berns, and Dierk Schleicher  »View Author Affiliations

JOSA A, Vol. 24, Issue 6, pp. 1516-1528 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (570 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(330.0330) Vision, color, and visual optics : Vision, color, and visual optics
(330.1730) Vision, color, and visual optics : Colorimetry

ToC Category:
Vision and Color

Original Manuscript: October 31, 2006
Revised Manuscript: January 22, 2007
Manuscript Accepted: January 23, 2007
Published: May 9, 2007

Virtual Issues
Vol. 2, Iss. 7 Virtual Journal for Biomedical Optics

Philipp Urban, Mitchell R. Rosen, Roy S. Berns, and Dierk Schleicher, "Embedding non-Euclidean color spaces into Euclidean color spaces with minimal isometric disagreement," J. Opt. Soc. Am. A 24, 1516-1528 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited