Skip to main content
SpringerLink
Log in

Differential geometry and color perception

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Summary

Combined with well known facts about the affine structure of color space P, a local homogeneity hypothesis implies that P=G/K is equivalent to either R + xR +xR+ or R + SL(2, R)/SO(2). The G-invariant metrics yield in the first case Stiles' generalization of Helmholtz' color metric, in the second a new color metric with respect to which P is not isometric to a Euclidean space. The paper contains an extensive historical introduction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abney, W. de W.: Researches in Colour Vision. London: Longmans, Green 1913.

    Google Scholar 

  2. Abney, W. de W., Festung, E. R.: Colour photometry. Phil. Trans. Royal Soc. London 177, 423–456 (1886).

    Article  Google Scholar 

  3. Braun, H., Koecher, M.: Jordan Algebren. Berlin-Heideiberg-New York: Springer 1966.

    MATH  Google Scholar 

  4. Cornsweet, T. N.: Visual Perception. New York: Academic Press 1970.

    Google Scholar 

  5. Cornsweet, T. N., Pinsker, H. M.: Luminance discrimination of brief flashes under various conditions of adaptation. J. Physiology 176, 294–310 (1965).

    Google Scholar 

  6. Descartes, René: Météores 1638.

  7. Ekman, Gösta: Weber's Law and related functions. J. Psych. 47, 343–352 (1959).

    Google Scholar 

  8. Euler, Leonhard: Essai d'une explication physique des couleurs engendrées sur des surfaces extrêmement nunces. Mém. de l'académie des sci. de Berlin 8, 262–282 = Opera Omnia, Ser. (3), 5, No. 209, 156–171 (1752).

    Google Scholar 

  9. Fechner, G. T.: Sachen der Psychophysik. Leipzig 1877.

  10. Goethe, Johann Wolfgang von: Theory of Colours, 1810. (English trans., reprinted.) Cambridge: The MIT Press.

    Google Scholar 

  11. Graham, Clarence H., ed.: Vision and Visual Perception. New York: J. Wiley 1965.

    Google Scholar 

  12. Grassmann, Hermann Günter: Zur Theorie der Farbenmischung. Ann. Physik und Chemie 89. 69–84 (1853) = Phil. Mag. 7, 254–284 (1853).

    Article  Google Scholar 

  13. Hecht, S.: A theory of visual discrimination. J. General Physiology 18, 767–789 (1935).

    Article  Google Scholar 

  14. Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962.

    MATH  Google Scholar 

  15. Helmholtz, Hermann von: Über die Theorie der zusammengesetzten Farben. Ann. Physik und Chemie 87, 45–66 (1852).

    Article  Google Scholar 

  16. Helmholtz, Hermann von: Treatise on Physiological Optics. 1866. English trans. of first German edition. Optical Society of America, 1924.

  17. Helmholtz, Hermann von: On Goethe's scientific researches. 1876a. Popular Lectures, English trans. London: Longmans, Green and Co. 1908.

    Google Scholar 

  18. Helmholtz, Hermann von: On the relation of optics to painting. 1876b. Popular Lectures on Scientific Subjects, second series, English trans. London: Longmans, Green and Co. 1908.

    Google Scholar 

  19. Helmholtz, Hermann von: Versuch einer erweiterten Anwendung des Fechnerschen Gesetzes im Farbensystem. Z. Psychol. Physiol. Sinnesorg. 2, 1–30 (1891).

    Google Scholar 

  20. Helmholtz, Hermann von: Versuch, das psychophysische Gesetz auf die Farbenunterschiede trichromatischer Augen anzuwenden. Z. Psychol. Physiol. Sinnesorg. 3, 1–20 (1892a).

    Google Scholar 

  21. Helmholtz, Hermann von: Kürzeste Linien im Farbensystem. Z. Psychol. Physiol. Sinnesorg. 4, 108–122 (1892b).

    Google Scholar 

  22. Hering, E.: Zur Lehre vom Lichtsinne. Wien: Carl Gerold's Sohn 1878.

    Google Scholar 

  23. Hooke, Robert: Micrographia, Reprint of edition of 1665. New York: Dover Pub. 1961.

    Google Scholar 

  24. Jacobson, N.: Lie Algebras. New York: Interscience Pub. 1962.

    MATH  Google Scholar 

  25. Koecher, M.: Jordan Algebras and their Applications. Lecture notes, Univ. of Minnesota, 1962.

  26. Land, Edwin, McCann, John J.: Lightness and retinex theory. J. Opt. Soc. Amer. 61, 1–11 (1971).

    Google Scholar 

  27. MacAdam, David L.: Color discrimination and the influence of color contrast on visual acuity. Rev. d'Optique 28, 161–173 (1949).

    Google Scholar 

  28. MacAdam, David L.: Loci of constant hue and brightness determined with various surround colors. J. Opt. Soc. Amer. 40, 589–595 (1950).

    Article  Google Scholar 

  29. MacAdam, David L.: Sources of Color Science. Cambridge: The MIT Press 1970.

    Google Scholar 

  30. Mach, Ernst: The Principles of Physical Optics. 1913. Reprint of English trans. New York: Dover Pub.

    Google Scholar 

  31. Maxwell, James Clerk: Theory of the perception of colors. Trans. Royal Scottish Soc. Arts 4, 394–400 (1856).

    Google Scholar 

  32. Maxwell, James Clerk: The diagram of colors. Trans. Royal Soc. Edinburgh 21, 275–298 (1857).

    Google Scholar 

  33. Maxwell, James Clerk: Theory of compound colors, and the relations of the colors of the spectrum. Proc. Royal Soc. London 10, 404–409 (1860).

    Article  Google Scholar 

  34. Maxwell, James Clerk: On color vision. Proc. Royal Inst. Great Britain 6, 260–271 (1872).

    Google Scholar 

  35. Newton, Isaac: New theory about light and colors. Phil. Trans. Royal Soc. London 80, 3075–3087 (1671).

    Google Scholar 

  36. Newton, Isaac: Opticks. London: Smith and Walford 1704.

    Google Scholar 

  37. Resnikoff, H. L.: A generalization of the psychophysical laws of Fechner and Stevens, to appear.

  38. Riemann, Bernhard: Über die Hypothesen, weiche der Geometrie zu Grunde liegen. 1854, in: Werke, reprint. New York: Dover Pub. 1953.

    Google Scholar 

  39. Riggs, L. A., Ratliff, F., Cornsweet, J. C., Cornsweet, T. N.: The disappearance of steadily fixated test objects. J. Opt. Soc. Amer. 43, 495–501 (1953).

    Article  Google Scholar 

  40. Schier, Donald S.: Louis Bertrand Castel, Anti-Newtonian Scientist. Cedar Rapids: The Torch Press 1941.

    Google Scholar 

  41. Schrödinger, E.: Grundlinien einer Theorie der Farbenmetrik in Tagessehen. Ann. Physik 63, 397–426, 427–456, 481–520 (1920).

    Article  Google Scholar 

  42. Stevens, S. S.: On the psychophysical law. Psych. Rev. 64, 153–181 (1957).

    Article  Google Scholar 

  43. Stevens, S. S.: Neural events and the psychophysical law. Science 170, 1043–1050 (1970).

    Article  Google Scholar 

  44. Stiles, W. S.: A modified Helmholtz line-element in brightness-colour space. Proc. Phys. Soc. London 18, 41–65 (1946).

    Google Scholar 

  45. Stiles, W. S., Wyszecki, Günter: Color Science. New York: J. Wiley and Sons 1967.

    Google Scholar 

  46. Vos, J. J., Walraven, P. L.: An analytical description of the line element in the zone-fluctuation model of colour vision. — I. Basic concepts. — II. Derivation of the line element. Vision Research 12, 1327–1344, 1345–1365 (1972).

    Article  Google Scholar 

  47. Whittaker, E. T.: A History of the Theories of Aether and Electricity, Vol. 1: Classical Theories. 1910. London: Thomas Nelson and Sons, revised 1951.

    Google Scholar 

  48. Yilmaz, H.: On color perception, in: Evidence for Gravitational Theories, pp. 239–251. New York: Academic Press 1962.

    Google Scholar 

  49. Young, Thomas: On the theory of light and colors. Phil. Trans. Royal Soc. London 92, 20–71 (1802).

    Google Scholar 

  50. Young, Thomas: A Course of Lectures on Natural Philosophy and the Mechanical Arts. London: Taylor and Walton 1845.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rice University, 77001, Houston, TX, USA

    H. L. Resnikoff

Authors
  1. H. L. Resnikoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Resnikoff, H.L. Differential geometry and color perception. J. Math. Biology 1, 97–131 (1974). https://doi.org/10.1007/BF00275798

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00275798

Keywords

Search

Navigation