Geometry of color perception. Part 1: structures and metrics of a homogeneous color space

The Journal of Mathematical Neuroscience - 2020
Edoardo Provenzi1
1CNRS, Bordeaux INP, IMB, UMR 5251, Université de Bordeaux, Talence, France

Tóm tắt

AbstractThis is the first half of a two-part paper dealing with the geometry of color perception. Here we analyze in detail the seminal 1974 work by H.L. Resnikoff, who showed that there are only two possible geometric structures and Riemannian metrics on the perceived color space $\mathcal{P} $ P compatible with the set of Schrödinger’s axioms completed with the hypothesis of homogeneity. We recast Resnikoff’s model into a more modern colorimetric setting, provide a much simpler proof of the main result of the original paper, and motivate the need of psychophysical experiments to confute or confirm the linearity of background transformations, which act transitively on $\mathcal{P} $ P . Finally, we show that the Riemannian metrics singled out by Resnikoff through an axiom on invariance under background transformations are not compatible with the crispening effect, thus motivating the need of further research about perceptual color metrics.

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Tài liệu tham khảo

Resnikoff HL. Differential geometry and color perception. J Math Biol. 1974;1:97–131.

Resnikoff HL. On the geometry of color perception. In: Some mathematical questions in biology. VI; 1974. p. 217–32. (Lectures on mathematics in the life sciences; vol. 7).

Resnikoff HL. On the psychophysical function. J Math Biol. 1975;2(3):265–76.

Resnikoff HL. The illusion of reality. Berlin: Springer; 2012.

Niall KK, editor. Erwin Schrödinger’s color theory: translated with modern commentary; 2017.

Schrödinger E. Collected papers on wave mechanics. Providence: American Mathematical Society; 2003.

Wyszecky G, Stiles WS. Color science: concepts and methods, quantitative data and formulas. New York: Wiley; 1982.

Schanda J. Colorimetry: Understanding the CIE system. New York: Wiley; 2007.

Rudd ME, Zemach IK. Quantitive properties of achromatic color induction: an edge integration analysis. Vis Res. 2004;44:971–81.

Gronchi G, Provenzi E. A variational model for context-driven effects in perception and cognition. J Math Psychol. 2017;77:124–41.

Dubois E. The structure and properties of color spaces and the representation of color images; 2009. (Synthesis lectures on image, video, and multimedia processing; vol. 4).

Faraut J, Koranyi A. Analysis on symmetric cones. Oxford: Clarendon Press; 1994.

Schrödinger E. Grundlinien einer Theorie der Farbenmetrik im Tagessehen (Outline of a theory of colour measurement for daylight vision). Available in English in: MacAdam DL, editor. Sources of colour science. Cambridge: MIT Press; 1970. p. 134–182. Ann Phys. 1920;63(4):397–456; 481–520.

Newton I. Opticks, or, a treatise of the reflections, refractions, inflections & colours of light. North Chelmsford: Courier Corporation; 1952.

Grassmann H. Zur theorie der farbenmischung. Ann Phys. 1853;165(5):69–84.

von Helmholtz H, Southall JPC. Treatise on physiological optics. Vol. 3. North Chelmsford: Courier Corporation; 2005.

Munkres J. Topology. 2nd ed. Upper Saddle River: Pearson; 2000.

Ashtekar A, Corichi A, Pierri M. Geometry in color perception. In: Black holes, gravitational radiation and the universe; 1999. p. 535–50.

Wallach H. Brightness constancy and the nature of achromatic colors. J Exp Psychol. 1948;38(3):310–24.

Helgason S. Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press; 1979. (Pure and applied mathematics; vol. 80).

Warner FW. Foundations of differentiable manifolds and Lie groups. Berlin: Springer; 2013. (Graduate texts in mathematics; vol. 94).

Gonzales RC, Woods RE. Digital image processing. New York: Prentice Hall: 2002.

Lie S. Theorie der transformationsgruppen III. Leipzig: Teubner; 1893.

Komrakov B, Churyumov A, Doubrov B. Two-dimensional homogeneous spaces. Matematisk Institutt, Universitetet i Oslo (1993).

Doubrov B, Komrakov B. Low-dimensional pseudo-Riemannian homogeneous spaces. Matematisk Institutt, Universitetet i Oslo (1995).

Yilmaz H. Color vision and a new approach to general perception. In: Biological prototypes and synthetic systems; 1962. p. 126–41.

Martelli B. An introduction to geometric topology. arXiv:1610.02592 (2016).

Amari S. Differential-geometrical methods in statistics. Berlin: Springer; 2012. (Lecture notes in statistics; vol. 28).

Calvo M, Oller JM. A distance between multivariate normal distributions based in an embedding into the Siegel group. J Multivar Anal. 1990;35(2):223–42.

Siegel CL. Symplectic geometry. Amsterdam: Elsevier; 2014.

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