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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 22, Iss. 5 — May. 1, 2005
  • pp: 820–827

Statistical properties of color-signal spaces

Reiner Lenz and Thanh Hai Bui  »View Author Affiliations


JOSA A, Vol. 22, Issue 5, pp. 820-827 (2005)
http://dx.doi.org/10.1364/JOSAA.22.000820


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Abstract

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron–Frobenius (and Krein–Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing.

© 2005 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(330.1690) Vision, color, and visual optics : Color

History
Original Manuscript: October 5, 2004
Manuscript Accepted: November 10, 2004
Published: May 1, 2005

Citation
Reiner Lenz and Thanh Hai Bui, "Statistical properties of color-signal spaces," J. Opt. Soc. Am. A 22, 820-827 (2005)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22-5-820


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References

  1. H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behaviour, W. H. Thorpe and O. L. Zangwill, eds. (Cambridge U. Press, Cambridge, UK, 1961) pp. 331–360.
  2. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987). [CrossRef] [PubMed]
  3. J. J. Atick, N. Redlich, “What does the retina know about natural scenes,” Neural Comput. 4, 449–572 (1992). [CrossRef]
  4. C. A. Párraga, G. Brelstaff, T. Troscianko, I. R. Moorehead, “Color and luminance information in natural scenes,” J. Opt. Soc. Am. A 15, 563–569 (1998). [CrossRef]
  5. P. Reinagel, S. Laughlin, “Natural stimulus statistics,” Network Comput. Neural Syst. 12, 237–240 (2001). [CrossRef]
  6. A. Levy, J. Rubinstein, “Hilbert-space Karhunen–Loeve transform with application to image analysis,” J. Opt. Soc. Am. A 16, 28–35 (1999). [CrossRef]
  7. F. R. Gantmacher, Matrizentheorie (Springer-Verlag, Berlin 1986). [CrossRef]
  8. N. Dunford, J. T. Schwartz, Part III, Spectral Operators (Interscience, New York, 1988).
  9. J. Toland, “Self-adjoint operators and cones,” J. London Math. Soc. Ser. 2 53, 167–183 (1996). [CrossRef]
  10. R. D. Nussbaum, “Eigenvectors of order-preserving linear operators,” J. London Math. Soc. Ser. 2 58, 480–496 (1998). [CrossRef]
  11. K. Yosida, Functional Analysis (Springer-Verlag, Berlin, 1978). [CrossRef]
  12. S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1999).
  13. R. Lenz, “Estimation of illumination characteristics,” IEEE Trans. Image Process. 10, 1031–1038 (2001). [CrossRef]
  14. R. Lenz, “Two stage principal component analysis of color,” IEEE Trans. Image Process. 11, 630–635 (2002). [CrossRef]
  15. H. Yilmaz, On Color Perception, Vol. XX of International School of Physics, Enrico Fermi (Academic, New York, 1962), pp. 239–251.
  16. H. L. Resnikoff, “Differential geometry and color perception,” J. Math. Biol. 1, 97–131 (1974). [CrossRef]
  17. M. Brill, G. West, “Group theory of chromatic adaptation,” Farbe 31, 4–22 (1983/1984).
  18. H. Fairman, M. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004). [CrossRef]
  19. J. Worthey, M. Brill, “Principal components applied to modeling: Dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004). [CrossRef]
  20. M. Brill, “A non-PC look at principal components,” Color Res. Appl. 28, 69–71 (2003). [CrossRef]
  21. S. Nascimento, F. Ferreira, D. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484–1490 (2002). [CrossRef]
  22. E. Oja, “A simplified neuron model as a principle component analyser,” J. Math. Biol. 15, 267–273 (1982). [CrossRef]
  23. J. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989). [CrossRef]

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