Abstract

The characterization of light by its tristimulus coordinates is briefly reviewed. The vector-transformation properties of these coordinates are shown, and the interpretation of change of reference stimuli as a change of basis in a 3-dimensional vector space is mentioned. The transformation coefficients are then examined as inner products, in a vector space over a basis of the order of the continuum. In this larger system, all possible sets of tristimulus coordinates are associated with a common 3-dimensional subspace. The vector analogy is thereby extended and a generalization is made from tristimulus coordinates to systems based on any number of arbitrary weighting functions. Each such system has an associated subspace, of the appropriate number of dimensions (not necessarily 3), not necessarily significant for human vision. Applications to spectrum characterization and electro-optical detector-response estimation are given.INDEX

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  1. W. D. Wright, The Measurement of Color (The Macmillan Co., Inc., New York, 1958), Chaps. 3–5.
  2. J. W. T. Walsh, Photometry (Constable & Co., London, 1958), 3rd revised ed., Chap. X and Ref. 3, p. 44.
  3. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1963), p. 51.
  4. Jozef Cohen and W. A. Gibson, J. Opt. Soc. Am. 52, 692 (1962).
  5. Paul R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1950).
  6. C. C. Macduffee, Vectors and Matrices (The Mathematical Association of America, George Banta Publishing Company, Menaska, Wis., 1943), pp. 56–58.
  7. See, for instance, Ref. 3, p. 52, pp. 274–5, and pp. 308–13. (On page 311 there is a misprint in equation 2.41. The quantity a2 is written as a2.)
  8. Although the detector response function has different units from those of source functions, this presents no difficulty, because these vectors appear together only in inner products. For ease in visualizing the space, we may think of all the wavelength functions as dimensionless multiples of appropriate fundamental units.
  9. G. Hadley, Linear Algebra (Addison Wesley Publishing Co., Inc., Reading, Mass., 1961), p. 46.
  10. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1963), pp. 309, 310.
  11. Note that negative portions of the F spectra may be constructed, if necessary, by the use of "differentially connected" detectors in a dc system.
  12. "cof(jk)" is the cofactor of element jk in the determinant detcjk.

Cohen, Jozef

Jozef Cohen and W. A. Gibson, J. Opt. Soc. Am. 52, 692 (1962).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1963), pp. 309, 310.

Gibson, W. A.

Jozef Cohen and W. A. Gibson, J. Opt. Soc. Am. 52, 692 (1962).

Hadley, G.

G. Hadley, Linear Algebra (Addison Wesley Publishing Co., Inc., Reading, Mass., 1961), p. 46.

Halmos, Paul R.

Paul R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1950).

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1963), p. 51.

Macduffee, C. C.

C. C. Macduffee, Vectors and Matrices (The Mathematical Association of America, George Banta Publishing Company, Menaska, Wis., 1943), pp. 56–58.

Walsh, J. W. T.

J. W. T. Walsh, Photometry (Constable & Co., London, 1958), 3rd revised ed., Chap. X and Ref. 3, p. 44.

Wright, W. D.

W. D. Wright, The Measurement of Color (The Macmillan Co., Inc., New York, 1958), Chaps. 3–5.

Wyszecki, G.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1963), p. 51.

Other

W. D. Wright, The Measurement of Color (The Macmillan Co., Inc., New York, 1958), Chaps. 3–5.

J. W. T. Walsh, Photometry (Constable & Co., London, 1958), 3rd revised ed., Chap. X and Ref. 3, p. 44.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (John Wiley & Sons, Inc., New York, 1963), p. 51.

Jozef Cohen and W. A. Gibson, J. Opt. Soc. Am. 52, 692 (1962).

Paul R. Halmos, Introduction to Hilbert Space (Chelsea Publishing Co., New York, 1950).

C. C. Macduffee, Vectors and Matrices (The Mathematical Association of America, George Banta Publishing Company, Menaska, Wis., 1943), pp. 56–58.

See, for instance, Ref. 3, p. 52, pp. 274–5, and pp. 308–13. (On page 311 there is a misprint in equation 2.41. The quantity a2 is written as a2.)

Although the detector response function has different units from those of source functions, this presents no difficulty, because these vectors appear together only in inner products. For ease in visualizing the space, we may think of all the wavelength functions as dimensionless multiples of appropriate fundamental units.

G. Hadley, Linear Algebra (Addison Wesley Publishing Co., Inc., Reading, Mass., 1961), p. 46.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1963), pp. 309, 310.

Note that negative portions of the F spectra may be constructed, if necessary, by the use of "differentially connected" detectors in a dc system.

"cof(jk)" is the cofactor of element jk in the determinant detcjk.

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