Hans Irtel

Computing Data for Color Vision Modelling

Abstract

The program PCSF uses published mathematical equations which approximate color sensitivity functions for computing data for simulations of color vision. It can compute tables of receptor absorptions, ocular media density, fundamental response and color matching functions, and chromaticity coordinates. It also may be used to evaluate spectral radiance data with respect to these sensitivity functions. The source code is available and may easily be incorporated into other programs.

Computing Data for Color Vision Modelling

Color vision modelling usually starts from color matching functions. A set of such functions has been standardized by the CIE for the CIE 1931 Standard Colorimetric System and may be found tabulated in several sources (e. g. Wyszecki & Stiles, 1982). The CIE color matching functions include V(lambda), the luminous efficiency function for the CIE 1924 standard observer. Later data have shown deficiencies of the original V(lambda) data and lead to a revision by Judd (1951, cited in Wyszecki & Stiles, 1982) which now is generally referred to as the "Judd 1951 modified V(lambda)". Judd's modification of V(lambda) also leads to a modification of the CIE 1931 standard observer color matching function. This new system is called the "Judd 1951 modified" CIE standard observer and has been the basis of most work in color vision theory involving color matching functions.

An example is the derivation of cone fundamental sensitivity functions by Smith and Pokorny (1975). This derivation is based on color and brightness matching data of dichromats and the hypothesis of reduction dichromacy. The Smith and Pokorny results have found rather strong confirmation by recent results of Baylor, Nunn and Schnapf (1987) who were able to directly measure response functions of primate cones. They found a close similarity between their physiological measurements of the cones' spectral sensitivities with the psychophysically derived fundamental sensitivities of Smith and Pokorny. This comparison has to take into acount the optical density of the lens and macular pigmentation since these modify absorption (Pokorny & Smith, 1986).

All the spectral functions mentioned may be found tabulated somewhere. Computer simulations of color vision models, however, are easier to program with functional descriptions of the sensitivity data. Examples are Brainard and Wandell's Color Analysis Package (cited in Wandell, 1987) using discrete tables of the Smith and Pokorny sensitivity functions to analyze the colors of complex surfaces, and Siminoff (1991) who does a simulation of lateral interactions of color selective cells in the human retina. He uses piece-wise polynomial approximations of sensitivity functions.

PCSF contains modules which may be used for similar purposes. They are based on functional descriptions of the Smith and Pokorny fundamentals which have been derived by Boynton and Wisowaty (1980) and of the cone spectral sensitivity functions of Baylor et al. (1987). The program may be used as a stand-alone program for computing and printing the built-in functions or its single modules may be incorporated into other programs. The following functions are provided:

1. The Smith and Pokorny (1975) long- (LWS), middle- (MWS), and short-wave (SWS) sensitive cone spectra according to the equations derived by Boynton and Wisowaty (1980) (Fig. 1.A).

2. The Judd 1951 modified V(lambda) function which is the sum of the LWS and the MWS functions.

3. A smooth combined optical media absorption function based on an equation also derived by Boynton and Wisowaty (1980) (Fig. 1.B).

4. Long-, middle- and short-wave sensitive cone sensitivities according to an equation derived by Baylor et al. (1987) (Fig. 1.C).

5. The Baylor et al. (1987) cone sensitivities corrected for absorption by the optical media absorption function.

6. The Judd 1951 modified CIE color matching functions (Fig. 1.D).

7. Chromaticity coordinates for the Judd 1951 modified CIE chromaticity diagram.

8. Spectral coordinates for the cone excitation diagram proposed by MacLeod and Boynton (1979).

PROGRAM USAGE

Printing Function Tables
The program's functions may either be requested directly from the command line by specifying Unix style command line options or they may be selected from a menu that is printed on the screen if no function generating command line options have been given. The command

pcsf -rgb -s 1

will print three consecutive tables containing the Smith and Pokorny long-, middle-, and short-wave sensitive cone fundamentals respectively. Wavelength step size is 1 nm. Simply typing

pcsf

will print the menu which is shown in Fig. 2 and then expect the user to type in option letters. Input is finished by the ENTER key. An input line which is equivalent to the above command line example is

rgb1

Note that no special character separators are necessary in menu mode. Sequences of option letters on the command line have to be preceded by a minus sign as shown in the example.

Computing Spectrum Coordinates
In addition to printing values of sensitivity functions at arbitrary wavelength steps the program may also be used to evaluate stimulus spectra by computing color coordinates. The radiance spectrum has to be tabulated in equal wavelength steps in an input file. This has to be an ASCII text file with a single pair of wavelength and radiance data per line. The spectrum data are read and the products of each radiance value with the requested color matching function value for the corresponding wavelength are summed according to Wyszecki and Stiles (1982, sect. 3.3.8) and printed to the output. To get stimulus coordinates instead of function tables it suffices to give one or more names of stimulus spectrum files in the command line when calling PCSF. The type of coordinates computed is determined by the options given in Fig. 2. Thus the command line

pcsf -rgb STIMULUS.DAT

will compute the long-, middle-, and short-wave sensitive cone coordinates for the stimulus spectrum that is tabulated in the file STIMULUS.DAT. Suppose STIMULUS.DAT contains the following spectrum values:

435.8 1.0

546.1 1.38

700 72.1

then the command line

pcsf -C STIMULUS.DAT

will return the Judd modified CIE chromaticities of the stimulus: (0.3440, 0.3317). There will also be a warning message in this case, since the spectral data are not equally spaced and thus color matching coordinates may be distorted.

Output Control
Figure 2 shows that most of the options are used to request special sensitivity functions. Some options are used to specify output modes. The option "n" results in normalized output values. This only affects sensitivity functions and color matching coordinates, but not chromaticity coordinates. If sensitivity function tables are requested, then normalization of the functions is with respect to their maximum. If color matching coordinates for a radiance spectrum are requested, then normalization is such that Y, V, and (R + G) approximate ordinary luminance in cd*m^2 if the input spectrum is given in watt * sr^-1 * m^-2 * nm^-1 and is equally spaced. This is not possible if the spectrum is given in terms of a spectral density function. Chromaticity coordinates are independent of the scaling of the radiance data. A lower case letter "l" leads to log10 output values instead of the usual linear values. This option does not affect spectrum coordinates computed from radiance data. The option "u" scales the blue sensitive fundamental such that it is identical to the _z'(lambda) function. In command line mode the option "s" followed by a number will set the wavelength step size. In menu mode the letter "s" need not be given, a number is sufficient. If PCSF prints more than a single table it may be useful to add a mark at the end of each table. In command line mode a marker string may be given after the option letter "a". In menu mode it is sufficient to enter a string which is enclosed between double quotes. This string will be printed at the end of the last line of each table. Note that double quotes on the command line have to be preceded by a backward slash (\), otherwise they will not be passed to PCSF by the operating system.

The output of PCSF is written to the standard output device and it may easily be redirected into a file by command line redirection: The command

pcsf > OUT.TAB

will call PCSF in menu mode and write the resulting data output into the file OUT.TAB.

SYSTEM REQUIREMENTS
The program is available both in source and in executable form. It is written in ANSI C and may be compiled with any ANSI C compiler for any operating system. Executable versions have been created with Microsoft C 6.0 and Turbo C 2.0 for MS-DOS and with the non-ANSI UNIX C compiler cc. No special hardware is required. The single functions are written in a modular manner and may easily be incorporated into other programs. The files are available from the author via electronic mail by sending a request to: irtel@psychologie.uni-mannheim.de.

REFERENCES
Baylor, D. A., Nunn, B. J., & Schnapf, J. L. (1987). Spectral sensitivity of cones of the monkey Macaca Fascicularis. Journal of Physiology, 390, 145-160.

Boynton, R. M., & Wisowaty, J. J. (1980). Equations for chromatic discrimination models. Journal of the Optical Society of America, 70, 1471-1476.

MacLeod, D. I. A., & Boynton, R. M. (1979). Chromaticity diagram showing cone excitation by stimuli of equal luminance. Journal of the Optical Society of America, 69, 1183-1186.

Pokorny, J., & Smith, V. C. (1986). Colorimetry and color discrimination. In K. R. Boff, L. Kaufman, & J. P. Thomas (Eds.), Handbook of perception and performance. Volume I. Sensory processes and perception (chapter 8). New York: Wiley.

Siminoff, R. (1991). Simulated bipolar cells in fovea of human retina. Biological Cybernetics, 64, 497-504.

Smith, V. C., & Pokorny, J. (1975). Spectral sensitivity of the foveal cone pigments between 400 and 500 nm. Vision Research, 15, 161-171.

Wandell, B. A. (1987). The synthesis and analysis of color images. IEEE Transactions on Pattern Analysis and Machine Intelligence. PAMI-9, 2-13.

Wyszecki, G., & Stiles, W. S. (1982). Color science: Concepts and methods, quantitative data and formulae, 2nd ed. New York: Wiley.

Footnotes
1 Both Boynton and Wisowaty (1980, p. 1472, Eq. 8) and Smith and Pokorny (1986, p. 8-34, Eq. 62) have a typographical error in the equation for converting from quantum catch to energy based sensitivity.

2 There is an ambiguity in Boynton and Wisowaty's description of their equation for the optical media function (their Eq. 13). They do not give density, but the power of density which corresponds to absorption. Thus optical density is the base 10 logarithm of their Eq. (13).

Figure Captions
Figure 1. Graphical displays of tables produced by PCSF. A: Output resulting from the option line "-rgbl". These are the Smith and Pokorny (1975) fundamental sensitivity functions on a logarithmic scale. B: Output from "-m", the smooth optical media density function. C: Output of "-DPTl", the Baylor et al. (1987) cone sensitivity functions on a logarithmic scale. D: Output of "-xyz", the Judd 1951 modified CIE color matching functions x, y, and z.

Figure 2. This screen menu is shown if PCSF is called from the command line without giving one of the function generating options. All but the last five options generate functions. The last five options control the output mode. After showing the menu PCSF will wait until some option characters are typed in and the input is finished by pressing the ENTER key.


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