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Normalization of the basis functions

We need one more step to turn the three SL functions into a convenient basis for a color space. Since I consider the and functions to be color opponent functions, I would like each function's two phases to cancel each other out in response to ``white'' (flat spectrum) light, i.e., to give a zero response. This corresponds to perceptual experience: white light does not seem to contain any color, either red, green, blue, yellow (the four primary colors), or any other one. I want the color model to reflect this directly. From Figure (p. ), we can see that the phases of the opponent functions are not equal in size. I therefore apply a normalization step as follows:

where is any of the six SOL response functions as discussed in the previous section, is its normalized version, is relative radiance, is wavelength as before, is the sigmoidification function as in equations ff (p. ff), is an energy equalization constant, and is the integral over positive values of only. The constant is discussed in more detail in Section (p. ). This normalization makes sure that the color-space dimensions are ``square'', since the maximum response of each function is obtained in response to a stimulus like

where is the zero-crossing wavelength of the function, i.e., the wavelength at which the response changes from inhibition to excitation. The normalization step makes sure that these maximum responses are the same for all functions. When the normalized functions are added pairwise, we get the normalized versions of the SL basis functions that I will refer to as the SLN functions :

with symbols as above, and and representing the maximum negative and positive response, respectively (see Section p. ). The normalization applied to the L and D functions is slightly different because the --L (or D) function is an inhibitor only, and has a much more limited range than the other five. The overall scaling factors , , and are set such that the maximum response of the function is 2, and the maximum for each opponent function is 1. The response to an equal-energy ``white'' spectrum (defined by ) must therefore be close to . It is not exactly equal to that because of the effect of the inhibitory phases of the opponent functions. The gray axis of the color space (the path defined by the coordinates of equal-energy spectra of increasing radiance) must therefore be close to the line through (black) and (white). Later I will examine the shape of the gray axis in more detail. Figure shows 3D plots of the resulting functions.