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Reducing six dimensions to three

As discussed in Section (p. ), [De Valois \& De Valois 1975] suggest that the six types of LGN cells found can be grouped into three dimensions, when mirror image pairs are combined into one dimension. That results in a red-green, a blue-yellow, and a brightness dimension which can be arranged in a ``double cone''-type color space. We now turn to the construction of such a 3-dimensional color space based on the six response functions described above. In view of the hypothesis that mirror-image coding of signals in the nervous system is a way to increase the dynamic range of the signal [Slaughter 1990], and following the suggestion of [De Valois \& De Valois 1975], I will assume that the responses of mirror-image pairs of functions can be added to give one composite function:

where is a new composite green--red opponent function, a new composite blue--yellow opponent function, a new composite brightness function (non-opponent), and represent the six response functions , , , , , and , respectively. The members of the pairs have to be subtracted rather than added because they are 180 degrees out of phase relative to each other, and we want the corresponding phases to add up rather than cancel each other. The order of the terms determines the sign of the phases of the composite functions, and is arbitrary. I will always use the order of equations -- as the convention. Figure (upper half) shows the resulting opponent functions at a relative radiance of 0.5.

Some interesting properties of these functions are:

1. The zero crossing (upper left) at 605 nm is close to the maximum of the (positive) phase of the function (upper middle) at 609 nm, and the zero crossing of the function at 503 nm is close to the maximum of the (negative) phase of the function at 528 nm. If we interpret the composite functions as perceptual opponent functions in the style of [Jameson \& Hurvich 1955] (Figure lower), these zero crossing wavelengths accord relatively well with their estimates of the wavelengths of unique yellow (578 nm) and unique green (498 nm), i.e., wavelengths at which one of the two opponent functions is zero and the other has a non-zero response. By this definition, there is no unique red or unique blue based on our opponent functions, since the other opponent function is non-zero over the entire red and blue phases. This differs from the model of [Jameson \& Hurvich 1955] where there is no unique red, but there is unique blue.

2. The wavelength of maximum response of the function, 563 nm, is close to the wavelength of the CIE Y (photopic luminosity) function maximum, 555 nm.

3. The absolute maximum response of the green phase of the function is identical to the absolute maximum response of the yellow phase of the function, and the absolute maximum response of the red phase of the function is very similar to the absolute maximum response of the blue phase of the function. This is without any scaling on the 6 component functions or the resulting composite functions, other than the assumption stated above that all cell types will saturate at the same absolute firing rate, and the responses being scaled relative to that rate (150 spikes/sec). These maxima are different from the function maximum, but since we consider that to be an independent channel from the two color opponent functions, that is no problem.

In my opinion, these observations lend support to the assumption of one global maximum firing rate, and to the method of combining the six component functions into three. I therefore feel confident in using the three composite functions obtained as a neurophysiological basis for a 3-dimensional color space with a color opponent organization, interpreting the and functions as color opponent dimensions and the function as the brightness dimension. Later I will investigate the usefulness of this color space for the color naming problem. For now, I would like to note that this approach may provide an interesting bridge between the neurophysiology and the psychology of color perception (see Section , p. ).