As Maxwell, Helmholtz and many others have shown, a variety of spectral distributions of light can produce perceptions of color which are indistinguishable from one another. The visual perceptual system is thus mapping a high--dimensional input, the distribution of energy values of the photons arriving at every point on the retina, onto a low--dimensional output where each point in the visual scene is assigned one color. Obviously, information is being lost in the process, but it seems reasonable that the visual system is attempting to preserve as much of the information as can be made useful to the system.
The distributional characteristics of the spectrum of light have been closely and carefully studied. However, when it comes to perception of color, there are three fundamental mathematical questions that suggest themselves.
It is apparent that these three questions are interdependent and that they do not have simple answers. For instance, it has already been noted that a variety of distributions of energy quanta of photons will be perceived as the same color. This means that many points in the space representing the physical distribution of photon energy quanta can be mapped onto a single point in perceptual color space. In other words, the mapping between the physical spectrum and the perceptual color space can be many--to--one.
Figure 8. This diagrammatic simulation shows in (A) the blue rectangle under bright illumination and in (B) the blue rectangle under a lower illumination. Although the left half of each blue rectangle seems to remain the same perceptual color, actually the right half of the rectangle in (A) contains the same distribution of energy quanta photons as does the left half of the blue rectangle in (B).
But consider the problem posed by color constancy [see Jameson 1989 for a review]. Color constancy is that property of visual perception which allows one to ascribe an object a constant color under a range of lighting conditions. Suppose one is looking at a light blue rectangle which is half covered by a neutral gray filter and is illuminated by bright light. As one reduces the illumination provided by the light, at some point the reflected spectrum from the uncovered half of the rectangle will match that which had formerly been transmitted through the filter under the bright illumination. Therefore a single point in the space representing the physical distribution of photon energy quanta can be mapped onto several points in perceptual color space. In other words, the mapping between the physical spectrum and the perceptual color space can be one--to--many.
It is truly a tangled mapping which is both one--to--many and simultaneously many--to--one. When faced with this problem it seems natural to ask, ``Is there a higher dimensional representation of both the physical spectra of stimuli in the world and/or a higher dimensional representation of the perceptual color space which will restate the domain and range of the mapping function so that either the one--to--many or the many--to--one reduces to a one--to--one relationship?''
This line of reasoning brings up fundamental mathematical questions, but is not a line which I have found directly addressed in the literature on the perception of color. That being the case, I will attempt to work through these issues one by one.