Perceptual Coherence : Hearing and Seeing

The Perception of Quality: Visual Color 307
Now we can substitute equation 7.6a for [E(λ)] and equation 7.6b for
[R b (λ)] to create the rather complex equation:
Excitation cone M =Σ[S m(λ)][ε 1E 1(λ) +ε 2E 2(λ)
+ε 3 E 3 (λ)][αR 1 (λ) +βR 2 (λ) +γR 3 (λ)]. (7.8)
The basis functions for illumination multiplied by the basis functions for
reflectance create the excitation for each cone system at each wavelength.
The excitation is multiplied by the absorption curve to yield the firing rate.
There would be analogous equations for the two other cones.
To estimate the surface color, we need to derive the surface descriptors
α, β, and γ. But to estimate the color, the observer simultaneously must
figure out the illumination. The difficulty is that there are more unknowns
than data points. There are three unknowns for the illumination (one value
for each of the three basis functions), and there are three unknowns for
the body reflectance (one value for each basis function). However, there are
only three known values, one for the excitation of each cone system. The
indeterminacy problem gets worse if more basis functions are needed to
model reflectance or if the surface is not uniform. Now, the indeterminacy
occurs at each point. All color models basically begin with three equations
like equation 7.8, one for each cone system. Each model described below
makes slightly different simplifying assumptions to achieve a solution.
Before considering specific color models, it is worthwhile to consider
the advantage of thinking about color constancy in terms of the linear sums
of excitation and reflectance basis functions. Linear models represent a priori
hypotheses about the likely set of inputs and, to the extent that the visual
system has evolved to pick up those likely inputs, the perceptual
inference problem becomes more tractable. Given that the illumination can
be described with three basis functions, then we can calculate the relative
amounts of each illumination basis function from the excitation of the
three cones. Similarly, given that the reflectances of most objects can be
adequately described with three basis functions, we can calculate the relative
amounts of each reflectance basis function from the excitation of the
three cones. We are replacing the direct calculation of the continuous illumination
and reflectance functions with the calculation of the relative
amounts of the respective basis functions.
Moreover, to the extent that the basis functions appear to be analogous to
physiological or perceptual functions, we can argue indirectly that the basis
functions underlie color constancy. For example, three basis functions have
been found to represent surface reflectance for a set of Munsell colors (Cohen,
1964). The first function is relatively flat across the wavelengths. The
second function contrasts green with red, while the third function contrasts
yellow mainly with blue and less strongly with red. Each basis function has
one more reversal, mimicking basis functions described previously. Thus,