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Color naming also belongs in the domain of psychology. I define
color naming as a mapping from visual stimuli to pairs of color
terms (or symbols or names) and ``confidence'', ``goodness'', or
``typicality'' measures.
More precisely, since I am not interested in spatial characteristics of
visual stimuli, I will represent stimuli as spectral distributions
associated with single points in the visual field only. The
domain of
is thus the set of all such distributions:
where represents wavelength in nm, and
represents
a spectral distribution. The functions
will not be further defined.
They need not be continuous or differentiable, for instance. A pure
monochromatic stimulus would be represented as an impulse function, and
mixtures of several monochromatic primaries would be represented as
multi-modal distributions. The mapping
can then be defined as
where is an enumerable set of undefined terms, and
is
the closed unit interval
.
Informally, the preferred interpretation of the model is that
represents the set of spectral distributions that function as
input to the visual system,
represents a set of basic color
terms, e.g.
, which is the set of basic color terms described in
[Berlin \& Kay 1969], and
represents the set of ``confidence'',
``goodness'', or ``typicality'' measures.
If we ignore the part of the
, we might think of
as representing a partition of the
set of all possible color percepts
into a number of equivalence
classes, one for each color term. If we do take the
part into account,
we could model the response to any given stimulus
, i.e.
, as a fuzzy partition of the set of color terms
[Kay \& McDaniel 1978][Zadeh 1971], which constrains the numbers
of the pairs
to sum to 1, or
where is the selector of the second element of an n-tuple, for any
. The mapping
then defines a set of membership or
characteristic functions
, one for each color category
,
on the universe
. It is not clear what the advantage of such a model
would be, however, or how well it fits the data on human color
categorization.
The definition of implies that I will not be concerned with
recovering
or
from
, as noted above.
A practical consequence of this is that changing the lighting of a scene
may yield a different value of
. I don't consider this a problem
as long as the change is consistent with human performance on the same
task.
From the brief discussion of the physiology of color vision above, it
follows that if we want to model the relation between the domain of
and a set of color terms, i.e. if we want to model color
naming, it is not sufficient to define the extensions of color terms as
intervals on the wavelength range between 380 and 770 nm. In particular,
this approach could only work for pure monochromatic stimuli, which are
very rare in real-world situations, and it would not explain the typical
graded membership functions one finds in anthropological and linguistic
research when subjects are asked to identify best examples and maximal
extensions of color terms with respect to a set of color chips with known
properties [Berlin \& Kay 1969]. This approach would also leave out non-spectral
colors like purple or brown altogether.
But
perhaps the biggest objection against such an approach would be that it
would constitute a system-external semantic model of color names,
while our interest is in system-internal semantics, to be explained
below. I claim that to model human color naming it is necessary to
take human color perception into account, just as [Ronchi 1957]
claims that it is necessary to take human physiology and psychology into
account when studying optics, if that is defined with respect to visible
light, i.e. implicitly with respect to an observer.