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[De Valois \& De Valois 1975] suggest that in the 3D color space derived from the six LGN cell response functions, hue would be coded in a circular fashion (ranging through blue, green, yellow, red, and back to blue), saturation as distance from the center of the hue circle (making hue and saturation specifiable with a polar-coordinate system in the plane), and brightness along an axis perpendicular to the hue circle. This kind of color space is well known from early work of, e.g., Munsell [Birren 1969a][Munsell 1946] and Ostwald [Birren 1969b], and, as I have noted in the preceding section, the NPP space is similar to those in it overall organization. It is relatively easy to define psycho-physical measures on the NPP space, if one considers the axes to be orthogonal and one uses the standard Euclidean distance metric.
I define saturation in the NPP space in the traditional way, as the distance between a color point and the gray axis in an equal-brightness plane going through the point:
where represents the saturation of the color represented by
NPP coordinates
, and
is a
parametric equation defining the gray axis in SLN coordinates (before
transformation via transform
given in equation
p.
):
These linear equations produce an RMS error of fit of over 11 equally-spaced points along the gray axis, or
normalized to a unit cube,
which is quite reasonable. When using a second order polynomial fit:
the RMS error is reduced to , or
when normalized to a unit cube.
I define hue also in the usual way, as the angle between a color point and
some reference point, with the gray axis at the origin, in a projective
plane of constant brightness. The reference hue is defined as 0 degrees. As
the reference point, I have chosen the hue of unique green, i.e., that
wavelength at which the function is zero, and the
function has a
non-zero green response. This is a good reference, since it is easy to
define in an unambiguous way, and the corresponding wavelength does not
change with intensity. Unique green in SLN coordinates corresponds to a
wavelength of 504.934 nm, which corresponds to NPP coordinates
(Figure
).
The equation for hue in the NPP space is thus:
where is the hue of the color point
, and
is the reference hue. The resulting
values are in the range
.
Brightness, finally, is just the third dimension of the NPP space:
which gives us a complete transform from NPP space to a psychophysical HSB
(Hue-Saturation-Brightness) space based on it. Since the computation of hue
is inherently unreliable with low saturation values, a threshold is applied
to saturation, below which all hues are considered undefined.