**CHRISTIAN WEHRHAHN**

**Development of a cone space for
obtaining a simple representation of receptor and post receptor mechanisms.**

We
wanted to investigate spatio-temporal interactions between chromatic and
achromatic surfaces, such as chromatic induction or chromatic motion. Most
attempts to investigate these issues are limited to just a few colors. We
developed a set of stimuli that include the gamut of color available on a color
video monitor. These colors are equiluminant and equally detectable.

**How did we construct the
Teufel colors**

We
first constrained the parameter luminance by determining a plane of equal
luminance in SML space. We chose a luminance of 42 cd/m^{2}. By
definition this plane also contains the color white with that luminance. Next
we constrained detection by determining detection thresholds for a range of
colors situated in the plane of equal luminance. We found that the locus of
colors having the five-fold detection threshold can be described well as a
circle around white in a modified SML space. By going around the circle all
colors available within these constraints are included. The radius of the
circle is given by five times the value of detection thresholds. We then
defined 16 colors that are situated on this circle and separated by segments of
equal width on the circle.

**What are they good for?**

We now
have 16 colors on a circle around white of luminance 42 cd/m^{2} in SML
space. The position of these colors can be described by 3 sinusoidal functions.
These functions quantify the cone excitations of our colors. From these functions
we can compute the excitations of the postreceptoral mechanisms. If we now
measure psychophysical thresholds in an induction task using these colors as inducers,
we can predict how they depend on cone contrasts according to these mechanisms.

**Detailed description of
Teufel colors**

These
are 16 isoluminant, equally detectable and perceptually equidistant colors.
They cover the hues available on a standard color monitor and provide a
convenient representation for illustrating a difference between receptoral and
postreceptoral mechanisms with respect to chromatic induction and adaptation.

The
position of these colors in the space spanned by the excitations of S-, M- and
L-cones (cone space) has been determined using psychophysical methods such as
flicker fusion photometry and detection experiments. In conventional cone space
(MacLeod & Boynton 1979) these colors form an ellipse situated on a plane
of equal luminance.

Using
flicker fusion photometry we first determined the plane of equal brightness for
5 color normal observers. Due to a small contribution of S-cones to the
perceived brightness the plane of equal brightness is tilted with respect to
the sum of L- and M-cone signals (see Figure a). Generally, scaling of the
three base vectors s, m, l of cone space (the cone fundamentals) is set
arbitrarily to s=m=l=1 (Kaiser & Boynton 1996). Rescaling cone fundamentals
as

s =
.06

m =
.37

l =
.63

yields
a simple representation of colors using the polar coordinates r (radius) and f
(azimuth angle) in cone space or cone difference space. Equally detectable
colors lie on a circle with radius r being equal to the detection threshold in
a plane of equal luminance in cone difference space (see figure a). Increase of
azimuth angle f corresponds to going counterclockwise around the circle. The 16
colors shown at the end of this page are generated by advancing in steps of
equal increments of f around a circle with a radius of five times the detection
threshold (as measured in five observers). These results are described in
detail in Teufel & Wehrhahn (2000).

The
best-fitting sinusoidal functions in cone contrast space that approximate the
detection circle are

c_{S}
= 0.1006 sin (f - 289.1)

c_{M}
= 0.0098 sin (f)

c_{L}
= - 0.0040 sin (f - 48.8).

Following
the thin arrows in the figure - going from a to b - illustrates how the cone
excitations of two of the colors situated at azimuth angles 180¡ and 270¡ are
transformed into S-cone contrasts (c_{S}), M-cone contrasts (c_{M})
and L-cone contrasts (c_{L}) at corresponding azimuth angles f. To test
the validity of this representation the results of many detection experiments
of five subjects are plotted (see figure at the right hand side above).
Observers had to indicate the occurrence of small rectangular chromatic and
isoluminant stimuli to the left or right of the fixation point. Thresholds were
computed at 75% of psychometric functions. Each point (c_{S}, c_{M},
c_{L}) is the result of 300 stimulus presentations. These data are well
approximated by the cone contrast functions as shown below.

These
functions are shown in the figure (panels a – c) below. They describe the
contributions of the three receptoral mechanisms to the detection of the
respective colors indicated by stars. Using (c_{S}, c_{M}, c_{L})
the contributions to the detection of the 16 colors can be expressed in terms
of the postreceptoral chromatic mechanisms blue- yellow { c_{S} - (c_{M}
+ c_{L})} and red-green { c_{M} - c_{L} }. These are
are shown in panels d and e in the figure below. Since the colors are
isoluminant, the receptoral mechanisms S-cone contrast plotted in a. and the
post receptoral mechanism blue-yellow plotted in d. are indentical for the 16
colors considered here.

We used these colors to
investigate the shift in color of a gray test field surrounded by either of the
16 colors. The three monitor guns (R,G,B) are calibrated using a
spectrophotometer. Psychophysical thresholds can then be transformed into
values of cone contrasts (c_{S}, c_{M}, c_{L}) and
plotted as a function of the azimuthal angle f. The Figure above shows that
from the cone contrasts (the receptoral mechanisms) the contributions of the
postreceptoral mechanism, red-green and blue-yellow can be computed.

As
mentioned above use of these colors in psychophysical experiments requires a
calibrated monitor and an otherwise well defined setting. Monitors used for browsing the internet
generally do not satisfy these conditions.

With
these caveats the picture below qualitatively mirrors the detectability and hue
range of the Teufel colors.

Cone
contrasts as well as CIE coordinates (x, y, Y) of these colors are in the table
below (Teufel & Wehrhahn 2000).

This
paper also descibes the derivation from flicker fusion photometry and detection
experiments and the geometrical considerations leading to the color space
described above.

**Literature:**

Kaiser
PK Boynton RM (1996) Human Color Vision. Optical Society of America, Washington
D.C.

MacAdam
DL (1942) Visual sensitivities to color differences in daylight. J Opt Soc
America 32: 247-274

MacLeod
DIA Boynton RM (1979) Chromaticity diagram showing cone excitation by stimuli
of equal luminance. J Opt Soc America 69: 1183-1186

Teufel
HJ Wehrhahn C (2000) Evidence for the contribution of S-cones to the detection
of flicker brightness and red-green. J Opt Soc America A 17: 994-1006

Teufel
HJ Wehrhahn C (2004) Chromatic induction in humans: How are the cone signals
combined to provide opponent processing? Vision Research 44: 2425-2435.