Rational Model of the Human Mechanism of Color Discrimination
Elena
Tolkova, Alexey
Chernyshov
Abstract
The problem of obtaining spectral sensitivity functions of the receptors of human eye retina (cones) is customary considered as an experimental task involving measurements with dichromatic observers. In the present paper, it is solved theoretically by approaching a human eye as an optimum receiver of the color signal. Standard RGB coordinates of monochromatic lights and MacAdam equicontrast diagram of chromaticity are all experimental data requested for the formal solution. Cone sensitivities derived theoretically agree closely with known empirical data.
Keywords:
cone sensitivity, color matching
functions, human color vision.
1. Introduction
According to the trichromatic theory of color vision formed
as long ago as in 19th century [1], color perception is based on the excitation
of three types of photosensitive receptors on the human eye retina, called the cones,
differing in spectral sensitivities. The sensitivity of a cone pigment as a
function of wavelength 
may be determined by
the number of molecules excited (or photons absorbed) when monochromatic
radiation of unit intensity at wavelength comes to the eye. Let us denote the cone sensitivities with
the peak in the region of long-, medium- and short-wavelength respectively as 
. Cone responses to radiation with the spectral power
distribution 
, or the numbers of photons absorbed
,
(1)
determine
the color for the observer. Cone signals 
may be considered the coordinates of the radiation in the
psychophysical color system (PCS).
Obviously, neither cone signals nor cone sensitivity
functions can be approached directly at the living eye. However, color
coordinates of any radiation in a system of real primary lights may be measured
in color matching experiments [2-4]. The coordinates of monochromatic
(spectral) light of unit intensity as functions of the wavelength are known as
color matching functions (CMFs) of this color system. As (1) shows, the cone
sensitivities are the color matching functions of PCS. Color coordinates of the
same light in all color systems, including PCS, are linearly related to each
other. Therefore, cone sensitivities may be calculated from known CMFs of some
color system of primaries 
as
, (2)
where 
is the vector of the
CMFs, and the columns of the matrix represent sets of color
coordinates or the primaries 
in PCS. In its turn, the columns of the inverse
matrix may be thought of as color coordinates of primaries of PCS, called
psychophysical primaries (PPs), in the space of the primaries . PPs are imaginary lights, and each of them excites only its
own type of cones.
So, to obtain cone sensitivities over all spectral range, one
only needs to find nine elements of the matrix representing three
primaries of any known color system in PCS, or the matrix 
representing PPs in
some known color space. This problem is customary considered an experimental
task, although neither these missing nine values nor entire sensitivity
functions can be measured directly without prior assumptions. In present paper,
we have derived the matrix theoretically by using
a new hypothesis about rational arrangement of the eye as a receiver of a color
signal. Our solution agrees closely with spectral sensitivity curves
constructed from numerous experimental data for normal and dichromatic color
vision. For a purpose of further
comparison of newly derived and existing data, let’s look briefly at known
empirical approaches.
The technique of search for PPs is based on the color
matching data for dichromatic observers and so-called pigment-loss hypothesis
[1]. This hypothesis considers dichromacy as reduced form of trichromacy due to
the loss of one pigment type. The other two pigment types, which dichromate
possesses, are the same as in normal observer. According to this hypothesis,
three types of dichromacy are possible: tritanopia, the result of the loss of the
«blue» pigment sensitive to the short-wave end of spectrum; deuteranopia, the
result of the loss of the «green» pigment with the pick of sensitivity in the
middle wavelength region; and protanopia, the result of the loss of the «red»
pigment sensitive to the long-wave end of spectrum. Loss hypothesis also makes
no distinction between cones and photosensitive pigment in them, assuming that
each type of cones contains only its own type of pigment.
Two colors 
and 
represented by the points
and 
in PCS are
discriminated by normal observer, but not discriminated by dichromate
(protanope in this case), who lacks the receptor associated with the first
color coordinate. The difference of color vectors not discriminated by
protanope 
determines the
direction (chromaticity) of the first psychophysical primary. Therefore, on the
trichromatic chromaticity diagram, both colors and the primary fall on a
straight line. So, the primary can be found on that diagram in the intersection
of lines joining chromaticities not discriminated by dichromate – so-called confusion
point. Therefore, PPs may be obtained by determining the confusion points of
the all three types of dichromates in the trichromatic chromaticity diagram.
Up to the middle of 20th century this technique of PPs
reconstruction seemed most reliable [5]. However it turned out, that this
technique gives reasonably stable location for «red» and «blue» psychophysical
primaries but results in a wide scatter of location of psychophysical «green»
[5-7]. It points, by the way, that at least deuteranopia is not limited by the
loss of one receptor. Nowadays, direct measurements of eye sensitivity in
adapting field are used to obtain cone spectral sensitivities [7-10].
Usually an eye sensitivity is determined as a reciprocal of
the test radiance (energy units) or the relative number of photons per target
(quantum units), that is just enough to enable an observer to see the target. When the target is showed against black
background, all three cone types contribute in full into formation of its
image. This state of eye is known as dark-adapted. It is generally taken a
priori, that strong background radiation, which excites only two types of pigment,
tends to suppress the responses of these two types and to «isolate» the third
pigment. If so, when pigment types are segregated, one to a cone, spectral
sensitivity of «isolated» cone type can be measured as sensitivity of
color-adapted eye. So, it is agreed, that bright yellow light containing all
wavelengths longer than 0.55 m suppresses both red and green receptors and isolate
the blue-cone; adaptation to purple light, containing wavelengths in both
violet and red spectral ranges, isolates the green-cone; adaptation to blue
light isolates the red-receptor [8].
However, it is difficult to expect absolute isolation of
a single cone type via this technique. Only the sensitivity curve of
blue-receptor can be measured more or less completely, because it lies far from
others. Separating the curves of the green- and red-cones takes place only
approximately, because these curves overlap in wide range. This problem can be
overcome by carrying out the measurements with dichromates. Spectral
sensitivity of violet-adapted eye of protanope or deuteranope is sensitivity of
the only pigment in the long-wavelength range, which this dichromate possesses.
This technique is based on two unproved suggestions, specifically on isolation
of a single receptor type by colored background, and on the loss hypothesis.
In the present paper, formal solution for the cone
spectral sensitivities is proposed. We use only color matching data and color discrimination
data for normal observers and the single additional suggestion about the
rational arrangement of the human eye as a color receiver. Cone sensitivities
derived here theoretically agree closely with spectral sensitivity curves
obtained from numerous experimental data for normal and dichromatic color
vision. The good agreement with known
solutions shows
that the rationalistic approach can be applied to so complexly built system as
human eye. We consider the last conclusion as the most interesting result of
our work.
2. Deriving Psychophysical Primaries
Let us start looking for PPs with brief description of the
color space arrangement and basic terms used below (section 2.1). The basic
idea how to find PPs is discussed in section 2.2. PPs and cone sensitivities
are derived in section 2.3.
2.1 Color space arrangement.
Color coordinates and intuitive color characteristics in
terms of brightness, color hue and saturation are in quite certain quantitative
relations. It is accepted, that color hue and saturation (the chromaticity of
light) are independent of brightness and determined only by the direction of
color vector in color space. For given chromaticity, the brightness varies
proportionally to the length of color vector. Thus, the color body, that is,
the region of color space containing real lights, fills the geometric cone with
vertex in the origin of coordinate system. Colors of spectral lights and purple
colors lie on the border of color body [2,3]. Because all real lights have
strictly non-negative coordinates in PCS, PPs appear either on border or
outside the color body, being imaginary in the last case.
Obviously, the cone of real colors is
completely defined by its border at the cross-section by some plane, for
example, by unit plane 
. The section of the color body represents so-called
chromaticity diagram. The chromaticity of any color vector is uniquely
determined by the position of its trace on the diagram. Chromaticities of real
lights are bounded on the chromaticity diagram by the curve of spectral colors
(locus) and by the line of purple colors joining the red and violet ends of the
locus. Real or imaginary colors with non-negative coordinates in given color
system are enclosed in a triangle (a Maxwell triangle), whose vertexes
represent primaries on the chromaticity diagram.
2.2 How to find PPs
According to the computational theory of vision by David Marr, the perception of visual information is based on a set of operations similar to algorithms of image processing in computer applications [11]. Continuing the analogy with artificial systems, we shall assume that the human visual system not only effectively processes the signal, but also effectively receives it.
Let us look at the problem of recognizing a color from excitation of three receptor types taking the human eye as an optimal receiver of a color signal. Obviously, to be optimal, our receiver has to provide the maximal dynamics range. It means that the interval of all possible signal values has to fall within the scale of our receiver, occupying most of the scale. If unused part of the scale remains, it should be minimal. For color perception, this condition means that PPs can be found on the chromaticity diagram of any color system (RGB, XYZ) as the vertexes of the triangle, which encloses the color body (because color coordinates of real lights in PCS are strictly positive) and at the same time contains minimal area outside the color body (imaginary chromaticities). From here on, standard colorimetric system XYZ will be used.
A cross-section of the color body by any plane, not only by
the unit plane, can be used as a chromaticity diagram. We shall continue with
MacAdam diagram of equal color contrast (ECC) generally used in computer
graphics. This diagram has the important property that the apparent distinction
between two lights is proportional to the distance between points representing
them [2,12]. Therefore, on the ECC diagram, the amount of chromaticities within
any given region can be expressed in terms of the area of this region.
The transformation
, 
(3)
obtained
as a result of experiments on color discrimination is known to describe ECC space
of variables (u,v) [2]. Because x and y in (3) vary over XYZ unit plane, we can rewrite (3) as
, 
, (4)
where 
are following vectors
, 
, 
, (5)
and
the tip of vector r lays in the XYZ
unit plane. Equations (4) tell us that pair of coordinates (u,v) has been selected in the plane 
, which is the central projection of XYZ unit plane from the
origin onto the plane with normal vector n.
So, vector n determines the ECC
plane. The amount of recognizable colors within any cell of the cross-section
of the color body by the plane
(6)
is
proportional to the cell area.
There exist several sets of conversion coefficients 
[2,12-14] describing
close ECC planes. Our final results depend very little on what set to select
and even less on coefficients 
not contributing into
vector n. Hereafter we will use MacAdam
set of coefficients [12]: 
, 
, 
.
So, we are looking for primary chromaticities of PCS as the
vertexes of the triangle of the least area in the ECC plane in XYZ space, which
encloses the color body.
2.3 A Maxwell triangle for PCS
Because the locus includes the large rectilinear section at the long-wavelength part of spectrum, this section would make one side of the enclosing triangle of the least area. Thus, obviously, the position of psychophysical «red» (PR) should coincide with the red end of locus, psychophysical «dark blue» (PB) should lay on the line connecting the red and dark blue ends of locus, psychophysical «green» (PG) should be located at the intersection of the tangents drawn from PR and PB to the locus (see fig.1). Therefore, the location of PR is known, whereas locations of PG and PB are determined entirely by the position of the point where the line connecting PB and PG touches the locus.
To specify the color body surface, Judd-Vos CMFs for 2-deg
vision field have been used [15]. Our calculations show that PB-PG side of the
desired triangle touches the locus at 
m.
The triangle is shown on the fig.1. Its vertexes give directions of axes of
PCS, which makes an optimal (in the sense discussed above) basis in the color
space. The PCS axes cross the XYZ unit plane at points:
, 
, 
. (7)
Expressions (7) define PPs within the factors of
brightness. These factors can be found using
standard normalization condition, that equienergy white color with
coordinates (1,1,1) in XYZ system should have the same coordinates in PCS. Now desired functions of spectral sensitivity
of three receptors of human eye can be calculated via equations (2) as follow:
,
(8)
where
vector denotes XYZ CMFs.
Obtained spectral sensitivity curves are shown on fig.2.
3. A
comparison with experimental data
Let us imagine that we measure the CMFs of tritanope – a dichromate, who has no pigment sensitive to the short-wave end of the spectrum. Let radiations of wavelengths 0.65 m and 0.48 m are taken as the matching stimuli. In the coordinate system of PR and PG (the third dimension is absent in the tritanope’s color space) the first stimulus is parallel to the PR axis because the green receptor sensitivity is zero from 0.65 m and the second stimulus is parallel to the PG axis because, for our data, the red receptor sensitivity goes to zero in the neighborhood 0.48 m. Therefore, if our data are true, radiations 0.65 m and 0.48 m are the psychophysical primaries of two-dimensional color system of tritanope. So, the sensitivities of the red cone and of the green cone can be measured directly (within normalizing factors) from color matching experiment with tritanopes with these primaries.
The measurements of CMFs of tritanopic observer with primaries
0.65 m
and 0.48 m
were done by W. D. Wright and are described in detail in paper [16].
Our spectral
sensitivity curves for green- and red-cone and Wright experimental data for
average tritanope are presented on fig.3, all curves are normalized to unity in
the maximum. The necessity of common normalization is caused by wide spread in
the height of the different curves, probably, because of individual
distinctions in absorption inside the eye of different wavelength [16]. We see
that our curves agree closely with experimental data, with an exception of
short-wavelength range for red-curve. However, Wright noticed also that tritanopes
discriminate wavelengths at the violet end of the spectrum and even do it
better than a normal observer [16]. It would not take place, if one of two
primaries contributed nothing to perception of short wavelength. Wright came to
the conclusion that the red receptor renews its activity at short wavelength,
that agrees with behavior of our red curve.
The
measurements of protanopes and deuteranopes spectral sensitivity functions
allow to obtain directly the sensitivities of the red- and green-cone – under
assumptions discussed above. Our curve 
in comparing with data of
measurements of protanope’s spectral sensitivity presented in papers [17-19] is
shown on fig.4. Experimental data are taken with normalizing factor to minimize
their deviation from the curve . We see
the good agreement between experimental curve and our curve.
Such agreement
with experimental data does not take place for red-cone sensitivity derived
here. However, as a number of facts shows, deuteranopia is caused by
superposition of responses of red- and green-cones rather than loss of the
green receptor. These facts are:
1.
Many deuteranopes possess a nearly normal spectral sensitivity [8],
whereas the perception of luminosity is caused by common response of red- and
green-cone [18,20].
2.
The
location of the confusion point on the chromaticity diagram is stable enough
for the different observers – protanopes and tritanopes. As to deuneranopes, locations
of their confusion points differ widely between individual observers and
between data of different investigators [5,6]. However, all these points fall
on the tangent to the long-wavelength part of the locus. This scatter of
confusion point position within the straight line can be explained by
superposition of responses of red and green channels in proportion, which is
individual for each observer, probably due to individual differences in
quantities of red and green pigment on the retina. (see Appendix).
So let us assume,
that deuteranope spectral sensitivity is the sum of sensitivities of both
“long-wavelength” pigments (perhaps, with the weigh coefficients), rather than
the sensitivity of the “red” pigment. Data of different investigators and the
curve 
are presented on
fig.5. We see small enough disagreement between our theoretical curve and existing
experimental data.
Spectral
sensitivity data for protanopes and deuteranopes together with loss hypothesis
allow estimating of the elements of the conversion matrix from XYZ CMFs to
cones spectral sensitivities (see Introduction). The most known version of
matrix belongs to Smith and
Pokorny [21]. Cone sensitivities derived with this matrix and our curves 
, 
and , after
common normalization to unity in the maximum, are shown on fig.6. Both groups of curves agree
very closely.
Thus, there are reasons to believe
that we have derived the cone spectral sensitivities thepretically,
investigating the color space of humans with normal color vision. We have found
PPs approaching the human eye as an optimal receiver of color signal, though
this rationalistic idea seems improbably simple for so complicated system as human
eye. We have found as well one more verification of the fact that deuteranopia is caused by integration of responses of both long-wavelength
channels, instead
of loss of one of them. What is
more interesting, however, it is not the cone sensitivity functions themselves
(they are already known), but the rationality of the human color mechanism,
which we have discovered here.
Appendix
About the location of the deuteranope confusion
point on the chromaticity diagram
The intersection point of
isochromatic lines for protanopes (or tritanopes) appears stably near the red
(or blue) end of locus. On the contrary, deuteranope confusion points found in
different experiments are scattered over the tangent to long-wavelength end of
the locus between 
and 
. Average cone sensitivities, derived by different authors
for different locations of PG on the chromaticity diagram, correspond to PG in the point 
[5].
We will show
that the scatter of PG position can be explained by suggestion that
deuteranopia is caused by superposition
of responses of red and green channels in proportion 
, which is individual for every deuteranopic observer. If so,
two colors 
and 
, represented by points 
and 
in trichromate PCS,
have coordinates 
and 
in a dichromate color
space. The deuteranope does not distinguish these colors, when their
coordinates at his color system are equal, that is
.
Therefore, the colors and are not discriminated
by the deuteranope when these colors differ by the vector 
with following
components at PCS:
,
or
.
The trace of vector on the chromaticity
diagram determines the location of the deuteranope confusion point. As the last
expression shows, this point falls on the straight line joining PG and PR
chromaticities. Variation of x-coordinate of the confusion point within 1.2-1.8
observed experimentally [5] corresponds to integration of red and green
channels in proportion within 
. Channels integration in proportion 1:1 gives the confusion
point location in 
.
Acknowledgments
Judd-Vos color matching functions,
Smith-Pokorny cone fundamentals and most empirical data have been found on-line
on the site of Color and Vision Research Laboratories at the
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Fig.1.
Prospective PCS color triangle (solid line) and XYZ color triangle (dotted
line) on ECC plane in XYZ color space. There are shown the section of color
body (filled) and the traces of white color, psychophysical primaries, primary
Y and primary Z (primary X is outside of the figure).
Fig.2.
Spectral cone sensitivities , and 
.
Fig.3.
Curves and for
sensitivities of “red” and “green” cones
(solid lines) and CMFs of average tritanope with the matching lights of
wavelengths 0.65m
(circles) and 0.48m
(squares).
Fig.4.
Curves (solid line)
and data of measurements of protanope spectral sensitivity according to Pitt
(crosses), Smith and Pokorny (circles), Stockman, MacLeod and Johnson
(squares).
Fig.5.
Curves (solid line)
and data of measurements of deuteranope spectral sensitivity according to Pitt
(crosses), Smith and Pokorny (circles), Stockman, MacLeod and Johnson
(squares).
Fig.6.
Cone spectral sensitivities derived by Smith and Pokorny (solid lines) and
samples of functions , and taken at
intervals of 0.005m
(circles).