Brightness and Darkness as Perceptual
Tony Vladusich1¤a*, Marcel P. Lucassen2¤b, Frans W. Cornelissen1
1 Laboratory of Experimental Ophthalmology & BCN NeuroImaging Centre, School of Behavioural and Cognitive Neurosciences, University Medical Centre Groningen,
University of Groningen, Groningen, The Netherlands, 2 Department of Human Interfaces, TNO Human Factors, Soesterberg, The Netherlands
A common-sense assumption concerning visual perception states that brightness and darkness cannot coexist at a
given spatial location. One corollary of this assumption is that achromatic colors, or perceived grey shades, are
contained in a one-dimensional (1-D) space varying from bright to dark. The results of many previous psychophysical
studies suggest, by contrast, that achromatic colors are represented as points in a color space composed of two or
more perceptual dimensions. The nature of these perceptual dimensions, however, presently remains unclear. Here we
provide direct evidence that brightness and darkness form the dimensions of a two-dimensional (2-D) achromatic color
space. This color space may play a role in the representation of object surfaces viewed against natural backgrounds,
which simultaneously induce both brightness and darkness signals. Our 2-D model generalizes to the chromatic
dimensions of color perception, indicating that redness and greenness (blueness and yellowness) also form perceptual
dimensions. Collectively, these findings suggest that human color space is composed of six dimensions, rather than the
Citation: Vladusich T, Lucassen MP, Cornelissen FW (2007) Brightness and darkness as perceptual dimensions. PLoS Comput Biol 3(10): e179. doi:10.1371/journal.pcbi.
It is well-known that our perception of achromatic colors,
or grey shades, depends on the contrast between adjacent
surfaces . A nearby surface induces either brightness or
darkness into the target depending on the polarity of the
inducing contrast (Figure 1). Brightness is perceived if the
target region has higher luminance than the background
(increment), whereas darkness is perceived if the target has
lower luminance (decrement). It is also known that brightness
or darkness is proportional to the contrast magnitude at the
inducing luminance edge . In more complex or naturalistic
displays, both brightness and darkness may be simultaneously
induced on a single surface. Some computational models of
achromatic color perception [2–5] posit that brightness and
darkness subtract to determine the perceived grey shade. This
is equivalent to stating that bright and dark constitute the
endpoints of a one-dimensional (1-D) achromatic color space
containing all possible grey shades.
The 1-D computational model is contradicted by the
findings of several psychophysical studies in which subjects
attempt to match achromatic colors associated with different
image regions (reviewed in ). In a typical achromatic color-
matching experiment, subjects adjust the luminance of a
matching surface such that the appearance of the surface
matches the appearance of a reference surface. Many
researchers have observed that, when such matches are
completed, residual differences in the color appearance of
reference and matching surfaces often persist. Such residual
unmatched differences are consistent with a computational
model in which achromatic color space is composed of two or
more dimensions [6–8]. According to one such model [6,7],
the achromatic color space is composed of one dimension
corresponding to surface reflectance and the other dimen-
sion corresponding to illumination intensity. One key
problem with this model is that it assumes that subjects can
independently estimate reflectance and illumination, even
though the eye receives only a mixture of these two
information sources—a difficult, though not insurmountable,
problem known as color constancy [9,10]. More immediately,
it remains unclear how such a model can explain any
difficulty subjects may have in matching achromatic colors
in flat, computer-generated displays. As computer screens
only emit light, it becomes difficult or impossible to mean-
ingfully parse the visual display into reflectance and
illumination components (just as, for example, the impres-
sion of depth is critically impoverished in the absence of
The present work represents an alternative approach to
identifying the dimensions of achromatic color space. Our
approach is motivated by the possibility, foreshadowed above,
that brightness and darkness may themselves constitute
perceptual dimensions . As detailed in the Discussion
section, this approach offers certain advantages over the
reflectance–illumination theory of achromatic color repre-
sentation [6,7], including neural plausibility. Indeed, attempts
have previously been made [8,11,12] to link perceived
brightness and darkness to the respective properties of the
parallel bright (ON) and dark (OFF) visual pathways.
Although the ON and OFF pathways run in parallel from
Editor: Karl J. Friston, University College London, United Kingdom
Received November 8, 2006; Accepted July 30, 2007; Published October 19, 2007
Copyright: ? 2007 Vladusich et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
* To whom correspondence should be addressed. E-mail: firstname.lastname@example.org
¤a Current address: Department of Cognitive and Neural Systems, Boston
University, Boston, Massachusetts, United States of America
¤b Current address: Lucassen Colour Research, Amsterdam, The Netherlands
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the primate retina to the second processing area (V2) of
visual cortex , direct evidence linking them to brightness
and darkness perception is lacking .
Here we endeavor to directly test the hypothesis that
brightness and darkness form the perceptual dimensions of
achromatic color space. We devise some simple visual displays
in order to demonstrate that brightness and darkness signals
remain separated at the highest levels of processing, rather
than interacting by subtraction. In one such display, for
example, a grey reference ring is bordered by a black disk and
a white background (Figure 2A). The inner edge between ring
and disk, and the outer edge between ring and background,
have equal contrast ratios but opposite polarities. The black
disk thereby induces brightness into the ring, whereas the
white background induces darkness. In a second display, the
grey reference ring is bordered by a white disk and a black
background (Figure 2B), with the complementary consequen-
ces in terms of induction.
We model the grey shade associated with the reference ring
as a point in a 2-D ‘‘grey space’’ formed by the induced
brightness and darkness signals. Likewise, we model the grey
shade associated with a matching ring—which differs from
the reference ring in terms of either luminance, contrast, or
both—as a point in grey space. In two psychophysical
experiments, we quantify the perceptual differences between
many pairs of reference and matching rings. We find that
grey shades are identical only if the ring, disk, and back-
ground all possess the same luminance values in both
reference and test displays (the displays are identical). The
residual perceptual differences between non-identical display
pairs are well-predicted by our 2-D model, even when the
details of the experimental design differ markedly.
Experiment One: Simultaneous Presentation, Variable
The rationale of Experiment One is perhaps best illus-
trated through an example: consider the above-mentioned
stimulus in which a black disk induces brightness and a white
background induces darkness into a reference ring (Figure
2A). The subject attempts to adjust the luminance of the
matching ring to make it appear the same grey shade as the
reference ring. Now suppose that the magnitude of brightness
and darkness induced into the matching ring is much less
than that induced into the reference ring (Figure 2A). In
other words, the matching background is light grey (rather
than white) and the matching disk is dark grey (rather than
black). Assuming that brightness and darkness constitute
perceptual dimensions, the subject can never adjust the
luminance of the matching ring to make the reference and
matching rings appear exactly the same grey shade. To match
the brightness dimension, for example, the subject must
increase the contrast of the brightness-inducing edge in the
matching display to equal the high contrast of the brightness-
inducing edge in the reference display. Conversely, the
subject must increase the contrast of the darkness-inducing
edge in the matching display to equal the high contrast of the
darkness-inducing edge in the reference display. These two
options are mutually incompatible, as increasing the contrast
of the brightness-inducing edge must necessarily decrease the
contrast of the darkness-inducing edge, and vice versa. Thus,
in attempting to decrease the mismatch in the brightness
dimension, the subject inadvertently increases the mismatch
in the darkness dimension, and vice versa.
The above reasoning implies that only partial color
matches are possible in such a display. It further implies that
the ease with which subjects can make matches will increase
as the contrast difference between reference and matching
displays decreases. To estimate the quality of achromatic
color matches in experiment one, we had subjects rate the
‘‘possibility of making a perfect match’’ on a 1–10 scale, after
they had completed each setting (10 ¼ perfect match; 1 ¼ no
possible match; intermediate values ¼ partial matches of
The results of Experiment One indicate that the possibility
of making achromatic color matches declines with the
contrast difference between the reference and matching
rings (Figure 3). When the reference ring was bordered by a
dark disk and bright background, subjects set the luminance
of the matching ring closer to the luminance of the bright
background. This implies that subjects placed more weight on
Figure 1. Brightness and Darkness Induction
Viewed from left to right, the grey disks appear to vary from dark to
bright, even though they share the same luminance value. This induction
effect arises because the background luminance varies along a gradient,
leading to a change in the polarity and magnitude of contrasts formed
against the disks. Brightness refers to the perceived luminance, or grey
shade, of the disk when the luminance of the disk is greater than that of
the background. Conversely, darkness is defined as the perceived
luminance of the disk when the luminance of the disk is less than that of
the background. The conventional way of thinking about this perceptual
effect is that darkness is simply the negative of brightness, meaning that
all the grey shades above are contained in a 1-D continuum, like real
numbers along a number line.
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Vision scientists have long adhered to the classic opponent-coding
theory of vision, which states that bright–dark, red–green, and blue–
yellow form mutually exclusive color pairs. According to this theory,
it is not possible to see both brightness and darkness at a single
spatial location, or an extended set of locations, such as a uniform
surface. One corollary of this statement is that all perceivable grey
shades vary along a continuum from bright to dark. At first glance,
the notion that brightness and darkness cannot coexist on a single
surface accords with our common-sense notion that a given grey
shade cannot be simultaneously both brighter and darker than any
other grey shade. The results presented here suggest that this
common-sense notion is not supported by experimental data. Our
results imply that a given grey shade can indeed be simultaneously
brighter and darker than another grey shade. This seemingly
paradoxical conclusion arises naturally if one assumes that bright-
ness and darkness constitute the dimensions of a two-dimensional
perceptual space in which points represent grey shades. Our results
may encourage scientists working in related fields to question the
assumption that perceptual variables, rather than sensory variables,
are encoded in opponent pairs.
Brightness and Darkness as Perceptual Dimensions
matching darkness (induced by the brighter background)
rather than brightness (induced by the darker disk). When the
contrast of the darkness-inducing edge was low on the
reference side of the display, for example, subjects set the
luminance of the matching ring close to the luminance of the
matching background in order to match this low contrast.
More generally, we can say that the luminance settings fell
close to theoretical predictions based on an assumption of
perfect darkness matching: subjects attempted to match the
contrast of the darkness-inducing edge across reference and
matching rings, largely ignoring the contrast of the bright-
ness-inducing edge. The results were rather different when
the reference ring was bordered by a bright disk and dark
background. In this case, subjects always set the luminance of
the matching ring close to the luminance of the reference
ring. This implies that subjects placed roughly equal weight
on matching brightness and darkness in these conditions.
Nonetheless, subjects rated the task of setting perfect color
matches as progressively less possible as the contrast differ-
ence between match and reference displays increased,
independently of the type of background.
To better understand the nature of achromatic colors, we
modeled grey shades as points in a 2-D achromatic color
space consisting of brightness and darkness dimensions.
According to this model, subjects can perfectly match either
brightness or darkness in our displays, but generally cannot
match both simultaneously. We estimated the weights
associated with brightness and darkness for each stimulus
display by fitting the model to the luminance settings made by
subjects (Materials and Methods). These fits were extremely
accurate, explaining more than 96% of the variance in
subjects’ settings. The weights were allowed to vary with
background and disk luminance to simulate gain control [4,5]:
the influence of gain control was, however, quite small (;5%
of the mean weight values) and did not affect our results
greatly. We then attempted to predict the possibility ratings
made by subjects based on these fitted weights. Within the 2-
D perceptual space, we constructed a simple dissimilarity
metric based on a modified version of the city-block method
(Materials and Methods). These predictions agree remarkably
well with subjects’ possibility ratings (Figure 3).
We next plotted grey shades associated with matching rings
as points in a 2-D grey space (Figure 4). We explain the
impossibility of setting perfect matches by noting that
subjects could only set grey values in the reference ring
along the dotted colored lines (or directly along the
horizontal and vertical axes, assuming that both edges share
the same polarity) indicated in Figure 4. This is because
adjusting the luminance of the matching ring to increase
brightness (darkness) involves a simultaneous decrease in the
amount of darkness (brightness). Subjects could therefore
only adjust achromatic color of the reference ring along a
single dimension—corresponding to luminance—within the
2-D grey space. Subjects were simply unable to tap into the
rich gamut of achromatic colors available in the entire 2-D
The values of fitted brightness and darkness weights in our
model reveal that darkness is about four times stronger than
brightness in displays containing bright backgrounds (Figure
4). Why did subjects place more weight on matching the
darkness dimension with bright backgrounds? We explain this
behavior in terms of the well-known observation that dark-
ness is inherently stronger than brightness [11,15,16],
combined with the unequal circumferences of inner and
outer ring edges. With bright backgrounds, the circumference
of the outer darkness-inducing edge was three times the
circumference of the inner brightness-inducing edge. Dark-
ness was therefore weighted more heavily than its inherent
value. With dark backgrounds, however, the circumference of
the inner darkness-inducing edge was one-third of the
circumference of the brightness-inducing edge. In this case,
brightness and darkness weights were roughly equal. From
these considerations, we calculate that edge weights are
related to circumference by a power function with exponent
;0.63 and that darkness is inherently about twice as strong as
brightness (see Materials and Methods). This estimate is
consistent with available data [11,15].
The dominance of darkness induction with bright back-
grounds also indicates that possibility ratings were not based
on the computation of Euclidian vectors in the 2-D space.
Unlike the case with dark backgrounds (Figure 4)—where
brightness and darkness weights were equally balanced—
subjects did not minimize the vector between reference and
matching rings. We instead found that the city-block metric
Figure 2. Stimuli Used in the Experiments
We are interested in the question of how to describe, for example, the
achromatic color percept of a ring surrounded by a dark disk inducing
brightness and a white background inducing darkness. Previous work,
summarized in the main text, suggests that achromatic color space is
composed of two or more dimensions. Here we argue that the
achromatic colors of the above rings are described by separate
brightness and darkness dimensions.
(A) Experiment One. Stimuli composed of a background brighter than
both the reference (left side) and matching (right side) rings. A horizontal
luminance gradient was rendered along the midline of the background
such that the contrast formed by disk and background was different on
the two sides (the gradient did not extend to the rings). Note that the
contrast formed by the ring and disk on the reference side was of the
opposite polarity to the contrast formed by the ring and background.
(B) Similar to (A) except that the background was darker than the rings,
whereas the disks were brighter than the ring. In each stimulus, the
background and central disk induced brightness or darkness into the
rings by means of simultaneous contrast.
(C,D) Experiment Two. The polarity relationships reversed between
successive presentations of reference and matching displays. See text for
details of the experiments.
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Brightness and Darkness as Perceptual Dimensions
—in which residual color mismatches in the brightness and
darkness dimensions are summed—is far more suitable for
modeling the rating data (Materials and Methods). This result
further implies that brightness and darkness pathways remain
physically separated even at the highest processing levels (see
Experiment Two: Successive Presentation, Variable Edge
In Experiment Two, we presented two grey rings in rapid
succession, rather than simultaneously (Materials and Meth-
ods). Subjects first viewed the reference display (one second
duration), followed immediately by the matching display (also
one second duration). We employed a similarity rating procedure
in which subjects rated how well the grey shade of the
matching ring matched the grey shade of the reference ring
(10 ¼ perfect match; 1 ¼ as different as black and white;
intermediate values¼partial matches of variable quality). We
kept the overall contrast the same between the reference and
matching displays but varied the relationship between edge
polarities. In one condition (Figure 2C), the reference ring
was bordered by a black background and a white disk. The
subsequent matching ring was bordered by a white back-
ground and a black disk. A second condition was composed of
the complementary set of polarity relationships (Figure 2D).
In two control conditions, the polarity relationships of the
reference and matching displays (disks and backgrounds)
remained the same over the interval. In each trial, we varied
the luminance of the reference ring between the maximum
and minimum luminance values associated with the disk and
background (see Materials and methods). The luminance of
the matching ring remained constant throughout. The aim of
the experiment was to determine the luminance value of the
reference ring that produces the best color match with the
matching ring. Note that subjects did not actually adjust any
variables in the experiment, so the terms ‘‘reference’’ and
‘‘matching’’ are used in the heuristic sense in the context of
the present experiment.
By systematically manipulating edge polarity across refer-
ence and matching displays—but keeping the overall contrast
the same—we sought to take advantage of the finding in
Experiment One that brightness and darkness induction
depend on relative edge circumference. All else being equal,
the ring-background edge of a given display induced stronger
brightness or darkness than the ring-disk edge. This implies
that the achromatic color of the ring changes when edge polarity
reverses across reference and matching displays. Critically, we
used the same ratio of edge circumferences in Experiments One
and Two (Materials and Methods). This allowed us to make a
set of theoretical predictions—based on the estimated
brightness and darkness weights obtained in Experiment
One—to test against the data from Experiment Two. For each
of the four conditions of Experiment Two, we inserted the
brightness and darkness weights derived from Experiment
Figure 3. Possibility of Match Plotted against the Contrast Difference between Reference and Matching Displays
Average data from six subjects (error bars indicate standard errors of the mean). With bright backgrounds (left side), subjects adjusted the luminance of
the matching ring to be either much higher (yellow data points in white region) or much lower (blue data points in white region) than the luminance of
the reference ring (always at 30 cd/m2). The dotted grey lines denote perfect darkness matches, indicating that subjects weighted darkness more
heavily than brightness in this situation. With dark backgrounds (right side), subjects set the matching luminance close to the reference luminance (red
and green data points in white region). In both cases, however, subjects rated matches as progressively less possible as the contrast difference between
reference and matching sides of the displays increased (data points joined by continuous lines in grey region). This implies that brightness and darkness
constitute dimensions of achromatic color space. We modeled this 2-D space by estimating brightness and darkness weighting factors from the
luminance data (model fits are the continuous lines in the white region) and then predicting the possibility ratings from the fitted weights (dotted lines
in the grey region). The model predictions agree reasonably well with the data. Symbols representing the stimuli are included to assist understanding of
the data and should not be considered as realistic representations.
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Brightness and Darkness as Perceptual Dimensions
One into the equations governing the 2-D grey space (Figure
To understand the resultant predictions, consider first a
control condition in which the disk and background
luminance values are identical for reference and matching
displays (Figure 5A). It is clear that the best match (similarity
rating ¼ 10) occurs when the reference and matching rings
share identical luminance values. In terms of Figure 5A, the
black square representing the matching ring coincides exactly
with the middle green disk. As the reference- to matching-
ring luminance ratio varies in either direction, however, we
expect systematic and symmetric deviations from the perfect
match. In other words, we expect only partial matches for the
remaining green disks in Figure 5A.
A different scenario arises when the polarity relationships
between reference and matching rings are reversed. Consid-
er, for example, the black square representing the matching
ring in Figure 5B. It is clear that none of the red disks
corresponding to the reference rings coincide with the
position of the black square. This implies that subjects
cannot choose a luminance value of the reference ring to
ensure an exact match with the matching ring. The closest
possible match, in this case, coincides not with the middle red
disk but with one of the disks nearer the bottom of the line of
red disks. This observation implies that, in addition to rating
the resultant similarity as less than perfect, subjects should
choose a luminance value for the reference ring that deviates
systematically from the luminance value of the matching ring.
The model predicts, in other words, a skewed relationship
between similarity ratings and the reference- to matching-
ring luminance ratio.
The mean ratings of six subjects are plotted in Figure 6
along with the model predictions. To a first approximation,
the model accurately predicts many aspects of the data. In
particular, the model correctly predicts the roughly symmet-
ric rating profiles in the control conditions (Figure 6A and
6C). The model also correctly predicts the skewed nature of
the data curves in the reversed-polarity conditions (Figure 6B
and 6D). The model does fall down somewhat, however, in
predicting the magnitude of the deviation from perfect
matches in these conditions. More specifically, the model
predicts close to perfect similarity ratings for certain
luminance ratios in the condition corresponding to the white
reference background and black disk (Figure 6D). The data
curve does not support this prediction but rather appears to
peak at a level comparable to the peak associated with the
curve associated with the black reference background and
white disk (see Discussion). In summary, the 2-D model does a
reasonable job of quantitatively predicting the results of an
experiment differing in several details from the original
experiment from which the model was derived.
In Experiment One, we showed that as the contrast
difference between reference and matching displays in-
creased, subjects were progressively less able to produce
perfect achromatic color matches. Subsequent modeling of
Figure 4. Brightness and Darkness as Perceptual Dimensions
(A–D) Achromatic color space consisting of brightness and darkness dimensions. For each luminance value of the matching ring, we plot the
corresponding grey shade as a point in the 2-D grey space. Horizontal dotted lines denote perfect darkness matches. Dotted colored lines represent the
gamut of grey shades available in the single dimension of luminance space along which subjects can physically adjust the matching ring. Solid grey
lines are the approximate vector projections of the grey shade associated with the reference ring onto all matching displays. The intersections of these
projection lines with the lines-of-adjustment roughly indicate the grey shades that subjects would set if they were minimizing the vector between
reference and matching rings. Subjects did not appear, however, to minimize this vector. This is particularly evident with brighter backgrounds (A,C),
where subjects placed far more weight on matching the darkness component than the brightness component. The different scales for brightness and
darkness in (A) and (C) provide further evidence that subjects weighted darkness more heavily than brightness.
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Brightness and Darkness as Perceptual Dimensions
these data supported the conclusion that brightness and
darkness form perceptual dimensions, with darkness being
weighted twice as strongly as brightness. One problem with
this interpretation, however, was that decreasing the ring
contrast led to a ‘‘foggy’’ appearance resembling trans-
parency [17,18]. Could this fogginess have caused the
reported difficulty in making perfect matches? In Experiment
Two, we generated stimuli in which overall contrast was high
in both reference and match displays. Thus, we reasoned, any
difficulty in matching achromatic colors could not be
attributable to differences in overall contrast. The similarity
ratings of subjects were reasonably consistent with the
quantitative predictions of our 2-D model. Perhaps the most
important discrepancy was that the model predicted perfect
color matches in cases where subjects actually perceived
residual color differences. This discrepancy may result from
gain control acting on edge signals [4,5,19–21]—which would
act to curve nominally straight lines in achromatic color
space [6,8]—thereby distorting our estimates of points in grey
space. Although we found a small effect of gain control in
Experiment One, we chose to omit this factor in our
predictions for Experiment Two in order to cap the degrees
of freedom of our model. Alternatively, it may be that
achromatic color space is composed of even more than two
dimensions, with luminance offering a possible candidate for
the extra dimension [22,23]. Further experiments are clearly
required to resolve this issue.
What might be the functional advantage of the 2-D grey
space introduced here? Relative to a 1-D space, the range of
achromatic colors available in a 2-D space is enormous.
Whereas a 1-D space encodes (n) grey shades (just noticeable
differences), a 2-D space contains
to a 1-D space with n ¼ 1,000, for example, the number of
discriminable grey shades in a 2-D space equals 250,000. Such
a 2-D representation may play a key role in the encoding of
achromatic colors in natural environments, where object
surfaces form both increments and decrements with respect
to variegated backgrounds . The preservation of bright-
ness and darkness information in a 2-D space ensures that the
visual system is sensitive to the variance and skewness of
luminance pixels bordering an object, rather than just the
mean. Such sensitivity may facilitate object detection at low
contrasts and play a role in texture discrimination [25–30].
Our framework is consistent with the recent finding that
the visual system uses skewness as an image cue to classify
bright and dark image regions into ‘‘object’’ and ‘‘light’’
properties . The computation of object and light proper-
ties is not, however, required to explain our data. A
perceptual space composed of brightness and darkness
dimensions explains most of our findings in a parsimonious
and quantitative manner. Indeed, our results provide a
challenge to anchoring theories of achromatic color percep-
tion [10,31] and to the proposal that the dimensions of
achromatic color space correspond to object and light
properties [6,7]. Several studies have shown that subjects
often need to be explicitly instructed to interpret flat,
computer-generated displays in terms of object and light
properties [21,32,33]. In the present study, subjects were
explicitly instructed to match grey shades they saw, rather
than putative object or light properties. Importantly, some
recent psychophysical findings suggest that judgments of
object and light properties may be based on spatial
? ?2grey shades. Compared
Figure 5. Model Predictions for the Reversed-Polarity Experiment (Experiment Two)
The model predicts that subjects will rate ring pairs as identical only when the grey shades associated with reference (colored disks) and matching
(black squares) coincide in grey space (A,C). Otherwise, the model predicts that perfect matches will not be possible (B,D). The black square in (D) does
not coincide exactly with the position of a yellow disk, although it is very close.
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Brightness and Darkness as Perceptual Dimensions
comparisons of surface brightness (and, by extension here,
darkness) across different regions of a 3-D scene [34,35]. In
our displays, it remains problematic to describe how object
and light properties might vary with the difference in
contrast magnitude (Experiment One) or polarity (Experi-
ment Two) between reference and matching displays. How,
for instance, should one interpret a change in edge polarity
between displays? The polarity reversal would not itself seem
to carry conflicting information about object and light
properties, as both local contrast magnitude and luminance
remain the same.
It is well-known that the edges directly bordering a target
surface (local edges) and edges distant from the target surface
(remote edges) both contribute to achromatic color [3–
5,20,21]. The present study helps to clarify the results of our
previous study  in which we found that subjects had greater
difficulty making achromatic color matches with opposite-
polarity (relative to same-polarity) combinations of local and
remote edges. This puzzle is clarified by noting that opposite-
polarity edge combinations simultaneously induce both
brightness and darkness into a surface. As in the present
study—in which we simultaneously induced brightness and
darkness at different local borders—this would lead to a
situation in which only partial color matches are possible.
Consider, for example, a disk-ring reference configuration in
which the local edge induces brightness into the disk, whereas
the remote edge induces darkness into the disk. In our
previous experiment, subjects adjusted a matching disk on a
uniform background to appear the same grey shade as the
reference disk. This means that the polarity of the disk
relative to the background could be either an increment or a
decrement. Subjects could therefore match either brightness
or darkness, but not both.
It is perhaps illuminating to speculate on the phenomeno-
logical aspects of simultaneously perceiving brightness and
darkness. Anstis  studied the ‘‘metallic’’ or ‘‘lustrous’’
appearance of a surface region composed of both brightness
and darkness, generated either through monocular or
binocular fusion. The binocular effect is generated by fusion
of a dark disk (decrement) presented to one eye with a bright
disk (increment) presented to the other eye. The monocular
version is obtained by rapidly modulating (in time) the
polarity of a disk with respect to a steady background . A
third (monocular) version of the effect is generated by
embedding a grey disk in a black Ehrenstein pattern on a
white background . In all cases, edges of opposite contrast
polarity combine to give the overall impression of lustre to
the surface. Anstis  argued that lustrous surfaces often
shimmer, leading him to conclude that lustre is associated
with competition between local bright (ON) and dark (OFF)
channels . Our results suggest precisely the opposite
conclusion. We claim that ON and OFF channels remain
separate at the highest levels of processing, giving rise to
percepts of simultaneous brightness and darkness, which can
be interpreted as appearing lustrous. A weak impression of
lustre may be seen in Figure 2A and Figure 2B, where the
higher-contrast rings appear ‘‘sharper’’ or more ‘‘metallic’’
than the ‘‘softer-appearing’’ low-contrast rings. This con-
clusion implies that lustre can be dissociated from shimmer.
How much evidence is there to link the properties of ON
and OFF pathways with brightness and darkness perception
[11,12]? A recent neurophysiological report reveals that gain
control operating in early ON and OFF pathways is sensitive
to the variance, but not the skewness and kurtosis, of
background pixels . Demonstrations of perceptual sensi-
tivity to skewness  thus further highlight the mismatch
between the properties of early ON and OFF pathways and
achromatic color perception [12,14]. As indicated previously,
Figure 6. Results of the Reversed-Polarity Experiment (Experiment Two)
Average data from six subjects (error bars indicate standard errors of the mean). The model correctly predicts the data curves associated with the two
control conditions (A,C). The data curves for the reversed-polarity conditions are consistent with the predicted skewness (B,D), and to a lesser degree
with the amount of mismatch.
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Brightness and Darkness as Perceptual Dimensions
separate ON and OFF pathways are maintained even at the
level of V2 in monkey visual cortex . Researchers have
claimed that some neurons in V1 and V2 encode achromatic
colors [38–41]. These claims have, however, been disputed
[42–44]. To correlate with achromatic color perception in our
displays, a visual area would need to combine local edge
information into a global surface representation [4,5,21]. The
computational nature of this brain area remains to be
As a final twist, we propose that redness and greenness—as
well as blueness and yellowness—may form perceptual
dimensions , rather than canceling [45–47]. Ekroll et al.
 have shown that subjects can only match certain
combinations of local and remotely induced colors (Figure
7). Subjects can, for example, adjust a matching disk on a red
background to appear the same hue and saturation as a green
reference disk on a grey background. Subjects can also make a
suitable match when the reference disk is a more saturated shade of
red than the matching background. Subjects cannot, however,
perform suitable matches when the reference disk is a shade of red
less saturated than the matching background. Ekroll et al. described
similar results for blueness and yellowness perception.
We explain these paradoxical results in terms of the
assumption that redness and greenness (blueness and yellow-
ness) form perceptual dimensions. Following Brenner et al.
, we propose that local color signals (within a surface
region) and remote color signals (induced at edges) combine
to give rise to the color percept. If the local color signal is red,
for example, and the induced color signal is green, then the
resultant percept will be composed of both redness and
greenness. In the above example, the matching disk can be
adjusted to match the reference disk when the reference disk is
green. To understand why, consider a subject attempting to
match only the local green color. Matching the local green
component alone will not be sufficient because the red
matching background also adds greenness to the matching
disk. In order to make a perfect match, the subject therefore
dials down the physical greenness of the matching disk by an
appropriate amount. Similarly, subjects can perfectly match
the reference disks whose red shades are more red than the
matching background. Again consider a subject setting a match
for the local redness alone. This match will not be perfect
because the red matching background subtracts redness from
the matching disk. A perfect match thus requires dialing the
physical redness of the matching disk up by a certain amount.
No match can be made, however, when the reference disk is
less red than the matching background. Consider a subject
matching only the local redness. This requires setting the
matching disk as a physical shade of red slightly less saturated
than the red of the matching background. The key point here
is that the red matching background induces greenness into
the match disk. The 2-D model advocated here implies that
Figure 7. Redness and Greenness as Perceptual Dimensions
(A) Stimulus conditions giving rise to mixed color percepts composed of complementary local and edge-induced colors.
(B) The x- and y-axes correspond to the CIE-designated redness and greenness of the reference and matching displays, respectively. The matching (or
adjustable) disk is matchable to the reference disk when the reference disk is green (the red matching background adding greenness to the matching
disk) or more red than the matching background (the red matching background subtracting redness from the matching disk. No match can be made,
however, when the disk is less red than the matching background. We claim that this is because the matching background adds greenness to the
matching disk, which remains separate from the local redness. As there is no corresponding color induced from the grey reference background into the
reference disk, only a partial color match is possible. Data adapted from .
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Brightness and Darkness as Perceptual Dimensions
this induced greenness does not subtract from the local
redness. As there is no corresponding greenness induced
from the grey reference background into the reference disk,
matching only the local redness will not give rise to a perfect
color match. In such cases, subjects display a wide range of
behavioral strategies in a vain attempt to solve an insoluble
Materials and Methods
Experiment One. Monitor calibration and experimental methods/
procedures were similar to those documented elsewhere . The
stimuli consisted of a reference ring of constant luminance (30 cd/
m2). The starting value of the matching ring varied randomly between
10 and 90 cd/m2. The reference disk and background adopted
luminance values of either 10 and 90 cd/m2(high contrast) or 25 and
36 cd/m2(low contrast). The matching disk and background adopted
luminance values of either 10 and 90, 15 and 60, 20 and 45, or 25 and
36 cd/m2. In total, 16 reference-match combinations of these
luminance values were used. The diameters of the inner and outer
ring edges were respectively 28 and 68, and the centers of matching
and reference disks were 14.48 apart. The gradient in background
luminance was restricted to be within 6.28 about the midline of the
monitor. Order of stimulus presentation was pseudo-randomized.
Computational model. We seek to characterize the residual
perceptual difference between reference and matching rings, as
follows. Let R, B, D correspond to the luminance values of the ring
(R), the bright (B) contiguous surface, and the dark (D) contiguous
surface (either disk or background). Following our previous approach
, we define the respective brightness and darkness signals
associated with the ring as xB¼ wB½logR
parameters wBand wDrepresent the weights applied to the increment
and decrement signals computed in the ON and OFF pathways of
early visual processing [11,12]. The operator ½ ?þ¼ max loga
represents half-wave rectification, implying that only non-negative
brightness and darkness values are possible. By writing separate
equations for matching and reference displays with appropriate
scripting (superscripts m and r for match and reference), we define
the matching task as the task of setting xm
solve these equalities separately for logRm, giving logRm
estimates of logRm, giving the generic solution for the log of the
matching ring luminance as logRm¼ w
the superscript j ¼ 1,2 representing the inner and outer edges. We
estimated the weights using a nonlinear least-squares optimization
routine. Note that we identify the values of the estimated weights with
the theoretical weights associated with brightness and darkness
even though the theoretical weights cancel during calculation of the
separate brightness and darkness solutions, logRm
argument is that subjects attempt to compromise between perfect
darkness and brightness matches by setting log Rmbetween the values
and darkness weights, such that wj
We then separately calculated absolute differences between
reference and matching rings in the brightness and darkness
dimensions of achromatic color space:
D?þand xD¼ wD½logB
Br . We then assume that subjects weight the two
Dr þ w
Br , with
D. We adopt this interpretation
B, meaning that we can estimate the relative brightness
Note that a perfect match is only possible for ‘‘identity
matches’’ (Bm¼ Brand Dm¼ Dr). Match possibility was computed
using a modified city-block metric suitable for bounded data,
based on the Euclidian metric, mpj
measure did not, however, correspond closely with the data, a
conclusion consistent with previous results . In practice, we
allowed weights to vary linearly with the luminance of the matching
background and disk. This factor simulated gain control of edge
signals [4,5]. The ensuing model had eight free parameters, two for
each of the colored curves in Figure 3. We also tested a model in
CB¼ 10ð1 ? asinððDxj
DÞ0:5ÞÞ. We also constructed a measure
E¼ 10ð1 ? ðDxj2
which the weights remained constant with contrast, giving a total of
four free parameters. This model performed only marginally worse.
In other words, edge weights did not vary greatly with edge contrast.
The results shown in Figure 3 were derived using the more complex
model. We assumed throughout the modeling that subjects obeyed
one of the constraints, Bm. Rm. Dmor Bm, Rm, Dm, depending
on the stimulus. This was true for all but the leftmost data point in
Figure 4C. All computations were performed in Matlab (version
7.0.4, The MathWorks) using standard and customized functions.
Experiment Two. The design of Experiment Two was the same as
that of Experiment One, except for the changes below, which were
tailored to counter specific criticisms of the reviewers. The changes
were as follows: (1) the reference and matching rings were presented
in rapid sequence, rather than simultaneously, in order to minimize
the role of eye movements and adaptation (simultaneous presenta-
tion would also have required a very steep, therefore noticeable,
luminance gradient across the screen); (2) the subjects’ task was to
judge the similarity in grey shades of the successive rings, thereby
minimizing the use of elaborate strategies during a matching task.
Even though there was no matching task, we adopt the nomenclature
of calling the first display the reference and the second the matching
display; (3) the polarity of the reference and matching disks and
backgrounds either stayed the same or flipped in polarity, whereas
the total contrast, logDrBm
trial, the luminance of the reference ring had one of 19 luminance
values, defined as constant steps in log space between the minimum
and maximum values of the disk and background (15 and 60 cd/m2).
The value of the matching ring always remained constant at 30 cd/m2.
The experiment had four conditions, with trials being randomized
across conditions. In the two control conditions, the polarity of the
disks and backgrounds stayed the same (Dr¼Dm¼60, Br¼Bm¼15, Dr
¼Dm¼15, Br¼Bm¼60). In the other two conditions, the polarity of
the disks and backgrounds flipped (Dr¼ Bm¼ 60, Br¼ Dm¼ 15, Br¼
Dm¼ 15, Dr¼ Bm¼ 60). Subjects performed one entire run of the
experiment in order to generate an internal scale for the similarity
ratings. The ratio of diameters of the inner and outer ring edges was
the same as in Experiment One (1:3). The rings were, however, much
larger than in Experiment One, with the ring and disk subtending
approximately 3.58 and 10.58, respectively.
Proof: 1-D space implies perfect achromatic color matching.
Letting xB¼ wB½logR
early brightness and darkness signals subtract at a cortical processing
stage , then net brightness and darkness signals are computed as
WB¼ [XB? XD]þand WD¼ [XD? XB]þ. Since WBand WDcannot be
simultaneously positive, all possible grey shades can only contain
either brightness or darkness, but not both at the same time. Thus, all
grey shades can be specified by a single number, as seen most easily by
removing the half-wave rectification constraint on WB and WD,
implying that WB¼ ? WDand ?WB¼ WD. Thus the corollary of the
subtraction assumption is that all possible grey shades are contained
within a single dimension. To see how a 1-D space implies perfect
matching, let Wrand Wmequal either the net brightness or darkness
associated with the reference and matching rings, respectively. Letting
Wm¼ Wr, we derive logRm¼ w
The existence of this solution shows that the luminance of the
matching ring can always be set to achieve a perfect match. We have
confirmed this result using a 1-D model of the data presented here.
Calculation of darkness–brightness weight ratio. Let the true
darkness–brightness weight ratio be represented by
BrDm, remained constant throughout; (4) in each
D?þand xD¼ wD½logB
R?þagain, assume that the
Dr þ w
wB. We know
the two circumference-biased estimates,
/log(3) ’ log
/log(3) ’ 0.63, givingwD
. We solve, n¼log
We thank Eli Brenner, Vebjo ¨rn Ekroll, and Frans Faul for several
Author contributions. TV and MPL conceived and designed the
experiments and performed the experiments. TV analyzed the data.
TV, MPL, and FWC contributed reagents/materials/analysis tools and
wrote the paper.
Funding. This work was supported by grant 051.02.080 from The
Netherlands Organization for Scientific Research (NWO).
Competing interests. The authors have declared that no competing
PLoS Computational Biology | www.ploscompbiol.orgOctober 2007 | Volume 3 | Issue 10 | e179 1857
Brightness and Darkness as Perceptual Dimensions
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Brightness and Darkness as Perceptual Dimensions