Contrast gain-control in stereo depth and cyclopean contrast

Journal of Vision (2013) 13(8):3, 1–19
http://www.journalofvision.org/content/13/8/3
1
Contrast gain-control in stereo depth and cyclopean contrast
perception
Laboratory of Brain Processes, Department of
Psychology, The Ohio State University,
Columbus, OH, USA
$
Key Laboratory of Behavioral Science, Institute of
Psychology, Chinese Academy of Sciences, Beijing, China
$
Ju Liang
Vision Research Laboratory, School of Life Science,
University of Science and Technology of China,
Hefei, Anhui, China
$
Yifeng Zhou
Vision Research Laboratory, School of Life Science,
University of Science and Technology of China,
Hefei, Anhui, China
$
Laboratory of Brain Processes, Department of
Psychology, The Ohio State University,
Columbus, OH, USA
$
Fang Hou
Chang-Bing Huang
Zhong-Lin Lu
Although human observers can perceive depth from
stereograms with considerable contrast difference
between the images presented to the two eyes (Legge
& Gu, 1989), how contrast gain control functions in
stereo depth perception has not been systematically
investigated. Recently, we developed a multipathway
contrast gain-control model (MCM) for binocular phase
and contrast perception (Huang, Zhou, Lu, & Zhou,
2011; Huang, Zhou, Zhou, & Lu, 2010) based on a
contrast gain-control model of binocular phase
combination (Ding & Sperling, 2006). To extend the
MCM to simultaneously account for stereo depth and
cyclopean contrast perception, we manipulated the
contrasts (ranging from 0.08 to 0.4) of the dynamic
random dot stereograms (RDS) presented to the left
and right eyes independently and measured both
disparity thresholds for depth perception and perceived
contrasts of the cyclopean images. We found that both
disparity threshold and perceived contrast depended
strongly on the signal contrasts in the two eyes,
exhibiting characteristic binocular contrast gain-control
properties. The results were well accounted for by an
extended MCM model, in which each eye exerts gain
control on the other eye’s signal in proportion to its
own signal contrast energy and also gain control over
the other eye’s gain control; stereo strength is
proportional to the product of the signal strengths in
the two eyes after contrast gain control, and perceived
contrast is computed by combining contrast energy
from the two eyes. The new model provided an
excellent account of our data (r2 ¼ 0.945), as well as
some challenging results in the literature.
Introduction
Depth perception is an amazing achievement in
evolution, allowing 3-D perception of the world and
judgment of distance. Predators take advantage of the
ability to swoop down onto their prey, and tree-living
animals exploit it to accurately judge distances when
rapidly jumping from branch to branch. One important
cue to depth arises because the retinas of the two eyes
receive projections of the world from slightly different
angles. These differences or horizontal disparities
between the views of identical features in the left and
right eyes provide disparity cues to depth perception.
How images from the two eyes combine to generate
cyclopean perception, including perceived depth and
appearance of the cyclopean image, has been one of the
Citation: Hou, F., Huang, C.-B., Liang, J., Zhou, Y., & Lu, Z.-L. (2013). Contrast gain-control in stereo depth and cyclopean contrast
perception. Journal of Vision, 13(8):3, 1–19, http://www.journalofvision.org/content/13/8/3, doi:10.1167/13.8.3.
doi: 10 .116 7 /1 3. 8. 3
Received December 21, 2012; published July 2, 2013
ISSN 1534-7362 Ó 2013 ARVO
Hou et al.
most important inquiries in the study of visual
perception (Cormack, Stevenson, & Schor, 1991; Ding
& Sperling, 2006, 2007; Howard & Rogers, 2012a,
2012b; Huang, Zhou, Zhou, & Lu, 2010; Julesz, 1971,
1986; Kaufman, 1964; Legge & Rubin, 1981; Marr &
Poggio, 1979; Poggio & Poggio, 1984; Qian, 1997).
In this paper, we investigated effects of image
contrast and interocular contrast difference on stereo
depth and cyclopean contrast perception by independently manipulating the contrasts of the two monocular images in random dot stereograms (RDS) and
measuring both disparity thresholds for depth perception and perceived contrasts of the cyclopean images.
The multipathway contrast gain-control model
(MCM), previously developed for binocular contrast
and/or phase combination (Ding & Sperling, 2006,
2007; Huang, Zhou, Lu, Feng, & Zhou, 2009; Huang,
Zhou, Lu, & Zhou, 2011; Huang et al., 2010), was
elaborated to simultaneously account for disparity
thresholds and perceived contrasts of the cyclopean
images. The model also accounted for some challenging
data on stereo depth perception in the literature
(Cormack et al., 1991; Ding & Levi, 2011; Legge & Gu,
1989).
Effects of image contrast on stereo depth
perception
Using narrow-band stimuli, Halpern and Blake
(1988) and Legge and Gu (1989) found that the slope of
disparity threshold versus (equal) image contrast in the
two eyes was around 0.5 on a log-log plot, indicating
an inverse-square-root dependency of disparity threshold on image contrast. Later, Cormack et al. (1991)
found that the log-log slope for high contrast was
0.33, comparable with previous findings. At low
contrast, however, the slope became closer to 2, which
indicates an inverse-square dependency of disparity
threshold on image contrast. The different slopes in
different contrast conditions imply that the relationship
between disparity threshold and image contrast is more
complex than a simple inverse-square-root or inversesquare law.
Effects of interocular contrast difference on
stereo depth perception
Halpern and Blake (1988) also investigated effects of
interocular contrast difference on stereo depth perception. They found that disparity threshold increased
monotonically when the contrast of the image in one
eye was maintained at a fixed high value while the
contrast of the image in the other eye decreased
successively from that value. For low spatial frequency
2
12
reduced in two eyes
reduced in one eye
10
Disparity Threshold
Journal of Vision (2013) 13(8):3, 1–19
8
6
4
2
0
0.08
0.16
0.24
0.32
0.4
Contrast
Figure 1. Schematic plots to illustrate the stereo contrast
paradox. Disparity thresholds are shown as functions of image
contrast in one eye when the contrast in the other eye is fixed
at 0.4 (squares) or equal image contrast in both eyes (circles).
Threshold increased more when image contrast in only one eye
was reduced than when image contrasts in both eyes were
reduced by the same amount.
stimuli, when the contrast of the image in one eye was
maintained at a fixed low value while the contrast in the
other eye gradually increased from the low value,
disparity threshold also increased. Using sine wave
gratings, Legge and Gu (1989) found that disparity
threshold for stereo depth perception increased as
interocular contrast ratio increased. Similar observations have also been reported by Ding and Levi (2011),
Schor and Heckmann (1989), and Simons (1984).
In addition, several studies (Halpern & Blake, 1988;
Legge & Gu, 1989; Schor & Howarth, 1986) reported
that disparity threshold rose more when image contrast
was reduced in only one eye than when image contrasts
in both eyes were reduced by the same amount. Given
that the former condition (image contrast was reduced
only in one eye) had more contrast energy than the
latter one (image contrast was reduced in both eyes, see
Figure 1), Cormack et al. (1991) coined the term
‘‘stereo contrast paradox’’ to refer to the counterintuitive observations. Take the left most two data points in
Figure 1 as an example; disparity threshold is greater
when the image contrasts in the two eyes were 0.08 and
0.40 than when the contrasts in the two eyes were both
at 0.08.
These results, including the stereo contrast paradox,
suggest a complex relationship between stereo depth
perception and image contrasts in the two eyes.
Existing models on stereo depth perception
Cormack et al. (1991) proposed a cross-correlation
model on stereo depth perception. In their model, the
strength of the stereo depth signal is proportional to the
cross-correlation of the images in the left and right
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
eyes. The model predicts that disparity threshold is
inversely proportional to the product of the contrasts
of the images in the two eyes, an inverse-square law
when image contrasts are the same in the two eyes. It
cannot however explain the disparity threshold function under unequal contrast conditions observed in
some experiments (Halpern & Blake, 1988; Legge &
Gu, 1989; Schor & Heckmann, 1989).
In an attempt to explain the observed effects of
interocular contrast difference on disparity threshold,
Legge and Gu (1989) developed a ‘‘peak’’ model, in
which (a) the standard deviation of stimulus location
encoding is inversely proportional to the square root of
the contrast of the stimulus, (b) the encoded stimulus
locations in the two eyes are correlated, and (c)
disparity threshold is proportional to the standard
deviation of the joint distribution of encoded stimulus
peak locations in the two eyes:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
2q
D r2L þ r2R 2qrL rR ¼
þ
pffiffiffiffiffiffiffiffiffiffiffiffi; ð1Þ
CL CR
CL CR
where D is disparity threshold, rL and rR are the
standard deviations of the distributions of the encoded
locations of p
the
ffiffiffiffi input images in the left and right eyes
with r 1/ C, q is the correlation between the
encoded locations in the two eyes, and CL and CR are
image contrasts in the two eyes.
The peak model is consistent with the inversesquare-root relationship between disparity threshold
and image contrast and provided a good account for
the U-shaped disparity threshold versus contrast ratio
(DVR) curves (Legge & Gu, 1989). However, the
correlation parameter q in the model is not specified
theoretically, and the predicted disparity threshold
saturates at large contrast ratios (Kontsevich & Tyler,
1994). Moreover, the correlation parameter might vary
with image contrast (Cormack et al., 1991).
Kontsevich and Tyler (1994) proposed an alternative
model, which shares the notion of the peak model
(Legge & Gu, 1989) that stereo depth is derived from
the encoded positions of the images in the two eyes, and
the standard deviation of the encoded image location in
each eye is proportional to the square root of the
contrast of the image. Instead of assuming correlation
between the encoded locations of the images in the two
eyes (Legge & Gu, 1989), Kontsevich and Tyler (1994)
used a weighted binocular summation rule (Schrodinger, 1926) in computing disparity threshold:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
;
ð2Þ
þ
D r2L þ r2R ¼
wL CL wR CR
where wL ¼ CpL /(CpLþ kCpR ) , wR ¼ CpR /(CpR þ kCpL ) , and p
is the exponent of the power function.
Although Kontsevich and Tyler’s model is superior
to the peak model, there are still some notable issues.
3
Similar to the peak model, it cannot explain the
different slopes of the disparity threshold versus
contrast function in low and high contrast conditions.
Moreover, the model predicts a disparity threshold
even when the contrast of the image in one eye is zero,
i.e., only one eye receives an image, and there is
obviously no disparity in the stimulus.
Cormack, Stevenson, and Landers (1997) later
proposed a hybrid model based on Legge and Gu
(1989) and Kontsevich and Tyler (1994). The model
incorporates both correlation between the encoded
positions in the two eyes and a weighted binocular
summation rule:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D r2L þ r2R 2qrL rR
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
2q
¼
þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;
ð3Þ
wL CL wR CR
wL CL wR CR
where wL ¼ CpL /(CpLþ kCpR ) , wR ¼ CpR /(CpR þ kCpL ), and p
is the exponent of the power function.
Although the hybrid model provided better fits to
their data, Cormack et al. (1997) commented that the
hybrid model inherited the shortcomings of its parents:
In addition to the frequency dependence implicit in the
models (Cormack et al., 1997; Legge & Gu, 1989), the
free parameters q and k could vary non-systematically
with spatial frequencies.
Binocular contrast and phase combination and
the multipathway contrast gain-control model
(MCM)
Ding and Sperling (2006) developed a paradigm to
measure perceived phase of cyclopean images as a
function of the contrasts of the images presented to the
two eyes. In this paradigm, two suprathreshold
horizontal sine wave gratings of the same spatial
frequency but different spatial phases are presented to
the left and right eyes of the observer. The perceived
phase of the binocularly combined cyclopean image is
measured as a function of the contrast ratio between
the images in the two eyes. Because the perceived phase
of the cyclopean image is determined by the relative
amplitudes of the component sine wave gratings at the
stage of binocular combination, one can infer the
monocular contrast transfer function (how grating
contrast is transmitted through each eye) from these
measurements. In addition, Ding and Sperling (2006,
2007) proposed a contrast gain-control model of
binocular combination in which each eye (a) exerts gain
control on the other eye’s signal in proportion to the
contrast energy of its own input and (b) additionally
exerts gain control on the other eye’s gain control. With
a single parameter, the model successfully accounted
Journal of Vision (2013) 13(8):3, 1–19
a)
b)
Futher Processes
in Separated
Pathway
‘
Hou et al.
Contrast
‘
‘
‘
c)
A2
Phase
A1
A3
TCE
‘
TCE
‘
d)
Stereo
‘
‘
4
excellent account of data from all the experimental
conditions. The MCM has also been used to successfully characterize deficits in binocular combination in
anisometropic amblyopia as a joint mechanism of both
attenuation of signals in the amblyopic eye and
increased inhibition from the fellow eye to the
amblyopic eye (Huang et al., 2011).
In the multipathway contrast gain-control model
(MCM) (Figure 2), signals in the two eyes first go
through interocular contrast gain control, in which
each eye exerts gain control not only on the other eye’s
signal (direct inhibition, Path A2 and its counterpart in
Figure 2a), but also on the incoming gain-control signal
from the other eye (indirect inhibition, Path A3 and its
counterpart in Figure 2a), with both effects in
proportion to an eye’s own signal contrast energy.
After contrast gain control, signal strengths in the two
eyes are:
0
Figure 2. A schematic diagram of the multipathway contrast
gain-control model (MCM). In the MCM (a), the signals in the
two eyes first go through interocular contrast gain control, in
which each eye exerts gain control not only on the other eye’s
visual signal but also on the incoming gain control signal from
the other eye, with both effects in proportion to an eye’s own
total contrast energy (TCE). The perceived phase, contrast and
depth information are computed in separate pathways (b, c, d).
CL ¼ CL
1
eR ;
1 þ 1þe
L
where
c
eL ¼ qCL1
ð4aÞ
1
eL ;
1 þ 1þe
R
0
CR ¼ CR
where
for 95% of the variance for 48 combinations of phases
and contrasts in their experiments. The paradigm and
the model were adopted by Huang et al. (2009) to study
suprathreshold cyclopean perception in anisometropic
amblyopia, finding that stimulus of equal contrast was
weighted much less in the amblyopic eye relative to the
fellow eye in binocular combination.
Huang et al. (2010) adopted the Ding-Sperling
paradigm but measured both perceived contrast and
phase of cyclopean images in 90 combinations of base
contrast, interocular contrast ratio, eye origin of the
probe, and relative phase. They found that the
perceived contrast of cyclopean images was independent of the relative phase of the monocular sine wave
gratings (but see Baker, Wallis, Georgeson, & Meese,
2012), although the perceived phase depended on the
relative phase and contrast ratio of the monocular
images. Based on these results, they developed a new
multipathway contrast gain-control model (MCM) that
elaborates the Ding-Sperling binocular combination
model in two ways: (a) phase and contrast of the
cyclopean images are computed in separate pathways,
although with shared contrast gain-control processes;
and (b) phase-independent local energy from the two
monocular images are used in contrast combination.
With three free parameters, the model yielded an
c
eR ¼ qCR1 ;
ð4bÞ
where CL and CR are the contrasts of the input images
in the left and right eyes, q is the contrast gain-control
efficient, and c1 is the exponent of the power-law
nonlinearity in the gain-control pathway.
The perceived phase and contrast of the cyclopean
percept are computed in separate pathways (Huang et
al., 2010):
!
1þc1
1þc1
C
þ
qC
C
qC
h
L
R
L
R
hperceived ¼ 2tan1
tan
1þc1
1þc1
2
CL þ qCL þ CR þ qCR
ð5aÞ
0
c
0
c
Cperceived ¼ ðCL2 þ CR2 Þ1=c2
c2
c
CL ð1 þ qCL1 Þ
¼
c
c
1 þ qCL1 þ qCR1
c
CR ð1 þ qCR1 Þ
þ
c
c
1 þ qCL1 þ qCR1
c2 !1=c2
;
where c2 is the exponent of the power function in
binocular contrast combination.
ð5bÞ
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
Elaboration of the MCM to model stereo depth
perception
The observed functional relationship between stereo
depth perception and image contrast in the literature
(Cormack et al., 1991; Halpern & Blake, 1988; Legge &
Gu, 1989; Schor & Heckmann, 1989; Simons, 1984)
and results from physiology (Anzai, Ohzawa, &
Freeman, 1999a, 1999b; Cumming & DeAngelis, 2001;
Grossberg & Howe, 2003; Ohzawa, 1998; Ohzawa &
Freeman, 1986a, 1986b) suggest that interocular
contrast gain control may play a central role in stereo
depth perception. Here, we elaborate the MCM in an
attempt to simultaneously model stereo depth perception and binocular contrast combination.
Our formulation of the relationship between disparity threshold and image contrasts in the two eyes is
based on the cross-correlation model, which has
received support from many psychophysical experiments (Allenmark & Read, 2010, 2011; Banks, Gepshtein, & Landy, 2004; Cormack et al., 1991; Filippini &
Banks, 2009; Harris, McKee, & Smallman, 1997;
Nienborg, Bridge, Parker, & Cumming, 2004), and has
found many successful applications in computer vision
(Clerc & Mallat, 2002; Donate, Liu, & Collins, 2011;
Heo, Lee, & Lee, 2011; Kanade & Okutomi, 1994). The
product rule in the cross-correlation model is consistent
with the prevailing disparity energy models of V1
neurons (Anzai et al., 1999a, 1999b; Chen & Qian,
2004; Filippini & Banks, 2009; Ohzawa, 1998; Ohzawa,
DeAngelis, & Freeman, 1990, 1996; Qian, 1994, 1997)
when the outputs of binocular complex cells with
opposite preferred polarities of disparity are compared
to extract the polarity of the perceived disparity. In the
original cross-correlation model (Cormack et al., 1991),
stereo strength is proportional to the product of signal
contrasts in the two eyes. Here, we postulate that stereo
strength is proportional to the product of the signal
strengths in the two eyes after contrast gain control,
and disparity threshold is inversely proportional to the
product:
Ds ¼
k
CL0 CR0
¼
CL 1þ1eR
1þeL
k
1
CR 1þ eL
k
10
¼0
B
@CL
1þeR
1
c
qC 1
1þ Rc1
1þqC
L
CB
A@CR
c
¼
1
1
c
qC 1
1þ Lc1
1þqC
R
C
A
c
kð1 þ qCL1 þ qCR1 Þ2
c
c ;
CL CR ð1 þ qCL1 Þð1 þ qCR1 Þ
ð6Þ
where k is a scaling constant, and q and c1 are the
contrast gain-control efficient and the exponent of the
5
power-law nonlinearity in the gain-control pathway,
respectively.
Overview
In this study, we independently manipulated the
contrasts of the two monocular images of random dot
stereograms (RDS) and measured both disparity
thresholds for depth perception and perceived contrasts
of the cyclopean images. The data provided a strong
test of the newly elaborated MCM in its ability to
simultaneously account for disparity threshold and
perceived contrast in cyclopean perception. In addition,
we applied the model to account for some challenging
results on stereo depth perception in the literature
(Cormack et al., 1991; Ding & Levi, 2011; Legge & Gu,
1989).
Methods
Observers
Three observers, S1 (the first author), S2, and S3,
aged 23 to 30 years, participated in the study. All had
normal or corrected-to-normal vision and normal
stereo acuity (Titmus Fly stereo test, Stereo Optical
Co., Inc., Chicago, Illinois). S2 and S3 were naive to
the purpose of the study. Written informed consent was
obtained before the experiment.
Apparatus
Gamma-corrected stimuli were generated by a PC
running Matlab (The Mathworks Corp., Natick, MA)
and PsychToolbox subroutines (Brainard, 1997; Pelli,
1997), and displayed on either a Dell 19-in M993s
monitor (mean luminance 37.3 cd/m2) for S1 or on a
Sony 17-in G220 monitor (mean luminance 22.8 cd/m2)
for S2 and S3. Both monitors had 2048 · 1536 pixel
resolution and a vertical refresh rate of 60 Hz. The
viewing distance was 2.44 m for S1 and 2.22 m for S2
and S3. Different viewing distances were used to match
the visual angles of the stimuli on different monitors. A
special circuit was used to achieve 14-bit grayscale
resolution (Li & Lu, 2012; Li, Lu, Xu, Jin, & Zhou,
2003). Stereoscopes (Geoscope, Standard Mirror Stereoscope) were used to deliver the left and right images
of the random dot stereograms to the corresponding
eyes. A chin/forehead rest was used to minimize head
movement during the experiment.
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
6
Figure 3. Experimental procedures. The 2-IFC depth detection task is depicted in (a) and (b). (a) Dichoptic stimuli presented to the two
eyes; (b) cyclopean percept of (a). The first interval contains a stereo stimulus: The lower half of the RDS patch is in front the upper
half. The display sequence and the cyclopean percept of the stereo contrast matching task is shown in (c) and (d). (c) Dichoptic
stimuli. The probe is only presented to the left eye; (d) cyclopean percept of (c).
Stimuli
Dynamic random dot stereograms (RDS) were used
in the experiment. Binary random dot patches with a
50% dot density were embedded in a 3.33 degree
diameter circular aperture with a black background.
Each dot consisted of 10 ·10 pixels and subtended 2.5
· 2.5 minutes of arc of visual angle. The dot patterns in
the two eyes were identical except for a relative
horizontal shift (disparity) in the target regions in the
two eyes. The random dot images in each eye could
independently take five possible contrasts: 0.08, 0.16,
0.24, 0.32, and 0.4, leading to a total of 25 possible
contrast combinations. The contrast range was carefully chosen to minimize adaptation (Ohzawa &
Freeman, 1994). A pilot study showed that the 81.6%
correct contrast detection threshold of the dynamic
RDS was , 0.06 for all observers and eyes. No one
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
exhibited significantly different detection thresholds in
the two eyes.
Procedure and experimental design
A two-interval forced choice (2-IFC) task was used
to measure disparity threshold. Each trial began with a
fixation image (Figure 3a), consisting of a dark fixation
dot (diameter of 6.25 min) in the center of a randomdot background (Michelson contrast ¼ 0.2). The
vertical and horizontal positions of the two monocular
images could be adjusted manually to achieve better
convergence in the beginning of each block. After
successfully combining the two monocular images into
one steady cyclopean image, observers pressed the
space bar on the computer keyboard to start the
presentation of a 1300-ms dynamic RDS movie. The
movie contained two 400-ms stimulus intervals, each
consisting of eight frames of independently sampled
RDS and delimited by a brief tone in the beginning and
a 500-ms interstimulus interval. Dynamic RDS with
zero disparity was presented during the 500-ms
interstimulus interval. The RDS in one of the randomly
chosen intervals contained a horizontal disparity in the
lower half of the images in the last six frames, which, if
above threshold, could be perceived as being closer to
the observer than the upper half of the images; The
RDS in the other interval had no horizontal disparity.
The two zero disparity frames were introduced in the
beginning of each interval to help observers achieve
better fusion and to eliminate the difference (if any)
between the first and second intervals. Observers were
instructed to indicate in which interval they saw stereo
depth (i.e., the lower half is in front of the upper half of
the images) by pressing a key on the computer
keyboard. A static RDS with zero disparity was shown
in the response period. No feedback was provided. The
w method (Kontsevich & Tyler, 1999) was used to
measure the disparity threshold at 81.6% percentage
correct in each contrast combination with 50 trials, for
a total of 1,250 trials (25 Combinations · 50 Trials/
Combination) per observer. The slope of the psychometric functions was set to 2.2 in the w procedure based
on the data from a pilot study. All conditions were
randomly intermixed.
Perceived cyclopean contrast was estimated using a
contrast matching task and a staircase method.
Specifically, each trial began with a fixation image and
was followed by six 50-ms dynamic RDS frames that
were split into upper and lower halves. The upper half
of each RDS frame was a monocular probe that was
presented to either the left or right eye, with a blank
image with background luminance in the other eye. The
lower half of each RDS frame contained images in both
eyes. The random dot image in the lower half of each
7
eye was independently assigned one of five possible
contrasts: 0.08, 0.16, 0.24, 0.32, and 0.4. The disparity
of the random dot stereogram in the lower half was
either crossed or uncrossed and set at around twice of
the individual’s largest disparity threshold measured
using stereograms with a 0.08 contrast in both eyes.
Observers were instructed to first report whether the
lower half of the RDS was in front of (crossed) or
behind (uncrossed) the fixation dot, and then which
part, the upper or lower half of the image, had a higher
contrast. The fixation point, present in the middle of
the screen throughout the entire trial, served as the
reference for depth discrimination.
The contrast of the probe was controlled by a
truncated staircase that converged to 50%. Only trials
with correct depth judgment were tracked by the
staircase. The staircases for different contrast combinations were interleaved. There were 50 Trials · 25
Contrast Combinations · 2 Eye Origin of Probe · 2
Depths (crossed and uncrossed), with 5,000 trials in
total for each observer. The perceived contrast of
cyclopean image could thus be expressed by the
matched probe contrast in the left or right eyes, which
was calculated as the average of the probe contrasts in
the upper half, taken at a minimum of 10 reversal
points in the staircase.
Data analysis
Within-observer analysis of variance (ANOVA)
was used to test whether matched probe contrast
depended on the contrasts of the images in the two
eyes, eye origin of the probe, depth polarity (crossed
and uncrossed), and their interactions, and whether
disparity threshold depended on contrasts in the two
eyes.
All the model-fitting programs were implemented in
Matlab (MathWorks). The summed squares of the
differences between
P model predictions and observed
values, SSE ¼ (yi ŷi)2, where yi and ŷi represented
the observed and predicted values, respectively,
and the
P
sum of total squared errors, SST ¼ (yi ȳ)2, where ȳ
is the mean of yi, were computed separately for
disparity thresholds and perceived contrasts in term of
matched probe contrasts in the left and right eyes. We
define r2 as:
X
SSE
r ¼
¼
SST
2
ðyi ŷi Þ2
X
:
ð7Þ
2
ðyi ȳÞ
The average r2, (r2d þ r2CL þ r2CR )/3, where r2d , r2CL , and
r2CR are r2s for disparity threshold, matched probe
contrasts in the left and right eyes, was used as the cost
function in a modified least-squares procedure in fitting
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
8
Figure 4. Matched probe contrasts of the cyclopean images as functions of image contrasts in the left and right eyes. Red dots:
matched contrast with probe in the left eye. Blue squares: matched contrast measured with probe in the right eye. Upper row:
matched contrasts of uncrossed depth planes. Bottom row: matched contrasts of crossed depth planes. Error bars represented 61
SD.
the model to the data. The goodness of fit of the MCM
was also evaluated by the average r2.
An F test was used to statistically compare the
performance of nested models:
Fðdf1 ; df2 Þ ¼
ðr2full r2reduced Þ=df1
ð1 r2full Þ=df2
;
ð8Þ
where df1 ¼ kfull kreduced, df2 ¼ N kfull; kfull and
kreduced are the numbers of parameters of the full and
reduced models, respectively, and N is the number of
data points.
Results
Effects of image contrasts on perceived
cyclopean contrast
Matched probe contrasts of the cyclopean images for
the three observers are plotted as bivariate functions of
image contrasts in the left and right eyes in Figure 4.
An analysis of variance (ANOVA) revealed that
matched probe contrast of cyclopean images varied
significantly with test image contrasts in both eyes, F(4,
8) ¼ 506, p ¼ 1.20 · 109, for the probe in the
nondominant eye; F(4, 8) ¼ 140, p ¼ 1.95 · 107, for the
probe in the dominant eye. There was also a significant
interaction between the test contrasts in the two eyes,
F(16, 32) ¼ 56.1, p ¼ 2.23 · 10308. There was no
significant effect of depth polarity, F(1, 2) ¼ 4.22, p ¼
0.176. The matched contrasts were averaged across the
uncrossed and crossed depth conditions in subsequent
analyses. As we noted in Figure 4, the matched probe
contrasts in the left eye were systematically deviated
from those in the right eye for all observers, there was a
significant effect of the eye origin of the probe, F(1, 2) ¼
44.0, p ¼ 0.022.
Because the same stimuli were used in the lower half
of the stereograms to construct the cyclopean images,
different resulting matches obtained with probes in the
two different eyes suggest that image contrasts in the
two eyes were not represented equally. To characterize
the difference of the matched probe contrasts for the
cyclopean images between two eyes, a straight line
through the origin:
Cmatched; probe in R ¼ bCmatched; probe in L ;
ð9Þ
where b is the slope, was fitted to the matched probe
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
S3
S2
S1
Disparity threshold (arcmin)
9
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0.08
0
0.08
0
0.08
0.16
0.08
0.16
0.24
Co
ntr
as
tR
0.24
0.32
0.4 0.4
0.32
ast
ntr
Co
L
0.16
0.08
0.16
0.24
Co
ntr
as
tR
0.24
0.32
0.4 0.4
0.32
Co
ast
ntr
L
0.16
Co 0.24
ntr
as 0.32
tR
0.4 0.4
0.08
0.16
0.24
0.32
Co
ntr
ast
L
Figure 5. Measured disparity threshold as a function of the contrasts in the left and right eyes for observers S1, S2, and S3. Error bars
represented 61 SD.
contrasts in the left eye versus those in the right eye for
all observers. For S1, the slope of the best fitting line
was 1.11, averaged across the crossed and uncrossed
conditions, significantly larger than one, t(48) ¼ 5.39, p
¼ 1.05 ·106. So for S1, signals in the left eye were
attenuated compared to the right eye by a factor of
0.902. For S2, the slope of best fitting line was 0.900,
significantly deviated from identity, t(48) ¼ 4.00, p ¼
1.00 · 104. For S3, the slope was 0.934, t(48) ¼ 3.41, p
¼ 6.00 · 104. For S2 and S3, signals in the right eye
were attenuated by 0.900 and 0.934, respectively,
relative to those in the left eye. Interestingly, for S1, the
direction of attenuation was not consistent with his eye
dominance measured by the hole in the card test. Taken
together, these results indicate that the effectiveness of
the same contrast was not equal in the two eyes,
consistent with imbalanced representation of the two
eyes found in others studies (Huang et al., 2010; Legge
& Rubin, 1981). In the rest of the paper, eye dominance
was defined based on the direction of attenuation in the
cyclopean contrast combination task.
Effects of contrast on disparity threshold
The observed disparity thresholds are plotted as
functions of image contrasts in the left and right eyes
for each observer in Figure 5. Generally, when image
contrast in one eye was low (e.g., 0.08), measured
disparity threshold dropped rapidly as image contrast
in the other eye increased; when image contrast in one
eye was high (e.g., 0.4), disparity threshold dropped
slowly as image contrast in the other eye increased.
Disparity thresholds depended significantly on image
contrasts in both the dominant, F(4, 8) ¼ 46.5, p ¼ 1.40
· 105, and nondominant, F(4, 8) ¼ 29.4, p ¼ 7.81 ·
105, eyes. The interaction between image contrasts in
the two eyes was marginally significant, F(16, 32) ¼
1.87, p ¼ 0.064.
To explore the relationship between disparity
threshold and image contrast, we graphed disparity
threshold as a function of the product of the contrasts
in the two eyes on a log-log plot (Figure 6). The data
points with image contrast in the left eye equal to,
greater than, or less than that in the right eye are
plotted with different symbols. Two straight lines of
slopes 1 and 0.25, which represent the inverse-square
and inverse-square-root dependency on contrast, are
overlaid on the graph.
Generally, disparity threshold decreased as contrast
product increased. When the contrasts in the two eyes
are equal (black squares), the slope of the threshold
versus contrast product curve is neither 1 nor 0.25
for all observers, suggesting that disparity threshold
cannot simply be described by an inverse-square or
inverse-square-root function on contrast. Taking all
data points (with equal and unequal contrasts in the
two eyes) into account, the overall slope of the
threshold versus contrast product function was 0.725,
0.509, and 0.594 for S1, S2, and S3, respectively.
In addition, for S1, the imbalance between the left
and right eyes revealed in the binocular contrast
combination task was also found in the disparity
threshold data: The threshold was higher when the
contrast in the left eye was greater than that in the
opposite direction.
Earlier studies (Cormack et al., 1991; Halpern &
Blake, 1988; Legge & Gu, 1989; Schor & Heckmann,
1989) have found that disparity threshold depended on
image contrasts in both eyes as well as interocular
contrast difference between the two eyes. To illustrate
the effect of unequal image contrasts in the two eyes,
we replotted the observed disparity threshold as
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
S3
S2
S1
Disparity Threshold (arcmin)
10
Slope=−0.725
5
4
3
5
4
3
5
4
3
2
2
2
1
1
1
0.005
L=R
L>R
L<R
Regression
Slope=−0.509
0.025
0.125
Contrast product in two eyes
0.005
0.025
0.125
Contrast product in two eyes
Slope=−0.594
0.005
0.025
0.125
Contrast product in two eyes
Figure 6. Disparity threshold versus product of image contrasts in the two eyes. Squares: image contrasts in the left and right eyes
were equal. Circles: image contrast in the left eye was greater than that in the right eye. Triangle: image contrast in the left eye was
less than that in the right eye. Dash lines represent the predictions of an inverse-square and inverse-square-root dependency. Solid
line: linear regression of all data points. Error bars represented 61 SD.
functions of interocular contrast ratios in Figure 7
using the format developed in Legge and Gu (1989).
The disparity threshold versus contrast ratio functions
(DVR) are U-shaped. Increasing contrast ratio increased disparity threshold. The results are consistent
with Legge and Gu (1989).
Several studies (Cormack et al., 1997; Halpern &
Blake, 1988; Legge & Gu, 1989; Schor & Heckmann,
1989) have found a phenomenon called stereo contrast
paradox using narrow band stimuli, in which disparity
threshold rose more when image contrast in only one
eye was reduced than when image contrasts in both
eyes were reduced by the same amount. In Figure 8, we
plotted the observed disparity thresholds as a function
S1
Disparity Threshold (arcmin)
5
of image contrast in one eye while keeping image
contrast in the other eye fixed at 0.4, along with the
disparity thresholds when image contrasts were reduced
in both eyes by the same amount. Inconsistent with
those studies in the literature, we found that disparity
threshold rose more rapidly when image contrasts were
reduced in both eyes than when contrast was reduced in
only one eye. In other words, we did not observe the
stereo contrast paradox. The absence of stereo contrast
paradox with RDS has also been reported by Cormack
et al. (1997) who used RDS and high spatial frequency
narrow-band stimuli. We discuss potential explanations
in the Discussion.
S2
5
0.08
0.16
0.24
0.32
0.40
4
4
4
3
3
2
2
2
1
1
1
5
3
2
1
2
3
L/R, Contrast Ratio, R/L
5
5
3
2
1
S3
5
2
3
L/R, Contrast Ratio, R/L
5
3
5
3
2
1
2
3
5
L/R, Contrast Ratio, R/L
Figure 7. Disparity threshold as a function of image contrast ratio between the left and right eyes and the right and left eyes for
observers S1, S2, and S3. The higher contrast was fixed at 0.08 (red circle), 0.16 (green square), 0.24 (blue asterisk), 0.32 (cyan cross),
or 0.4 (purple triangle). Error bars represented 61 SD.
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
S2
S1
S3
5
5
Disparity Threshold (arcmin)
11
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0.08
0.16
0.24
0.32
0.4
0
0.08
Contrast in one eye
0.16
0.24
0.32
0
0.4
reduced in both eyes
reduced in Left eye
reduced in right eye
0.08
Contrast in one eye
0.16
0.24
0.32
0.4
Contrast in one eye
Figure 8. Black circles: disparity threshold as a function of (equal) image contrasts in the two eyes. Red triangles: disparity threshold
as a function of image contrast in the left eye while image contrast in the right eye was kept at 0.4. Blue triangles: disparity
threshold as a function of image contrast in the right eye while image contrast in the left eye was kept at 0.4. Error bars represent
61 SD.
Model fit
Cperceived
Because of the observed asymmetry of the two eyes
in our data, we adopted an elaborated MCM developed
by Huang et al. (2011) in the context of amblyopic
vision that modeled imbalanced processing of information in the two eyes with three mechanisms: (a)
attenuation of signal contrast in the nondominant eye
(g), (b) stronger direct inhibition from the dominant
eye (a), and (c) stronger indirect inhibition (b) from the
dominant eye to the gain control signal coming from
the nondominant eye. For simplicity, here we assume
the left eye is the dominant eye in these equations. In
cases where the right eye dominates, we simply reverse
the symbols.
In the elaborated model, after contrast gain control,
signal strengths in the two eyes are:
CL0 ¼ CL
1
1þ
eR
1þbeL
;
CR0 ¼ gCR
!c2 !1=c2
1
¼
CL
þ gCR
eR
aeL
1 þ 1þbe
1 þ 1þe
L
R
c2
c1
CL ð1 þ bqCL Þ
¼
c
1 þ bqCL1 þ qðgCR Þc1
c2 !1=c2
gCR ð1 þ qðgCR Þc1 Þ
;
ð11Þ
þ
c
1 þ aqCL1 þ qðgCR Þc1
where c2 is the transducer nonlinearity for the power
summation in binocular contrast combination. Thus,
when the probe is presented to different eyes, we have:
Cmatched; probe in L ¼
1
aeL ;
1 þ 1þe
R
c
eR ¼ qðgCR Þc1 ;
c
CL ð1 þ bqCL1 Þ
c
1 þ bqCL1 þ qðgCR Þc1
c2
gCR ð1 þ qðgCR Þc1 Þ
þ
c
1 þ aqCL1 þ qðgCR Þc1
where
eL ¼ qCL1 ;
!c2
1
ð10Þ
CL and CR are the contrasts in the left and right eyes, q
is the contrast gain-control efficiency, g is used to
model contrast attenuation in right eye, a, b are used to
model the stronger inhibition to the right eye from the
left eye, and c1 is the transducer nonlinearity in the
gain-control pathway.
The perceived contrast of the cyclopean image
becomes (Huang et al., 2011):
c2 !1=c2
;
ð12aÞ
and
Cmatched; probe in R
1
¼
g
c
CL ð1 þ bqCL1 Þ
c
1 þ bqCL1 þ qðgCR Þc1
c2
gCR ð1 þ qðgCR Þc1 Þ
þ
c
1 þ aqCL1 þ qðgCR Þc1
c2!1=c2
:
ð12bÞ
Journal of Vision (2013) 13(8):3, 1–19
Observer
Model
S1
Reduced1
Reduced2
Full
Reduced1
Reduced2
Full
Reduced1
Reduced2
Full
S2
S3
k
Hou et al.
q
0.013 36.8
0.012 40.3
0.022 28.0
0.017 6.22
0.016 6.71
0.018 7.22
0.017 6.55
0.016 6.51
0.016 6.45
c1
c2
1.56
1.57
2.23
1.39
1.40
1.63
1.25
1.25
1.22
0.828
0.826
1.28
0.902
0.909
1.02
0.910
0.912
0.880
g
a
b
12
2
r
Comparison
- 0.942 Reduced1, Reduced2
0.903 - 0.976 Reduced1, Full
0.916 1
4.47 0.980 Reduced2, Full
- 0.887 Reduced1, Reduced2
0.888 - 0.919 Reduced1, Full
0.891 1
1.72 0.920 Reduced2, Full
- 0.931 Reduced1, Reduced2
0.950 - 0.938 Reduced1, Full
0.950 1.14 1
0.940 Reduced2, Full
F
F(1,
F(3,
F(2,
F(1,
F(3,
F(2,
F(1,
F(3,
F(2,
70)
68)
68)
70)
68)
68)
70)
68)
68)
¼
¼
¼
¼
¼
¼
¼
¼
¼
p
99.2
43.1
6.80
27.7
9.35
0.425
7.90
3.40
1.13
4.77 · 1015
9.99 · 1016
0.002
1.48 · 106
3 · 105
0.656
0.006
0.023
0.328
Table 1. Parameters of best fitting MCM models for all observers and comparisons between them.
The disparity threshold is computed as:
k
10
D¼0
@CL
1
qðgCR Þc1
c
1þbqC 1
L
1þ
A@gCR
nondominant eye in addition to monocular attenuation. The predictions of the best fitting models are
plotted in Figure 9.
1
1
c
aqC 1
1þ1þqðgCL Þc1
A
R
c
c
k 1 þ bqCL1 þ qðgCR Þc1 1 þ aqCL1 þ qðgCR Þc1
¼
:
c
gCL CR ð1 þ bqCL1 Þ 1 þ qðgCR Þc1
ð13Þ
The elaborated MCM was fitted to the observed
disparity thresholds and perceived contrasts by a
nonlinear least-squares method to accommodate the
evident eye dominance in our data. To explore whether
the attenuation mechanism alone is enough to account
for the interocular imbalance in the current data, or
whether the inhibition mechanism is also required,
three models, reduced Model 1 (with four parameters k,
q, c1, and c2), reduced Model 2 with only attenuation
coefficient g, and a ¼ b ¼ 1 (with k, q, c1, c2, and g, a
total of 5 parameters) and a full model with both
attenuation and inhibition mechanisms (with k, q, c1,
c2, g, a, and b, a total of seven parameters) were
compared by the nested model test.
The details of the fitting results are listed in Table
1. For all observers, the reduced Model 2 and full
model were significantly better than the most reduced
Model 1 (all p , 0.05, see Table 1), suggesting that
additional parameters (g, and/or a, b) were necessary
to account for the observed imbalance between the
two eyes in our data. The elaborated MCM with the
attenuation mechanism was statistically comparable
to the full model with all three mechanisms for
Observers S2 and S3, accounting for 91.9% and
93.8% of the variance in the data. On the other hand,
the full model with three mechanisms is superior to
its reduced versions, F(2, 68) ¼ 6.80, p ¼ 0.002, and
accounts for 98.0% of the data for S1, suggesting
stronger inhibition from the dominant eye to the
Fitting the MCM to some challenging data in the
literature
Cormack et al. (1991) measured disparity threshold
of RDS as a function of image contrasts in the two
eyes. The data were extracted from their paper and
replotted in Figure 10. In their study, the RDS were
presented with image contrasts at multiples of the
detection threshold, so an MCM with four parameters
(k, q, c1, and contrast threshold s) was fitted to the
data. The MCM accounted for 95.9% and 90.4% of the
variance of the data for observers LKC and CMS,
respectively (for detail of best fitting parameters, see
Table 21). Generally speaking, the MCM provided
better fits than the inverse-square or inverse-squareroot functions, although there is some misfit in the high
contrast conditions (see Discussion).
Legge and Gu (1989) measured disparity threshold
of sinusoidal gratings as a function of interocular
contrast ratio. We extract the data from their paper and
replotted them in Figure 11. There were two sets of
data measured at 0.5 cycles/8 and 2.5 cycles/8. The
higher contrast in one eye was fixed at either 0.125 or
0.25. The full MCM with attenuation and inhibition
mechanisms was used because of the evident imbalance
of the two eyes in the data (see figure 6 in Legge & Gu,
1989). To further constrain the MCM, the same set of
parameters (k, q, c1, g, a, b) were used to account for all
the data in one spatial frequency condition for each
observer. The best fitting results are plotted in Figure
11. For DR, r2 is 0.929 and 0.970 in the 0.5 cycles/8 and
2.5 cycles/8 conditions, respectively. For KD, r2 is 0.974
and 0.965 in the 0.5 cycles/8 and 2.5 cycles/8 conditions,
respectively (Table 2).
Ding and Levi (2011) measured phase disparity
threshold for amblyopic observers using Legge & Gu’s
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
Perceived Contrast in Left eye
Disaprity threshold (arcmin)
S1
S2
S3
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
0.08
0.16
0.08
0.16
Co 0.24
0.24
ntr 0.32
tL
0.32
as
tras
t R 0.4 0.4
Con
0.08
0.16
0.24
0.32
0.4 0.4
0.32
0.24
0.16
0.08
0.16
0.24
0.32
0.08
0.4 0.4
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.24
0.16
0.08 0.08
0.16
0.24
0.32
0.32
0.4
0.24
0.16
0.08 0.08
0.16
0.24
0.32
0.24
0.16
0.08 0.08
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.32
Co 0.24
0.24
ntr 0.16
as
0.16
tL
t R 0.08 0.08
tras
Con
0.4
0.08
0.16
0.24
0.32
0.4
0.4
0.4
0.32
0.16
0.32
0.4
0.6
0.4
0.32
0.24
0.4
0.4
0.4
0.32
Perceived Contrast in Right eye
13
0.32
0.24
0.16
0.08 0.08
0.16
0.24
0.32
0.4
0.32
0.24
0.16
0.08 0.08
0.16
0.24
0.32
0.4
Figure 9. Disparity threshold and matched probe contrast in the left and right eyes as functions of monocular image contrasts are
plotted in the upper, middle, and lower rows, respectively. The symbols represent the observed data. The continuous lines are the
predictions of the best fitting MCM. For S1, the full model is plotted. For S2 and S3, the reduced Model 2 is shown.
procedure (1989). The data were extracted from their
paper for observer GD, consisting of three conditions
with the higher contrast in one eye fixed at 0.24, 0.48,
and 0.96. The full MCM including attenuation and
inhibition was adopted for the amblyopic observer,
with a total of six parameters. The same set of
parameters was used to account for the three data sets
for GD. The MCM provided very good accounts of the
data (r2 ¼ 0.859, Figure 12, Table 2).
Discussion
In this study, we measured disparity threshold and
perceived contrast of dynamic RDS with different
image contrast combinations in the two eyes. We found
that both disparity threshold and cyclopean contrast
depend strongly on test image contrasts in the two eyes,
exhibiting characteristic binocular contrast-gain control properties, consistent with previous findings with
narrow-band stimuli (Halpern & Blake, 1988; Legge &
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
14
CMS
50
40
40
Stereoacuity (arcsec)
Stereoacuity (arcsec)
LKC
50
30
20
10
0
0
30
20
10
0
10
20
30
40
Contrast (threshold multiples)
0
10
20
30
40
Contrast (threshold multiples)
Figure 10. MCM fits to the disparity threshold data reported in Cormack et al. (1991) (replotted with permission).
Gu, 1989; Schor & Heckmann, 1989; Simons, 1984).
We extended the MCM (Huang et al., 2011; Huang et
al., 2010) to account for the full dataset. The new
MCM adopted the idea that cyclopean perception of
contrast and depth shared the same front-end binocular
contrast gain control, in which each eye exerts gain
control on the other eye’s signal in proportion to its
own signal contrast energy, and also gain controls over
the other eye’s incoming gain control. After interocular
contrast gain control, the outputs are processed by
different pathways to compute perceived phase, contrast, and depth. The extended MCM provided very
good accounts for the observed disparity thresholds
and cyclopean contrasts with the same set of parameters.
the quadrature pairs of its subunits to produce featureselective but phase-invariant response is very successful
in accounting for edge/line detection, motion detection,
and disparity extraction (Adelson & Bergen, 1985;
Emerson, Bergen, & Adelson, 1992; Morrone & Burr,
1988; Ohzawa, 1998; Ohzawa et al., 1990). Also,
contrast constancy, first introduced by Georgeson and
Sullivan (1975), is an important function of visual
system. Contrast constancy has been found in stimuli
with coherent and incoherent phase spectra (Brady &
Field, 1995), and is independent of size, spatial
frequency, bandwidth of the stimuli (Cannon & Fullenkamp, 1988; Georgeson & Sullivan, 1975; Tiippana
& Nasanen, 1999). In our experiment, only depth
polarity was manipulated. Future studies with a wide
range of disparities are necessary to examine whether
perceived contrast is independent of the magnitude of
stereo depth.
Independent pathway for contrast perception
The MCM was originally proposed to model
perceived phase and contrast during binocular combination (Huang et al., 2010) with separated pathways
for contrast and phase perception. Here, we found that
perceived cyclopean contrast did not depend on the
disparity polarity of the RDS stimuli. An independent
pathway for contrast perception is consistent with the
literature. For example, the response of complex cell in
V1 is phase invariant (Carandini et al., 2005; Hyvarinen, Hurri, & Hoyer, 2009). The prevailing energy
model of complex cells, which combines inputs from
Data
Cormack et al. (1991)
Legge and Gu (1989)
Ding and Levi (2011)
Observer
k
LKC
CMS
DR, SF ¼ 0.5
DR, SF ¼ 2.5
KD, SF ¼ 0.5
KD, SF ¼ 2.5
GD
0.026
0.011
8.00 · 104
4.00 · 104
0.020
0.002
2.13
Stereo contrast paradox
Consistent with the literature, we found that the
relationship between disparity threshold and image
contrast (when contrasts are equal in two eyes) couldn’t
be simply described by an inverse-square or inversesquare-root function (Cormack et al., 1991), and the
disparity threshold versus contrast ratio function was
U-shaped (Ding & Levi, 2011; Legge & Gu, 1989).
There are however some discrepancies.
q
5.66
426
4.80
1.22
364
3.57
16.1
· 104
· 105
· 106
· 105
Table 2. Parameters of best fitting MCM models for data in the literature.
2
c1
s
g
a
b
r
4.32
2.13
3.85
3.20
2.11
2.29
1.56
0.012
0.012
-
1
1
1
1
1
1.65
1
1.24
1
1.24
-
0.959
0.904
0.929
0.970
0.974
0.965
0.859
1
1.05
1
2.24
1
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
Disparity threshold (arcmin)
DR
KD
40
40
30
30
20
20
10
10
0
0
4
15
2
1
2
4
(L/R) Contrast ratio (R/L)
SF=0.5, Cst=0.125
SF=0.5, Cst=0.25
SF=2.5, Cst=0.25
SF=0.5, MCM fit
SF=2.5, MCM fit
4
2
1
2
4
(L/R) Contrast ratio (R/L)
Figure 11. Fits to the disparity threshold reported in figure 6 of Legge and Gu (1989) (replotted with permission). When the contrast
ratio varied, the higher contrast in one eye was always fixed at 0.125 or 0.25.
Some previous studies showed that, with narrowband stimuli, disparity threshold increased more when
image contrast in one eye was reduced than when image
contrasts in both eyes were reduced by the same
amount (Cormack et al., 1997; Halpern & Blake, 1988;
Legge & Gu, 1989; Schor & Heckmann, 1989), a
phenomenon referred to as stereo contrast paradox
(Cormack et al., 1997). However, using RDS in this
study, we did not observe stereo contrast paradox.
Cormack et al. (1997) also did not find stereo contrast
paradox with RDS and high spatial frequency narrowband stimuli. They concluded that the paradox is
restricted to low spatial frequencies and suggested the
absence of the paradox was due to different processing
of low- and high-spatial frequency information by the
visual system (Cormack et al., 1997; Stevenson &
Cormack, 2000).
In the MCM with balanced processing of information from the two eyes (Equation 6), disparity threshold
is Deq ¼ [k(1 þ 2qCc1 )2]/[C2(1 þ qCc1 )2] when image
contrasts in the two eyes are equal; disparity threshold
c
c
is Dineq ¼ [k(1 þ qCc1 þ qC01 )2]/[C0C(1 þ qCc1 )(1 þ qC01 )]
when image contrast is C in one eye and C0 in the other
eye. Stereo contrast paradox results when Dineq Deq .
0, and therefore:
c
ð1 þ qCc1 þ qC01 Þ2 ð1 þ 2qCc1 Þ2
.
:
c
Cð1 þ qCc1 Þ
C0 ð1 þ qC01 Þ
ð14Þ
In other words, in a given condition (C, C) and (C,
C0), whether stereo contrast paradox occurs is determined by two parameters, q and c1, in the MCM.
In Figure 13, we plot the magnitude of stereo
contrast paradox (Dineq Deq) as a function of q and c1
when image contrasts in the two eyes are (0.08, 0.08)
and (0.08, 0.4). The area with positive threshold
difference indicates parameter combinations that pro-
η = 1, α = β = 1
Paradox
Dineq - Deq > 0
200
GD
700
600
Cst=0.24
Cst=0.48
Cst=0.96
ρ
Stereo threshold (arcsec)
800
500
100
0
400
300
50
200
25
100
0
0.125 0.25 0.5
1
2
4
Contrast ratio (NDE/DE)
1
0.51
8
Figure 12. Fits to the disparity threshold data reported in Figure
4 (observer GD) of Ding and Levi (2011) (replotted with
permission).
2
4
γ1
8
D ineq - Deq < 0
No paradox
Figure 13. Predicted threshold difference (Dineq – Deq) between
conditions when image contrasts in the two eyes are (0.08, 0.4)
and (0.08, 0.08) as a function of q and c1 for an observer with
balanced processing of information from two eyes.
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
ρ = 76.5, 1γ = 1.11
α, β
200
Paradox
Dineq - Deq > 0
from the two eyes may be imbalanced even in people
with normal vision (e.g., S1 in our experiment and also
see Huang et al., 2010; Legge & Rubin, 1981). The
imbalance is much more severe for people with visual
disorder like amblyopia (Huang et al., 2011).
The MCM
100
0
50
25
1
0.1 0.2
16
0.4
0.6
η
0.8
1 D ineq - Deq < 0
No paradox
Figure 14. Predicted threshold difference (Dineq – Deq) as a
function of g and a ( ¼ b) between conditions when image
contrasts in the two eyes are both 0.08 and when image
contrast is 0.08 in the nondominant eye and 0.4 in the
dominant eye for an observer with q ¼ 76.5 and c1 ¼ 1.11.
duce stereo contrast paradox: q . 49 and 1.8 , c1 , 6.
One possible reason that some studies found stereo
contrast paradox and some did not is that different
experimental conditions, e.g., spatial frequency content
and duration of the stereograms, may have invoked
different q and c1.
In the MCM with imbalanced processing of information from the two eyes (Equation 13), whether stereo
contrast paradox occurs is determined by q and c1 as
well as monocular attenuation (g) and interocular
inhibition (a and b). Stereo contrast paradox could
occur because of imbalanced processing even in
combinations of q and c1 for which stereo contrast
paradox would not occur if the two eyes were balanced.
In Figure 14, we plot the magnitude of stereo contrast
paradox as a function of g and a ( ¼ b) between
conditions when image contrasts in the two eyes are
both 0.08 and when image contrast is 0.08 in the
nondominant eye and 0.4 in the dominant eye for an
observer with q ¼ 76.5 and c1 ¼ 1.11, which are the best
fitting parameters of the average normal observer in
(Huang et al., 2010) and would not have resulted in
stereo contrast paradox if the two eyes were balanced
(Figure 13). As shown in the figure, stereo contrast
paradox occurs when g ¼ 0.1 and a and b are greater
than 1.56 and when g ¼ 1 and a and b are greater than
9.68. This exercise suggests that another possible reason
for different results in the literature is differences
between observers: Some observers may have more
imbalanced processing of information from the two
eyes than others. Indeed, processing of information
The MCM provided an excellent account of the data
from three normal observers. It also provided good fits
to some very challenging data in the literature
(Cormack et al., 1991; Ding & Levi, 2011; Legge & Gu,
1989). It’s noteworthy that there are some slight but
systematic misfits when image contrast was 15 times or
more above threshold (Figure 11). When stimulus
contrast increases, the contrast response function of
neurons in V1 exhibits compression and saturation
(Albrecht & Hamilton, 1982). In the high contrast
range, the gain parameter q in the MCM may start to
vary as a function of contrast (Albrecht & Geisler,
1991; Heeger, 1992; Ohzawa & Freeman, 1994), before
the binocular interaction stage. That is why we
carefully chose the range of contrast (0.08 ; 0.4) in the
current study. The MCM may need a more sophisticated contrast response function in high contrast
conditions.
There is evidence that, in addition to decreased
monocular visual acuity and contrast sensitivity (Ciuffreda, Levi, & Selenow, 1991; McKee, Levi, &
Movshon, 2003), people with amblyopia also exhibit
abnormal stereoacuity (Ciuffreda et al., 1991; Goodwin
& Romano, 1985; Walraven & Janzen, 1993). However,
whether the abnormal stereopsis of amblyopia is due to
decreased contrast sensitivity or interocular inhibition
or both is an interesting question. Research on stereo
depth perception in amblyopia based on the MCM may
shed important light on the mechanisms of deficits in
stereo vision in that population.
Taken together, the extended MCM has successfully
accounted for the empirical data on stereo depth and
cyclopean contrast perception in a wide range of
contrast conditions from the current and previous
studies. The hypothesis of multiple pathways with
shared front-end binocular interaction for phase,
contrast, and stereopsis is compatible with our knowledge of the visual system. Investigation on other
phenomena in binocular vision, i.e., binocular rivalry
(Blake, 2001; Blake & Fox, 1974; Lee, Blake, & Heeger,
2007; Logothetis, Leopold, & Sheinberg, 1996) and
interocular masking (Baker & Graf, 2009) could
provide further tests for the MCM.
Keywords: depth perception, binocular vision, contrast, gain-control, disparity
Journal of Vision (2013) 13(8):3, 1–19
Hou et al.
Acknowledgments
The work was supported by National Eye Institute
grants (R01 EY017491 and R01 EY021553 to ZLL),
the Knowledge Innovation Program of the Chinese
Academy of Sciences, Institute of Psychology
(Y1CX201006 to CBH), the National Basic Research
Program of China (973 Program: 2009CB941303 to
YZ), and National Natural Science Foundation of
China (81261120562 to YZ).
Commercial relationships: none.
Corresponding author: Zhong-Lin Lu.
Email: lu.535@osu.edu.
Address: Laboratory of Brain Processes, Department
of Psychology, The Ohio State University, Columbus,
OH.
Footnote
1
Large qs are not a problem—in the MCM, a large q
means that the contribution of contrast energy is much
greater than one; in that case, the q would drop out of
the equations.
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