0042-6989/91$3.00+ 0.00
Copyright 0 1991Pergalnon FVessplc
Vision Res. Vol. 31, No. 12, pp. 2195-2207, 1991
Printed in Great Britain. All rights reserved
INTEROCULAR CORRELATION, LUMINANCE
AND CYCLOPEAN PROCESSING
CONTRAST
LAWRENCE K. CORMACK, Sco3-r B. STEVENSONand CLIFTON M. SCHOR
School of Optometry,
University of California, Berkeley, CA 94720, U.S.A.
(Received 27 August 1990; in revised form 17 January 1991)
Abstract-We
have investigated the nature and viability of interocular correlation as a measure of signal
strength in the cyclopean domain. Thresholds for the detection of interocular correlation in dynamic
random element stereograms were measured as a function of luminance contrast, a more traditional
measure of stimulus strength. At high contrasts, correlation thresholds were independent of contrast. At
low contrasts, correlation thresholds were inversely proportional
to the square of contrast.
Stereothresholds were also measured as a function of both contrast and interocular correlation. At low
contrasts, stereoacuity was inversely proportional to both interocular correlation and the square of
contrast. These results are consistent with an inherently multiplicative mechanism of binocular combination, such as a cross-correlation of the two eye’s inputs.
Interocular correlation
Contrast
Stereopsis
INTRODUCTION
correlation
can be
In general, interocular
thought of as the degree to which the images in
the two eyes match one another. Intuitively,
interocular correlation, if reasonably defined,
should provide a measure of signal strength in
the cyclopean domain. That is, if the two eye’s
views are almost identical, as is the case with
binocular fixation of a flat surface, interocular
correlation is very close to the maximum possible. On the contrary, if the two eye’s views of
the world are predominantly non-overlapping
then nothing can be predicted about the right
eye’s image given only the left eye’s image and
vice versa. In this case there would be zero
interocular correlation and, hence, no cyclopean
information available.
Formally,
interocular
correlation
can be
defined as the cross-correlation
of the image
pair comprising the right and left eye’s views of
the world. For the simple one-dimensional case
the interocular correlation at some disparity d is
given by:
IOC(d) =
f(x)h(x + d) dx
(1)
s
where f(x) and h(x) represent the intensity
profiles (or some derivative of them) along the
horizontal meridian of the right and left eye’s
retinae.
This definition is simple, quantitative, and
works for any given image pair. For “natural”
images, with their generous variety of colors,
luminances, etc. interocular correlation is somewhat difficult to intuit, regardless of the definition employed. It would be unclear, for
example, how to change one member of an image
pair in order to reduce the interocular correlation by some desired amount. In the laboratory however, where one can restrict the visual
environment to one-bit random dot stereograms
of 50% density, the notion of interocular correlation is quite intuitive. Under these conditions,
the interocular correlation for a given disparity
is simply a linear function of the proportion of
dots which match at that disparity, i.e.
IOC(d) = 2P, - 1
(2)
where Pd is the proportion of matching dots at
disparity d.
Examples of different interocular correlations
are shown in Fig. 1. The top stereo pair
illustrates an interocular correlation of + 1 (all
dots match at 0 disparity) and, when fused,
the percept is that of a flat plane. In the middle
and bottom panels, the interocular correlation
has been reduced to +OS and 0 respectively,
accompanied by a degradation of the perceived
quality of the plane. It should be noted that the
phenomenology of these static examples differs
somewhat from that of dynamic displays such as
were employed in our experiments. Specifically,
the dynamic displays of interocular correlations
less than unity give rise to a percept much like
2195
2196
K.
C.
Fig.
Examples of random noise with various amounts
of interocular
correlation.
These examples can
either be free-fused or viewed through a stereoscope.
In (a). the interocular
correlation
is 100% at 0 dn,
parity, and the percept is one of a Aat plane. In (b) the interocular
correlation
has been reduced to SO” ,I
while a flat plane is still perceived, it appears less robust and accompanied
by dots both in front of :III~!
behind the plane of fixation. In (c), the interocular
correlation
is 0, and no percept of a coherent plani.
extant. In dynamic versions of these stimuli, such as were employed in the experiments,
the phenomtr;
ology is somewhat
different.
Lower interocular
correlations
in particular
appear as semi-transparct!t
volumes when dynamic, whereas. when static, they appear as opaque surfaces with chaotic topograph?
dirty window embedded in fog; as correlation
increases, the window grows dirtier and the fop
grows thinner. Absent in the dynamic displays
a
was the appearance
of’s pstchwork
01 cc)-plan;lr
dots imbedded
in a ri~Hlrousllustr_c,ab:~l\
cv-ound, such as is seen
t‘
lnterocular correlation,
luminance contrast and cyclopean processing
Yet Fig. 1 does illustrate a basic point, which
is that as one decreases the interocular correlation of the stimulus, the salience of the flat
plane also decreases. In this sense, interocular
correlation could represent a metric of signal
amplitude in the cyclopean domain analogous
to the manner in which luminance contrast
provides a metric of signal amplitude in the
spatial domain. * The generation of a cyclopean
signal does not occur in parallel with the generation of a spatial (contrast) signal however; the
presence of a contrast signal is a necessary
precursor to the generation of a cyclopean
signal. Given this ordinal relationship, and the
fact that contrast is already known to influence
such hypercyclopean functions as stereoacuity
(Halpern & Blake, 1988; Legge & Gu, 1989;
Heckman & Schor, 1989), it might be possible
to control the amplitude of a cyclopean signal
by manipulating
luminance
contrast.
For
example, if the luminance contrast of a cyclopean stimulus is reduced, it might be possible
to compensate for the resulting decrease in
signal strength by increasing the interocular
correlation, thereby maintaining a constant (e.g.
threshold) level of performance on some task.
Thus, given a threshold level of performance on
said task, a trading relation would be expected
between contrast and interocular correlation.
Moreover, the form of this trading relation
would reflect the manner in which cyclopean
signals are derived from monocular contrast
signals. Accordingly, we measured the effect
of contrast on the detection of interocular
correlation in the first experiment. Based on
the results of this experiment, a model was
developed to generate predictions concerning
hypercyclopean functions such as stereoacuity.
Predictions of this model were then tested
in Experiment 2 by measuring stereoacuity as
a function of both contrast and interocular
correlation.
GENERAL METHODS
The experiments were performed using dynamic random-element stereograms of 50% element density. A diagram of the basic apparatus
is shown in Fig. 2. A random noise signal was
hardware generated via shift registers running at
*In this paper, we will use the term “cyclopean” to refer to
the site and/or processes of binocular combination itself
and the term “hypercyclopean” to refer to processes
occurring after or beyond the cyclopean stage.
2197
Fig. 2. Schematic illustration of the experimental apparatus.
Random bit streams were hardware generated and sent to
a pair of video monitors, which were viewed through a
mirror haploscope. Disparities were created by delaying the
sync to one monitor. Interocular correlation was manipulated by driving the monitors with a single dot generator,
independent dot generators, or a combination thereof (see
text for details). The psychophysics (stimulus presentation,
data aquisition, etc.) were all under computer control.
7 MHz and was displayed on a pair of matched
TSD monitors
(p4 phosphor,
60 Hz noninterlaced) viewed through a mirror haploscope.
The viewing distance was 53.7 cm and the
haploscope mirrors were adjusted for each
subject to the corresponding
convergence
angle, thus obviating any mismatch between
convergence and accommodation
or “higher
level” distance cues. Mean luminance was
80cd/m2. The displays were viewed through
7 deg circular apertures in an otherwise black
surround.
Horizontal disparities were produced by delaying the horizontal video sync to one monitor.
This was accomplished via a programmable
delay chip (Digital Delay Devices model PDU13256-0.5), which allowed us to delay the noise
stimulus to one eye in 0.5 nsec (corresponding to
2 arc set) increments.
Interocular correlation was simply the proportion of the dots which were “forced” to
match in the two images; the remainder of the
dots then had a 50% chance of matching. Thus,
in a display which had an interocular correlation of 0, half of the dots in the right image
were matched by dots in the left image. In a
display which had an interocular correlation of
- 1 (“anticorrelation”),
the right image was
simply the opposite contrast version of the left
image and no matches existed. For an interocular correlation of + 1, of course, the two
images were identical.
2198
K.
Thus, rearranging equation (2), the proportion of matching dots is, on average, simply
(IOC + 1)/2; the two are linearly related, as is
somewhat demanded by intuition.
Interocular correlation was varied through
the use of two independent noise generators. To
produce an interocular correlation of + 1, the
output of a single noise generator was sent
to both monitors. To produce an interocular
correlation of 0, each monitor was driven by a
separate noise generator. Intermediate interocular correlations were produced by switching
between the above two conditions at a sufficiently high rate (1 kHz) to allow the spatiotemporal integration of the visual system to
render the stimulus identical with one in which
interocular
correlation
is statistically determined on a dot-by-dot basis. It should be noted
that this switching pulse was not synched to the
video signal, nor was the switching rate sufficiently close to an even multiple of the frame
rate. lest stationary or drifting bands of correlation be visible. Using this method. the interocular correlation is simply the duty cycle of the
(rectangular wave) switching pulse. and could
*Two control experiments were done to Insure that this
method was adequate. First, we had subjects attempt to
distinguish between an intermediate correlation stimulus
produced by a 1 kHz switching pulse (as used in our
experiments) and a 100 kHz switching pulse. All subjects
failed to make the distinction. On the assumption that
100 kHz (100 switching cycles every msec) is fast enough.
then this experiment shows that
kHz is also fast
enough.
Second, we had subjects attempt to dtscriminate between “true” zero correlation (right and left eyes’ images
produced by independent random bit streams) and an
intermediate correlation as produced by a 1 kHz mixture
of
and - 1 interocular correlation. The independent
variable was the duty cycle of the switching pulse. As the
duty cycle of the switching pulse went to SO%, the ability
of the subjects to make the discrimination fell to chance
levels.
tlndividual monocular contrast detection thresholds were
measured for our dynamic random element stimuli on
the same apparatus using the same 2 aft/constant stimuli
paradigm. Contrast detection thresholds were between 2
and 4% (Michelson contrast for a single stimulus frame).
These relatively high thresholds are to be expected
because of the effective contrast reduction which the
temporal integration of the visual system imparts on
dynamic stimuli. The lowest contrast at which a particular subject could perform either the correlation detection
task (Experiment 1) or the stereoacuity task (Experiment
2) corresponded to the calculated contrast at which both
eyes were reliably detecting the stimulus [i.e. P(RE
detection AND LE detection) = 0.751, as obtained by
multiplying the psychometric functions for the right and
1ett eve&,
be placed under software (Experiment 1) or
hardware (Experiment 2) control.*
Thus, to create an interocular correlation of
0.75, the duty cycle of the rectangular wave
switching pulse was such that the subject was
viewing a fully correlated display 75% of the
space/time, and viewing an uncorrelated display
for the remainder. Given the 1 kHz switching
pulse frequency and the 60 Hz non-interlaced
frame rate, however, the perception was one of
a continuously present intermediate correlation;
no perception of the correlation switching was
present during the experiments.
Luminance contrast was controlled by adjusting the peak-to-trough value of the video signals
using customized
hardware
and software.
The response functions of the two monitors
were matched and calibrated prior to both
experiments using a Photo Research Spectra
Spotmeter photometer.
Experiment 1: The Detection of Interocular Correlation as a Function of Luminance Contrast
Method
The 3 authors served as subjects. All had
normal or corrected to normal acuity, normal
contrast sensitivity and good stereopsis.
A temporal 2AFC-method of constant stimuli
paradigm was employed. The subjects fixated a
12 arc min wide “ +” which, to insure accurate
convergence, was flanked above and below by
48 x 5 arc min nonius lines. A block of trials
consisted of 1 trial at each of 4-6 correlation
levels, chosen to bracket the expected threshold
based on pilot data. The order of trials was
randomized within blocks. A run was composed
of 30 blocks of trials at a single stimulus contrast. A threshold correlation for a particular
contrast level was defined as 75% correct on a
psychometric function fit to the data from 3
(subject CMS) or 5 (subjects SBS and LKC)
such runs. Each subject ran at between 9 and
11 contrast levels, which were even multiples
of that subject’s contrast threshold for our
dynamic random-element stimuli.7
A trial was composed of two stimulus intervals, 1.2 set in duration, which were delimited
by audible tones. The dynamic noise was continuously present but, during one of the two
intervals, switched from 0 interocular correlation to some positive interocular correlation
for 200msec. The plane of positive correlation
was always presented at 0 disparity. i.e. the
plane of stimulus presentation was coincident
Interocular correlation,
luminance contrast and cyclopean processing
with the plane of fixation as defined by the
nonius horopter. The subject’s task was to
signal, by means of a key press, which interval
contained some non-zero interocular
correlation. The subject’s response was followed by
auditory feedback and initiated the next trial.
Results
The results for the 3 subjects are shown
superimposed in Fig. 3. The axis of ordinates
represents correlation threshold while the axis
of abscissas represents contrast expressed in
threshold multiples. Both axes are logarithmic.
As can be seen from the figure, correlation
threshold is independent of contrast at relatively
high contrasts. At relatively low contrasts, however, correlation
thresholds decrease rapidly
as stimulus contrast increases. Thus, in the
low contrast region, a decrease in stimulus
effectiveness due to a contrast reduction can be
compensated for by an increase in interocular
correlation in order to restore a criterion (e.g.
threshold) level of performance.
Generally, the data can be described as consisting of two regions, a high contrast region in
which the data asymptote to a line of slope 0,
and a low contrast region in which they asymptote to a line of slope - 2. The solid line in Fig. 3
is of slope -2, plotted for reference. The
short vertical lines at the top of the figure
show representative error bars for the low (lefthand line) and high (right-hand line) contrast
regions.
e
+
u
z!
sbs
ems
10
Fig. 3. Threshold for the detection of interocular correlation
as a function of luminance contrast, expressed in threshold
multiples, for the 3 subjects. Both axes are logarithmic. The
data asymptote to a log-log slope of 0 at high contrasts and
- 2 at low contrasts. A line of slope - 2, indicating a trading
relation between interocular correlation and the square of
contrast, is plotted for reference. The left- and right-hand
vertical lines at the top of the figure represent typical SD for
low and high contrast judgments respectively.
2199
Discussion
can be reasonably assumed that the detection of a stimulus in a psychophysical task
occurs when a signal-to-noise ratio reaches
some critical value at the relevant site (or in
the relevant functional unit) of the visual system
(cf. Green SC Swets, 1966). The data from
this experiment indicate that, at higher contrasts, the relevant signal-to-noise ratio remains
constant as contrast changes; this is reflected
by the fact that correlation thresholds are independent of contrast at high stimulus contrasts.
At lower contrasts, however, this is not the
case. As stimulus contrast is reduced, eventually
contrasts are reached which render a previously
threshold level correlation
indistinguishable
from uncorrelated noise. Threshold level performance can be restored, however, by increasing the interocular correlation of the stimulus
by some amount. This indicates that, at low
contrasts,
the relevant signal-to-noise
level
is changing with stimulus contrast and that
this change can be compensated
for by
appropriate adjustment of the interocular correlation. Further, the relative effectiveness of
contrast and interocular correlation on the
signal-to-noise ratio (i.e. the trading relation
between the two variables) is given by the
log-log slope to which the data asymptote at
low contrast levels. From Fig. 3, it can be seen
that this slope is roughly -2, which indicates
that the relevant signal-to-noise ratio is proportional to the square of stimulus contrast. A
possible means of binocular combination by
which this behavior could be realized is a simple
cross-correlation
of edge information in the
stimulus.
A cross-correlation
of two signals produces
a cross-correlation
function, the height of
which at some relative displacement reflects the
degree to which the two input signals match at
that displacement. When a horizontal crosscorrelation
is done on two retinal images
(assuming correct vertical alignment of the
images), the resulting function can be thought of
as representing the number of matching stimulus elements as a function of retinal disparity. In
order to determine the behavior of such crosscorrelation functions in response to various
stimulus parameters,
particularly
luminance
contrast and interocular correlation, a model of
binocular combination was developed which
incorporated the operation of cross-correlation
as the “engine” of the model.
2200
A truly global cross-correlation
would be
undesirable because such a model would have
difficulty distinguishing between transparent
stimuli and stimuli with local depth variations.
For example, a transparent stimulus consisting
of one surface in front of a second surface and
a “checkerboard”
stimulus in which alternate
squares lie at different depths would both lead
to a double peaked cross-correlation function.
This problem can be avoided by doing “local”
cross-correlations
on smaller patches of the
visual scene. The result is a set of crosscorrelations functions whose members correspond to different visual directions. Thus,
transparent stimuli would give rise to double
peaked cross-correlation
functions at each
visual direction
along which transparency
exists, whereas depth variations across a visual
scene would give rise to single peaked crosscorrelation functions, with the location of the
peak depending on the visual direction to which
the function corresponds.
A cartoon representation
of the model is
shown in Fig. 4, which illustrates the various
stages of the model along with the physiological
properties which they were intended to mimic.
The input to the model was a series of image
arrays, representing video frames of the stimulus, the contrast and interocular correlation of
which could be varied. Other image parameters,
such as the size and density of the elements
:omprising the stereograms, were set to match
the conditions of our experiments. The input
image arrays were “blurred” by the optics of the
:ye via convolution with a gaussian representation of the point spread function of the
:mmetropic eye (0.84 arc min space constant).
The blurred image frames were then sampled by
“retinae” with a 30 arc set inter-receptor distance, and averaged over space-time to simulate
reasonable values for spatial and temporal integration of the visual system. The amount of
integration was varied within sensible limits to
insure that the general conclusions from the
modeling did not depend on the particular
values chosen; the primary effect of increased
spatio-temporal integration is simply to reduce
the relative amplitude of noise from spurious
matches [i.e. the “ghost” matches of Cogan
1978) or the “phantom” matches of Julesz
(1971)] in the cross-correlation
function. The
images were then differentiated with respect to
space (edge extraction) to yield an image representation assumed to exist at an immediately
pre-binocular stage in the visual system.
These filtered left and right eye image pairs
were then cross-correlated to produce a crosscorrelation function such as is illustrated in
Image
Fig. 4. An illustration representing the flow of processing in the model. The input to the model is a series
of one bit arrays, analogous to our stimuli. The arrays are then blurred by an amount typical of an
emmetropic eye. Spatio-temporal averaging is then done to simulate the spatial and temporal integration
of the visual system. Edge information is extracted by means of differentiation. Finally, the left and right
eye “images” are cross-correlated to yield a function on which the binocular operations of correlation
detection and localization can be performed. Not illustrated is the addition of noise, assumed to be present
and intrinsic to the visual system. to the cross-correlation function
Interocular correlation,
luminance contrast and cyclopean processing
2203
i
h
f
Fig.
the interocular correlation of an image pair with
some fixed contrast is reduced, the signal amplitude is
decreased proportionally while the noise level remains constant. Shown in this figure is a family of cross-correlation
functions, the members of which differ in the interocular
correlation of the input image pair. The particular interocular correlation of each input image pair is displayed to
the right of the corresponding function. The contrast of the
input image pair was 20% in all cases, and the functions are
displaced vertically for clarity.
If one reduces contrast, however, one eventually reaches a point where the intrinsic noise
(which is assumed not to vary with contrast) is
of much greater amplitude than the noise produced by spurious matches. At this point, overall noise level is effectively constant since its
amplitude is determined almost exclusively by
the amplitude of the intrinsic noise. Signal-tonoise ratio, then, will be determined solely
by the peak height, which varies linearly with
correlation but as the square of contrast. It
follows that the interocular correlation required
to produce a given signal-to-noise ratio (e.g.
that corresponding
to threshold) would be
proportional to the square of the stimulus contrast. For example, if one starts with a crosscorrelation function with a signal-to-noise ratio
corresponding to threshold, and then reduces
the contrast by a factor of 2, the signal-to-noise
ratio will decrease by a factor of 4. To restore
the threshold level signal-to-noise ratio, the
interocular correlation would then have to be
increased by a factor of 4. Thus, in the low
contrast region, we would expect a log-log
plot of correlation threshold as a function of
contrast to show a slope of -2.
Combining the above reasoning from the low
and high contrast regions, we expect the data to
take the form shown in Fig. 8. At high contrasts,
Fig. 8. Schematic diagram showing the effect which intrinsic
noise of constant amplitude would have on the detection of
interocular correlation as a function of stimulus contrast.
When contrast is sufftciently high, the amplitude of the false
target noise is much greater than that of the internal noise.
Thus the effective noise amplitude is simply the amplitude
of the false target noise. Since the signal height and the false
target noise vary conjointly as contrast is changed, the
signal-to-noise ratio of the cross-correlation function remains constant and no change in correlation detection
threshold is anticipated. As contrast is reduced, however,
eventually a point is reached where the amplitude of the
intrinsic noise is much greater than that of the false target
noise. Thus the effective noise amplitude is simply the
amplitude of the intrinsic noise, which is constant. Under
these conditions, signal-to-noise ratio will vary directly with
signal amplitude which, in turn, varies with the square of
contrast (Fig. 7). But since signal-to-noise ratio is linearly
related to interocular correlation (Fig. 8), one anticipates a
square relation to determine the correlation/contrast combinations which will yield a given (e.g. threshold level) signalto-noise ratio. This is reflected by the log-log slope of -2
in the low contrast region of the figure.
where the signal-to-noise ratio is independent of
contrast, we expect the data to asymptote to a
log-log slope of 0. At lower contrasts, where the
signal-to-noise ratio varies as the square of
contrast, we expect the data to asymptote to a
log-log slope of -2.
Since the data (Fig. 3) conform to this expectation, it would seem that for dynamic random
element stereograms, a cross-correlation of the
two eye’s inputs after accounting for such factors as optical low pass filtering (“blurring”)
and spatial and temporal summation, provides
an adequate description of the stimulus at
cyclopean levels of the visual system, i.e. the
earliest level of the visual system where information from the two eyes is integrated. But how
far past the stage of binocular combination
can this analysis be continued? That is, can
downstream binocular processes be treated as
an analysis of, or operations performed on, a
cross-correlation
function? With this question
in mind, we attempted to extend our experimentation into the hypercyclopean domain.
2204
In Experiment
2, we measured
stereothresholds (stereoacuity) as a function of both
luminance contrast and interocular correlation.
The predictions of the cross-correlation model
are as follows.
Consider a cross-correlation function such as
illustrated in Fig. 5. The certainty with which
the “true” position of the function can be
determined depends on the signal-to-noise ratio
of the function. For a given signal-to-noise ratio
however, a sharp peak can, in principle, be more
precisely localized than a broad peak, so it is
important to know how the second derivative of
the cross-correlation function behaves. Further,
the zero-crossings or loci of steepest slope
could be used, along with assumptions of symmetry, to determine the position of the function (cf. Legge & Gu, 1989) so it is also of
import to determine the behavior of the first
derivative.
As it turns out, the height of both the first and
second derivatives of the cross-correlation function is simply a linear function of the height of
the cross-correlation
function (as is the case
with sinusoids). This is convenient, because it
means that in order to express the precision with
which the position of the cross-correlation function can be localized (without invoking disparity
domain interactions), it is necessary only to
specify the signal-to-noise ratio of the crosscorrelation function.
As we already know, both the height of the
cross-correlation
function and the signal-tonoise ratio grow linearly with increasing interocular correlation (Fig. 7). Thus, we predict that
stereoacuity should simply be a linear function
of interocular correlation.
The predicted slope for stereoacuity as a
function of contrast however, depends on the
absolute contrast level. At higher contrasts,
where false target noise is the dominant source
of noise, both the noise amplitude and the peak
signal height are proportional
to the square
of contrast such that the signal-to-noise ratio
remains constant (Fig. 6). At lower contrasts,
where intrinsic noise is the dominant source
of noise, both the peak signal height and
the signal-to-noise ratio vary as the square of
contrast, since noise amplitude is effectively
constant. Thus, at low contrasts, the crosscorrelation
model predicts the stereoacuity
will vary in proportion
to the square of con-
trast, whereas at higher contrasts the model
predicts that stereoacuity will be independent of
contrast.
The methods of this experiment are essentially the same as those of the first experiment.
The 3 authors served as subjects and a temporal
ZAFC-method of constant stimuli paradigm
was employed. The observers task, however,
was to indicate which of the two temporal
intervals contained a horizontal step change in
disparity approximately
halfway down the
stimulus field. For determining stereoacuity as a
function of contrast, interocular correlation
set at 80%, and contrast was varied over a
broad range. For determining stereoacuity as a
function of interocular correlation, contrast was
set at 12% (Michelson definition, as measured
on a static pattern) and correlation
was
varied from either 30% (LKC and SBS) or 60%
(CMS) up to 90%. Thresholds for each run
(where a “run” is the same as defined in Experiment 1) were defined as 75% correct on the
psychometric function, and the thresholds from
5 runs per subject at each contrast/correlation
combination were obtained.
Stereothresholds as a function of contrast are
shown for all 3 subjects in Fig. 9. The high
contrast portion of the data (above 5 contrast
threshold multiples, say) is essentially a replication of both Halpern and Blake (1988) and
contrast
(threshold
Fig. 9. Stereoacuity as a function of contrast for the 3
subjects. Both axes are logarithmic and contrast is expressed
in threshold multiples. Over almost a full log unit of
contrast, the data can be described by a cube root law
contrast dependence, roughly in accordance with the data of
previous investigators. At lower contrasts however, the data
more closely follow a square law dependence
channels centered at higher spatial frequencies,
stands only to increase stereoacuity via probability summation. This possibility need not be
seriously entertained, however, given the second
line of evidence against a significant contribution of high spatial frequency information.
This is simply that both Halpern and Blake
(1988) and Legge and Gu (1989) found a square
root dependence of stereoacuity on contrast
using narrow band targets (the specific targets
employed were tenth derivatives of gaussians,
mercifully known as DlOs, and truncated sinusoids, respectively).
A second mechanism which could subserve
the improvement of stereoacuity with increasing
contrast is a compressive contrast response
function inherent in the stereopsis “mechanism”. In its current form, our model predicts the
same shape for the data in both the correlation
detection and stereoacuity tasks, because performance on both tasks is assumed to reflect
the signal-to-noise ratio of the same crosscorrelation function. If, however, an appropriate compressive contrast response function was
inserted at some cyclopean stage prior to stereo
localization, the high contrast branch of the
model would asymptote more gradually, yielding a curve resembling the data of Fig. 9. This
would indicate that either the mechanism of
correlation detection and stereoacuity each have
a different contrast response and operate in
parallel or that the cross-correlation
function
is passed through a compressive transducer
function after the stage at which correlation
detection occurs but before the stage at which
stereo localization occurs.
The later alternative is perhaps the more
plausible one on the grounds that stereolocalization is probably the primary reason that a
cross-correlation
operation
would be performed. A second cross-correlation
operation
occurring in parallel would, therefore, be superfluous in all but artificial tasks of correlation
detection such as in the present experiments
(it is conceivable
that a parallel crosscorrelation operation could be occurring in
the eye movement pathway, but it is unlikely
that this would be a factor in a psychophysical
task).
A third mechanism which could bring about
an improvement of stereoacuity with increasing
contrast is inhibition across disparity tuned
“units”, be they the individual cells of Poggio
and Poggio (1984), the channels of Stevenson
i’t nl. (1990).
the detectors of Tyler (1983).
Consider the possibility that cross-correlation
functions are encoded as relative activity in an
array of such units located along the disparity
axis. Without disparity domain inhibition, a
graph of unit activity vs peak disparity tuning of
the unit for various contrasts would simply look
like Fig. 6. With the addition of inhibition
between the units, however, the signal-to-noise
ratio would continue to improve as contrast
increased, rather than remaining constant. The
nature of this inhibition could be “tweeked” in
one’s model to yield whatever contrast dependence was desired. The argument at low stimulus contrasts, however, need not be altered by
the addition of inhibition. At low stimulus
contrasts, where intrinsic noise seems to effectively determine the overall noise amplitude, the
activity of the disparity-tuned units outside of
the stimulus plane could be so low that their
influence through inhibitory pathways would be
negligible.
The presence of some sort of disparity domain inhibition is also supported by the phenomenology of random element stereograms.
As illustrated in Fig. 7, noise in the crosscorrelation function due to false dot matches is
unaffected by stimulus correlation. Thus, when
disparity domain inhibition is not considered,
the activity of a unit tuned to a disparity other
than that of, or immediately adjacent to, the
stimulus plane, would be unaffected by changmg the interocular correlation. Perceptually.
however, this is not the case. When viewing
;+ random element stimulus of 0 interocular
correlation, a voluminous “swarm” of dots is
perceived. But when interocular correlation is
raised to 100% at some disparity, the only thing
that is seen is a flat plane at that disparity,
no false matches are seen at all. Therefore, it’
it is assumed that some process like a crosscorrelation occurs at the site of binocular combination, then there must also be some process
to “clean up” the spurious disparities before a
site is reached where the neural activity IS
reflected perceptually.
Since cross-correlation is an inherently multiplicative mathematical operation, our model
is somewhat at odds with additive models of
binocular contrast combination (e.g. Leg@ &
1989). Dynamic random element stimuli,
however, by their very nature isolate a highly
specialized subset of the visual system. It would
be premature, therefore, to place too much
weight on apparent contradictions with results
&tained with other sorts of stimuli
Interocular correlation,
luminance contrast and cyclopean processing
In conclusion, our psychophysical results indicate that interocular correlation is an important controlling variable at both the cyclopean
and hypercyclopean stages of visual processing.
Further, a model which utilizes the operation of
cross-correlation
as the means of binocular
combination provides a good description of the
stimulus in both the cyclopean and hypercyclopean domains insofar as it accounts for
and/or predicts psychophysical data from both
of these domains.
Acknowledgements-This
work was supported in part by
grant # EYO 06045 to SBS and CMS and grant # ROlEY03532 to CMS. The authors with to thank Gordon Legge
for his invaluable comments and J. Malik and D. Jones for
helpful conversation.
Barlow, H. B. (1964) The physical limits of visual discrimination. In Giese, A. C. (Ed.), Photophysiology (Vol. 2,
pp. 163-201). New York: Academic Press.
Cogan, A. I. (1978) Fusion at the site of the “ghosts”. Vision
Research, 18, 657-664.
DeValois, R. L. & DeValois, K. K. (1988) Spatial vision.
New York: Oxford University Press.
Green D. M. & Swets, J. A. (1966). Signal detection theory
and psychophysics. New York: Wiley.
2201
Halpem, D. L. & Blake, R. (1988) How contrast affects
stereoacuity. Perception, 17, 483-495.
Heckman, T. & Schor, C. M. (1989) Is edge information for
stereoacuity spatially channeled? Vision Research, 29,
593-607.
Julesz, B. (1971) Foundations of cyclopean perception.
Chicago: University of Chicago Press.
Legge, G. E. & Gu, Y. (1989). Stereopsis and contrast.
Vision Research, 29, 989-1004.
Nishihara, H. K. (1987). Hidden information in transparent
stereograms. Proceedings of the Twenty-first Asilomar
Conference on Signals, Systems & Computers, 21,695-700.
Poggio, G. F. & Poggio, T. (1984). The analysis of stereopsis. Annual Review of Neuroscience, 7, 379-412.
Rogers, B. J. & Anstis, S. M. (1975). Reversed depth from
positive and negative stereograms. Perception, 4, 193-201.
Schor, C. M., Wood, I. C. & Ogawa, J. (1984). Spatial
tuning of static and dynamic local stereopsis. Vision
Research, 24, 573-578.
Stevenson, S. B., Cormack, L. K. & Schor, C. M. (1991).
Depth
attraction
and repulsion in random
dot
stereograms. Vision Research, 31, 805-813.
Stevenson, S. B., Cormack, L. K., Schor, C. M. & Tyler, C.
W. (1990). Disparity tuned channels in human stereopsis.
Investigative Ophthalmology and Visual Science (Suppl.),
31, 95.
Tyler, C. W. (1983) Sensory processing of binocular disparity. In Schor, C. M. & Ciuffreda, K. J. (Eds), Vergence
eye movements: Basic and clinical aspects (pp. 199-295).
London: Butterworth.
Westheimer, G. & Levi, D. M. (1987). Depth attraction and
repulsion of disparate fovea1 stimuli. Vision Research, 27,
1361-1368.