Binocular disparity and
Stereopsis
Bruce Cumming
Laboratory of Sensorimotor Research,
National Eye Institute,
National Institutes of Health
Put red
lens over
left eye,
blue lens
over right
eye
Stereo
anaglyph by
Prof.
Michael
Greenhalgh,
Australian
National
University
(with
permission).
stereopsis
L
R
correspondence problem
left eye’s image
right eye’s image
random-dot patterns
•
•
•
•
a completely unnatural stimulus
image changes every few ms
no recognisable objects e.g. faces
each dot has dozens of identical potential
matches
• and yet a clear perception of depth!
Neurons and depth perception
• A simple model to generate disparity
signals.
• How neurons reflect this.
• Some psychophysical limits this
explains.
• Further processing.
head image from Royal Holloway
University of London Vision Research
Group (with permission)
Left retina
Right retina
Receptive Field
Fovea
Disparity-selective neuron
Right RF



*
Left RF




*
R
L
basic building-block
• inner product of image with receptive field

v   dxdy. x, y I x, y 

Pos(v)
….
-0.1 + 1 + 1
….S
= response
Left RF
=l
out  (l  r )2
+
+
Right RF
=r
S
(l  r )2  l 2  r 2  2lr
l1
r1
l2
r2
l2
r1
Input (membrane V)
Circuitry for complex cell
left
RF1
RF2
right
binocular simple cells
BS 1
BS 2
complex cell
Cx
BS 3
If RF2 = -RF 1 in both eyes, then half
squaring then summing is equivalent to
simply squaring.
BS 4
energy model
square the result
n

C   vLj  vRj

2
j 1
sum over
many such
subunits
convolution of left convolution of right
eye’s image with jth eye’s image with jth
left receptive field right receptive field
add together
R
L
R
Right Stimulus Position
L
Complex cell
Model
Left Stimulus Position
Ohzawa et al, 1990
Disparity-selective neuron
Right RF
R
Left RF
L
R
L
R
L
R
Right Stimulus Position
L
L
R
Complex
cell
Model
Left Stimulus Position
Ohzawa et al 1990
Disparity-selective neuron
Right RF



*
Left RF




*
R
L
Left RF
-d
Right RF
d
0
Correlation
1
0.5
0
-0.5
-1
-50
0
Disparity (pixels)
50
1
Patern 1
Patern 2
Correlation
0.5
Patern 3
Patern 4
0
Patern 5
Mean
-0.5
-1
-50
0
Disparity (pixels)
50
-d
0
Disparity
d
Left RF
Right RF
Correlation
1
0.5
0
-0.5
-1
0
Disparity
DeAngelis, Ohzawa and Freeman, (1991)
Cat simple cell RF maps
For single subunits (simple)
• Odd symmetric disparity tuning implies phase
disparity
• Even symmetry around non-zero disparity
implies position disparity
True for complex cells if:
• All subunits have same phase disparity
• All subunits have same position
disparity.
Monkey complex cells
Firing rate (spikes/s)
duf043
duf065
60
40
20
0
-1.4
-0.9
-0.4
0.1
0.6
1.1 1.4
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Disparity (degrees)
So far:
• Energy model measures crosscorrelation after filtering.
• V1 contains a bank of filters measuring
these correlations after displacements
of both phase and position.
Disparity-selective neuron
Right RF



*
Left RF




*
R
L
response
[spikes/sec]
50
50
the neuronal response
cf: 0.06 cpd
80
60
40
20
0
0 0.5 1 1.5 2
0
time [sec]
0
250
5
10
15
200
-2
0
0
response
[spikes/sec]
50
100
150
cf: 0.06 cpd
2
20
25
the neuronal response
80
60
40
20
0
0 0.5 1 1.5 2
cf: 0.5 cpd
80
60
40
20
0
0 0.5 1 1.5 2
time [sec]
relative modulation
1
response
[spikes/sec]
relative modulation
cf: 0.06 cpd
0.5
0
0.1
1
10
corrugation-frequency
[cpd]
80
f1
60
40
f0
20
0
0 0.5 1 1.5 2
cf: 0.5 cpd
80
60
40
20
0
0 0.5 1 1.5 2
time [sec]
-2
0
2
0
50
100
150
200
0
250
5
10
15
20
25
corrugation cutoff [cpd]
output exponent:
1
2
2
1.5
1
0.5
0
n=19
r=0.45
0
0.5
1
1.5
2
1/(2*π*SD of RF height)
[degree-1]
4
Predicted from
mean V1 response
(mean ecentricity 3.7º)
Temporal impulse response (LGN)
10ms
Reppas, Usrey and Ried (2000)
Temporal frequency tuning for
contrast and disparity
1.5
1
40
0.5
0
1
10
0
100
temporal frequency [Hz]
tf cutoff for
drifting luminance grating [Hz]
drifting luminance grating
80
response [spikes/sec]
relative modulation [f1/f0]
disparity modulation
40
20
n=27
0
0
20
40
tf cutoff for
disparity modulation [Hz]
Summary
• We don’t solve the correspondence
problem dot-by-dot.
• Is this enough?
×
=
Monocular response
Correlation
1
0.5
0
-0.5
-1
-50
0
50
RF Disparity (pixels)
Correlation
1
0.5
0
-0.5
-1
-50
0
RF Disparity (pixels)
50
Disparity is two-dimensional
P’
P
direction of gaze
fovea

nodal
point
Epipolar line
Y
Z
X

Y
Z
X

Y
Z
X

probability density function for disparities
encountered during natural viewing
vertical disparity (degrees)
-15
-10
-5
0
5
10
15
-10
0
10
20
horizontal disparity (degrees)
30
probability density function for disparities
encountered during natural viewing
vertical disparity (degrees)
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
horizontal disparity (degrees)
1
-6
vertical disparity
-4
-2
0
2
4
6
-15
-10
-5
0
5
10
15
horizontal disparity
20
25
30
Preferred 2-D Disparity
0.6
Vertical Disparity (degrees)
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Horizontal Disparity (degrees)
0.6
Stevenson and Schor (1997)
Schreiber et at (2001)
Searching for Matches
• Not a 2-D problem.
• Vertical extent of RF may be enough to
deal with most epipolar lines.
correlation
1
0.5
0
-0.5
-1
-50
0
RF Disparity (pixels)
50
correlation
1
0.5
0
-0.5
-1
-50
0
RF Disparity (pixels)
50
Size-disparity correlation
Spatial Period of Center
Frequency
Disparity range (min)
1/(threshold contrast)
Smallman and MaCleod (1994)
Binocular phase
Center spatial frequency
Size-disparity correlation (2)
Prince and Eagle (1999)
R
L
Stimulus Disparity
correlation
1
0.5
0
-0.5
-1
-50
0
RF Disparity (pixels)
50
Stimulus Disparity
Stimulus Disparity
1
correlation
.5
0
-.5
-1
-50
0
50
RF Disparity (pixels)
0
50
Threshold
2 cpd
0.5 cpd
2 cpd + 0.5 cpd
2 cpd, half cycle
Farell, Li and McKee (2004)
correlation
1
0.5
0
-0.5
-1
-50
0
Disparity (pixels)
50
Tsai and Victor (2003)
anti-correlated stimuli
left eye’s image
right eye’s image
black  white
1
correlation
.5
0
-.5
-1
-50
0
50
RF Disparity (pixels)
0
50
energy model simulation
simulated firing rate
4
3.5
3
correlated
stimuli
2.5
2
1.5
anti-correlated
stimuli
1
0.5
0
-50 -40 -30 -20
-10
0
disparity
10
20
30
40
50
Firing Rate (spikes/s )
120
200 Cell rb332
180
160
140
120
100
80
60
40
20
Cell rb313
100
80
60
40
20
0
0
-0.4
-0.2
0.0
0.2
0.4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
Disparity (degrees)
Correlated disparity
Anticorrelated disparity
Energy model prediction
120
Firing Rate (spikes/s )
100
80
60
40
20
0
-0.4
-0.2
0.0
Correlated disparity
Anticorrelated disparity
0.2
0.4
120
Cell rb313
Firing Rate (spikes/s )
100
80
60
weaker response
for anti-correlated
stimuli
40
20
0
-0.4
-0.2
0.0
Correlated disparity
Anticorrelated disparity
0.2
0.4
what the energy model gets wrong
 quantitative response to anticorrelation
– real cells respond more weakly to
anticorrelated stimuli
35
firing rate (spikes / s)
30
25
20
15
10
5
0
monocular stimuli
left
right
35
firing rate (spikes / s)
30
25
20
15
10
5
0
“this cell is monocular”
left
right
disparity tuning curve
35
left
right
left
right
left
right
firing rate (spikes / s)
30
25
20
15
10
5
0
-1.5
-1
-0.5
0
0.5
disparity (degrees)
1
left
eye
has
purely
inhibitory
effect
35
firing rate (spikes / s)
30
25
20
-
15
10
5
0
-1.5
-1
-0.5
0
0.5
disparity (degrees)
1
Disparity Discrimination Index (DDI)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Ocular Dominance Index
1
but -!
• this isn’t possible in the energy model.
the energy model says that each eye sends
both excitatory and inhibitory input
receptive
fields
=ON region of RF
=OFF region of RF
BS
the energy model says that each eye sends
both excitatory and inhibitory input
receptive
fields
=ON region of RF
=OFF region of RF
BS
disparity tuning curve
35
left
right
Inhibition from
left eye
25
20
left
right
left
right
15
uncorrelated
firing rate (spikes / s)
30
10
5
0
-1.5
-1
-0.5
0
0.5
disparity (degrees)
1
Response rates to random dots
140
Ideal
monocular
neuron
monocular
120
100
80
Ideal
binocular
neuron
60
40
20
0
0
20
40
60
80
100
binocular uncorrelated
120
what the energy model gets wrong
 quantitative response to anticorrelation
– real cells respond more weakly to
anticorrelated stimuli
 cells where one eye always inhibits firing
– not possible within the energy model
energy model:
• disparity tuning curve is the crosscorrelation of the left and right eye’s
receptive fields.
C = [vL+vR]2 = vL2 + vR2 + 2 vLvR
D = 2 L * R
left eye’s receptive field
right eye’s receptive field
0.6
-0.6
0.4
-0.4
0.2
-0.2
0.35
0
0
-0.2
0.2
-0.4
0.4
-0.6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.6
0.6
-0.4
-0.2
0
disparity tuning curve
2
1.5
0.35
1
0.5
0
-0.5
-0.25
0
0.25
0.5
0.2
0.4
0.6
how to test
• measure receptive fields?
S
Cx
S
S
how to test
• measure receptive fields?
• not possible for complex cells.
• make the comparison in Fourier space.
• this works for simple and complex cells.
S
Cx
S
S
energy model:
• disparity tuning curve is the cross-correlation
of the left and right eye’s receptive fields:
D = 2 L* R
• the Fourier power spectrum of the disparity
tuning curve is the product of the Fourier
amplitude spectra of the left and right eye’s
receptive fields:
FT2(D) = 2 FT(L)FT(R)
spatial frequency
tuning curve
receptive field (RF)
0.6
firing rate
0.4
0.2
0
-0.2
-0.4
0 0
-0.4
-0.2
0
0.2
0.4
Fourier
transform
1
0
-1
-2
-0.4
-0.2
0
0.2
position
0.4
6
8
Fourier spectrum
-4
2
-0.6
4
spatial frequency
0.6
RF cross-section
x 10
2
0.6
amplitude
-0.6
-0.6
0 0
2
4
6
spatial frequency
8
if the energy model is right:
• then by obtaining the cell’s spatial
frequency tuning….
• we obtain the Fourier amplitude
spectrum of the RF profile.
0.6
firing rate
0.4
0.2
0
-0.2
-0.4
-0.6
-0.6
0
2
4
6
spatial frequency
8
-0.4
-0.2
0
0.2
0.4
0.6
monocular spatial frequency tuning curves
left eye
right eye
2
2.5
2
1.5
x
1
0.5
00
1.5
1
0.5
2
4
6
00
8
spatial frequency
2
4
6
8
spatial frequency
Fourier spectrum of disparity tuning curve
4
3
=
2
1
00
2
4
6
spatial frequency
8
spatial frequency tuning
left eye
right eye
firing rate (spikes/s)
60
50
40
30
20
10
0
0.1
1
10
spatial frequency (cycles per degree)
0
0.1
1
10
spatial frequency (cycles per degree)
product of fitted monocular
spatial frequency tuning curves
60
0.05
50
0.04
40
30
0.03
20
0.02
10
00
0.01 0.1
0.2
0.5
1
2.5
5
10 15
spatial frequency (cycles per degree)
firing rate
(spikes/s)
disparity tuning curve
100
90
80
70
60
50
40
30
20
10
0
-1.5
-1
-0.5
0
0.5
disparity (degrees)
1
1.5
0.05
firing rate (spikes/s)
firing rate (spikes/s)
60
50
0.04
40
30
0.03
20
0.02
10
0
0.1
1
10
100
90
80
70
60
50
40
30
20
10
0
-1.5
normalized units
spatial frequency (cycles/degree)
-1
-0.5
0
0.5
disparity (degrees)
0.05
0.04
0.03
0.02
0.01
0
0.02
0.05
0.1
0.2
1
0.5
1
2.5
5
spatial frequency (cycles per degree)
10 15
1.5
Peak frequencies differ
normalized units
0.05
product of fitted spatial
frequency tuning
curves
0.04
Fourier transform of
fitted disparity tuning
curve (minus baseline)
0.03
0.02
0.01
0
0.01 0.02 0.05 0.1 0.2
0.5
1
too much power at DC
2.5
5
10 15
Firing rate (spikes/s)
40
duf065
Right
Left
35
60
30
25
40
20
15
20
10
5.0
0
-1.4
-0.9
-0.4
0.1
0.6
Disparity (degrees)
1.1 1.4
0.0
0.05 0.10
1.0
10
Spatial Frequency (cpd)
30
Firing rate (spikes/s)
duf043
50
70
60
40
Right
Left
40
20
20
0
0
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Disparity (degrees)
0.05 0.10
1.0
10
Spatial Frequency (cpd)
30
what the energy model gets wrong
 quantitative response to anticorrelation
– real cells respond more weakly than predicted to
anticorrelated stimuli
 suppressive effect from one eye
– not possible within the energy model
 mismatch between disparity frequency and
response to gratings
– real disparity tuning curves have more power at
low frequencies than predicted
how can we fix the problem?
• one simple modification to the energy
model.
• keeps all the successes of the energy
model.
• but fixes all these problems at a stroke!
energy model
n

C   vLj  vRj

2
j 1
images
receptive fields
binocular
simple cell
BS
disparity-selective
complex cell
Cx
Read’s modified version


C   PosvLj   PosvRj 
n
j 1
images
receptive fields
monocular
simple cells
2
= half-wave rectification
binocular
simple cell
disparity-selective
complex cell
MS
BS
MS
Cx
Read’s modified version


C    PosvLj   PosvRj 
n
2
j 1
images
receptive fields
monocular
simple cells
binocular
simple cell
disparity-selective
complex cell
MS
BS
MS
Cx
suppression from one eye


C   PosvLj   PosvRj 
images
receptive fields
monocular
simple cells
2
binocular
simple cell
disparity-selective
complex cell
MS
BS
Cx
MS
input purely
inhibitory
cell never
fires
problems our model solves

• suppressive effect from one eye
– inhibitory synapse after monocular simple cell
firing rate (spikes/s)
0.05
50
0.04
40
30
0.03
20
0.02
10
0
0.1
1
10
100
90
80
70
60
50
40
30
20
10
0
-1.5
spatial frequency (cycles/degree)
normalized units
firing rate (spikes/s)
60
-1
-0.5
0
0.5
disparity (degrees)
0.05
0.04
0.03
0.02
0.01
0
0.02
0.05 0.1
0.2
1
0.5
1
2.5
5
spatial frequency (cycles per degree)
10 15
1.5
firing rate (spikes/s)
0.05
50
0.04
40
30
0.03
20
0.02
10
0
0.1
1
10
100
90
80
70
60
50
40
30
20
10
0
-1.5
spatial frequency (cycles/degree)
normalized units
firing rate (spikes/s)
60
-1
-0.5
0
0.5
disparity (degrees)
0.05
0.04
0.03
0.02
0.01
0
0.02
0.05 0.1
0.2
1
0.5
1
2.5
5
spatial frequency (cycles per degree)
10 15
1.5
C  vL  vR 
2
 v  v  2vL vR
2
L
2
R
when vL and vR are
negatively correlated, this
tends to be negative
100
firing rate (spikes/s)
90
80
70
pulling the response down
below the uncorrelated level
60
50
40
30
20
10
0
-1.5
-1
-0.5
0
0.5
1
1.5
C  PosvL   PosvR 
2
 PosvL   PosvR   2PosvL PosvR 
2
2
when vL and vR are
negatively correlated,
this is zero
100
firing rate (spikes/s)
90
80
70
pushing the response up closer
to the uncorrelated level
60
50
40
30
20
10
0
-1.5
-1
-0.5
0
0.5
1
1.5
disparity
tuning curve
energy model
0
-50
0
our modified version
0
50
-50
Fourier power
spectrum
disparity
0
50
disparity
no
power
at DC
0
0
0.02 0.04 0.06
spatial frequency
increased
power at
DC
0
0
0.02 0.04
0.06
spatial frequency
threshold at zero
monocular simple cells
receptive fields
binocular
simple cell
complex cell
MS
BS
MS
Cx
increased threshold
monocular simple cells
receptive fields
binocular
simple cell
complex cell
MS
BS
MS
Cx
energy model
our modified version
disparity
tuning curve
zero threshold
0
-50
0
50
0
-50
Fourier power
spectrum
disparity
0.02 0.04 0.06
spatial frequency
50
0
-50
disparity
no
power
at DC
0
0
0
high threshold
0.02 0.04
50
disparity
increased
power at
DC
0
0
0
0.06
spatial frequency
maximum
power at
DC
0
0
0.02 0.04
0.06
spatial frequency
problems our model solves
• suppressive effect from one eye
• mismatch
between disparity frequency
and response to gratings
– inhibitory synapse after monocular simple cell
– threshold boosts power at low frequencies
anticorrelation
C  v  v  2vL vR
2
L
2
R
 image in one eye replaced with negative
 one of the convolutions changes sign
 disparity-modulated term inverts; amplitude unchanged:
C  v  v  2v v
2
2
L
R
L R
 a consequence of the linearity of the model
modified model
C  PosvL   PosvR   2PosvL PosvR 
2
2
anticorrelation: convolution changes sign
C  PosvL   PosvR   2PosvL Pos vR 
2
2
clearly disparity-modulated term
no longer simply inverts
MS
Energy model
Modified model
0
-40 -20 0 20 40
Disparity
-40 -20 0 20 40
Disparity
MS
Response
2
1.5
1
0.5
-40-20 0 20 40
Disparity
problems our model solves



• suppressive effect from one eye
– inhibitory synapse after monocular simple cell
• mismatch between disparity frequency and
response to gratings
– threshold boosts power at low frequencies
• quantitative response to anticorrelation
– with high enough thresholds, arbitrarily low
amplitude ratios can be obtained
heterogeneity
• real neurons vary greatly in behavior.
• some well-described by energy model.
• complex cells have many binocular
subunits:
• perhaps some are like the energy model
– linear binocular combination
• others are like our modified version
– threshold prior to binocular combination
heterogeneity
some
binocular
subunits
as in our
model…
MS
MS
BS
Cx
…others
as in the
original
energy
model
BS
complex cells
receive input from
many binocular
subunits.
summary
• the energy model gives a good
qualitative account of disparity-tuned
neurons.
• it has been widely used in
computational models.
• there are a number of discrepancies
when it is compared with quantitative
data.
summary
• A simple, plausible modification
removes these discrepancies.
• Consequences for models of later
processing largely unexplored.
Extrastriate cortex
• V2, V4, MT, and MST all show responses
to anticorrelated RDS, like V1.
• IT does not response to anticorrelated
RDS.
• Does the solution have to be
represented explicitly?
conclusion
• Good understanding of the mechanisms
of disparity selectivity in primary visual
cortex, without invoking complex
network interactions.
• provides a firm basis for understanding
the computations enabling stereo
vision.
Put red lens
over left eye,
blue lens over
right eye
Stereo anaglyph by
Prof. Michael
Greenhalgh,
Australian National
University (with
permission).
plus… a prediction
• Consider case where convolutions are
equal and opposite: vL=-vR
• Original energy model: they cancel out
C  vL  vR   0
2
• Our version: no cancellation
C  PosvL   PosvR   PosvL 
2
2
disparate drifting grating
right
eye
left
eye
typical simple cell response
firing rate
• one burst of firing per cycle of the
stimulus.
time (one stimulus cycle)
phase difference 0o
right
eye
MS
BS
left
eye
MS
phase difference 0o
…half a cycle later
right
eye
MS
BS
left
eye
MS
phase difference 180o
right
eye
MS
–
+
BS
MS
left
eye
phase difference 180o
…half a cycle later
right
eye
MS
+
BS
left
eye
MS
–
60o
120o
180o
240o
300o
interocular phase difference
0o
energy model modified version
time (2 stimulus periods)
modified version
60o
120o
180o
240o
300o
interocular phase difference
0o
energy model
time (2 stimulus periods)
hg226.0
40
10
20
20
10
5
0
0
0
80
60
40
20
30
10
1
0
0
0
80
60
40
20
0
20
3
20
100
50
2
10
0
0.50
90o
60
40
60
40
40
20
20
0
0
20
0
8
40
6
20
20
0
spikes / s
-1.00o
-180
o
-0.50
-90
0.00
0o 
hg212.0
0
1.00
180o
interocular phase difference
hg136.0
4
10
0.2
0.4
0.6
0.8
1
0
2
0.2
0.4
0.6
0.8
1
time (1 stimulus period)
0
0.2
0.4
0.6
0.8
1
summary
• we postulate that some binocular
simple cells receive input via monocular
simple cells.
• straightforward, physiologically plausible
mechanism.
• extends our repertoire so that we can
account for all known observations.
• even predicted something before it was
observed!
Stereo anaglyph by
Michael Greenhalgh,
Australian National
University.
Put red lens over left
eye, blue lens over right
eye
long-term goal of our work
to understand:
• the algorithm the brain uses for
stereoscopic depth perception.
• how this algorithm is implemented
physiologically.
• where this occurs within the brain.
The stereo correspondence problem
1
Patern 1
Patern 2
Correlation
0.5
Patern 3
Patern 4
0
Patern 5
Mean
-0.5
-1
-50
0
Disparity (pixels)
50
Right RF
Left RF
-90
90
0
Correlation
1
0.5
0
-0.5
-1
-90
0
Disparity
90
Y
“straight
ahead”
yL
X
“straight
ahead”
yR
xL
L
Z
xR
R
1
Patern 1
Patern 2
Correlation
0.5
Patern 3
Patern 4
0
Patern 5
Mean
-0.5
-1
-50
0
Disparity (pixels)
50
basic building-block
• inner product of image with receptive field

v   dxdy. x, y I x, y 

“ON” region
Pos(v)
basic building-block
• inner product of image with receptive field

v   dxdy. x, y I x, y 

“OFF” region
Pos(v)
Left RF
0
Right RF
d
simulated firing rate
energy model simulation
correlated
stimuli
uncorrelated stimuli
-50 -40 -30 -20
-10
0
disparity
10
20
30
40
50
shape of disparity tuning
• a key prediction of the energy model.
• demonstrating this result would be
strong evidence for the energy model.
Preferred Grating Frequency (cpd)
5.0
4.0
3.0
2.0
1.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Disparity Frequency (cpd)
Preferred Grating Frequency (cpd)
5.0
4.0
3.0
2.0
1.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Disparity Frequency (cpd)
0.05
normalized units
Peak
0.04
0.03
0.02
0.01
low
0
0.01 0.02 0.05 0.1 0.2
0.5
1
2.5
5
10 15
Response at 0.05cpd
Monocular grating
Peak frequency
Response at peak
5.0
1.0
4.0
0.8
3.0
0.6
2.0
0.4
1.0
0.2
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.2
0.4
FT of disparity tuning
0.6
0.8
1.0
70
firing rate (spikes/s)
firing rate (spikes/s)
45
40
35
30
25
20
15
10
5
0
0.1
1
60
50
40
30
20
10
0
10
normalized units
spatial frequency (cycles/degree)
-1.5
-1
-0.5
0
0.5
disparity (degrees)
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
spatial frequency (cycles per degree)
4
1
1
correlation
.5
0
-.5
-1
-50
0
50
RF Disparity (pixels)
0
50