Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
S. A. Klein and D. M. Levi
1543
Position sense of the peripheral retina
Stanley A. Klein and Dennis M. Levi
College of Optometry, University of Houston, Houston, Texas 77004
Received March 26, 1987; accepted April 8, 1987
Position acuity was measured over a wide range of eccentricities, from 3 min to 10 deg in the horizontal meridian, by
using both a three-dot bisection task and a three-dot vernier task. A foveal fixation dot appeared for 1 sec before an
outer pair of test dots was flashed for 200 msec. Bisection and vernier tasks were used to measure position acuity in
the radial and tangential directions, respectively. The vernier data were well fitted by a straight line on linear axes
of offset threshold versus eccentricity. The bisection data, on the other hand, were poorly fitted by a single straight
line. However, a double-line fit worked very well. The line segment at large eccentricities (>0.5 deg) had an x
intercept of about 0.6 deg, in good agreement with previous estimates based on cortical magnification and on
hyperacuity in the presence of flanks. These results imply that three-dot vernier thresholds are set by a single
orientation mechanism at all eccentricities and that three-dot bisection thresholds are set by a pair of mechanisms.
For eccentricities less than 15 min, thresholds are in good agreement with calculations based on spatial-frequency
filters. For larger eccentricities, the bisection thresholds agree with scaled anatomical modules that are presumed
to exist in the human visual cortex. The thresholds for position acuity in the tangential direction are as low as 0.005
times the eccentricity. In the radial direction, thresholds are poorer, implying that additional cortical factors may
further constrain performance.
INTRODUCTION
Position acuity is often called hyperacuity
1
because under
amine one particular scheme, based on cortical sampling, to
degrade the absolute position cue in the periphery in order
to bring the model calculations into agreement with results
appropriate conditions observers can localize the relative
of experiments in which widely spaced stimuli are used.
position of a line about 10 times better than they can resolve
a pair of lines. This high precision of spatial localization is
It should be pointed out that a class of spatial-frequency
models (as opposed to local-filter models) does a good job in
predicting the linear relationship between threshold and
separation even for separations greater than 15 min. Three
of the appendixes of Ref. 2 showed that bisection thresholds
found only in the presence of nearby reference features.
When the separation between lines is greater than a few
minutes, the position threshold is approximately proportional to the separation. Until recently it has not been
obvious how to account for either the magnitude
of the
optimal hyperacuity thresholds or the linear dependence on
separation.
During the past few years there has been progress in modSeveral ideal-observer calculations 2' 3
eling hyperacuity.
based on outputs of localized spatial filters are in agreement
with bisection experiments for closely spaced lines. 2 For
example, Fig. 1 shows the prediction of our viewprint mod-
el.2 For separations less than 1.5 min the stimulus lines
become blurred, and the displacement threshold is based on
detecting a thin-black-line
cue. For separations between 2
and 15 min the most sensitive mechanisms for detecting the
displacement
are mechanisms
whose zero crossings have
about the same spacing as the line separation.2 The model
provides a good fit to the data 2 for separations of less than 15
min. As shown by Fig. 1, for separations greater than 15 min
the model no longer predicts thresholds that are linearly
proportional to separation; rather, the predicted thresholds
are constant. This result is to be expected, since the idealobserver model can use an absolute position cue for detect-
ing the displacement, causing the threshold to be independent of separation. It is interesting that the absolute posi-
of about 1/30 of the separation can be derived from simple
Fourier considerations. However, although the Fourier approach seems to work for isolated two-line and three-line
stimuli, it is hard to imagine how a pure Fourier approach
would give threshold estimates that are not degraded by the
presence of randomly positioned perturbations. 5 In order
to discount irrelevant features, it would seem that models
based on filters localized in both space and spatial frequency
are needed.
An attractive modification of filter-based models for wide-
ly spaced stimuli invokes aspects of retinal and cortical sampling such as cortical magnifications'7 that may limit performance. Since the spatial grain of the cortex (i.e., M-') is
approximately linearly proportional to eccentricity, it may
not be surprising that position thresholds are also linearly
proportional to eccentricity. The rationale is that the location of the peripheral stimulus is increasingly uncertain as
its eccentricity increases. The purpose of the present experiments is to measure the intrinsic position uncertainty of
the periphery by using tiny dots that are briefly flashed.
An important issue for quantifying the spatial metric of
the cortex is whether the mapping from retina to cortex is
locally isotropic; that is, does a tiny circle imaged onto the
tion cue predicted by the model is about 20 sec, similar to the
peripheral retina map into a circle in the cortex (each point
retinal line-spread function. This predicted threshold is
also similar to the foveal resolution limit and to the thresh-
on the circle being equally discriminable from the center) or
old for motion displacement.
4
In the present paper we ex0740-3232/87/081543-11$02.00
does it become an elongated ellipse? Suppose that the mappings from the retina to the cortex in the radial direction
© Optical Society of America
1544
J. Opt. Soc. Am. A/Vol. 4, No. 8/August 1987
S. A. Klein and D. M. Levi
100
in the periphery by varying the stimulus separation with
both test and reference dots in the periphery.6 ' 9 The goal of
the present experiments is to determine the position uncertainty of the peripheral visual field in both the radial and
tangential directions with only a single test dot per hemi-
tjJ
field.
(IU)
Specifically, we want to know how well the visual
system knows the coordinates of a peripheral test dot with
10
-n
only a foveal reference point.
tuJ
I-
I I
0.1
Fig. 1.
. . . . . . ,.
1
. . . . . . . ..
10
SEPRRRTION
(MIN)
. . 1
100
Predictions by the viewpoint model of Klein and Levi 2 for a
three-line bisection task. The horizontal axis is the separation
between the middle test line and the outside lines. The vertical axis
is the calculated offset threshold. Cauchy-3 filters were used, and
thresholds were taken at the point where d' = 0.675, as was done for
the data of Klein and Levi.2 The data points indicate the separations that were calculated. The thresholds were obtained by an
iterative procedure, since the computer calculated d' with the offset
given rather than vice versa. The offset, x, was divided equally
between the middle line (which was given an offset of x/2) and the
reference lines (which were given offsets of -x/2). This was done to
minimize the absolute position cue. Further details on the model
are found in Ref. 2. For separations
less than 15 min, the model
predictions are in good agreement with the data, except that the
data show an anomalous region of elevated thresholds between 2.0and 2.5-min separations. 2 Above 15-min separations the model
fails, since the predicted thresholds are constant whereas the actual
thresholds continue to rise proportional to the separation.
(directed away from the fovea) and tangential direction
(perpendicular to the radial direction) are given by
ADr = krAE/(E2r + E),
t
ADt = ktEAO/(E2 + E),
(la)
The present experiments extend past work in four ways.
First, most studies of three-dot tasks in peripheral vision
had both the test and reference dots in the periphery, as
shown in Fig. 2A. The present study has one dot (the fixation dot) in the central fovea, and the test dots are varied in
eccentricity from about 4 min to 10 deg as shown in Figs. 2B
and 2C. This wide range of separations provides useful
constraints on models. For separations (eccentricities)
greater than 1-2 deg, the spatial-frequency models for position discrimintion are implausible, and the role of the spatial
grain of the periphery must be taken into account.' 0 Second, previous work focused on either bisection acuity7 or
vernier acuity,6 "1'using rather different stimuli. Here, we
investigate both vernier acuity and bisection acuity under
identical conditions. Third, the roles of reference and test
dots are reversed. In our previous studies6 the peripheral
reference stimuli were on continuously, and the central test
stimulus was exposed briefly. In the present experiments,
on the other hand, a dot in the fovea acts as the reference,
and a pair of symmetrically placed test dots is flashed on for
0.2 sec. This allows the high precision of the fovea to serve
as a ruler against which to measure the position sense of the
periphery. Since the peripheral test dots are flashed briefly,
there is insufficient time for either averaging or eye move-
ments, so the results are informative about the raw positionprocessing capability of the peripheral field. Fourth, the
results of our bisection experiments clarify the interrelation-
(ib)
E
where D is the cortical distance in millimeters, k is a scale
factor in millimeters, E is the eccentricity of the stimulus in
degrees, and E2 is the eccentricity in degrees at which the
inverse cortical magnification is twice the foveal value. The
superscripts r and t are to remind us that the scaling in the
radial and tangential directions need not be equal. The
quantities AE and EAO are the changes in the retinal position in the radial and tangential directions. If kr = kt and
E2r = E 2t, the mapping becomes isotropic and can be written
as a complex log transform 8 :
D = k log(1 + E/E 2 ),
A
X
Fixation
(3)
This is the linear relationship needed to account for the
growth of hyperacuity thresholds as the separation increases. In order to test the isotropy assumption on which
the complex log transform is based, we have measured posi-
tion thresholds in both the tangential and radial directions.
Previous studies have investigated relative position acuity
Test
*I
B
Test
where D and E become complex numbers representing the
AE = const. X (E + E 2 ).
I
k-
E
(2)
two-dimensional cortical and retinal maps. According to
this transform, if the position threshold has a fixed value
when expressed in cortical millimeters (AD = constant),
then the position threshold in retinal coordinates becomes
I
I
H
Fixation
Test
E
C
±
I-
Test
Fixation
Test
Fig. 2. A schematic depiction of two ways to measure bisection
acuity in the periphery. A, The method used by Yap et al.,6 in
which both the test line and the reference lines are in the periphery.
B, The method used in the present study for measuring radial
position thresholds, in which the foveal fixation dot acts as the
reference and the two test dots are flashed to the periphery. C, The
stimulus in the three-dot vernier task for measuring tangential
position thresholds.
Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
S. A. Klein and D. M. Levi
ship of two widely held models of visual localization, namely,
spatial-filter models and spatial-grain models.
PROCEDURE
Stimuli
The stimuli were presented on the monitor of a Commodore
PET computer with green phosphor. A fixation dot appeared for one second. A pair of dots placed to the right and
left of the fixation point was then turned on, and 0.2 sec later
all three dots were turned off.
Position acuity was tested in both the radial (bisection)
and the tangential (vernier) directions in counterbalanced
runs. For the radial task the three dots were actually tiny
vertical lines 4.0 mm high and 0.5 mm wide, and the two
flashed dots were displaced slightly to the right or the left of
the symmetry position. The observer made a bisection
judgment about the direction in which the fixation dot appeared to be displaced from the midpoint of the peripheral
dots and also about the magnitude of this displacement.
For the tangential task the three dots were actually tiny
horizontal lines 3.7 mm wide and 0.5 mm high, and the two
flashed dots were displaced slightly up or down. The observer made a vernier judgment about whether the fixation
dot appeared to be aligned with the peripheral dots. The
1545
For E > 0.4 deg, both the intensity and the size of the
stimulus increased with eccentricity, since the observer was
positioned closer to the screen. Therefore control experiments were performed to test whether the intensity and
stimulus size affected thresholds. In one set of control experiments the flashed dots were 3.3 cm from the fixation
point rather than 6.7 cm, the separation used in the main
experiment. For a given eccentricity the viewingdistance in
this control experiment is half the viewing distance in the
main experiment, so the intensity times the line length of
each dot is increased fourfold. The results indicated that
the thresholds were unchanged, so the dot intensity is not a
relevant factor for the range of intensities in these experiments. As further evidence that stimulus size is not relevant, we will include data for a similar bisection task at small
separations that were reported earlier.2
The exposure duration, on the other hand, does affect
thresholds. In some control experiments we found that
thresholds improved about twofold for small eccentricities
when the dots were flashed for 1 sec rather than 0.2 sec. We
are concerned, however, that with the longer duration, eye
movements are possible, allowing the three-dot stimulus to
be scanned. Here, we report only the 0.2-sec data. The
relevance of the stimulus duration will be considered in the
Discussion section.
flashed dots were displaced -2, -1, 0, 1, or 2 units from the
Observers
symmetry points. The observer responded with rating values of -2, -1, 0, 1, and 2 to indicate the judged position of
the flashed dots.
Position jitter of the entire stimulus was introduced so
that the flashed dots could not be localized with respect to
the outside edges of the display. On each trial the fixation
point was shifted randomly by up to 24 pixels in both the
horizontal and vertical directions. This amount of jitter is
much larger than the position threshold, which was at most 4
Three observers with normal vision and with appropriate
refractive correction participated in these experiments.
Viewing was binocular with natural pupils. Two of the
observers were the authors, and the third subject was naive.
pixels.
For eccentricities greater than 0.4 deg the dots were
flashed a fixed distance (6.7 cm) from the fixation point, and
the viewing distance was varied to achieve the different
eccentricities. When the flashed dots had an eccentricity of
10 deg the viewing distance was 38 cm. At this distance the
stimulus lines were about 30 min in length, thereby providing multiple samples. This serves to provide the periphery
with sufficient samples to be comparable to the fovea, which
Analysis
The data were analyzed with our ROCFLEX program, which
executes a maximum-likelihood multicriterion probit analy4
SiS.
The threshold is defined as the offset giving a d' value
of 1.0 corresponding to 84%correct. In order to compare our
thresholds with those defined at the 75% correct level, our
thresholds and standard errors should be multiplied by
0.675.
Each threshold estimate (Th) is based on several runs of
125 trials per run. The multiple thresholds from different
runs, Thi, at a given eccentricity were combined by a weighted average (on a logarithmic scale), where the inverse variance of each individual run (1/SEi 2 ) is the weighting factor:
has multiple effective samples even for a point object be-
cause of the relatively broad point-spread function and the
overlap of cortical receptive fields.12 For eccentricities less
than 0.4 deg, the viewing distance was fixed at 10.8 m, and
the dot separation was varied on the screen.
The room lights were off. The only room illumination
came from a dim light bulb behind the display monitor.
This dim illumination provided a faint outline of the display
monitor that helped to stabilize accommodation and provided a vertical and horizontal reference that minimized the
effects of head tilt and cyclotorsional eye movements that
could interfere with vernier judgments. In addition, a chin
and forehead rest was used to stabilize the observer's head
position. When the viewing distance was less than 10 m, a
0.9-neutral-density filter was placed over the stimulus. The
intensity of the dots at all eccentricities was always greater
than 15 times the threshold.
Th =
E
(Thi/SEi2)/Z (1/SE2).
(4)
The standard error (SE) for each run (SEi) was determined
by the ROCFLEX fit to the data. Each datum is plotted with
a pair of SE's. The two different estimates for the SE were
obtained by two methods for combining the SE's of repeated
runs. The SE with the narrow crossbar is given by summing
the inverse variances of each separate run:
1/SE2
=
I1/SE 2 .
(5)
The SE with the wide crossbar is given by multiplying the
first SE [Eq. 5] by a heterogeneity factor that is based on the
between-runs variability of thresholds.13 The heterogeneity factor is given by [X2 /(n - 1)] 1/2, where n is the number of
.runs at a given eccentricity and the x 2 statistic is given by
1546
J. Opt. Soc. Am. A/Vol. 4, No. 8/August 1987
x2 =
E
(Th
-
S. A. Klein and D. M. Levi
Thi)2 /SE 2 ,
each observer are low, indicating that the variance of the
data points is fully accounted for. The expected x2 , as
indicated in the last column in Table 1, is the number of data
minus 4 (the number of parameters for a double-line fit).
The need for a double-line fit has a surprising interpretation
in terms of cortical magnification scaling (see the Discussion
(6)
where Th is given by Eq. (4). If the between-runs variability
is due solely to statistical fluctuations, then x2 is expected to
equal the number of degrees of freedom (df), which is n
-
1.
Any extra variability that is due, for example, to variations
in fatigue would increase x2 .
section).
A x2 statistic is also used for fitting smooth curves to the
averaged data, as will be discussed in the Results section.
For this purpose the SE of each datum is taken to be the
larger of the two SE's discussed in the preceding paragraph.
In Fig. 4 the data are replotted such that the vertical axis is
now the eccentricity Weber fraction, which is the threshold
5.0
4.5
RESULTS
4.0
Figures 3 and 4 show plots of radial (bisection) and tangential (vernier) thresholds in minutes versus eccentricity in
degrees for three observers.
2 3.5
3.0
Linear axes are used in Fig. 3;
C 2.5
logarithmic axes are used in Fig. 4. Only data for eccentricities below 2.8 deg are plotted in Fig. 3 to improve the clarity
of the data at low eccentricities. However, the lines shown
are derived by fitting the entire data set, which is shown in
Fig. 4. The SE of each individual run was found to be
between 10 and 15% of the threshold value. The wide and
narrow error bars are approximately the same size, implying
that the heterogeneity factor is close to unity. Thus there is
an insignificant amount of systematic between-runs variability.
,,, 2.0
1. 5
1.0
0.5
ECCENTRICITY
(DEG)
5.0
4.5
As is shown by Fig. 3, the data for the tangential thresholds were well fitted by a straight line of the form
Th = W (E + E2),
(7)
where E is the eccentricity, E 2 is a constant that indicates the
eccentricity at which the threshold Th doubles, and W is the
asymptotic eccentricity Weber fraction (threshold in degrees divided by eccentricity
in degrees for large E).
A
nonlinear-regression algorithms was used to find the bestfitting values of W and E2 and their SE's. Nonlinear regression was used rather than linear regression in order to have
the regression program calculate the SE of E2, the horizontal
intercept of the line. Table 1 summarizes the parameter
estimates and x2 values for the three observers.
For a good
4.0
l
3. 5
K3.0
2.5
L2.
I-
0
1.5
1.0
0.5
ECCENTRICITY
(DEG)
fit the x2 value should approximately equal df, which in this
case is the number of data points minus 2 (since there are
5.0
two free parameters,
4.0
W and E2 , for a straight line).
For
observers SK and WS the vernier data have x2 values that
are quite close to df. For observer DL, there is a good
explanation for the fact that the x2 value is a bit high. His x2
4.5
; 3. 5
K 3. 0
would have been 14.4 (df = 10) instead of 19.3 (df = 11) if the
-J 2. 5
data point at E = 2.8 min had been excluded. Observer DL
was the only observer tested at such small separations; the
lowest separation tested by SK and WS was 4.0 min. For
separations of less than 4 min, three-dot vernier acuity falls
off rapidly.6 Thus the single-line fit is valid for DL for E > 3
min. The low x2 values imply that Eq. (7) provides an
excellent summary of the data in the tangential direction.
The data for the radial direction, however, are not well
2. 0
JL,
fitted by a single straight line, since the x2 values are quite
high. For observers DL and WS the likelihood that the data
are fitted by a single straight line is less than p = 0.0001.
For observer SK the likelihood isppt 0.01. A double-line fit
to the data, however, shown by a solid line in Fig. 3, provides
a rather good fit. For the double-line fit the x2 values for
I- 1. 5
1.0
0.5
0.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
ECCENTRICITY
(DEG)
Fig. 3. Radial (bisection) and tangential (vernier) position thresholds for three observers SK (A), DL (B), and WS (C). The double
error bars represent two methods for calculating SE's, as discussed
in the text. The vernier data are fitted by a single straight line, and
the bisection data are fitted by a double straight line. Only data
below 2.8 deg are plotted here for clarity, but the fit is based on all
the data, as shown in Figs. 4 and 5. The axes are linear in order to
illustrate best the quality of the straight-line fits.
Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
S. A. Klein and D. M. Levi
10
10
1547
WS
0'
RADI[L
A ----
TRNGENT]IRL
C-)
._
a-
z
LU)
L)
U:
Uj
U_
CD
*L
CD
I
CD
LU
C)
-J
cr_
U')
LU
0I.
0.01
0.1
1
ECCENTRICITY(DEG)
ECCENTRICITY (DEG)
ITo
10
Fig. 4.
The data from Fig. 3 are replotted on logarithmic axes in
order to spread out the low-eccentricity data while still showing the
data at 10deg. The vertical axis is the threshold in degrees divided
by the eccentricity in degrees and times 100 (to convert to percent).
U)
UJ
LI
U
LU
U-I
CD
The open circles are bisection data for SK taken from Fig. 4 of Ref. 2,
since the earlier experiment was similar to the present one except
that stimulus lines 30 min long were used instead of the dots of the
present experiment. The data show that the line length is impor-
-j
CD
LU
tant only for separations less than 0.05 deg (3 min). The lines from
Fig. 3 are replotted here. The additional line for the bisection data
is based on the one-line fit that produced a large x2 value (see Table
0.1
0.1
1). This line is plotted to make explicit the nature of the poor fit.
1
ECCENTRICITY (DEG)
A, B, and C correspond to Figs. 3A, 3B, and 3C, respectively.
Table 1. Parameter Estimates and x2 Values for Three Observers
One-Line Fit
Direction
Weber
of
Offset
Large Eccentricity
Fraction
Observer
Two-Line Fit
Small Eccentricity
Weber
Fraction
(%)
E2
(deg)
x2
df
Tangential (vernier)
SK
DL
WS
0.52 + 0.02
1.01 4 0.03
1.00 ± 0.03
0.27 ± 0.02
0.093 ± 0.011
0.21 + 0.03
11.2
19.3
10.1
9
11
9
Radial (bisection)
SK
DL
WS
1.51 ± 0.07
1.27 I 0.06
1.90 + 0.07
0.15 + 0.02
0.31 + 0.03
0.19 + 0.02
23.4
51.6
39.0
10
10
9
offset divided by the eccentricity. This plot emphasizes
how the two branches of the bisection data correspond to
two different Weber fractions. The curves through the data
are the same straight-line fits shown in Fig. 3 (the curvature
of the lines is due to the logarithmic axes). The open circles
on the plot for SK are data from an earlier experiment 2 using
a test line that is flashed for 0.3 sec between a pair of reference lines (the data for a line luminance of 0.879 cd/m were
chosen from Fig. 4 of Ref. 2 because it matches the lumi-
nance of the present stimuli for small separations). The
data are shown here to point out that for separations between 3 and 6 min (the largest separation tested) there is no
difference between the bisection threshold for points and
that for lines. For separations less than 3 min the bisection
threshold for lines is lower than the threshold for dots (also
Weber
Fraction
(%)
E2
(deg)
1.36 + 0.08
1.07 + 0.07
1.58 + 0.10
0.28 1 0.06
0.64 + 0.08
0.60 + 0.15
2.55 + 0.47
3.45 ± 0.62
3.02 + 0.30
(%)
E2
(deg)
0.049
0.047
0.071
+
+
+
0.026
0.022
0.022
X2
df
8.2
10.1
15.4
8
8
7
see Fig. 3 of Ref. 2 for the effect of line length and separation
on bisection thresholds).
DISCUSSION
The results of our experiments were quite surprising. Our
original expectation, that position acuities in the tangential
and radial directions would have some simple fixed relation-
ship to each other, proved to be incorrect. We had anticipated that position thresholds would show a linear dependence on eccentricity of the form Th = W(E + E2). Instead,
we found that the single-line relationship held for the tangential direction but a two-line fit was necessary for the
radial thresholds. The ratio between radial and tangential
thresholds is not constant. Yap et al.a showed a similar
1548
J. Opt. Soc. Am. A/Vol. 4, No. 8/August 1987
S. A. Klein and D. M. Levi
Table 2. Parameter Values and E2 Estimates for Retinal and Cortical Structures and for Several Psychophysical
Thresholds
Parameter
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Spacing between ganglion cell receptive fields
Resolution: Cutoff spatial frequency: subject JM
subject PA
Cortical receptive-field size
Inverse magnification (deg/cortical module)
Inverse magnification (deg/psychophysical module)
Interference zone for vernier acuity
Single-line abutting vernier: subject JM
subject PA
Multiple-line abutting vernier: subject JM
subject PA
Brief, optimal three-dot bisection: subject YY
subject DL
Slow, one-dot radial (bisection) subject DL
Slow, one-dot tangential (vernier) subject DL
Present study, one-dot tangentiale: subject SK
subject DL
subject WS
Present study, one-dot radialef: subject SK
subject DL
subject WS
Eccentricity
Factor E 2
Eccentricity
Weber Fraction
Reference
(deg)a
W (%)-
E = 0 deg
15
6
6
16
17-19
z2.5
3.0 + 0.2
2.2 + 0.1
-2.5
-0.8
z0.6
-0.8
0.67 + 0.09
0.53 + 0.07
0.77 + 0.05
0.62 + 0.08
0.59 + 0.05
0.55 + 0.06
0.012 + 0.008
0.12 + 0.03
0.27 + 0.02
0.093 + 0.011
0.21 + 0.03
0.33
0.4
0.4
0.50
0.72
0.53
6
6
6
6
6
7
7
6
6
10
-10
P10
0.41
0.47
0.3
0.25
1.0
0.8
2.2
0.7
0.35
0.68
0.68
Threshold
W(E+ E 2) (min)c
4.8
3.6
4.8
0.16
0.15
0.14
0.09
0.35
0.26
0.016c
0.050
0.057
0.038
0.085
0.063
0.066
0.087
E= 10 deg
2.5
3.1
2.9
65
64
65
2.6
3.0
1.9
1.6
6.4
5.1
13.2d
4.3d
2.2
4.1
4.1
5.44
4.61
6.78
aThe values indicated by an approximation sign (j) are estimates by eye. The values of E 2 for the psychophysicaldata have SE's of less than 20%. The values
with SE's were obtained by fitting the data with y = W(E + E2).
b Thresholds are converted to minutes of arc in order to facilitate comparison with other studies.
c The expression W(E + E2) is not valid for eccentricities less than 1.5min because of the eye's blur function. Thus the optimal threshold would occur at about E
0.02 deg, giving a threshold of 0.042min.
d The predicted thresholds at 10deg are not reliable because the data went up only to 20 min of eccentricity.
I The Weber fraction from Table 1 is reduced by 0.675to permit comparisonof the present data with the results of other studies shownin the table, for which the
threshold was defined at d' = 0.675.
fValues of E2 and Ware not listed, since the radial direction requires a double-line fit (seeTable 1). Thresholds atE = Odeg are calculated from the small-eccentricity segment. Thresholds at E = 10deg are calculated from the large-eccentricitysegment.
anisotropy using a bisection task in both directions. A complex log transform of the retina-to-cortex maps therefore
can not account for the absolute position judgments of the
present study unless further assumptions are made. In retrospect, we are delighted with the present results, since they
will be shown to fit in well with previous findings.
Before we
discuss the implications of the present results, it is useful to
review previous findings on psychophysics and anatomy in
the periphery.
Table 2 presents estimates of E 2 (second column) for ana-
tomical structures in the retina and cortex and estimates of
E2 for several psychophysical thresholds (resolution position
acuity) measured across the visual field. The third column
shows the values of each of these parameters
given as a
mately the same value of E 2 (row 4 of Table 2) is the cortical
receptive-field size.'6 It is noteworthy that the scaling of
both ganglion and cortical receptive-field sizes are matched
to each other and to pyschophysical resolution scaling. The
match provides strong support for the usefulness of scaling
notions.
The mapping from the retina to the cortex changes faster
than the ganglion cell density.'7-1 9 Thus each foveal gangli-
on cell is associated with a larger cortical area than is each
peripheral ganglion cell. The cortical mapping is typically
described in terms of M-', the inverse magnification factor,
which specifies the degrees of visual field per millimeter of
cortex. The inverse magnification is approximately proportional to the effective eccentricity (E + E 2 ). Two values of
fraction of the effective eccentricity E*, where E* = E + E2.
In order to clarify the eccentricity dependence, the values of
each of these parameters at the fovea and at 10 deg are
E2 for cortical magnification are presented in Table 2. The
value of E 2 = 0.8 deg, used by Levi et al. 6 was based on Dow's
shown in the last two columns.
value of E 2 = 0.6 deg that is used in the present paper is
The first row of Table 2 specifies the angular distance
between adjacent P (Parvo) on-center ganglion cell receptive-field centers. Near the fovea this angular separation is
based on estimates of the psychophysical modules from several studies including those of Levi et al. 6 and Yap et al.7 and
on data from the present study. The precise value of E2 is
uncertain, but all current estimates lie between 0.3 and 0.9
deg. Van Essen et al.18 emphasize the variability of cortical
magnification among the monkeys that they studied. The
equal to the cone spacing.
The ganglion spacing is given by
(E + 2.5) X 0.0033,corresponding to 0.5 min in the fovea and
2.5 min at 10 deg in the periphery.'
5
Rows 2 and 3 of Table 2
show that the cutoff spatial frequency (where W corre-
data on the receptive-field centers of two monkeys. The
sponds to the width of a half-cycle) is in close agreement
uncertainty in E 2 is further illustrated by Dow's original
estimate of E 2 = 0.3 deg,17 which was based on his pooling of
with the ganglion cell spacing. Also shown with approxi-
the data from two monkeys. We obtained estimates of E2
=
S. A. Klein and D. M. Levi
Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
0.77 deg by doing linear regressions on the data of his two
monkeys separately. 6 All recent estimates of E 2 for cortical
magnification indicate that there is more than a threefold
decrease in the value of the E 2 in the primary visual cortex as
compared with the ganglion cell spacing.
The Weber fraction (W) for cortical modules (anatomical)
is more difficult to determine, since a multiplicity of anatomical structures are available. In Row 5 of Table 2 the
hypercolumn of Hubel and Wiesel20 (a pair of ocular dominance columns) is taken as the fundamental module, leading
to a value of W = 0.1.6 Other definitions (orientation columns, cytochrome oxidase blobs) might lead to slightly different values for the eccentricity Weber fraction.
1549
ments. The covariance can not be ignored, since the observer's uncertainty in gaze directions correlates the judged
locations of the two positions; that is, if the gaze direction
was judged to be 2 deg to the right of the actual gaze direction, then both dot locations would be misjudged 2 deg to the
right. Equation (8) seems to imply that the SE of the distance judgment (i.e., the threshold) is a constant. However,
as shown in Fig. 3 (also in Refs. 1, 2, 6, 7, 11, and 12),
thresholds are not constant but are instead proportional to
the distance d between relevant features.
There are at least three ways by which the visual system
could accomplish this Weber's law for position:
Row 7 of
Table 2 indicates that the region of interference of flanking
dots in a vernier task has about the same size as a single
cortical module. 6
Rows 8-13 of Table 2 show the behavior of two hyperacuity tasks across the visual field. The data for the singleline abutting vernier experiment were obtained from the
rightmost data points of Fig. 11 of Ref. 6. Both the abutting
vernier targets6 and the three-dot bisection target7 have
values of E 2 between 0.5 and 0.7. Levi et al. 6 point out the
close correspondence between E 2 for relative position and
the values obtained from recent estimates of cortical magnification.
Rows 14 and 15 of Table 2 are based on data from Fig. 3 of
Ref. 6. The experiments were three-dot bisection, two-dot
vernier, and three-dot vernier tasks with the middle dot
always centered on the fovea as in the experiments
of the
present paper. The main difference from the present experiments is that the test dot was on for 1 sec and the
reference dots for the bisection case were never more than 15
min from the central dot. The present research was an
outgrowth of this earlier study, with the intent of extending
the results to larger eccentricities. We were interested in
1. Size-Tuned Filters
Rather than measuring the individual positions of each dot
[as in Eq. (8)] the visual system may measure the difference
in position (similar to a differential amplifier21 ). This could
be accomplished by the size-tuned filters that are believed to
exist in the visual system. The Weber's-law behavior arises
as a natural consequence of the logarithmic connection between stimulus strength and response in the filter models.
The filter models are implausible at wide separations for
several reasons. First, it is unlikely that a single mechanism
would extend from the fovea to 10 deg.
Second, without
invoking additional nonlinearities, filter models would have
difficulty accounting for the robustness of thresholds to perturbations and filtering.5"10 Third, the filter models actually provide multiple cues. For separations of less than 15
min, our filter model2 predicts Weber's law by having the
optimal mechanism respond to the pair of lines or dots.
However, for widely separated targets, as shown in Fig. 1, the
small filters provide an absolute positive cue that is more
sensitive than the response of the large filters, whose size is
matched to the target size.
whether the low values of E 2 = 0.012 and E 2 = 0.12 (rows 14
2.
and 15 of Table 2) were valid at larger eccentricities and for
brief exposure durations.
It is possible to measure each point individually and to take
the difference between the two locations. Weber's law in
this case would mean that the differencing operation has an
uncertainty proportional to the distance. One way to
Rows 16-21 of Table 2 show a summary of the present
data with the thresholds multiplied by 0.675 in order to
facilitate comparison with the other results in Table 2. In
the other studies the threshold was defined as d' = 0.675,
whereas in the present study d' = 1.0 was used.
The following subsections explore the relationship between the present results on the intrinsic positional uncertainty of the visual system and the limitations imposed by
retinal and cortical sampling (magnification) as specified in
Table 2. Since the results for the radial and tangential
directions differed, they will be discussed separately.
Relationshipbetween Intrinsic Positional Uncertainty
Decorrelation
achieve Weber's law is to have the covariance in Eq. (8)
decrease with separation. For small separations the two
positions would be perfectly correlated, so the difference
thresholds would be very small. At large separations the
covariance would go to zero, and the threshold would be (a12
+ U22)1/2. This has the problem that at large separations
thresholds would be constant, as in Fig. 1. Another scheme
would be to postulate that large distances are calculated by
summing multiple smaller measurements made on a finer
grid. The Weber's-law behavior can be achieved by postulating systematic noise in the size of the underlying ruler.
and Weber's Law
Our task involvedjudging the distance or angle between two
dots at locations xi and x2. Here, we wish to consider the
effect of intrinsic positional uncertainty al and 02 on the
precision of this distance judgment. The distance between
3. Cortical Magnification
For eccentricities greater than about 0.5 deg the absolute
position uncertainty of a given point may be proportional to
the eccentricity of that point (the cortical magnification
the two locations is given by d
assumption);
=
x2
- x1 . The SE of the
judged distance is given by
SE
= [a, 2 + 22
-
cov(x1 , x 2 )]1/2,
(8)
where cov(xl, x2 ) is the covariance of the two position judg-
that is, if x 2 is in the periphery, then a2 is
proportional to x2. If x, is in the fovea so that its uncertainty is negligible compared with that of x2, then the uncertainty in the separation distance would be proportional to x2.
This scheme has the desirable property that it leads to We-
1550
J. Opt. Soc. Am. A/Vol. 4, No. 8/August 1987
S. A. Klein and D. M. Levi
ber's-law behavior at arbitrarily large separations. We elaborate on this idea below.
Intrinsic Positional Uncertainty
In our experiments a pair of test stimuli was flashed simultaneously to both hemispheres at locations E + A and -E +
A, where E is the eccentricity and A is the offset in either the
horizontal or the vertical direction. The observer's task was
to compare the average position of the flashed dots with a
foveal reference. Suppose that the intrinsic position uncertainty at the test point has a standard deviation of o-. The
cortex seems to be labeled for position with a 0.12-mm (l/8of
a cortical module) uncertainty.
The threshold value of 0.012 (E + 0.6) seems remarkable
not only because the value of E 2 - 0.6 deg is in agreement
with cortical magnification but also because the eccentricity
Weber fraction of 0.012 is in good agreement with the optimal peripheral bisection thresholds with nearby reference
points. For example, Yap et al. 7 found that the optimal
bisection acuity for three dots at 10 deg in the lower visual
field was about 6 min, in agreement with an eccentricity
Weber fraction of W = 0.01. This optimal acuity occurred
for dot separations of about 1 deg (one cortical module at 10
average position of the two flashes is [(E + A) + (-E + A)]/2
deg of eccentricity).
= A, and the position uncertainty of the average position is
(r 2 + a 2)'/2/2 = a/2. Thus the psychophysical thresholds
underestimate the intrinsic positional uncertainty by a constant factor of V/2. In other words, if the threshold is 1 min
(defined at d' = 1),then the intrinsic position uncertainty a
is 1.4 min. If the foveal reference dot were not present, then
the threshold would be rja for the two-dot separation task
rather than o/jF for the three-dot bisection task. In the rest
of the discussion we ignore the 14
factor and continue to
discuss the results in terms of the psychophysical data.
module, the bisection acuity was degraded. The agreement
between three-dot bisection acuity (closely spaced reference
dots) and the one-dot radial position acuity of the present
study (foveal reference dot) may be fortuitous. However, it
is possible that there is a fundamental absolute position
capability that limits bisection acuity (size judgments) to
about 0.1 mm of cortex.
Connection between Present Psychophysical Results and
Radial Direction at Large Eccentricities
The bisection data required a double-line fit. First, consider the data for large eccentricities (E > 0.5 deg). The average Weber fraction of the three observers is 1.25%, corre-
sponding to one part in 80. Thus, dots flashed 80 min in the
periphery would have a radial position threshold of 1 min.
This result is slightly better than previous findings for size
judgments of large objects.10'22
The values of E 2 = 0.28, 0.64, and 0.60 deg for the three
observers are compatible with recent estimates of the cortical magnification factor' 7 ' 9 and with previous psychophysical estimates using stimuli in which both the test and reference features were close together in the periphery as shown
in Table 2. Abutting line vernier acuity exhibited values of
E 2 between 0.5 and 0.7 deg.6 Three-dot bisection acuity also
had values of E 2 between 0.5 and 0.6 deg. 7 One of the main
factors that constrained the value of E2 to be about 0.6 deg in
these earlier studies was that strong masking occurred
(thresholds increased) when the stimulus features were separated by less than 10%of the effective eccentricity E* (E* =
E + E2). The region of masking, 0.1 E*, corresponds to 1
mm of cortex and can be considered a cortical module. For
When the dots were closer than one
Radial Direction at Small Eccentricities
For E < 0.5 deg a different set of rules applies. Here,
relative position provides the cue for the radial position
judgments. Thus, at small eccentricities, thresholds are
approximately a constant fraction of the separation of the
reference features, corresponding to low values of E 2, as we
discuss below. The Weber fractions for this relative position mechanism are 2.55, 3.45, and 3.02%for the three observers (column 7 of Table 1). Since this Weber fraction is a
factor of 2 to 3 worse than the Weber fraction for the absolute position cue, this relative position cue is used only for
small eccentricities at which the relative cue has greater
sensitivity than the absolute cue. This relative position cue
operates within the central 0.5 deg of visual field where the
magnification is relatively constant.
The Weber fraction of 3.45%for DL is slightly larger than
his previous results (-2.5%) reported for bisection with
small separations (see Fig. 3 and Table 2 of Ref. 6). The
previous experiments differed in two important respects.
First, the threshold was reported
for a d' of 0.675 (corre-
sponding to 75%correct), so that the present thresholds and
Weber fractions should be multiplied by 0.675 to compare
with the earlier results. Second, the previous test patterns
were on for 1 sec, compared with a display time of 0.2 sec for
not seem to be a Weber's law for cortical separation; that is,
the present data. A relative position cue of 3%of the separation for briefly flashed dots is in good agreement with the
recent results of peripheral three-dot bisection.7 For example, for observer DL the bisection threshold was about 3 min
for line separations of 100 min, independent of whether the
middle dot of the three-dot target was at 2.5, 5.0, or 10 deg in
the periphery.7 This relative position cue, which is found
for closely spaced dots near the fovea, seems to play a role in
position judgments in the periphery. The separations at
which the relative position cue becomes useful are larger in
the periphery (in proportion to 1 + E/0.6). The relative
position cue operates in the range from one to about seven
modules. When the three dots of the bisection target fall
within a single module, the bisection threshold degrades
the position of a dot that is 20 mm from the fovea (cortical
units) can be correctly judged to within 0.12 mm just as well
as a dot that is 5 mm from the fovea. Each location in the
rapidly. 7
The value of E 2 for small eccentricities was less than 0.1
deg for all three observers. This low value is reminiscent of
the fovea this module is about 3.6 min in size.
The results of the present experiments imply that for the
radial direction a constant fraction of the cortical module
sets an upper limit to the absolute position uncertainty even
without peripheral reference dots that could cause crowding.6 Since the radial position thresholds
(Table 1) can be
expressed as approximately 0.012 (E + 0.6) (for E > 0.5 deg),
and since each millimeter of cortex corresponds to a spatial
extent of 0.1 (E + 0.6) deg, then the radial position threshold
is 0.12 mm of cortex.
This is a most interesting result.
It
implies that a feature can be localized to an absolute position
with a constant uncertainty in cortical units. There does
S. A. Klein and D. M. Levi
Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
the remarkably low thresholds that are obtainable both experimentally and theoretically for bisection tasks with closely spaced long lines with long exposure durations.
2
These
thresholds can be written approximately as (E + 0.015)/60
(see Fig. 1 for the theoretical predictions). The thresholds
of the present study are larger, undoubtedly because the
stimuli were briefly presented dots. A study of bisection
acuity with long-duration dots gives thresholds of about (E
+ 0.012)/45 (see Fig 3 of Ref. 6). Burbeck 23 has found that
short stimulus durations can severely degrade width judgments for small separations. The bisection thresholds of
the present study (see also Ref. 6) are about (E + 0.065)/30.
In order to check whether the spatial-filter model is able to
account for these different dependencies on stimulus separation, one must first measure the contrast-sensitivity function and the contrast-response function under the different
line-length and temporal-duration conditions. It is clear
from Fig. 1 that the spatial-filter model fails dismally for
eccentricities greater than 0.2 deg, where the model predicts
a constant threshold rather than a threshold proportional to
E. The viewpoint of this paper suggests that as the stimulus
separation increases above 0.3 deg, cortical sampling should
be introduced to the model in order to degrade position
acuity. It is interesting to note the eccentricity at which the
transition between the large and small regimes occurs; i.e.,
the eccentricity of the inflection point of our two-line fits.
For our three observers the transition occurs between 0.2
and 0.4 deg. This transition marks the shift from the filter
regime to the cortical position regime (see the subsection
titled Hyperacuity Thresholds in Cortical Units, below).
1551
observers. As was mentioned earlier, the leftmost datum of
DL indicates a rise in the vernier threshold above the
straight-line fit. This is the point at a separation of 2.8 min,
which is within a cortical module for the fovea, where masking effects are expected. 6 A similar result was found for
three-dot vernier and bisection for long durations. With
long durations, vernier thresholds become larger than bisection thresholds for separation of less than 2.5 min.6 At these
long durations and small separations a luminance cue is
available for the bisection task that is not available in the
vernier case.2' 7 This luminance cue was not present for the
brief exposures of the present experiment, as can be seen by
comparing the open circles and the filled circles in Fig. 4 for
SK.
The validity of the straight-line fit to the vernier data was
tested by attempting to make two-line fits to the data. We
were unsuccessful in this attempt. The x2 value never
dropped by more than 2, which would be needed for an
improved fit with two extra parameters. In addition, the
optimal parameter values for this fit were radically different
among the three observers.
Hyperacuity Thresholds in Cortical Units
Figure 5 shows the data from Fig. 3 and 4 replotted using a
different vertical axis. In Fig. 4, thresholds were expressed
as a percentage of the eccentricity E. In Fig. 5, thresholds
are expressed as a percentage of the effective eccentricity E*
= E + 0.6. As we have discussed previously,
6
the effective
eccentricity is directly related to the inverse magnification
factor. The vertical axis in Fig. 5 can therefore be expressed
in units of millimeters of cortex, as is shown on the right-
hand ordinate. The lines fitted to the bisection data in Fig.
Tangential Direction
Detection of an offset in the tangential direction is a vernier
5 are constrained to have a value of E 2 = 0.6 deg for the large-
or orientation task.
eccentricity part of the fit; thus this portion of the fit is
perfectly horizontal. The value of 0.6 is similar to the value
As shown in Table 1, a single straight
line provides an excellent fit to the data. The x2 values of
11, 19, and 10 are very close to the expected values of 9, 11,
and 9, especially considering our earlier explanation of the
high x2 value for DL. Observers DL and WS had eccentrici-
ty Weber fractions of 1%, and observer SK had a Weber
fraction of 0.5%. These two Weber fractions correspond to
tilt angles of 0.5 and 0.25 deg with respect to the horizontal.
These values are in agreement with previous findings for
two-dot orientation judgments.2 4' 25 Our results imply that
three-dot vernier thresholds are determined by a constant
orientation cue from 0.3 to at least 10 deg.
The vertical axis in Fig. 4 is expressed in units of percent-
age of eccentricity. This scale is identical to an orientation
threshold in units of centrads (0.01 rad).
Figure 4 shows
that the orientation thresholds are not constant. The orientation thresholds rise as the eccentricity decreases below
about 0.5 deg. We believe that this gradual falloff is attributable to the brief test durations and the low visibility of the
test dots in our experiments. These conditions will have a
greater effect on the closely spaced stimuli, where the optimal thresholds are very low.
For all three observers thresholds in the tangential direction were better than radial thresholds. This result is compatible with those of other studies on peripheral hyperacuity. 9 At eccentricities of less than 0.2 deg the two thresh-
olds approach each other. This behavior is related to the
finding (Table 1) that E2 for the vernier task was larger than
E 2 for the bisection task (small eccentricities) for all three
of E 2 obtained for observers DL and WS. However, observer SK had E 2 = 0.28 deg. Thus, for observer SK, forcing E 2
to be 0.6 results in a shift of the transition from 0.2 to 0.8 deg.
When expressed as a fraction of (E + 0.6), both the tangential and radial thresholds are fairly constant. For small
separations the thresholds are about 0.3%of E*, similar to
the 1/40 of a cortical module found in previous experiments
for hyperacuity threshold at the optimal separation.' 24 At
large eccentricities the thresholds increase about fourfold to
about 1% of E*. Thus the absolute position thresholds are
about a factor of 4 worse than the best hyperacuity thresholds under the present test conditions. We are not aware of
any hyperacuity results in peripheral vision in which thresholds of less than 0.2% of E* have been found (1/50 of a
module). Thus nearby reference lines can enhance position
acuity by at most a factor of 5 over the absolute position
acuity of single lines. The absolute position acuity examined in the present study should therefore provide a general
ceiling on peripheral positional thresholds. The range of
position thresholds at a given eccentricity is small. The
foveally referenced thresholds of the present study provide
an upper limit. A lower limit, achieved with nearby reference lines, is about four times smaller than the upper limit.
It is interesting to compare the optimal values for hyperacuity thresholds in peripheral vision with ganglion cell
spacing and thresholds in resolution tasks. Table 2 shows
that both ganglion cell spacing and resolution have Weber
1552
J. Opt. Soc. Am. A/Vol. 4, No. 8/August 1987
S. A. Klein and D. M. Levi
lute position acuity for observer SK in the tangential direc-
10
tion was 0.35% of E*, as shown by Table 2 (ford'
=
0.675). It
is tempting to speculate that the ganglion-cellsampling provides the limiting factor to tangential absolute position acuity in the periphery. The similarity between position acuity
and ganglion-cellspacing at 10 deg is striking. The ganglion
spacing is about 2.4 min, and the vernier thresholds range
from 1.6 to 4.3 min. However, for bright, long-duration
stimuli, two-dot vernier acuity at 10deg can have thresholds
LU
CD
;
l
CD
U)
as low as 1 min.24 This threshold is about half of the gangli-
0.
0.1
on cell spacing. Multiple samples may account for the factor-of-2 improvement over ganglion sampling.'2 In the radial direction for all three observers and in the tangential
direction of observers DL and WS, the Weber fractions for
absolute position judgments are poorer than the ganglion
cell sampling, suggesting that further limitations must be
imposed by cortical processing. Some of these factors are
considered next.
1
ECCENTRICITY (DEG)
10
5
A---
*-
RRDIAL
TRNGENTIRL
DL
Zo
LU
I
-J
C)
TO'
LU
0.1 I0l
0.1
1
ECCENTRICITY (DEG)
.10
10
Z0
LU
LI
CD
'M
ed out, size judgments can be difficult because of size con-
I
Cif
n 1
V- *0.01
IntersubjectVariability and Cognitive Factors
A surprising feature of our data was the occurrence of systematic differences among observers. At large eccentricities
on the tangential task the orientation thresholds were 0.005
and 0.01 rad for observers SK and DL, respectively. At
small eccentricities observer DL was better than SK on the
same task. At large separations observer DL (very well
practiced) claimed that his difficulty was connected to his
perception that the entire three-dot pattern often appeared
tilted.
On the bisection task the situation was reversed. Observer DL was better than SK for large separations, and SK and
DL had similar threshold for small separations. Observer
SK complained that the size task was difficult for large
separations and required great concentration, whereas the
vernier task required minimal effort. As Burbeck has point-
0.1
1
ECCENTRICITY (DEG)
10
Fig. 5. The data in Fig. 4 are replotted in order to represent the
thresholds in cortical units. The vertical axis is 100 times the
threshold in degrees divided by the effective eccentricity. The
effective eccentricity equals the eccentricity plus 0.6 deg. A new
double-line nonlinear regression was made for the bisection data,
with the value of E2for the large-eccentricity branch constrained to
be E2 = 0.6 deg. The right-hand axis has the threshold expressed in
millimeters of cortex. We used the relationship that 1 mm of cortex
corresponds to approximately 10% of the effective eccentricity. A,
B, and C correspond to Figs. 4A, 4B, and 4C, respectively.
fractions of about 0.004 (about 2.4 min at 10 deg in the
periphery). The uniformly dashed curve in Fig. 5 for observer SK is the line given by 0.0033 (E + 2.5), taken from
Table 2 as the function expressing ganglion cell spacing.
This function decreases from 1.4% of E* in the fovea to 0.33%
stancy.2 6 She pointed out that the perceived angular size of
a target was extremely difficult to judge if the perceived
distance was not fixed. Accurate size judgments require
high stability of the perceived distance to the target. It
would not be surprising to find strong individual differences
in the ability to make use of perceived distance. These
effects would be expected to be strongest for large separa-
tions.
SUMMARY
We measured position thresholds along the horizontal meridian over a 200-fold range of eccentricities, from 3 min to
10deg, with only a fovealreference point. Thresholds in the
tangential direction were approximately
a factor of 2 lower
than those in the radial direction. Our nonlinear-regression
analysis, as reported in Table 1, showed that the tangential
data were well fitted by a straight line of the form W(E +
E2 ). This implies a constant orientation
cue from about
0.3 to at least 10 deg. The radial data required a doubleline fit. The large eccentricity portion of the radial data was
fitted by a line segment with E2
:
0.6 deg, a value close to
of E* in the periphery. It is interesting that this is the same
range of values as the hyperacuity thresholds, although, as
estimates of the inverse cortical magnification factor.6 If
hyperacuity thresholds are plotted in units of cortical milli-
Fig. 5 shows, the ganglion-cell curve changes in the opposite
meters, then, as shown in Fig. 5, the thresholds are constant
direction as a function of eccentricity. The single-dot abso-
(0.1 mm) for the radial direction at eccentricities greater
S. A. Klein and D. M. Levi
Vol. 4, No. 8/August 1987/J. Opt. Soc. Am. A
than about 0.5 deg. In cortical units this threshold is independent of the cortical separation, similar to our ability to
use a meterstick to measure a separation accurate to 0.1 mm
for a wide range of separations. We plan to incorporate the
absolute position uncertainty measured by these experiments into our future spatial-vision models. The present
model, 2 whose predictions
were shown in Fig. 1, requires
modification so that it may avoid the constant position
thresholds that are found for eccentricities above 15 min.
Inclusion of position uncertainty representing a fixed cortical interval of uncertainty should improve the model predictions. This modification to the model is quite different
from scaling the peripheral filters according to the reduced
peripheral contrast-sensitivity function. This latter change
would seem quite natural, since the filter sensitivity is set by
the contrast-sensitivity function. However, the implications of our measured values of E 2 (E 2
-
0.6 deg) for the
bisection task imply that cortical blurring must be present in
addition to the retinal factors that control the peripheral
scaling of the contrast-sensitivity function.
ACKNOWLEDGMENT
This study was supported by research grants R01 EY04776
and R01 EY01728 from the National Eye Institute.
REFERENCES AND NOTES
1. G. Westheimer, "Visual acuity and hyperacuity," Invest.
Ophthalmol. 14, 570-572 (1975).
2. S. A. Klein and D. M. Levi, "Hyperacuity
thresholds
1553
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