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Spatial information and uncertainty in anisometropic amblyopia
Vi.hn Res. Vol. 21, No. 4, pp. 661-4574, 1987 Printed in Great Britain. All ri@s reserved
SPATIAL
Copyright
0
0042.6989/87 S3.00 + 0.00 1987 Pergamon Journals Ltd
INFORMATION AND UNCERTAINTY ANISOMETROPIC AMBLYOPIA
IN
R. J. W~rr’ and R. F. HESS* ‘MRC Applied Psychology Unit, 15 Chaucer Road, Cambridge CB2 2EF and The Physiological Laboratory, University of Cambridge, Downing Street, Cambridge CB2 3EG, England (Received 24 July 1985; in revised form 15 May 1986)
Al&met-Anisometropic amblyopes were found to have a reduced sensitivity for shape discrimination. The introduction of positional jitter in the elements of the display had a profound effect on the performance of the normal eye, but not on that of the amblyopic eye. On the other hand the introduction of gaussian blur atkcts the performance of both eyes to the same degree. We conclude that rasied spatial uncertainty due to metrical scrambling is a suitable model for anisometropic amblyopia. Amblyopia
Vernier
Blur
Scrambling
INTRODUCTION
Anisometropic amblyopia is a relatively common condition in which one eye fails to develop because of unequal refractive error in early life. It is thought that this initial optical imbalance initiates abnormal cortical interactions that result in permanent disruption to the connections from the deprived eye. Psychophysical measures of this dysfunction included reduced contrast sensitivity especially at high spatial frequencies (see for e.g. Levi et al., 1981) and reduced positional sensitivity (Bedell and Flom, 1981, 1983; Flom and Bedell, 1985; Levi and Klein, 1982, 1983, 1985; Bradley and Freeman, 1985). It is of interest to discover how this condition can be modelled, because it has implications for understanding the development of normal vision and perhaps also for the treatment of anisometropic amblyopia when detected at an early age. In this paper we will examine three different models of the deficit in amblyopia. In all cases we are concerned with physical models, that is transformations of the input stimulus which mimic the degradation suffered in anisometropic amblyopia. The physiological realization of these models is a separate issue.
When Vernier acuity is expressed as a fraction of the period of the highest spatial frequency grating that can be resolved, this is the same in each eye. They suggest that orientation and position information are not markedly disturbed in anisometropic amblyopia, the implication being that space is simply scaled in size, and all stimuli with it. Thus their Vernier data are accounted for by scaling the spatial displacement thresholds by grating resolution. A second line of evidence was reported by Flom and Bedell (1985) who suggested that anisometropic vision can be mimicked by blurring normal vision. This is also sugested by Levi and Klein (1985). M2: weak signal The second model regards the condition as arising from a weak signal, particularly for high spatial frequencies. Blurring is, of course, formally equivalent to selective contrast reduction at high spatial frequencies, but undersampling is not. We can consider three different forms of this hypothesis, where internal signal C’ is related to physical signal strength C by a constant k,, k,, k,: Wl: C’=C-k,
(k, > 0)
w2: c+
(kz > 1)
W3: C’ = &
(k,’
M 1: undersampling The first model regards the condition as arising from undersampling or internal blurring of the stimulus. For example Levi and Klein (1983) found that although anisometropic amblyopes have reduced vernier acuity, the reduction is in the same proportion as their grating acuity.
1).
Of these W2 is normally used, and we shall refer to it as the (unqualified) weak signal. This weak signal model accounts for the 661
662
R J
WATT and R
finding of Levi and Klein (1983) by pointing out that grating acuity is determined by the contrast sensitivity to high spatial frequencies, and that Vernier acuity, at least for gratings, is contrast dependent. Bradley and Freeman (1985) also suggest that the Vernier acuity defect can be accounted for by this contrast scaling (i.e. setting the Vernier cosine targets to the same factor above contrast threshold in the two eyes). Some of the evidence for this claim is found in Fig. 1I of Bradley and Freeman (1985). which plots displacement thresholds versus contrast. The two eyes provided two parallel, straight, diagonal lines on the logarithmic axes, and so can be superimposed by contrast scaling, i.e. a horizontal shift of the amblyopic data with respect to the normal data. This finding rules out WI and W3 forms of the weak signal model, but supports W2. Of course, a vertical shift which corresponds to spatial displacement scaling, the proposal of Levi and Klein (1983), will also superimpose the two sets of data. Inspecting the data for subject R.S. in the Bradley and Freeman paper, their only anisometrope. in Figs 3, 4, 6 and 8 shows that the amblyopic eye always peaks at spatial frequencies a factor of between 4 and 8 lower than the normal eye. This also indicates spatial scaling as a viable alternative to contrast scaling. In summary, on the one hand we have the finding that Vernier acuity is the same fraction of grating resolution acuity in normal and anisometropic eyes, and on the other we have the finding that Vernier acuity is the same in the two eyes when the targets are set to the same contrast factor above ~sibii~ty threshold. The fundamental difficulty is that Vernier acuity is a similar function of increasing contrast and increasing spatial frequency (above about 4 c/deg). and these two variables are generally confounded.
F. HESS
signs, IV, with some error. e N + I -4-L’ where e is a random variable. The mean value of e corresponds to the constant error or ,fi.yed spatial disrorrion and the standard deviation of the distribution of values of e corresponds to the variable error or spatial uncertainty of the system. If the sampling of I on S is infinitesimally tine, then the spatial uncertainty is identical to blurring the image. We are interested. however. in the case where the sampling is of the order of about 0.5 arc min, in which case the spatial uncertainty is equivalent to a random jitter or scrambling of the target. Let us suppose that the error had a standard deviation of a few arc minutes. The effect of this on a cosine wave grating would depend upon the spatial frequency: high frequency gratings would be scrambled out so that they appeared like noise; low frequency gratings on the other hand would be hardly changed in appearance at all. Therefore a reduced grating acuity and reduced contrast sensitivity specifically for high frequencies would be expected. An error in spatial mapping would, of course, lead to a rise in Vernier acuity for a11 targets, but especially those whose contrast was low, or whose spatial frequency was high. In the present study we have set out to investigate these three hypotheses, specifically starting from the often repeated observation that Vernier acuity is much poorer in amblyopes. We begin with one of the simplest possible models for precision in Vernier acuity namely that for such stimuli the visual system has a internal error. a:, and to this is added the stimulus error or perturbation, ui, and that the overall threshold error, bT is related by a multiplicative constant, k, such that
M3: spar&l scr~~l~g The third model to be considered is that the metric of the visual space in anisometropic amblyopia is scrambled (Hess et al., 1978; Hess, 1982). The retinal image is a continuous function, S, of two spatial dimensions. In the expriments to be described, the stimuli were made of bright lines or dots whose size was conskkrably less than the point-spnad function of the human eye. The retinal image may thus be characterized formally as a set, 1, of stimulated loci, that are samples of S. This set is transfcrrcd into the system to produce a neural set of local
Let us consider what will happen for a normal visual system whose internal error is rr: when confronted with stimuli of increasing 0;. Threshold error will be initially determined largely by the internal error until the external error equals it in magnitude, there afk threshold should rise in proportion to &. In this argument so far, the error could be due to either blur, or scrambling. If the defect in anisometropic amblyopia is a raised internal blur, then as stimulus blur is increased in the
Anisometropic
normal eye beyond its own internal blur, threshold should rise and meet that for the amblyopic twin, at which point both should rise further together. Of course, this behaviour would not be expected, if the defect is scrambling, but increasing stimulus scrambling in each eye should behave in that fashion instead. The difficulty with positional experiments involving continuous and smoothly changing luminance waveforms, such as cosine gratings, is that the performance measures are affected in similar fashions by either blurring or contrast reduction (Watt and Morgan, 1983, 1984). This is principally because Vernier thresholds are limited by noise for these stimuli, and are generally rather higher than the finest thresholds of a few arc sec. obtained with bright line stimuli. In the experiments below we have employed thin bright lines in shape discrimination tasks, where the finest spatial performance is expected (Andrews ef al., 1973; Watt, 1984). The experiments demonstrate a spatial deficit that is characterized mainly by a local jumbling distortion of the spatial metric, and that cannot be accounted for by neural blurring or contrast reduction.
663
amblyopia MJZTHbDS
Subjects Seven anisometropic amblyopes were used in the experiments to be described. Their vision through their normal eye was corrected to 6/5 acuity. The clinical details of each subject is given in Table 1. Stimuli The stimuli were all drawn on the screen of a HP1336S display oscilloscope under the control of a Motorola VME/lO microcomputer: four DACS were used to drive the differential X & Y amplifiers of the oscilloscope at a rate of 60 Hz. The stimuli were presented at a luminance comfortable to the subject and at I .5 log units above threshold for the particular eye. Normal room lighting provided a background illuminance equivalent to lOOcd/m*. Subjects viewed the display from a distance of 2.3 m with natural pupils. The stimuli were always presented at a random orientation within the interval vertical + 4 deg. Exposure duration was unlimited (i.e. lasted until the subject had made a response).
Table 1 Subject
Age
N.N.
30
P.S.
50
A.C.
40
P.W.
46
J.S.
42
L.B.
62
D.C.
32
G.F.
34
Present clinical data Central fixation Acuity R 615 L 6120 R, R Plano L -0.75/-2.50 x 10 Central fixation Acuity R 616 L 6136 R, R -4.50/- 1.25 x 90 L -2.25 DS Central fixation Acuity R 616 L 6160 R, R -4.50/- 1.25 x 90” L -3.00 DS Central fixation Acuity R 6118 L 615 R, R +O.SO/-3.25 x 20 L -0.251-0.25 x 70” Central fixation Acuity R 2160 L 615 R,R -5.OODS L + 1.251-0.25 x 25” I’ Eccentric fixation Central scotoma Acuity R 2160 L 615 R, R +8.00 DS L +0.50 DS Central fixation Acuity R 6118 L 615 R, R +3.50 DS L plan0 Central fixation Acuity R 615 L 6/9+ R, R +0.75 DS L +4.75/- 1.25 x 50”
Clinical history First R, age 25 years
First R, age 6 years
First R, age 11 years
First R, age 6 years
First R, age 36 years
First R, age 21 years
First R, age 7 years Intermittent occlusion therapy First R, age 3 years Occlusion therapy Present R, worn constantly
R. J. WAIT and R. F. HESS
664
Procedure APE (Watt and Andrews, 1981), an adaptive method of constant stimuli was employed to measure thresholds. On each trial the subject was presented with two targets, one on each side, left and right, of the screen. One of these was the standard target without cue, and the other had the cue. The subject was requested to indicate which of the targets he judged to have the cue, and to guess if he could not decide. The amplitude of the cue was varied from trial to trial, as was the side (left or right of the screen) on which it was located, to collect a full psychometric function. Probit analysis (Finney, 1971) was used to assess the standard deviation of the normal cumulative psychometric function thus obtained, and is the measure of threshold reported. Experiment 1: Integration Region for Shape It is known that shape information is integrated over around 30 arc min along the length of a line (Andrews et al., 1973; Watt, 1984). It is possible that the raised thresholds obtained in anisometropic amblyopes are due to a reduced integration region. In the first experiment, this possibility was tested by measuring the effect of the space constant, S, on the threshold for detecting the presence of a curvature cue C(y) = m(1 - y2/s2) exp( -y2/2s2) -22.5’ < y < 22.5’ where m is the cue amplitude and was varied from trial to trial. The subject was presented with two lines, one was straight, the other curved, and required to identify the curved line. The side that the curved line was displayed at was varied randomly from trial to trial so that a full psychometric function could be collected. The lines were both 45 arcmin in length and comprised 90 dots spaced at intervals of 0.5 arc min.
amblyopia is normal, the rise in threshold must be due to an increase in the uncertainty or error of the data that is being integrated. In the second experiment, the influence of spatial uncertainty on Vernier acuity and the curvature task was assessed. In performing this task. the visual system presumably uses some derivative or integrative process on the raw contour locus set N, and errors in the mapping onto N are going to lead to poorer performance. If we suppose that the internal spatial uncertainty and stimulus positional jitter are additive independent sources of error, then one should expect that shape discrimination, ds. will be related to stimulus positional jitter, J,, by
where J, is a fixed factor for each eye (normal and amblyopic). It can be seen that: a3 =k,J,
for
J,>>J,
(2) and so the performance for the two eyes at high levels of stimulus positional jitter should converge. The stimuli for the experiment were once again made of a vertical row of dots, spaced at 0.5 arcmin intervals. To one of the two was added a shape cue which was: (a) a Vernier step offset 0’ < y < 22.5’ C(Y) = m C(Y) =
0
-22.5’
(b) a curvature cue C(y) = m(1 - y2/s2) exp(-y2/2s2) - 22.5’ < y < 22.5’ s=5arcmin. 200 r
Results The results for each eye of observer N.N. are shown in Fig. 1. It can be seen that the only difference between the eyes is a factor of 8 difference in al1 conditions. Integration occurs over the same distance in the two eyes. These results show the difference in sensitivity to shape in anisometropic amblyopia. Experiment 2: A Measure of Spatial Uncertainty Since the integration region in anisometropic
2L
1
I
I
1.25
2.5
5.0
I Ia0
Curvature space canslont (on mm ) Fig. 1. The effect of cwvatwe we apace conataat (hear extent) on detection thresholds. lntqmtion regions in the two eyes are essentially the same.
Anisometropic
Threshold amplitude m of the cue was measured as a function of the standard deviation of the imposed one dimensional (horizontal) pasitional jitter.
two eyes for the Vernier offset and the curvature cue respectively, for the both observers. A continuous line is drawn through each set of data. This line represents a least squares fit of equation (1) to the experimental results, and the figures aiso tabulate for each observer and each eye the two parameters: internal uncertainty or
Results Figures 2 and 3 show the thresholds for the Subject 0 .
Normal
P ,L,
,111
1
g
Sublect
N-N
Addlflve MultIpI Addltlve MultIpI.
Amblyoplc
665
amblyopia
error * 13.899” error 2 1.205 error = 45.558” error = 0.977
l,,ll
III1
IO
,111,
0
Normal
.Amblyoprc
I
u
l,l,
100
I 1
ICOO
Positional
jitter
I
: AC.
Addilwe Multi@ Addttwa MultIpI
l11lllll
error error error error
I
I lllllll
10
(arc
= 6.603” s 0.637 5 Vl6449” * 1 269
I 100
I l!J_Ll.u loo0
set)
Fig. 2. The effect of positional jitter on Vernier acuity. The amblyopic eye shows an elevated positional uncertainty, as indicated by the raised additive error. The multiplicative error for both eyes is similar, and so the two data sets converge at high positional jitter. Subject: NN 0 .
Normal Amblyoplc
Addlttve MultIpI. Addltivo Multlpl.
Subject:
error * 7.145 erlor ’ 0.794 ermr * 52.895” erm * 0. 824
0 l
Norma I Amblyoplc
GF
Addttwe Multtpl. Acklttrve Multipl.
errcf error error error
’ 15.979” = 0.662 m 41.405” = 0.550
3 $ cr
1-
, 1
I , ~~~~~~~ I 1~~11111I I IO
IllJJJJ
100
Positional
I
1000
jitter
I 1 lllllll
I 10
1
(arc
I1111111 100
I I
Illllu
loo0
secl
Fig. 3. The e%ct of positional jitter on curvature detection. The amblyopic eye shows an elevated positional uncertainty, as in Fig. 2.
R. J. WATTand R. F. HESS
666 Subwit Norma I 0 .
Amblyopx
dent, internal source of positional jitter such that a truly collinear row of dots has an effective base jitter of Ji, then the addition of stimulus jitter, J,, produces a base jitter of
NN
Addltwe
error
:
9258”
Mull~pl
emx
*
0865
Addttwt
error
= 47 106 ”
Mi.lltlol
srrw
=
0792
J = (Jf + JZ)‘f’. Substituting obtain
equation
(4) in equation
(4) (3), we
CU= k,(Jf + J;)‘,‘.
(9
We have J + dJ = [Jf -I- (.I, c dJ,)‘]‘,‘.
tf9
Substituting aY from equation (5) into quation (6) and solving for dJ,, we have dJe = [Ji + (ki + 2kj) (51+ J:)]“’ - Je. Positioml
jitter
(arc
set
I
Fig. 4. The effect of positional jitter on curvature detection for horizontal targets.
additive error, Ji, and the multiplicative error k,. The principle finding is that the amblyopic eye has higher internal uncertainty than the normal, by a factor of 3 for N.N. (Vernier); a factor of 24 for AC (Vernier); a factor of 7.5 for N.N. (curvature); and a factor of 2.5 for G.F. (curvature). Figure 4 shows a second set of curvature data for subject N.N., this time with horizontal targets. The result is very similar to the case with vertical targets. The results are consistent with a mode1 in which performance is related to the total positional error as given by equation (I). The largest difference between the system parameters of the two eyes is for Ji. This result is further evidence for a positional uncertainty deficit in anisometropic amblyopia.
(7)
In the third experiment, the discrimination of positional jitter was measured in the normal and amblyopic eyes. The stimulus was made of 90 dots spaced vertically at intervals of 0.5 arc min in a straight row. Each dot was independently displayed with an error left or rightwards with a Gaussian distribution of uncertainty. Two such stimuli were presented, and the subject was required to report which had the more jitter (the jitter of largest standard deviation). A jitter difference threshold was obtained as a function of base jitter. The threshold is the spatial standard deviation of the subject’s error response distribution. Results The results for the two observers are shown in Fig. 5. The continuous lines show the func~on expected by substitu~ng the known values of J, (from Experiment 1) into equation and finding a fit for the one parameter kj* The fits obtained in all cases are reasonable. It is further suggested that the ambiyopic eye has a raised spatial uncertainty.
Experiment 3: An Eflect of Raised Spatial Uncertainty
~~~~~s~on: spatial ~certainty
The judgement of positional jitter in a row of dots is determined by the precision with which the standard deviation of the jitter can be measured. which is proportional to that standard deviation, so that when an increment in positional jitter is added to a pedestal base of jitter. f. then the threshold level of i+ will be given by
The first three experiments have demonstrated that the visual system of the anisometropic amblyope is subject to a raised spatial uncertainty: the mapping from retinal image to some neural image is subject to random errors. There are several different transformations of an image that would lead to an increase in the uncertainty wirh which a point in that image could be localized:
dj=k,J
(3)
Suppose that there is an additive, indepen-
(1) Any operation which reduces the sharpness of the stimulus peak, such as blurring or
667
Anisometropic amblyopia Subjec?
l
SublocI
: NN
o Normal Amblyopu
Eye
AC
o Normal Eye
Eye
.
Pedestal
positionol
jitter
Amblyoplc
(arc set
Eye
1
Fig. 5. The results of Experiment 3 on positional jitter discrimination. It can be seen that there is a large difference between the normal and amblyopic eye for low degrees of positional jitter, but that the functions convergefor largedegrees. The amblyopiceye showsan elevatedpositionaluncertainty.The continuous lines are the predicted threshold functions according to equation (7).
contrast reduction will increase spatial uncertainty. (2) Any operation which weakens the power of the representation substrate, such as undersampling, will also tend to increase the spatial uncertainty of the system. (3) A random distortion of the representation substrate, by scrambling for example, will lead to an increased spatial uncertainty. (4) A raised level of intensive noise (equivalent to a higher level of luminance noise in the stimulus) will lead to larger positional errors. The first two are, of course closely related, since undersampling is defined with respect to a given spatial scale or degree of blurring. By analogy with the effects of positional jitter, one can measure the internal blurring by progressively blurring the stimulus, until one reaches a point at which the blur in the stimulus begins to affect thresholds, which is the point when it is commensurate with the internal blur. This should be markedly different in the two eyes, if the raised uncertainty in anisometropic amblyopia is due to an internal blur. The results of this are described in Experiment 4. In a similar fashion, the possibility that raised intensive noise is the cause of raised positional uncertainty can be tested by measuring Vernier acuity as a function of luminance level. The results of this are described in Experiment 5.
Experiment 4: The Efects of Gaussian Blur on Vernier acuity The possibility that the raised Vernier acuity found in the previous studies coud be due to an increased level of internal blur was tested in the fourth experiment by measuring the effect of one-dimensional (horizontal) gaussian blur in the stimulus on Vernier thresholds. Once again, we assume the standard deviation of the stimulus blur and internal blur to be independent and additive, according to B:=B;+B:
(8)
where, B, is the effective blur; Bi, the internal blur; and B, the stimulus blur. Equation (8) follows from convolution. As in Experiment 2 we suppose that Vernier acuity is proportional to the standard deviation of effective blur, so that ds = /C,(Bf + B;)“‘.
(9)
In fact, according to the MIRAGE model of Watt and Morgan (1985), the exponent in this expression should be 0.75, not 0.5. However for blur less than 10 arc min the difference is negligible, since a% = kbBf
for
B,sB,
(10)
and whatever value of a this is not a function of &
668
R. J. WAIT and R. F. HES
As in Experiment 2, we expect no rise in threshold with increasing blur until B, is near to Bi, when thresholds should rise. If internal blur in the two eyes is different, this point will be different. The stimulus was made up of two gaussian luminance profiles, one above the other, and the subject was required to judge the direction of their offset. The luminance profiles were created by varying the horizontal spacing between a set of parallel vertical lines in a gaussian density distribution. None of the individual lines were resolvable, and by this method non-linea~ties in the Z-axis amplifier circuit were bypassed. Results The effects of gaussian blur on vernier acuity for the two eyes of both subjects are shown in Fig. 6. The effects are similar in the two eyes, with blur having little effect if less than about 3 arc min in standard deviation. There was no level of blur in the experiment at which the two eyes performed with the same accuracy. The additive error for the two eyes is the same: the only difference is in the multiplicative error, k,,. A similar pattern of results were obtained for subject N.N. with optical defocus blur. The possibifity that some form of internal blurring or undersampling could be responsible for the spatial defect in anisometropic amblyopia is n&d out by the present results.
Ezcperiment 5: The Ejgecr of Luminance In the final experiment, the effect of luminance on the curvature detection task was measured. Bradley and Freeman (1985) in describing their version of the weak signal model, point out that normal eyes should deteriorate to the level of performance of anisometropic eyes if contrast is reduced sufficiently, and that. conversely, amblyopes should improve to the performance level of normal eyes if contrast is increased by the same degree. Results Figure 7 shows the effect of luminance on the curvature detection task in each subject for both eyes. For the normal eye, it seems that performance is essentially independent of luminance, provided that the target is actually visible. This is in confirmation of casual observations made by many workers and detailed measurements on normal eyes made by us. For the amblyopic eye, on the other hand, there is a relatively marked effect of luminance level on performance, with threshold not reaching an asymptotic level until luminance is at least one log unit above detection threshold. There is no luminance level at which the amblyopic eye equals the normal eye. In terms of the three weak signal models we introduced above, it is clear that contrast scaling (W2) is
.%&XINN
Subtect: NWttVJl e AmblyoDlc l
E
1
1
IO
100
moo Gaussian
blur
I
Additwe
ermt
= 335.167”
MulctpI
error
:
Addrtwe
error
= 327.089”
Muit~pl
error
=
I I I lilll
I 10
I fore
set
AC
QOZ? 0167
I I I lilll 100
I
I t
rru
-I
1ooo
J
Fi& 6. The effect of gati blur on Vernier acuity. Tbc c&t in tk two cyu is ca~~ially tk same, and thc~ is no dqrce of blur at which the two cycs have tk same thrcsboid. This is indiated by a rise in the multiplicative error but not in the additive error.
Anisometropic amblyopia
zooor
.
Subject : AC t 0
ambly0W normal
t
669
blyopic eye’s performance matched that for the normal eye at the lowest luminance tested: over this range there is no overlap of performance at all, and the normal eye at any luminance is better than the best obtained precision of the amblyopic eye. The two predictions of the Bradley and Freeman (1985) weak signal model fail in this set of data. The anisometropic eye is clearly sensitive to the stimulus contrast, but this sensitvity must arise from some other defect in anisometropic amblyopia. Figure 9 shows that spatial scrambling is a good model for the effects of luminance in an amblyopic eye. The figure plots, for two normal observers, both naive, the effect of stimulus luminance on curvature detection, just as in Fig. 7, but for a stimulus which had 48 arc set spatial jitter. The data are strikingly similar to the amblyopic eyes of N.N. and A.C. Our conclusions from this experiment are:
(i) Anisometropic amblyopes are sensitive to stimulus contrast. J 3 1 2 0 (ii) This sensitivity cannot, however, provide Log. luminance (above threshold) a simple explanation for the condition. In particular, contrast scaling does not predict the Fig. 7. The effect of target luminance level on curvature detection. The effect in the two eyes is rather different, but performance of the amblyopic eye. the functions do not converge. (iii) The sensitivity is itself explained by the spatial scrambling hypothesis. 5 t,=-y--,.
inapplicable. A form of model W3, with k, set to above 10 would possibly suffice, but a model C’ - CO.1 is contradicted by the data of Bradley and Freeman (1985). There is no weak signal model which can account for both the present data.and their results. Figure 8 shows a repeat of Experiment 2 at higher a luminance level for subject N.N. The performance of the amblyopic eye is improved, and its positional uncertainty is also apparently improved, although this was also true of the good eye and a substantial difference still exists between the two eyes. The results of this experiment suggest that some form of contrast or contrast sensitivity defect is not responsible for the positional uncertainty elevation in anisometropic amblyopia. There is no simple linear or power law scaling of contrast that will result in the amblyopic eye behaving like the normal eye. There is furthermore no stimulus luminance at which the amblyopic eye converges in performance on the normal eye. Within the 2 log units tested there was no stimulus luminance at which the am-
GENERAL DISCUSSION
In the experiments above a characteristic spatial deficit in anisometropic amblyopia has Subjal: NN
.
s f g
1
erw . 5.583” Multlpt. error = 0.795
Numal
Addltw.
Amblyopic
AddiIive
WIW
Mullipl.
error 1
0
* 29.636” 0.662
E, ,,,,,,,,, ,,,,,,,,, ,,,
,r,u
1
10
Position01
100
jitter
(arc
1000
set
1
Fig. 8. The effect of positional jitter on curvature detection at a high luminance level. This result is basically the same as that at lower luminance (see Fig. 3).
R. J. WATTand R. F. HFSS
670
2
L
0
3
I
I
I
1
2
3
Log lmunoncc (above ttmst&Y) Fig. 9. Tltc effect of target luminamx kwl on curvature detection for two naive nomtal subjects. The targets had 48
arc set positional jitter. The simikrity between these data and those of Fig. 7 arc striking.
been identified. How should it best be described? It is clearly due to spatial uncertainty, rather than integration failure, as the results of Experiment 1 show. Many factors could increase the spatial uncertainty of the visual system, and in this paper we have considered three. The first is a raised level of internal blur, but the results of Experiment 4 show that internal blur in anisometropic amblyopia is normal. A related possibility is the loss or malfunction of high frqucncy filters, a suggestion made by Levi and Klein (1982). For a simple, isolated Vernier target such as those we have employed, we can rule this out as a cause for low precision.
Watt and Morgan (1984,19%5)have shown that the low frequency filters (at 4c/deg) provide most of the high precision in Vernier acuity, although high frequency filters do contribute. The results of Experiment 4 afso indicate that anisomctropic amblyopes are worse by the same multiplicative factor even for highly blurred stimuli, where the high fraquency filters are not relevant. A related possibility is the hypothesis of undersampling. The spatial uncertainty of a sampling grid is proportional to the intersample
distance, and it is possible that this is the cause of the deficit in anisometropic amblyopia. Once again the results at high degrees of blur rule this out: a low frequency target, equated for visibility in the two eyes, will not be affected b> undersamplin~. The second possibility we have explored is the idea that a reduced contrast or weak signal can account for anisometropic amblyopia. This is unlikely, given the results of Experiment 5. where we showed that, although increasing the stimulus energy yielded lower thresholds for the amblyopic eye, and led to a small reduction in the spatial uncertainty of the eye, it could not approach the performance of the good eye. Our results are incompatible with weak signal model W2; Bradley and Freeman’s results are incompatible with weak signal models Wl and W3. We conclude that a weak signal is not the problem in anisometropic amblyopia. The sensitivity of the amblyopic eye to stimulus strength was accounted for by the third possibility, spatial scrambiing (Hess et al., 1978; Hess, 1982). The third possibility we have considered is raised spatial jitter or scrambling. The results of Experiments 2 and 3 are evidence in support of this hypothesis. In fact, the results of Experiment 2 are themselves sufficient to rule out both the internal blur and reduced contrast or weak signal hypotheses, neither of which predict the exact form of results obtained. The clearest demonstration of this is obtained by taking normal observers and imposing either + 3 D optical defocus or using a very low contrast dispiay to repeat Experiment 2. The results of these two manipulations are shown in Figs IO and I1 respectively. In each case the manipulation raises both the additive error and the multiplicative error, whereas we found in Experiment 2 that anisometropic amblyopia only raises the additive error. Of course it follows from simple algebra that a raised internal spatial scrambling would affect only the additive error. We now ask whether the spatial scrambling hypothesis can account for the other data concerning anisometropic amblyopia. A random spatial scrambling would obviously reduce Snellen acuity by degrading the letters; it would also reduce grating acuity for the same reason. The reduction in acuity in each case will be related to the degree of scrambling, and it is plausible that acuity would be proportional to scrambling. This would account for the report by Levi and Klein (1983) that Vernier
Anisometropic amblyopia Subject: NOrlTlOl 0 .
r3D0ptres
671 Subtect : CG
RJW
Addltwe
error
* 4. 702”
tilti@.
error
* 0.386
Addllwe
error
= 18.639”
Multipi.
error
= 1.988
I
I I Il~~~~l
I I 1 l~~~~l
1
10
100
Normal 0 .
Adduwe
error
MultIpI.
error .
Additwe
error
= 27.541
Multipl.
errcf
=
+2,5Dloptres
.
6.421” 0.333 ”
0727
I I I LLUJ
IODO
Position01
jitter
(arc set
1
Fig. 10. This shows the effect of stimulus blurring (by optical defocus) on the pattern of results obtained in Experiment 2, for two normal observers. Notia that the additive error and multiplicative error are
increased in proportion.
acuity in anisometropic amblyopes scales with grating acuity, but inverts the argument of
casuality. We argue that spatial scrambling causes low acuity and low Vernier acuity; they argued that the former was the cause of the latter; an argument we have disproved. Subject Normal 0
.
V&k
tavt
I 7
I
Bradley and Freeman (1985) showed that the amblyopic eye has a Vernier acuity similar to a good eye at lower contrast. Watt and Morgan (1983) have shown that contrast and blur behave in the same manner, and so the argument of Bradley and Freeman is inconclusive. How-
MC
Subject:
Additive
error
=
5 026”
Mult~pl
error
*
0.725
Add~llve
error
’
15.697”
11 111111 10
I
I 1111111
Norm
I
0 *Weak target
I
L
I lll~
100 Poeltconol
1ODO jitter
I
error
hlulrlpl
error * 0.664
AWllive
error
* 7643”
Munipl.
error
* 1792
I1111111
1
10 (arc
KA
Addltwe
I
= 4.646”
11111111 100
I
1
I lll~
-I
loo0
set)
Fig. 1I. This shows the efTectof a weak stimulus (luminance set to 1.5 times the threshold luminance for visibility: background luminance was 100 cd/m*) on the pattern of results obtained in Experiment 2, for two normal observers. Notice that the additive error and multiplicative error are increased in proportion.
R. J. WATT and R. F. HESS
672
.amblyope
~76936419296462412
6
3
0
3
6
12
24
48
96
192
364
766
0
Sub]rcf:NN . omblyope 0
I
1
1
1
1
1
1
7663841%2%462412
Gaussian
blur
(arc
1
1
6
3
I 0
3
)
set
I
II
I
I
normal
I
I,
I
6122448%192%4766
Positional
jitter
(arc set
1
Fig. 12
.
l\.
-1 ._ “I
III
$
II
X6384,92%
11 46
24
II 12
6
~IIIIlllll 3
0
1000
f
3
6
12
24
46
36
192 364
7%
E
1
1
1
7'66394192%48
Gaussian
1
1
I
2412 blur
(arc
1 6
1
3
I
0
II 3
Illlrl~
6
t2
hsitional
MC)
Fig. 13
24
48
jitter
96192389766 (arc
s(K)
673
Anisometropic amblyopia Subject : DC . omblyO00 0
normal
Subject:
I-
766364S266462412
6
blur (arc set
Goussion
3
0
3
6
PW
l
ombtyooo
0
normal
12
24
46
96
192 3or) 7%
Positional jitter (arc set I
f Fig. 14
1000
Subioct : PS l ambiyopo 0
;
Jj____/
k
~
III~II’~~ ?66364?92%492412 Gaussion
normal
blur (arc set
6
3
1 0
1
if'filf@'J 3
6
92
hsitional
24
46
jitter
96192364766 (arc
sac1
Fig. IS Figs t2-15. Summary data for seven subjects. On the left is shown the effectof blur ORVernier acuity, and in each case the normal and amblyopic data are parallel. On the right is shown the ei%ct of positional jitter on Vernier acuity, and in each case the normal and amblyopic data converge at high values of jitter.
ever, the spatial scrambling of a target is equiv- converge and increased blur does not. Figures alent to dispersing that target in space, which 12-l 5 show this effect for a total of 7 subjects effectively reduces contrast. Once again we con- with anisometropic amblyopia. Figure 12 is the clude that the reduced effective contrast is a data of Figs 2 and 6 replotted. Zero blur and consequence of spatial scrambling, rather than jitter is in the centre of the graphs, with increasing blur leftwards from centre and ina primary cause of amblyopia. The two critical experimental results are those creasing jitter rightwards from centre. In every of Experiments 2 and 4, that increased jitter case the jitter data converge and the blur data causes the performance of the two eyes to remain parallel.
R. J. W~rr and R. F. Hiss
614
CONCLUSIONS
(1) Shape discrimination, a task that requires the finest local positional information. can be an order of magnitude poorer in amblyopia. (2) This is not due to an inability to collate information along the line of the target, because the integration region is normal. (3) It is, however, due to an elevated positional uncertainty. (4) Anisometropic amblyopia cannot be mimicked or modelled by raised internal blurring or contrast scaling by the contrast threshold deficit. (5) The raised positional uncertainty is adequately modelled by a higher level of scrambling in the fovea1 metric of the amblyopic eye-brain than is normal. Aehowle&awnu-Thir cd Raeucb
Cod
work was supported by the Mcdiof Gc Britain and the Welk.omc Trust.
R.F.H. k a Welkomc Senior Lecturer.
REFEBENCES
Fmnev D. J. (1971) Pro611 AnahJis. 3rd edn. Cambridge Univ. Press. Flom M. C. and Bcdell H. E. (1985). lndentifying amblyopia usmg associated conditions, acuity. and nonacmty features. Am. J. Oprom. Physiol. 0p1. 62, 153-160. Hess R. F. (1982) Development sensory impairment amblyopia or tarachopia? Human Ncurobio/. 1, 17-29. Hess R. F.. Campbell F. W. and Gmnhalgh (1978) On the nature of the neural abnormality in human amblyopia: neural aberrations and neural sensitivity loss. *tigers Arch. ges. Physiol. 377. 201-207.
Lawden M. C.. Hess R. F. and Campbell F. W. (1982) The discriminability of spatial phase relationships in amblyopia. Vision Res. 22, 100~1016. Levi D. M.. Harwcth R. S.. Pass A. F. and Vcnvcrtoh J. (1981) Edge sensitive mechanisms in humans with abnormal visual experience. E.rp/ Brain Res. 43, 27&280. Levi D. M. and Klein S. (1982) LIifferencea in vernier discrimination for gratings between strabiamic and aniaometropic amblyopes. Incesr. Ophrhai. oisual Sci. 23, 39847. Levi D. M. and Klein S. A. (1983) Spatial locahsation in normal and amblyopic vision. Vision Res. 23, 1005-1017. Levi D. M. and Klein S. A. (1985) Vernier acuity. crowding and amblyopia: Vision Res. 25, 979-991. Rentschlcr 1. and Hilz R. (1985) Amblyopic processing of positional information. Part I: Vernier acuity. Exp/ Brain Res. 60, 270-278. Watt R. J. (1984) Towards a general theory of the visual acuitics for shape and spatial arrangement. Vision Res. 24, I377- 1386.
Andrews D. P.. Butcher A. K. and Buckley B. R. (1973) Act&tea for rpatial arrattmt in line figures: human and ideal obrmen compared. Vision Rcs. 13, W-620. Bcdcll H. E. and Rom M. C. (1981) Monocular spatial distortion in atrabiamic amblyopia. Incest. Ophthal. kuul Sci. Zo, 263-268. &dell
spaa
H. E. and Flom M. C. (1983) Normal and abnormal perception. Am. J. Oprom. Physiol. Opt. 0,
426435.
Bradley A. and Framan R. D. (1985) Is rcduad vernier acuity in amblyopia due to position, contrast or fixation d&its? Virion Res. 25. 5566.
Watt R. J. and Andrews D. P. (1981) APE: Adaptive probit estimation of psychomctrioe functions. Curr. Psrchol. Rev. I. 205-214.
Watt R. J. and Morgan M. J. (1983) The recognition and reprcscntation of cdp blur: evidence for spatial primitives m human vision. Vision Res. 23, 1457-1477. Watt R. J. and Morgan M. J. (1984) Spatial filters and the localization of luminance changes in human vision. Vision Res. 24, 1387-1397.
Watt R. J. and Morgan M. J. (1985) A theory of the primitive spatial code in human vision. Vision Res. 25. 1661-1674.
SPATIAL
Copyright
0
0042.6989/87 S3.00 + 0.00 1987 Pergamon Journals Ltd
INFORMATION AND UNCERTAINTY ANISOMETROPIC AMBLYOPIA
IN
R. J. W~rr’ and R. F. HESS* ‘MRC Applied Psychology Unit, 15 Chaucer Road, Cambridge CB2 2EF and The Physiological Laboratory, University of Cambridge, Downing Street, Cambridge CB2 3EG, England (Received 24 July 1985; in revised form 15 May 1986)
Al&met-Anisometropic amblyopes were found to have a reduced sensitivity for shape discrimination. The introduction of positional jitter in the elements of the display had a profound effect on the performance of the normal eye, but not on that of the amblyopic eye. On the other hand the introduction of gaussian blur atkcts the performance of both eyes to the same degree. We conclude that rasied spatial uncertainty due to metrical scrambling is a suitable model for anisometropic amblyopia. Amblyopia
Vernier
Blur
Scrambling
INTRODUCTION
Anisometropic amblyopia is a relatively common condition in which one eye fails to develop because of unequal refractive error in early life. It is thought that this initial optical imbalance initiates abnormal cortical interactions that result in permanent disruption to the connections from the deprived eye. Psychophysical measures of this dysfunction included reduced contrast sensitivity especially at high spatial frequencies (see for e.g. Levi et al., 1981) and reduced positional sensitivity (Bedell and Flom, 1981, 1983; Flom and Bedell, 1985; Levi and Klein, 1982, 1983, 1985; Bradley and Freeman, 1985). It is of interest to discover how this condition can be modelled, because it has implications for understanding the development of normal vision and perhaps also for the treatment of anisometropic amblyopia when detected at an early age. In this paper we will examine three different models of the deficit in amblyopia. In all cases we are concerned with physical models, that is transformations of the input stimulus which mimic the degradation suffered in anisometropic amblyopia. The physiological realization of these models is a separate issue.
When Vernier acuity is expressed as a fraction of the period of the highest spatial frequency grating that can be resolved, this is the same in each eye. They suggest that orientation and position information are not markedly disturbed in anisometropic amblyopia, the implication being that space is simply scaled in size, and all stimuli with it. Thus their Vernier data are accounted for by scaling the spatial displacement thresholds by grating resolution. A second line of evidence was reported by Flom and Bedell (1985) who suggested that anisometropic vision can be mimicked by blurring normal vision. This is also sugested by Levi and Klein (1985). M2: weak signal The second model regards the condition as arising from a weak signal, particularly for high spatial frequencies. Blurring is, of course, formally equivalent to selective contrast reduction at high spatial frequencies, but undersampling is not. We can consider three different forms of this hypothesis, where internal signal C’ is related to physical signal strength C by a constant k,, k,, k,: Wl: C’=C-k,
(k, > 0)
w2: c+
(kz > 1)
W3: C’ = &
(k,’
M 1: undersampling The first model regards the condition as arising from undersampling or internal blurring of the stimulus. For example Levi and Klein (1983) found that although anisometropic amblyopes have reduced vernier acuity, the reduction is in the same proportion as their grating acuity.
1).
Of these W2 is normally used, and we shall refer to it as the (unqualified) weak signal. This weak signal model accounts for the 661
662
R J
WATT and R
finding of Levi and Klein (1983) by pointing out that grating acuity is determined by the contrast sensitivity to high spatial frequencies, and that Vernier acuity, at least for gratings, is contrast dependent. Bradley and Freeman (1985) also suggest that the Vernier acuity defect can be accounted for by this contrast scaling (i.e. setting the Vernier cosine targets to the same factor above contrast threshold in the two eyes). Some of the evidence for this claim is found in Fig. 1I of Bradley and Freeman (1985). which plots displacement thresholds versus contrast. The two eyes provided two parallel, straight, diagonal lines on the logarithmic axes, and so can be superimposed by contrast scaling, i.e. a horizontal shift of the amblyopic data with respect to the normal data. This finding rules out WI and W3 forms of the weak signal model, but supports W2. Of course, a vertical shift which corresponds to spatial displacement scaling, the proposal of Levi and Klein (1983), will also superimpose the two sets of data. Inspecting the data for subject R.S. in the Bradley and Freeman paper, their only anisometrope. in Figs 3, 4, 6 and 8 shows that the amblyopic eye always peaks at spatial frequencies a factor of between 4 and 8 lower than the normal eye. This also indicates spatial scaling as a viable alternative to contrast scaling. In summary, on the one hand we have the finding that Vernier acuity is the same fraction of grating resolution acuity in normal and anisometropic eyes, and on the other we have the finding that Vernier acuity is the same in the two eyes when the targets are set to the same contrast factor above ~sibii~ty threshold. The fundamental difficulty is that Vernier acuity is a similar function of increasing contrast and increasing spatial frequency (above about 4 c/deg). and these two variables are generally confounded.
F. HESS
signs, IV, with some error. e N + I -4-L’ where e is a random variable. The mean value of e corresponds to the constant error or ,fi.yed spatial disrorrion and the standard deviation of the distribution of values of e corresponds to the variable error or spatial uncertainty of the system. If the sampling of I on S is infinitesimally tine, then the spatial uncertainty is identical to blurring the image. We are interested. however. in the case where the sampling is of the order of about 0.5 arc min, in which case the spatial uncertainty is equivalent to a random jitter or scrambling of the target. Let us suppose that the error had a standard deviation of a few arc minutes. The effect of this on a cosine wave grating would depend upon the spatial frequency: high frequency gratings would be scrambled out so that they appeared like noise; low frequency gratings on the other hand would be hardly changed in appearance at all. Therefore a reduced grating acuity and reduced contrast sensitivity specifically for high frequencies would be expected. An error in spatial mapping would, of course, lead to a rise in Vernier acuity for a11 targets, but especially those whose contrast was low, or whose spatial frequency was high. In the present study we have set out to investigate these three hypotheses, specifically starting from the often repeated observation that Vernier acuity is much poorer in amblyopes. We begin with one of the simplest possible models for precision in Vernier acuity namely that for such stimuli the visual system has a internal error. a:, and to this is added the stimulus error or perturbation, ui, and that the overall threshold error, bT is related by a multiplicative constant, k, such that
M3: spar&l scr~~l~g The third model to be considered is that the metric of the visual space in anisometropic amblyopia is scrambled (Hess et al., 1978; Hess, 1982). The retinal image is a continuous function, S, of two spatial dimensions. In the expriments to be described, the stimuli were made of bright lines or dots whose size was conskkrably less than the point-spnad function of the human eye. The retinal image may thus be characterized formally as a set, 1, of stimulated loci, that are samples of S. This set is transfcrrcd into the system to produce a neural set of local
Let us consider what will happen for a normal visual system whose internal error is rr: when confronted with stimuli of increasing 0;. Threshold error will be initially determined largely by the internal error until the external error equals it in magnitude, there afk threshold should rise in proportion to &. In this argument so far, the error could be due to either blur, or scrambling. If the defect in anisometropic amblyopia is a raised internal blur, then as stimulus blur is increased in the
Anisometropic
normal eye beyond its own internal blur, threshold should rise and meet that for the amblyopic twin, at which point both should rise further together. Of course, this behaviour would not be expected, if the defect is scrambling, but increasing stimulus scrambling in each eye should behave in that fashion instead. The difficulty with positional experiments involving continuous and smoothly changing luminance waveforms, such as cosine gratings, is that the performance measures are affected in similar fashions by either blurring or contrast reduction (Watt and Morgan, 1983, 1984). This is principally because Vernier thresholds are limited by noise for these stimuli, and are generally rather higher than the finest thresholds of a few arc sec. obtained with bright line stimuli. In the experiments below we have employed thin bright lines in shape discrimination tasks, where the finest spatial performance is expected (Andrews ef al., 1973; Watt, 1984). The experiments demonstrate a spatial deficit that is characterized mainly by a local jumbling distortion of the spatial metric, and that cannot be accounted for by neural blurring or contrast reduction.
663
amblyopia MJZTHbDS
Subjects Seven anisometropic amblyopes were used in the experiments to be described. Their vision through their normal eye was corrected to 6/5 acuity. The clinical details of each subject is given in Table 1. Stimuli The stimuli were all drawn on the screen of a HP1336S display oscilloscope under the control of a Motorola VME/lO microcomputer: four DACS were used to drive the differential X & Y amplifiers of the oscilloscope at a rate of 60 Hz. The stimuli were presented at a luminance comfortable to the subject and at I .5 log units above threshold for the particular eye. Normal room lighting provided a background illuminance equivalent to lOOcd/m*. Subjects viewed the display from a distance of 2.3 m with natural pupils. The stimuli were always presented at a random orientation within the interval vertical + 4 deg. Exposure duration was unlimited (i.e. lasted until the subject had made a response).
Table 1 Subject
Age
N.N.
30
P.S.
50
A.C.
40
P.W.
46
J.S.
42
L.B.
62
D.C.
32
G.F.
34
Present clinical data Central fixation Acuity R 615 L 6120 R, R Plano L -0.75/-2.50 x 10 Central fixation Acuity R 616 L 6136 R, R -4.50/- 1.25 x 90 L -2.25 DS Central fixation Acuity R 616 L 6160 R, R -4.50/- 1.25 x 90” L -3.00 DS Central fixation Acuity R 6118 L 615 R, R +O.SO/-3.25 x 20 L -0.251-0.25 x 70” Central fixation Acuity R 2160 L 615 R,R -5.OODS L + 1.251-0.25 x 25” I’ Eccentric fixation Central scotoma Acuity R 2160 L 615 R, R +8.00 DS L +0.50 DS Central fixation Acuity R 6118 L 615 R, R +3.50 DS L plan0 Central fixation Acuity R 615 L 6/9+ R, R +0.75 DS L +4.75/- 1.25 x 50”
Clinical history First R, age 25 years
First R, age 6 years
First R, age 11 years
First R, age 6 years
First R, age 36 years
First R, age 21 years
First R, age 7 years Intermittent occlusion therapy First R, age 3 years Occlusion therapy Present R, worn constantly
R. J. WAIT and R. F. HESS
664
Procedure APE (Watt and Andrews, 1981), an adaptive method of constant stimuli was employed to measure thresholds. On each trial the subject was presented with two targets, one on each side, left and right, of the screen. One of these was the standard target without cue, and the other had the cue. The subject was requested to indicate which of the targets he judged to have the cue, and to guess if he could not decide. The amplitude of the cue was varied from trial to trial, as was the side (left or right of the screen) on which it was located, to collect a full psychometric function. Probit analysis (Finney, 1971) was used to assess the standard deviation of the normal cumulative psychometric function thus obtained, and is the measure of threshold reported. Experiment 1: Integration Region for Shape It is known that shape information is integrated over around 30 arc min along the length of a line (Andrews et al., 1973; Watt, 1984). It is possible that the raised thresholds obtained in anisometropic amblyopes are due to a reduced integration region. In the first experiment, this possibility was tested by measuring the effect of the space constant, S, on the threshold for detecting the presence of a curvature cue C(y) = m(1 - y2/s2) exp( -y2/2s2) -22.5’ < y < 22.5’ where m is the cue amplitude and was varied from trial to trial. The subject was presented with two lines, one was straight, the other curved, and required to identify the curved line. The side that the curved line was displayed at was varied randomly from trial to trial so that a full psychometric function could be collected. The lines were both 45 arcmin in length and comprised 90 dots spaced at intervals of 0.5 arc min.
amblyopia is normal, the rise in threshold must be due to an increase in the uncertainty or error of the data that is being integrated. In the second experiment, the influence of spatial uncertainty on Vernier acuity and the curvature task was assessed. In performing this task. the visual system presumably uses some derivative or integrative process on the raw contour locus set N, and errors in the mapping onto N are going to lead to poorer performance. If we suppose that the internal spatial uncertainty and stimulus positional jitter are additive independent sources of error, then one should expect that shape discrimination, ds. will be related to stimulus positional jitter, J,, by
where J, is a fixed factor for each eye (normal and amblyopic). It can be seen that: a3 =k,J,
for
J,>>J,
(2) and so the performance for the two eyes at high levels of stimulus positional jitter should converge. The stimuli for the experiment were once again made of a vertical row of dots, spaced at 0.5 arcmin intervals. To one of the two was added a shape cue which was: (a) a Vernier step offset 0’ < y < 22.5’ C(Y) = m C(Y) =
0
-22.5’
(b) a curvature cue C(y) = m(1 - y2/s2) exp(-y2/2s2) - 22.5’ < y < 22.5’ s=5arcmin. 200 r
Results The results for each eye of observer N.N. are shown in Fig. 1. It can be seen that the only difference between the eyes is a factor of 8 difference in al1 conditions. Integration occurs over the same distance in the two eyes. These results show the difference in sensitivity to shape in anisometropic amblyopia. Experiment 2: A Measure of Spatial Uncertainty Since the integration region in anisometropic
2L
1
I
I
1.25
2.5
5.0
I Ia0
Curvature space canslont (on mm ) Fig. 1. The effect of cwvatwe we apace conataat (hear extent) on detection thresholds. lntqmtion regions in the two eyes are essentially the same.
Anisometropic
Threshold amplitude m of the cue was measured as a function of the standard deviation of the imposed one dimensional (horizontal) pasitional jitter.
two eyes for the Vernier offset and the curvature cue respectively, for the both observers. A continuous line is drawn through each set of data. This line represents a least squares fit of equation (1) to the experimental results, and the figures aiso tabulate for each observer and each eye the two parameters: internal uncertainty or
Results Figures 2 and 3 show the thresholds for the Subject 0 .
Normal
P ,L,
,111
1
g
Sublect
N-N
Addlflve MultIpI Addltlve MultIpI.
Amblyoplc
665
amblyopia
error * 13.899” error 2 1.205 error = 45.558” error = 0.977
l,,ll
III1
IO
,111,
0
Normal
.Amblyoprc
I
u
l,l,
100
I 1
ICOO
Positional
jitter
I
: AC.
Addilwe Multi@ Addttwa MultIpI
l11lllll
error error error error
I
I lllllll
10
(arc
= 6.603” s 0.637 5 Vl6449” * 1 269
I 100
I l!J_Ll.u loo0
set)
Fig. 2. The effect of positional jitter on Vernier acuity. The amblyopic eye shows an elevated positional uncertainty, as indicated by the raised additive error. The multiplicative error for both eyes is similar, and so the two data sets converge at high positional jitter. Subject: NN 0 .
Normal Amblyoplc
Addlttve MultIpI. Addltivo Multlpl.
Subject:
error * 7.145 erlor ’ 0.794 ermr * 52.895” erm * 0. 824
0 l
Norma I Amblyoplc
GF
Addttwe Multtpl. Acklttrve Multipl.
errcf error error error
’ 15.979” = 0.662 m 41.405” = 0.550
3 $ cr
1-
, 1
I , ~~~~~~~ I 1~~11111I I IO
IllJJJJ
100
Positional
I
1000
jitter
I 1 lllllll
I 10
1
(arc
I1111111 100
I I
Illllu
loo0
secl
Fig. 3. The e%ct of positional jitter on curvature detection. The amblyopic eye shows an elevated positional uncertainty, as in Fig. 2.
R. J. WATTand R. F. HESS
666 Subwit Norma I 0 .
Amblyopx
dent, internal source of positional jitter such that a truly collinear row of dots has an effective base jitter of Ji, then the addition of stimulus jitter, J,, produces a base jitter of
NN
Addltwe
error
:
9258”
Mull~pl
emx
*
0865
Addttwt
error
= 47 106 ”
Mi.lltlol
srrw
=
0792
J = (Jf + JZ)‘f’. Substituting obtain
equation
(4) in equation
(4) (3), we
CU= k,(Jf + J;)‘,‘.
(9
We have J + dJ = [Jf -I- (.I, c dJ,)‘]‘,‘.
tf9
Substituting aY from equation (5) into quation (6) and solving for dJ,, we have dJe = [Ji + (ki + 2kj) (51+ J:)]“’ - Je. Positioml
jitter
(arc
set
I
Fig. 4. The effect of positional jitter on curvature detection for horizontal targets.
additive error, Ji, and the multiplicative error k,. The principle finding is that the amblyopic eye has higher internal uncertainty than the normal, by a factor of 3 for N.N. (Vernier); a factor of 24 for AC (Vernier); a factor of 7.5 for N.N. (curvature); and a factor of 2.5 for G.F. (curvature). Figure 4 shows a second set of curvature data for subject N.N., this time with horizontal targets. The result is very similar to the case with vertical targets. The results are consistent with a mode1 in which performance is related to the total positional error as given by equation (I). The largest difference between the system parameters of the two eyes is for Ji. This result is further evidence for a positional uncertainty deficit in anisometropic amblyopia.
(7)
In the third experiment, the discrimination of positional jitter was measured in the normal and amblyopic eyes. The stimulus was made of 90 dots spaced vertically at intervals of 0.5 arc min in a straight row. Each dot was independently displayed with an error left or rightwards with a Gaussian distribution of uncertainty. Two such stimuli were presented, and the subject was required to report which had the more jitter (the jitter of largest standard deviation). A jitter difference threshold was obtained as a function of base jitter. The threshold is the spatial standard deviation of the subject’s error response distribution. Results The results for the two observers are shown in Fig. 5. The continuous lines show the func~on expected by substitu~ng the known values of J, (from Experiment 1) into equation and finding a fit for the one parameter kj* The fits obtained in all cases are reasonable. It is further suggested that the ambiyopic eye has a raised spatial uncertainty.
Experiment 3: An Eflect of Raised Spatial Uncertainty
~~~~~s~on: spatial ~certainty
The judgement of positional jitter in a row of dots is determined by the precision with which the standard deviation of the jitter can be measured. which is proportional to that standard deviation, so that when an increment in positional jitter is added to a pedestal base of jitter. f. then the threshold level of i+ will be given by
The first three experiments have demonstrated that the visual system of the anisometropic amblyope is subject to a raised spatial uncertainty: the mapping from retinal image to some neural image is subject to random errors. There are several different transformations of an image that would lead to an increase in the uncertainty wirh which a point in that image could be localized:
dj=k,J
(3)
Suppose that there is an additive, indepen-
(1) Any operation which reduces the sharpness of the stimulus peak, such as blurring or
667
Anisometropic amblyopia Subjec?
l
SublocI
: NN
o Normal Amblyopu
Eye
AC
o Normal Eye
Eye
.
Pedestal
positionol
jitter
Amblyoplc
(arc set
Eye
1
Fig. 5. The results of Experiment 3 on positional jitter discrimination. It can be seen that there is a large difference between the normal and amblyopic eye for low degrees of positional jitter, but that the functions convergefor largedegrees. The amblyopiceye showsan elevatedpositionaluncertainty.The continuous lines are the predicted threshold functions according to equation (7).
contrast reduction will increase spatial uncertainty. (2) Any operation which weakens the power of the representation substrate, such as undersampling, will also tend to increase the spatial uncertainty of the system. (3) A random distortion of the representation substrate, by scrambling for example, will lead to an increased spatial uncertainty. (4) A raised level of intensive noise (equivalent to a higher level of luminance noise in the stimulus) will lead to larger positional errors. The first two are, of course closely related, since undersampling is defined with respect to a given spatial scale or degree of blurring. By analogy with the effects of positional jitter, one can measure the internal blurring by progressively blurring the stimulus, until one reaches a point at which the blur in the stimulus begins to affect thresholds, which is the point when it is commensurate with the internal blur. This should be markedly different in the two eyes, if the raised uncertainty in anisometropic amblyopia is due to an internal blur. The results of this are described in Experiment 4. In a similar fashion, the possibility that raised intensive noise is the cause of raised positional uncertainty can be tested by measuring Vernier acuity as a function of luminance level. The results of this are described in Experiment 5.
Experiment 4: The Efects of Gaussian Blur on Vernier acuity The possibility that the raised Vernier acuity found in the previous studies coud be due to an increased level of internal blur was tested in the fourth experiment by measuring the effect of one-dimensional (horizontal) gaussian blur in the stimulus on Vernier thresholds. Once again, we assume the standard deviation of the stimulus blur and internal blur to be independent and additive, according to B:=B;+B:
(8)
where, B, is the effective blur; Bi, the internal blur; and B, the stimulus blur. Equation (8) follows from convolution. As in Experiment 2 we suppose that Vernier acuity is proportional to the standard deviation of effective blur, so that ds = /C,(Bf + B;)“‘.
(9)
In fact, according to the MIRAGE model of Watt and Morgan (1985), the exponent in this expression should be 0.75, not 0.5. However for blur less than 10 arc min the difference is negligible, since a% = kbBf
for
B,sB,
(10)
and whatever value of a this is not a function of &
668
R. J. WAIT and R. F. HES
As in Experiment 2, we expect no rise in threshold with increasing blur until B, is near to Bi, when thresholds should rise. If internal blur in the two eyes is different, this point will be different. The stimulus was made up of two gaussian luminance profiles, one above the other, and the subject was required to judge the direction of their offset. The luminance profiles were created by varying the horizontal spacing between a set of parallel vertical lines in a gaussian density distribution. None of the individual lines were resolvable, and by this method non-linea~ties in the Z-axis amplifier circuit were bypassed. Results The effects of gaussian blur on vernier acuity for the two eyes of both subjects are shown in Fig. 6. The effects are similar in the two eyes, with blur having little effect if less than about 3 arc min in standard deviation. There was no level of blur in the experiment at which the two eyes performed with the same accuracy. The additive error for the two eyes is the same: the only difference is in the multiplicative error, k,,. A similar pattern of results were obtained for subject N.N. with optical defocus blur. The possibifity that some form of internal blurring or undersampling could be responsible for the spatial defect in anisometropic amblyopia is n&d out by the present results.
Ezcperiment 5: The Ejgecr of Luminance In the final experiment, the effect of luminance on the curvature detection task was measured. Bradley and Freeman (1985) in describing their version of the weak signal model, point out that normal eyes should deteriorate to the level of performance of anisometropic eyes if contrast is reduced sufficiently, and that. conversely, amblyopes should improve to the performance level of normal eyes if contrast is increased by the same degree. Results Figure 7 shows the effect of luminance on the curvature detection task in each subject for both eyes. For the normal eye, it seems that performance is essentially independent of luminance, provided that the target is actually visible. This is in confirmation of casual observations made by many workers and detailed measurements on normal eyes made by us. For the amblyopic eye, on the other hand, there is a relatively marked effect of luminance level on performance, with threshold not reaching an asymptotic level until luminance is at least one log unit above detection threshold. There is no luminance level at which the amblyopic eye equals the normal eye. In terms of the three weak signal models we introduced above, it is clear that contrast scaling (W2) is
.%&XINN
Subtect: NWttVJl e AmblyoDlc l
E
1
1
IO
100
moo Gaussian
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I
Additwe
ermt
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MulctpI
error
:
Addrtwe
error
= 327.089”
Muit~pl
error
=
I I I lilll
I 10
I fore
set
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I I I lilll 100
I
I t
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J
Fi& 6. The effect of gati blur on Vernier acuity. Tbc c&t in tk two cyu is ca~~ially tk same, and thc~ is no dqrce of blur at which the two cycs have tk same thrcsboid. This is indiated by a rise in the multiplicative error but not in the additive error.
Anisometropic amblyopia
zooor
.
Subject : AC t 0
ambly0W normal
t
669
blyopic eye’s performance matched that for the normal eye at the lowest luminance tested: over this range there is no overlap of performance at all, and the normal eye at any luminance is better than the best obtained precision of the amblyopic eye. The two predictions of the Bradley and Freeman (1985) weak signal model fail in this set of data. The anisometropic eye is clearly sensitive to the stimulus contrast, but this sensitvity must arise from some other defect in anisometropic amblyopia. Figure 9 shows that spatial scrambling is a good model for the effects of luminance in an amblyopic eye. The figure plots, for two normal observers, both naive, the effect of stimulus luminance on curvature detection, just as in Fig. 7, but for a stimulus which had 48 arc set spatial jitter. The data are strikingly similar to the amblyopic eyes of N.N. and A.C. Our conclusions from this experiment are:
(i) Anisometropic amblyopes are sensitive to stimulus contrast. J 3 1 2 0 (ii) This sensitivity cannot, however, provide Log. luminance (above threshold) a simple explanation for the condition. In particular, contrast scaling does not predict the Fig. 7. The effect of target luminance level on curvature detection. The effect in the two eyes is rather different, but performance of the amblyopic eye. the functions do not converge. (iii) The sensitivity is itself explained by the spatial scrambling hypothesis. 5 t,=-y--,.
inapplicable. A form of model W3, with k, set to above 10 would possibly suffice, but a model C’ - CO.1 is contradicted by the data of Bradley and Freeman (1985). There is no weak signal model which can account for both the present data.and their results. Figure 8 shows a repeat of Experiment 2 at higher a luminance level for subject N.N. The performance of the amblyopic eye is improved, and its positional uncertainty is also apparently improved, although this was also true of the good eye and a substantial difference still exists between the two eyes. The results of this experiment suggest that some form of contrast or contrast sensitivity defect is not responsible for the positional uncertainty elevation in anisometropic amblyopia. There is no simple linear or power law scaling of contrast that will result in the amblyopic eye behaving like the normal eye. There is furthermore no stimulus luminance at which the amblyopic eye converges in performance on the normal eye. Within the 2 log units tested there was no stimulus luminance at which the am-
GENERAL DISCUSSION
In the experiments above a characteristic spatial deficit in anisometropic amblyopia has Subjal: NN
.
s f g
1
erw . 5.583” Multlpt. error = 0.795
Numal
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AddiIive
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E, ,,,,,,,,, ,,,,,,,,, ,,,
,r,u
1
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100
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1
Fig. 8. The effect of positional jitter on curvature detection at a high luminance level. This result is basically the same as that at lower luminance (see Fig. 3).
R. J. WATTand R. F. HFSS
670
2
L
0
3
I
I
I
1
2
3
Log lmunoncc (above ttmst&Y) Fig. 9. Tltc effect of target luminamx kwl on curvature detection for two naive nomtal subjects. The targets had 48
arc set positional jitter. The simikrity between these data and those of Fig. 7 arc striking.
been identified. How should it best be described? It is clearly due to spatial uncertainty, rather than integration failure, as the results of Experiment 1 show. Many factors could increase the spatial uncertainty of the visual system, and in this paper we have considered three. The first is a raised level of internal blur, but the results of Experiment 4 show that internal blur in anisometropic amblyopia is normal. A related possibility is the loss or malfunction of high frqucncy filters, a suggestion made by Levi and Klein (1982). For a simple, isolated Vernier target such as those we have employed, we can rule this out as a cause for low precision.
Watt and Morgan (1984,19%5)have shown that the low frequency filters (at 4c/deg) provide most of the high precision in Vernier acuity, although high frequency filters do contribute. The results of Experiment 4 afso indicate that anisomctropic amblyopes are worse by the same multiplicative factor even for highly blurred stimuli, where the high fraquency filters are not relevant. A related possibility is the hypothesis of undersampling. The spatial uncertainty of a sampling grid is proportional to the intersample
distance, and it is possible that this is the cause of the deficit in anisometropic amblyopia. Once again the results at high degrees of blur rule this out: a low frequency target, equated for visibility in the two eyes, will not be affected b> undersamplin~. The second possibility we have explored is the idea that a reduced contrast or weak signal can account for anisometropic amblyopia. This is unlikely, given the results of Experiment 5. where we showed that, although increasing the stimulus energy yielded lower thresholds for the amblyopic eye, and led to a small reduction in the spatial uncertainty of the eye, it could not approach the performance of the good eye. Our results are incompatible with weak signal model W2; Bradley and Freeman’s results are incompatible with weak signal models Wl and W3. We conclude that a weak signal is not the problem in anisometropic amblyopia. The sensitivity of the amblyopic eye to stimulus strength was accounted for by the third possibility, spatial scrambiing (Hess et al., 1978; Hess, 1982). The third possibility we have considered is raised spatial jitter or scrambling. The results of Experiments 2 and 3 are evidence in support of this hypothesis. In fact, the results of Experiment 2 are themselves sufficient to rule out both the internal blur and reduced contrast or weak signal hypotheses, neither of which predict the exact form of results obtained. The clearest demonstration of this is obtained by taking normal observers and imposing either + 3 D optical defocus or using a very low contrast dispiay to repeat Experiment 2. The results of these two manipulations are shown in Figs IO and I1 respectively. In each case the manipulation raises both the additive error and the multiplicative error, whereas we found in Experiment 2 that anisometropic amblyopia only raises the additive error. Of course it follows from simple algebra that a raised internal spatial scrambling would affect only the additive error. We now ask whether the spatial scrambling hypothesis can account for the other data concerning anisometropic amblyopia. A random spatial scrambling would obviously reduce Snellen acuity by degrading the letters; it would also reduce grating acuity for the same reason. The reduction in acuity in each case will be related to the degree of scrambling, and it is plausible that acuity would be proportional to scrambling. This would account for the report by Levi and Klein (1983) that Vernier
Anisometropic amblyopia Subject: NOrlTlOl 0 .
r3D0ptres
671 Subtect : CG
RJW
Addltwe
error
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tilti@.
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I
I I Il~~~~l
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.
6.421” 0.333 ”
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I I I LLUJ
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Fig. 10. This shows the effect of stimulus blurring (by optical defocus) on the pattern of results obtained in Experiment 2, for two normal observers. Notia that the additive error and multiplicative error are
increased in proportion.
acuity in anisometropic amblyopes scales with grating acuity, but inverts the argument of
casuality. We argue that spatial scrambling causes low acuity and low Vernier acuity; they argued that the former was the cause of the latter; an argument we have disproved. Subject Normal 0
.
V&k
tavt
I 7
I
Bradley and Freeman (1985) showed that the amblyopic eye has a Vernier acuity similar to a good eye at lower contrast. Watt and Morgan (1983) have shown that contrast and blur behave in the same manner, and so the argument of Bradley and Freeman is inconclusive. How-
MC
Subject:
Additive
error
=
5 026”
Mult~pl
error
*
0.725
Add~llve
error
’
15.697”
11 111111 10
I
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I
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I
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error
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1
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KA
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I
= 4.646”
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I
1
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-I
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Fig. 1I. This shows the efTectof a weak stimulus (luminance set to 1.5 times the threshold luminance for visibility: background luminance was 100 cd/m*) on the pattern of results obtained in Experiment 2, for two normal observers. Notice that the additive error and multiplicative error are increased in proportion.
R. J. WATT and R. F. HESS
672
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96192389766 (arc
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673
Anisometropic amblyopia Subject : DC . omblyO00 0
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Subject:
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Positional jitter (arc set I
f Fig. 14
1000
Subioct : PS l ambiyopo 0
;
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~
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Fig. IS Figs t2-15. Summary data for seven subjects. On the left is shown the effectof blur ORVernier acuity, and in each case the normal and amblyopic data are parallel. On the right is shown the ei%ct of positional jitter on Vernier acuity, and in each case the normal and amblyopic data converge at high values of jitter.
ever, the spatial scrambling of a target is equiv- converge and increased blur does not. Figures alent to dispersing that target in space, which 12-l 5 show this effect for a total of 7 subjects effectively reduces contrast. Once again we con- with anisometropic amblyopia. Figure 12 is the clude that the reduced effective contrast is a data of Figs 2 and 6 replotted. Zero blur and consequence of spatial scrambling, rather than jitter is in the centre of the graphs, with increasing blur leftwards from centre and ina primary cause of amblyopia. The two critical experimental results are those creasing jitter rightwards from centre. In every of Experiments 2 and 4, that increased jitter case the jitter data converge and the blur data causes the performance of the two eyes to remain parallel.
R. J. W~rr and R. F. Hiss
614
CONCLUSIONS
(1) Shape discrimination, a task that requires the finest local positional information. can be an order of magnitude poorer in amblyopia. (2) This is not due to an inability to collate information along the line of the target, because the integration region is normal. (3) It is, however, due to an elevated positional uncertainty. (4) Anisometropic amblyopia cannot be mimicked or modelled by raised internal blurring or contrast scaling by the contrast threshold deficit. (5) The raised positional uncertainty is adequately modelled by a higher level of scrambling in the fovea1 metric of the amblyopic eye-brain than is normal. Aehowle&awnu-Thir cd Raeucb
Cod
work was supported by the Mcdiof Gc Britain and the Welk.omc Trust.
R.F.H. k a Welkomc Senior Lecturer.
REFEBENCES
Fmnev D. J. (1971) Pro611 AnahJis. 3rd edn. Cambridge Univ. Press. Flom M. C. and Bcdell H. E. (1985). lndentifying amblyopia usmg associated conditions, acuity. and nonacmty features. Am. J. Oprom. Physiol. 0p1. 62, 153-160. Hess R. F. (1982) Development sensory impairment amblyopia or tarachopia? Human Ncurobio/. 1, 17-29. Hess R. F.. Campbell F. W. and Gmnhalgh (1978) On the nature of the neural abnormality in human amblyopia: neural aberrations and neural sensitivity loss. *tigers Arch. ges. Physiol. 377. 201-207.
Lawden M. C.. Hess R. F. and Campbell F. W. (1982) The discriminability of spatial phase relationships in amblyopia. Vision Res. 22, 100~1016. Levi D. M.. Harwcth R. S.. Pass A. F. and Vcnvcrtoh J. (1981) Edge sensitive mechanisms in humans with abnormal visual experience. E.rp/ Brain Res. 43, 27&280. Levi D. M. and Klein S. (1982) LIifferencea in vernier discrimination for gratings between strabiamic and aniaometropic amblyopes. Incesr. Ophrhai. oisual Sci. 23, 39847. Levi D. M. and Klein S. A. (1983) Spatial locahsation in normal and amblyopic vision. Vision Res. 23, 1005-1017. Levi D. M. and Klein S. A. (1985) Vernier acuity. crowding and amblyopia: Vision Res. 25, 979-991. Rentschlcr 1. and Hilz R. (1985) Amblyopic processing of positional information. Part I: Vernier acuity. Exp/ Brain Res. 60, 270-278. Watt R. J. (1984) Towards a general theory of the visual acuitics for shape and spatial arrangement. Vision Res. 24, I377- 1386.
Andrews D. P.. Butcher A. K. and Buckley B. R. (1973) Act&tea for rpatial arrattmt in line figures: human and ideal obrmen compared. Vision Rcs. 13, W-620. Bcdcll H. E. and Rom M. C. (1981) Monocular spatial distortion in atrabiamic amblyopia. Incest. Ophthal. kuul Sci. Zo, 263-268. &dell
spaa
H. E. and Flom M. C. (1983) Normal and abnormal perception. Am. J. Oprom. Physiol. Opt. 0,
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Bradley A. and Framan R. D. (1985) Is rcduad vernier acuity in amblyopia due to position, contrast or fixation d&its? Virion Res. 25. 5566.
Watt R. J. and Andrews D. P. (1981) APE: Adaptive probit estimation of psychomctrioe functions. Curr. Psrchol. Rev. I. 205-214.
Watt R. J. and Morgan M. J. (1983) The recognition and reprcscntation of cdp blur: evidence for spatial primitives m human vision. Vision Res. 23, 1457-1477. Watt R. J. and Morgan M. J. (1984) Spatial filters and the localization of luminance changes in human vision. Vision Res. 24, 1387-1397.
Watt R. J. and Morgan M. J. (1985) A theory of the primitive spatial code in human vision. Vision Res. 25. 1661-1674.