Regional distribution of the mechanisms that underlie spatial localization

vision Res. Vol. 30, No. 7, pp. 102~~1031, 1990 Printed in Great Britain. All rights nscrvcd

REGIONAL

0042-6989/W $3.00 + 0.00 Copyright 0 1990 Pergamon Press plc

DISTRIBUTION OF THE MECHANISTS UNDERLIE SPATIAL LOCALIZATION

THAT

R. F. HESS’and R. J. WAX-~ ‘The Physiological Laboratory, Downing Street, Cambridge, CB2 3EG, England and 2Department of Psychology, University of Stirling, Stirling, Scotland, U.K. (Received 6 April 1988; in &al revised fomt 20 December 1989)

Abstract-In order to understand the regional distribution of the mechanisms which underlie localization accuracy we (1) chose a task which is known to involve localization accuracy (2) optimized stimulus parameters for eccentric loci and (3) determined how two key spatial factors which affect localization accuracy vary as a function of eccentricity. These involve Gaussian blur and Gaussian jitter. These results suggest that there are three different functions with eccentricity for the mechanisms underlying this task which we mlate to the spatial properties of the retina, namely mean cone density, receptoral convergence and regularity. Localization

Hyperacuity

Eccentricity

Filters

INTRODUCTION

Over the long history of vision research a limited number of basic visual measures have emerged from which we have obtained most insight into the visual process. High up on this list are acuity, contrast sensitivity and localization accuracy. Each has given a rather different insight into visual processing, however their interrelationship is still obscure. This lack of understanding of the relationship between the mechanisms underlying these three basic visual measures is particularly evident whenever fovea1 and peripheral visual processing is compared. This has in turn impeded our understanding of one of the most important properties of functional organization, namely how the visual system processes information at different eccentricities. The question of how the mechanisms which underlie our vision vary with eccentricity has long been recognized as an important issue. Acuity measurements have traditionally been applied to answer this question since this measure tells us how the smallest visual filter varies as a function of eccentricity (Wertheim, 1894). Since there is evidence for a range of different sized filters subserving vision (Campbell & Robson, 1968; Blakemore & Campbell, 1969; Stromeyer & Julesz, 1972) it was seen to be important to know how their individual sensitivity varied with eccentricity. Robson and Graham (1981) and more recently

Cones.

Pointer and Hess (1989) showed that for stimuli above 1 c/leg, the fall-off in sensitivity is equivalent when eccentricity is considered in units of the spatial wavelength of the stimulus. Thus there is a linear scaling that relates how the ~nsiti~ty falls-off across eccentricity (in degrees) for stimuli of different spatial frequency. One interpretation of this scaling is that filters of different size might have scaled distributions across the visual field. Another potentially important piece to the puzzle of how visual function varies with eccentricity

involves

the measure

localization

accu-

racy. However, to date the approaches that have been adopted to provide this information have been disappointing. Most studies have used stimuli that provide several different cues for the subject to use. For example, Vernier acuity (Levi, Klein & Aitsebaomo, 1985) has orientation, spatial offset, phase and shape cues, and perhaps others. There is no reason to suppose that these all co-vary with eccentricity, or all relate to the same determining factors in performance. This means that it is difficult to reach conclusions about anything except how performance itself on the specific task varies with eccentricity. If we are to understand how the rn~~~~~~ that underlie localization accuracy vary with eccentricity then it is important to determine not how localization accuracy itself varies with eccentricity but how the most relevant factors that affect it vary as a function of eccentricity. For example, it is just as

1021

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R. F. His

and R. J. WATT

meaningless to ask how localization accuracy varies across the visual field as it is to ask how contrast sensitivity varies across the visual field. Each in turn is dependent on spatial factors which are fundamental to an unders~nding of the functional organization of the underlying mechanisms. It is for this reason that it is the spatial frequency dependence of the contrast sensitivity fall-off with eccentricity that is of importance. Using this approach, the way in which the factors that inffuence localization accuracy (and not localization accuracy per se) need to be examined at a range of different eccentricities and by comparison with known acuity measu~ments, it should be possible to assess how the organization of the filters that subserve this task vary across the field of vision. One instance where determining factors rather than ~r~r~~ce itself have been assessed is reported by Westheimer (1982). Westheimer measured sensitivity to orientation specified by the relative positions of two dots as a function of their separation. Whilst optimum hyperacuity thresholds increased 1O-fold between the fovea and 10 de8 eccentricity, visual resolution thresholds rose 4-S-fold over the same distance, and optimum separation between targets for h~rac~ty rose only 2-3-fold. We chose to assess localization accuracy by using a curvature task for line stimuli of high contrast since this has been recently shown to involve a highly efficient use of a positional (s~ifically orthoaxial area-see Watt, Ward & Casco, 1987) rather than an orientation cue (Watt, 1984). Our approach has been to first determine the optimal stimulus parameters for measuring localization accuracy at each of a number of eccentricities. These parameters include curvature space constant, luminance and exposure duration. Secondly, for stimuli with optimal values of the above parameters we assessed the influence of two spatial factors that we consider relevant to revealing key properties of the underlying mechanisms subserving this task at each eccentricity. These spatial stimulus factors are Gaussian positional jitter and Gaussian blur from which we infer the underlying visual factors associated with this task. By measuring, at each eccentricity, curvature sensitivity as a function of Gaussian positional jitter and Gaussian blur, we estimate the equivalent internal spatial error and internal blur. The variations in these properties are compared with acuity measures across eccentricity, to give a more complete characterization of the spatial

info~ation available at various sites in the visual field. We have already reported results for similar experiments on anisometropic amblyopes (Watt & Hess, 1987). Our present results may be summarized thus: (i) internal blur rises by a factor of ca 15 from the fovea1 out to 32deg eccentricity (around 2-0.2 c/deg); (ii) internal spatial error rises by a factor of ca 100 from the fovea out to 32deg eccentricity (around 0.1-10 min); (iii) resolution rises by a factor of cu 40 over the same distance (around OS-20min); (iv) but, from 2 deg outwards resolution and spatial error are proportional to each other, differing by a factor of about 2).

Subjects The authors acted as subjects in the experiments and have normal or well-corrected (6,/S) vision.

Stimuli The stimuli were all drawn on the scrmn of a HP133368 display osc&soope under the control of a VME/f33 rni~~ornpu~%~ at a rate of 6OHz. Low noise in the rnul~pl~n~ DACs gave a positional accuracy of 1 aeo arc. The stimuli were presented at a huninance_comfortable to the subject and at least 3 log units above threshold. NormaI room Iig!Mg provided a background ilIumi~ equiv&nt to 100 cd/m*. For measurements at the fovea and 2deg eccentricity, subjects viewed the &play from a distance of I .8 m with naturaf puipils. Each subsequent doubling of eccentrioity was achieved by halving the viewing d&ance. Targets were always presented vertically below fixation. For the foveal testing of stimulus space constants exceeding 80 tin arc the viewing distance was proportionately reduced so as to frilly represent the stimulus. The stimuli were constructed from a row of dots spaced at intervals of 0.3 min arc. The target was presented at an o~en~tiun s&acted randomly from the interval vertical f 4 dog. Th stimuli had negligible width (the spot sixe of the HP1 336 is quoted as 0.1 mm) and so it is important to set the intensity appropriately. Exposure duration was unlimited (i.e. until the sub@ had made a response), except in the experiment which assessed the effects of exposure duration.

Mechanisms utxkrlying spatial Iocalization

1023

(A)

Procedure

APE (Watt & Andrews, 1981), an adaptive method of constant stimuli was employed to measure thresholds. On each trial the subject was presented with a target, which contained a perturbation cue whose magnitude and sign were set according to a quasi-random sequence. The subject was requested to indicate the sign of the cue, k and to guess if he could not decide. The amplitude of the cue was varied from trial to trial, as was its sign, to collect a full psychometric function. Probit analysis was used to assess the standard deviation of the normal cumulative psychometric function thus obtained, and is the measure of threshold reported.

(6)

Curvature sensitivity

(C)

The stimulus used in these experiments con-

sisted of a line with a bump of the following form: P(y) = a(1 - yZ/s2)exp(-y2f2r2). The parameter, s, the space constant determines the length of the bump. The principal bump is 2F in length. It was presented in two augment. Either it was presented alone and the subject had to identify whether the bump occurred on the right or left hand sides of the line (unreferenced condition) or else it was paired with a straight line, in which case the task was to identify which of the lines contained the ~~urbation (referenced condition). The referenced stimulus configuration as well as the different stimulus manipulations used in this study are illustrated in Fig. 1. A fixation point was always provided and eccentricity was varied along the vertical superior meridian of the visual field. The refraction at central and eccentric loci was arranged to give optimal focus. Spatial acuity Spatial acuity was measured using a two spatial alternative forced choice procedure in which two small fields of square wave gratings of 100% contrast were generated side by side on the HP display. One was .horizontal and the other vertical. The task was to detect the horizontal stimulus. The procedures used to measure acuity were otherwise identical to those already described for localization.

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Fig. 1. IlIustration of the different types of stimulus manipulations used in tbc pfwcnt study. These in&de the OriginaI stimufus (A), Gaussian blur (B) and Gaussian positional jitter (C).

EXPERIMENT 1

We start by selecting, independently at each eccentricity, the curvature space constant, exposure duration and intensity at which the measured threshold is lowest. Our reasons for this are twofold. First, we thereby obtain the highest experimental precision, i.e. the least variable data. Secondly, we can be sure that at each eccentricity the visual system is operating at its best efficiency. The results in Figs 2 and 3 show the dependence of curvature threshold on exposure duration and line intensity at each of six loci in the visual field. As exposure duration is increased (Fig. 2), curvature threshold is reduced even for duration above 1 sec. The exponent of the function is around 0.3 for fovea1 viewing. The dependence on exposure duration does not vary as a function of eccentricity as one might have expected if Troxler fading. were operative.

1024

R. F. HESS and R. J. WATT

In Fig. 3, curvature threshold is plotted against the line intensity above threshold. For fovea1 viewing threshold is constant for stimuli that are a factor of 10 or more above threshold. A similar dependence is seen for peripherally located stimuli, although the magnitude of the effect is smaller for fovea1 stimuli. Neither of these two parameters play a crucial role in the assessment of l~ali~tion accuracy as a func-

tion of eccentricity. A more crucial parameter is curvature space constant (Watt et al., 1987). In Fig. 4, curvature threshold is plotted against space constant for two tasks, one in which an uncurved reference is provided (A) and one without an explicit reference (B). For fovea1 viewing, the lowest curvature thresholds are seen for stimuli having intermediate space constants of around 8 min arc. The position of this optimum increases monotonically with eccentricity. At 32 deg eccentricity it is now around 100min arc. This effect is similar for both tasks employed in this study although there are differences in the data between tasks at larger than optimal space constants. These results are important because they highlight the arbitrariness of attaching any significance to the slope at which localisation sensitivity falls off with eccentricity. This point is brought out more

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clearly in Fig. 5 where the results of Fig. 4 have been replotted ignoring the space constant of the curved target. The slope of the fall-off in sensitivity now depends on which data one chooses to use. Whether we use the same space constant or different space constants to compare eccentric loci is an arbitrary decision, and so it is possible to obtain many different variations in the relationship between curvature sensitivity and ~nt~~ty. This point has not been generally appreciated by previous workers. For this reason we choose to use parameters giving optimal performance at each eccentric loci and to attach no significance to how the curvature sensitivity per se varies with eccentricity. Instead we investigate how the spatial factors which affect it vary as a function of eocentricity as only this reveals the propcb~s of the underlying mechanisms. In the folIowing experiments the space constant used at each eccentricity corresponds to the optimum found in Fii. 4. As a consequence of the results of Figs 2 and 3, the exposure time was unlimited (in practice l-2 set) and the line luminance was set to 30 x threshold for each eccentric locus. ExPEmMEm

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(i) Gausskw blur. MO&; optical and neural factors ensure that even an iffily thin bright line has an appreciable degree of blur. This blur affects the accuracy with which it can be

Mechanisms underlying spatial localization

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Fig. 5. The results of Fig. 4(A) have been replotted as curvature threshold vs eccentricity with cue space constant as the parameter. The actual slope of the fall-off has no meaning because it depends on which of these points are used to derive it.

1983a, b). When blur is introduced stimulus, we have:

$2 deg ,16 deg

where B is the stancjard deviation of the Gaussian stimulus blur and F,, is the standard deviation of the internal blur of the visual process involved. The number of samples (or in this case area) depends upon, contrast, filter size and stimulus blur such that: n = kCF,/(B= + F;)“=;

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Fig. 4. Curvature threshold is plotted against cue space constant for a range of visual field eccentricities. The results in (B) are for the unreferenced task (subject RJW) while those in (A) are for the referenced task (subject RFH).

localized. Following Westheimer and McKee (1977) and Watt and Morgan (1983a, b, 1984) we assume that the centroid of the image or of the filtered image is measured. In the presence of white noise the standard deviation of localization, dL is given by: dL = KS/~“= where s is the standard deviation of the light dispersion and n is the total quantity of light or neural response (Krauskopf & Campbell, unpublished manuscript; Watt & Morgan,

dL = k(B= + i=;)3’4/CF:.5. The derivation of these formulae can be found in Watt (1988). These formulae have the effect that when the stimulus blur is less than the internal blur, curvature threshold should be nearly constant and largely determined by the internal blur of the visual filter responsible. When the stimulus blur exceeds this value then curvature threshold should rise in accordance with: dL = kB3’=. The knee point of this function gives an estimate of the blur of the largest visual filter involved in the task. The effects of one-dimensional Gaussian blur on curvature threshold at each of a number of eccentricities are shown in Fig. 6. The solid lines are best fits to the data according to the previously outlined model. Also tabulated

1026

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on the figure are the values linear fit of the model.

of I;, obtained by a

In Fig. 6A results are displayed for the curvature task with a reference and in Fig. 6B without a reference. The results display an ir@ial iiat region where curvature thresh&d is not aSWed by stir&i blw: and then a region where curvature threshold varies with blur. The point at which stimulus blur first affects curvature threshold is a measure of the largest filter underlying the task and it is seen to be systematically

displaced to larger blurs as eccentitity is increased. At 32deg it has inch by about a factor af 18 in A and by a factor of 14 in B. (ii) Gaussianpositiondj&;lar. Mod& WCstart

by assuming that the curvature threshold is proportional to the spatial error, E,, with which the stint&s is mapped in the visual system: T,=A*E, Now if the physical target is itself displayed

with an additional error E, then this will

Mechanisms underlying spatial localization

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add to the internal positional error E,, so that:

T, = k(E; + Et)“‘. For low values of E,, threshold will be dominated by the internal spatial sampling error and will only begin to rise when the external error is greater than the internal error. Initially, the threshold should be independent of stimulus jitter and later depend directly on it. The knee point gives an estimate of the value of the internal sampling error and can be assessed as a function of eccentricity.

Results The results of this experiment

are shown in Fig. 7. The solid lines are best fits to the data according to the model just outlined. The values for E, are tabulated on the figure. Curvature threshold is initially independent of the positional jitter of the dots making up the stimulus. As stimulus jitter increases there comes a point after which thresholds rise in proportion to stimulus jitter. The point at which one behaviour gives way to the other increases with increasing eccentricity. For the curvature

R. F. HESSand R. .I. WATT

1028

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jitter both have an exponent of around 1.3; equivalent blur has an exponent around 0.6, However, as we will now see the results for stimuli imaged within 2 deg eccentricity reveal a different dependence. Within 2 deg of eccentricity, resolution is more similar to equivalent blur than to equivalent jitter.

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RFH

(A)

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The smallest parafoveal eccentricity used in the previous experiment was 2 deg, and so we now report experimental measurements of resolution and jitter at smaller eccentricities. The methods were essentially the same except for two details that were modified to allow us to investigate smaller eccentricities. The target for the jitter measurements was now a single horizontal line, below fixation, and the subjects were required to judge whether it was curved up or down. Because a grating patch subtends an appreciable visual angle, we adopted instead the resolution task employed by Watt et al. (1987). The subject saw two lines of equal flux; one was effectively continuo~ (dots at 15 set arc) and the other had a dot spacing that was increased to measure resolution separation. The only cue to the subject was the highest frequency spatial detail (see Watt et al., 1987 for details). APE was used as before. Results

Eccentricity

(degarc)

Fig. 8. Derived spatial measuresare c~mpprrd with the amntricity dependence of spatial rcsoh~tion and mean cone separation expreaaed in terms of filter space constant (dashed line). The cone resolution data comw from the work of Perry and Cowcy (1985).

condition with reference (Fig. 7A) it varies by a factor of 70 from fovea1 to 32 deg eccentricity and by a factor of 300 for the unrefered condition (Fig. 7B). Reasons for this difference for the two procedures are not known. S-Y

OF MX%%‘IS

In Fig. 8 derived measures are pk>tted for the stimulus population of bIur and jitter as a function of euxmtricity at and beyond Zdeg. The main finding revealed in this figure is that beyond 2 deg of mtricity there are two different power law functions. Resolution and

The effects of stimulus jitter were measured at 0, f, 1 and 2 dcg eccentricity. These results are displayed for two subjects in Fig. 9, From these measurements we assessed the equivalent internal jitter as before. Although the changes in stimulus configuration a&c&d absolute curvature thresholds a little, the model provided acceptabie fits to all the data as previously, and accordingly we only show the &its of eccentricity on the equivalent jitter and resolution. Figure 10 shows that the two functions, although similar beyond 2deg (see Fig. 8) are very different within the parafovea. Resolution hardly varies over this raw, but spatial un~~nty changes by an order of ma~~tude.

We now turn to examine what may be learnt about the structure of the visual system, as seen through our too1 of curvature discrimination

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although it has poor resolution and hence is not suitable for complex images. Our present data show that the size of the largest filter subserving this task increases relatively slowly with eccentricity, and, in fact, at a similar rate to the increase in mean cone spacing (see Fig. 8, compare dashed line with blur data). This implies a constant or more likely slowly increasing convergence rate from cones to the receptive fields of the largest filters at each eccentricity. The diameter of the central region of the filter receptive field is approximately the same distance as 25 intercone separations. We speculate that this reflects an optimum integration extent beyond which there would typically be little signal modulation, and for smaller extents the effects of intrinsic noise would be manifest.

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Fig. 9. Curvature threshold is plotted against Gaussian positional jitter with parafoveal eccentricity as the paramcter. Results for two subjects are displayed, The curves through the data are best fits to the model outlined in the text.

and the effects of our stimulus manipulations various eccentricities.

at

Equivalent neural blur We start by considering equivalent neural blur data and its variation with eccentricity. We chose to measure the equivalent internal blur in our task because this gives an indication of the size of the largest filter involved in localization processes (see Watt & Morgan, 1984 for argument). When considering isolated edges, then the largest filter, because it has the greatest integration region, has the finest localization accuracy (Watt & Morgan, 1984),

Now we turn to consider the spatial resolution data. This can be related theoretically to the mean photoreceptor spacing as follows. Resolution is an awkward phenom~on to discuss because it has spatial and contrast elements. We shall take the simple view that at least three receptors in a row are involved and that resolution is determined by a level of imaged detail such that the central one of the three has a significantly lower response than its neighbours (for bright line stimuli). In this case, if the intercone spacing is d, then resolution, r, will be r = 2d. In a hexagonal ,.mosaic it is possible that a slightly lower resolution criterion could be a lied, giving a theoretical distance of: r = d p” 3. From the mean cone spacing data, we should expect a shallow rise in resolution with increasing eccentricity. In fact, this is what is found from the fovea out to about 2 deg eccentricity (see Fig. 10, compare dashed line with resolution data), but beyond that resolution falls very much steeper than should be expected (see Fig. 8, compare dashed line with resolution data). What is the cause of this? This is most likely due to the fact that the degree of receptoral convergent for the smallest filter undergoes a steady increase beyond 2deg of vertical eccentricity. But why should the degree of convergence and hence the size of the smallest filter need to vary so dramatically with eccentricity? In simple terms, the above view of resolution has an implied assumption that the topology of the retinal mosaic is the same as the topology of the image. We were discussing three cones in a row, a topological condition. Now, the receptor

R. F. HES and R. J. WAIT

1030

postreceptoral factors) at these eccentricities. At even larger eccentricities postreceptoral spatial sampling limits resolution (Anderson & Hess, 1990).

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Finally, we turn to consider the spatial uncertainty data. Suppose that a point of light is able to be imaged as a point onto a receptor mosaic.

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(deg

The mosaic, under these conditions, imposes a discrete set of possible measured locations for the point, the magnitude of this error, E,, is related to the distance between the centres of adjacent receptors (Andrews, Butcher & Buckley, 1973), and its standard deviation is given by: E, = 4245.

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and thus in this idealized case the ratio of resolution to spatial error is:

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r/E, = 6.0-6.9. This provides a useful figure with which to compare the data. In this analysis we are actually able to avoid making any ciaims about the constant d, the distance between adjacent receptors. It f&lows G m lo-’ that we can assess peripheral function without t% adding a great deal about the retin~~lay~ut. Beyond 2deg the measured ratios that are 2: 10-2J found are all smaller than 6, but within each 2 1 0 subject are remarkably constant (1.9 for RJW; 2.5 for RFH). That these numbers are less than Eccentrictty tdegwcl the theoretical value of six is no problem, msrely Fig. 10. Derived spatial measures from the par&veal data indicating that spatial error is worse than optiof Fig. 9 a.rc compared with the axwhity dq.whmx of mal. Between the fovea and 2deg eccentricity spatial resolution and mean cone qxuation cxprssccdin the ratio drops from being close to 6 at the fovea terms of filter spixc constant (dashtd line). The cone resolutiondata comes from the work of Ferry and Cowcy to around 2. (1985). We speculate that the reason for this lies in the tight packing of cones in the fovea and the mosaic is st~ct~ally two~ime~~onal and thus progressively larger gaps between them as echas the same topology as the image. But if it centricity increases. The position of a receptor is were connected via the optic nerve to post- very highly constrained in the dense conditions receptoral neurones with a mapping containing of the fovea so that it must lie more or less crossovers and other errors, then the equivalent midway between its neighbours. This ii less and topology of the mosaic would not be identical less the case further out into the perip&y, and to that of the image. Put another way, if our there is increasing scope for a reduction in the row of three receptors 1, 2 and 3 were wired regularity of the mosaic. Such reductions in up to three postreceptoral neurones ordered 2, regularity have been noted by Andrews et al. 1, 3, then resolution would be worse than the (1973), Yellott (1982), and Hirsch and Hylton ideal. We speculate that this is a reason for (1984). Unless the visual system knew exactly the need for larger degrees of receptoral cun- where each receptor was, spatial uncertainty vergence (and hence poorer resolution due to would be bound to increase.

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Mechanisms underlying spatial localization SUMMARY

We have reported three different functional variations across the visual field. These are: (i) equivalent internal blur, corresponding to the size of the largest filter used in localization tasks; (ii) resolution; (iii) equivalent internal spatial jitter or uncertainty. We account for these three different functions in terms of the spatial properties of the retina; mean cone density, convergence and regularity, and their interaction with the spatial demands of the tasks in our experiments and the visual mechanisms employed. Acknowledgements-We gratefully acknowledge financial support from the Medical Research Council and Wellcome Trust. RFH is a Wellcome Senior Lecturer. REFERENCES Anderson, S. J. & Hess, R. F. (1990). Postreceptoral undersampling in normal human peripheral vision. Vision Research, 30, in press. Andrews, D. P., Butcher, A. K. & Buckley, B. R. (1973). Acuities for spatial arrangement in line figures: Human and ideal observers compared. ViFion Reseurch, 13, 599-620. Blakemore, C, & Campbell, F. W. (1969). Gn the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images. Journal of Physiology, London, 203, 237-566. Campbell, F. W. & Robson, J. G. (1968). Application of Fourier analysis to the visibility of gratings. Journaf of Physiology, L.ondon, 197, 551-566. Hirsch, J. Bi Hylton, R. (1984). Orientation dependence of visual hyperacuity contains a component with hexagonal symmetry. Journal of ihe Opticul Society of America, A 1, MO-308. Levi. D. M., Klein, S. A. and Aitsebaomo, P. (1985). Vernier acuity, coding and cortical magnification. Vision Research, 25, 963-977.

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