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Acuity for spatial separation as a function of stimulus size
V~rton Research. Vol. IX. pp. 615-619 Q Pergamon Press Ltd 1978. Printed m Great Britain
ACUITY FOR SPATIAL SEPARATION AS A FUNCTION OF STIMULUS SIZE D. P. ANDREWS and D. T. MILLER Department
of Communication,
University of Keele, Keele, Staffordshire, ST5 SBG. England (Received
4 January
1977)
for comparing separations was measured using a pattern of 3 parallel lines. Line length and separation were varied within a maximum eccentricity of 3f”. Threshold was constant for lines of length 30’ arc or shorter; above 30’ performance improved with line length. Performance fell with increasing separation. When compared with an ideal processor, efficiency was much lower than has been recorded for discriminating orientation or shape in small patterns. Efficiency falls with line length, at first steeply but then gradually; the change occurs at a line length of order 30min arc. It is proposed that the positional specificity of cortical units is the effective information source for the task. and that this specificity is low. Abstract-Acuity
Andrews, Webb and Miller (1974) showed that disis more efficiently performed with continuous lines (Fig. 1A) than with the end parts The present study concerns a detail of a wider probalone (Fig. IS), although the end parts contain ail lem, how is the visual input organized into the unitary and apparently continuous visual world which we ex- the spatial information which is relevant to the task. A gap in a line appears always to depress spatial perperience? We can judge the spatial relation between formances. A similar result was observed by Andrews any two targets which stimulate the retina together. (1967b) and Andrews, Butcher and Buckley (1973) for It follows that there must be a connection (direct or orientation and curvature tasks. In these experiments, indirect) between every input to the visual pathway the effect of a gap could be described as precipitating and every other. How is this system of connections for small patterns the kind of processing losses engineered? usually found onty for large patterns. There was no Our approach to this problem is statistical: the geometrical statistics of the retina are known, so we evidence that gaps could depress performance below this base level. can calculate the best possible performance that the retina could support in a given visual task. This is These differences in efficiency can readily be underachieved by considering an ideal machine which stood in terms of a two-stage process, as proposed shares the same input limitations as the human retina, by Andrews. Webb and Miller (1974). At the first level but which integrates the positional information availare units which encode spatial parameters such as able to it without loss. For example, in judging the orientation, disparity, or shape; these units make full orientation of a line, the ideal machine would fit a use of the detailed positional information from the line of least squares’ through the positions of its reretina, but this is not available at any higher level. sponding receptors. The variance of the regression All spatial parameters not encoded at the first level coefficient can be calculated explicitly, and the result have to be derived from the positional specificity of compared with human psychophysical performance. the coding units, which is low. The ratio of these two variances is called efiiciency The primary coding process is intended to reflect (of use of positional information). the feature-specific behaviour of single units in the In applying this approach, some performances are visual cortex, and most of the specificities required found to be more efficient than others. Andrews of the coding process have been observed there. (1967a, b) showed that orientation in lines up to The present experiments extend the findings of 10min arc is perceived almost as accurately as the Andrews, Webb and Miller (1974). If estimation of retina would allow, but longer lines are perceived distance depends on secondary positional informawith progressively greater processing losses. Andrews, tion, as they suggest, we can expect that performance Butcher and Buckley (1973) showed that curvature wit1 be inde~ndent of stimulus size up to the field of Lines is just as eEcientIy processed as orientation, size of the units, just as the retina can not differentiate but the optimum efficiency is maintained for lines up signals which fall within one receptor. Likewise, patto 30min arc in length, so that for lines between terns which do extend more than one functional unit 10min and 30min arc, curvature is perceived more may be perceived more accurately than those which efficiently than orientation. The same is true for do not. detecting vernier offset. It appears therefore that The stimulus arrangement of Fig. l(C) was used to shape is coded independently of orientation. find whether our ability to equate separations is dependent on the length of the lines. The results should show whether there is integration of spatial ’ This is a simplified statement; see the section on e& information in the task, and if so, how efficiently the ciency for a fuller account. integration is accomplished. INTRODUCTION
tance comparison
615
616
D.
P.
ANVREWS and D. T. MILLJS
.
Two experienced subjects, DPA and DTM. were used. Both had normal vision, DTM after correction for slight astigmatism in the right eye. Figure l(B, C) shows the stimulus patterns used. The lines were horizontal and thus the distances to be compared were in the vertical meridian. The principal variable of interest was the length of the lines: the effect of varying the separation was also explored.
.
EXPERIMENT I The inter-line separation remained fixed at X2’ arc (except of course for the small variation required to
. A Fig. The and was
8
C
1. Stimulus con~gurations for distance comparisons. present paper is concerned primarily with line length line separation in configuration (C). A fixation spot shown at the centre of each figure for I set immediately preceding the stimulus presentation (2 set).
The procedure has been fully described by Andrews. Webb and Miller (1974) and is here only briefly summarized. The patterns (Fig. IB. C) were displayed on a DEC 338 computer-controlled display and viewed binocularly from a distance of f m. Line width and spot size were approximately 2 min arc. Screen luminance was 0.5 log !I L and stimuli were at 1.0 Iog ft L. Subjects were instructed to fixate on the fixation point (always at the centre of the pattern) during a pre-stimulus period of I set and the stimulus period of 2 sec. A blank period of 3 set completed a 6sec cycle. The method of constant stimuli was used; the subject gave a forced choice response after each presentation according to which separation appeared larger. Both separations varied between trials, the outer lines moving together and the central line remaining fixed. There were four separation values in force at any one time, with the computer selecting one of the four at random for each trial. A suitable range of stimulus values was set up by the computer from the responses during a short pre-run. The run proper consisted of 120 trials. Within the run the set of four stimuli was reset automatically from time to time in the light of the subject’s recent responses. The object of this procedure was to optimize measurement of the subject’s response error distribution; the optimum occurs when the stimulus range is centred on the response distribution, and encompasses about 2.7 standard deviations of the response error distribution. This condition was tracked as accurately as we could achieve within a simple computer program. (The statistical work to establish optimal stimulus range and optimal tracking procedure will be the subject of separate communications. Details of the working programme are available on request from the authors.) A probit analysis of the response counts (see Finney. 1952) gave estimates of the mean and SD of the response error distribution. The mean is a measure of the average or constant error when the separations appeared equal to the subject. (This value rarely corresponded to objective equality of separations.) The SD is a measure of sensitivity to separation change. Threshold for separation change is defined as the SD of the error distribution.
Ideal
\ IO spa*
50 l-one
length,
200
rn~narc
Fig. 2. Threshold for separation discrimination as a function of line length at a fixed line separation of 82’ arc. The upper line was fitted to the RMS value (4) of 6 observations. DTM (0) performed consistently better than DPA (0). Tire lower function shows the performance of an ideal machine in the same task. (Assumptions and calculation are described in the text.) The idea] machine has the same retinal input as the human observer. but makes rhe best possible integration of the available information. The spots were about 2min arc in length. The corresponding ideal performance is not shown, because it depends critically on the number of receptors covered. The central Function shows the ratio of observed to ideal performance on an efficiency scale (to the right of the figure). Efficiency is calculated as the ratio of ideal to observed variance. (Variance = threshold23
617
Acuity for spatial separation Line iength:164mmarc l -.... ... 26Zmm arc D
DPA
DTM
Ideal
/
................
..............
....... ........ .......... .” ..........
200
$0 Line
,’
SO
lb
separation,
2bO
mtn clrc
Fig. 3. Observed threshold for separation discrimination as a function of separation for each of two line lengths. The lower curves shows the performance of an ideal machine in the same tasks.
determine the threshoid). Six different line lengths between 2’ arc (i.e. spots) and 262’ arc were tested in the order 2, 33, 131,262, 65, 16 min arc. The series was then repeated in the reverse order. After Experiment II had been completed a third series was run to see if performance had changed. There was no evidence of a si~ific~t change in performance and the results of all three series are shown in Fig. 2. Performance was approximately constant up to a line length of about 30’ arc and then gradually improved. DTM’s performance was consistently better than DPA’s (by a factor of about 1.4). For both subjects the constant errors were very different from the null value of zero. The expected sampling variation of the constant error for a run of 120 trials is about 15% of the SD (see Finney, 1952); the observed magnitudes were up to fifteen times this value, when separations of 77.9 and 86.1’ appeared equal. DPA’s constant errors were consistently larger than those of DTM and the lower interval always seemed larger than the upper when objectively they were equal. For DTM, constant errors had this same polarity for lines up to about 65’ arc in length; small constant errors of the opposite polarity occurred among longer lines. Wolfe (1923) also reported large constant errors in midpoint estimation which varied between individuals. Most individuals see the lower interval as the larger. As in the present experiments, these differences were unrelated to visual acuity. Constant error in matching intervals has also been found to vary with fixation point and with exposure duration (Andrews and Butcher, in preparation).
EXPERIMENT II
Experiment II measured how performance varies with separation at a fixed line length. Four separations (20.5, 41, 82, 164’ arc) and two line lengths (16.4, 262’ arc) were used, the 8 combinations being tested in random order. The series was then repeated in the reverse order. The thresholds obtained are shown in Fig. 3. For a fixed line length, acuity for separation fell as the separation of the lines increased. There was an approximately linear relationship between SD and separation on a log-log plot. DTM’s performance was again better than DPA’s, but significantly so only for the greater separations. Constant errors again differed significantly from zero, and DPA’s were once more all in the same direction. RESULTS EXPRESSED AS EFFICIENCIES
Efficiency is the info~ation used by the human observer as a fraction of the information available. It is calculated by comparing human performance with that of an ideal machine. The ideal machine has a finite mosaic of inputs with the same statistics as the human retina. Given this limitation, the ideal machine makes a maximum likelihood estimate of the spatial parameter which the human observer was asked to judge. The resulting error of performance can be calculated. The case of separation judgement between 3 lines is particularlv simnle. If cones were uniformlv senar-
618
D. P. ANDREWS and D. T. MILLER
ated by a mean distance d. then a horizontal line containing II cones could be located vertically with a variance dz/12n. (The distribution of cone positions about the image is assumed rectangular.) For cones of unequal size, the best estimate of vertical position is given by
where the weight wi is proportional to l/d!. Intercone spacing is taken from Fig. 2 of Andrews, Butcher and Buckley (1973). The variance of separation judgements VT is given by VT = 2V, + 4V,, where V, is the variance of an outer line. V, the variance of the centre line. Various factors are not taken into account in the above calculation. During the 2sec of exposure duration. an ideal machine could reposition the stimulus pattern on its mosaic an arbitrary number of times. and so achieve a greater precision. The appropriate number should reflect the restraints shared by a human retina-finite integration time and limited variability of eye position during voluntary fixation. The burden of this is that Vr should be divided by an unknown factor n. the number of independent samples available to the ideal machine. Other factors have been considered by Andrews (1967b. p. 1007). and these also change the value of n. In drawing efficiency functions such as those of Figs. 2 and 3. n is taken as unity; the functions therefore represent relative efficiency rather than absolute efficiency. However, their shape remains the same for any value of n, and the conclusions we wish to draw mainly concern the shapes of the functions. Different perceptual tasks may be compared with each other in their efficiency of positional information use. The proviso for doinn so is that n mav be assumed constant: the present experiments may be compared with others in which there was voluntary fixation and a 2-set exposure duration. and preferably using the same subjects. (Where these conditions are not met, large differences in efficiency may still be established by allowing for variation of the relevant factor.) The best relative efficiency in estimating the separation of lines is about ly{ for closely spaced short lines. (For spots, efficiency might be still higher, but the calculation is suspect because it depends critically on the number of cones which the target is supposed to stimulate.) For lines of length 30’ and spaced 82’ arc, relative efficiency was about 0.12%, gradually falling with increasing separation or line length. In the worst case, relative efficiency was about 0.045’%. For targets of 30’ or over, overall efficiency fell by a factor of about 0.80 for each doubling of length or separation.
’ If curvature detection rested on a fixed central part of the target and the outer parts were ignored, then threshold (as defined) should rise as the square of line length. and the efficiency function should fall about four times as steeply as it does. It follows that the end parts of the curve are not ignored, though it remains possible that there is some differential weighting between retinal sites.
DISCUSSION
Figure 2 shows that threshold for separation judgement was nearly constant for short lines. For longer lines. threshold fell steadily, showing that the extra line length does benefit performance, though not greatly. For an ideal processor of course, all increase in line length would improve performance. Comparing human with ideal performance the most obvious difference is in level: relative efficiency is very low. Comparable values for discriminating orientations or detecting curvature are about 50% for small targets. (Andrews, Butcher and Buckley, 1973). The question arises whether this difference in et%ciency may be associated with retinal site rather than with perceptual task; might curvature detection for example be equally inefficient at retinal eccentricities of the order l-2”? This question can be answered by reference to data from Andrews, Butcher and Buckley (1973). Relative efficiency for curvature detection in a line 3” long was about IO?%, and falling at a rate of about xi-for each doubling of line length. Now, in this task, the central third of the pattern by itself provides less than 5% of useful information concerning curvature, so performance rests alsmot entirely on the outer parts of the pattern.’ In comparable retinal loci therefore. curvature detection is performed between 10 and 100 times more efficiently than distance comparison. The smaller patterns used in Experiment II support the same estimate. The shape of the efficiency function is also of interest. For short lines, efficiency fell sharply. Between a line length of 2’ arc (spots) and 30’ arc there was no change in human performance, though an ideal machine could reduce threshold tenfold given the same stimulus range. (Note that Fig. 2 does not show all of the corresponding efficiency drop, and the steep portion could justifiably be extended much higher on the left.) For lines over 30’ arc in length, efficiency falls much less sharply. The results do not permit the knee to be located very accurately, and a smooth curve might represent the data better, but there is no doubt that efficiency function falls first sharply and then shallowly as line length is increased. These results are consistent with the hypothesis that judgements of separation depend on a low-grade source of positional information. It is as though for the purpose of judging separations we had a retina with very few receptors. and all stimuli which fall within one ‘receptor’ are assigned the same location. Stimuli large enough to excite more than one ‘receptor’ can be positioned more accurately than by one alone. The retina is clearly not the true site of this resolution loss. However, the functional grain size of the visual pathway does appear to be about 30 times coarser for the purpose of distance estimation than for the best known visual performances. Loss of spatial resolution can be achieved by a convergence of many lines on a single output, and it seems a fair speculation that this does occur. With less confidence we would speculate on the site of the convergence; single unit studies show that cortical cells have response specificities for orientation, binocular disparity, spatial wavelength and the like. and these properties require extensive convergence of con-
Acuity for spatial separation nections. We propose that this is the convergence which precipitates the observed loss of positional specificity. This hypothesis has the virtue that no new entities need be proposed. It offers the further economy that no special class of ‘local sign bearers’ is required: all units share in the task of carrying positional information, albeit of low specificity. CONSTANT ERRORS
Separations judged equal were usually unequal, sometimes by as much as l@k of the distance to be judged. Large constant errors have been observed in judgements of parallelism, which were highly correlated between the two eyes (Andrews 1967a,b): Constant errors have been reported in other distance comparisons: patterns A and B (Fig. 1) elicit different constant errors from individual subjects (Andrew% Webb and Miller, 1974). All of these constant errors vary with location in the field and between subjects. These findings can be assimilated into the above hypothesis by supposing that primary coding units are mislabelled in the domains of orientation and location. For example, the responses of a cell which is orientation selective will be received higher in the pathway as si~~ying some particular o~en~tion #. The stimulus orientation 8 which elicits optimal response is not necessarily the same, and assignment errors are to be. expected unless the signal lines are perfectly ordered and scaled. Similarly, errors are to be expected in the scaling of stimulus location and stimulus size. SECOND-ORDER
INTEGRATION
The results show that separation comparison does improve with line length for long lines. It follows that
619
the positional information available within large patterns is partially integrated, and it can be seen in Figs. 2 and 3 that the secondary integration is a fairly efficient process. Efficiency falls, but only slightly. Unfortunately, this high efficiency prevents any hy pothesis about how the secondary integration is effected: all summation processes which do not involve heavy loss would serve equally well. The way forward is to look for circumstances in which processing is less efficient, when the kind of loss observed may favour one hypothesis rather than another. Stimuli larger or briefer than ours might show up the limits of efficient secondary inte~ation. Ac~owledgements-We thank the Medical Research Council for their support, and Dr. P. Hammond for helpful discussion of the manuscript.
REFERENCES
Andrews D. P. (1967a) Perception of contour orientation in the central fovea, Part I: Short lines. Vision Res. 7, 915-991. Andrews D. P. (1967b) Perception of contour orientation in the centrat fovea. Part II: Spatial integration. Vision Res. 7, 999-1013. Andrews D. P., Butcher A. K. and Buckley B. R. (1973) Acuities for spatial a~angement in line figures: human and ideal observers cornoared. Vision Res. 13. 599-620. Andrews D. P. and Butche; A. K. (In preparation). Acuity for separation of visual targets. Andrews D. P., Webb J. M. and Miller D. T. (1974) Acuity for length comparison in continuous and broken lines. Vision Res., 14, 757-166. Finney D. J. (1952) Probit Analysis. Cambridge University Press, Cambridge. Polyak S. L. (1941) The Retina. Univ. of Chicago press, Chicago. Wolfe H. K. (1923) On the estimation of the middle of lines. Am. J. Psychol. 34, 313-358.
ACUITY FOR SPATIAL SEPARATION AS A FUNCTION OF STIMULUS SIZE D. P. ANDREWS and D. T. MILLER Department
of Communication,
University of Keele, Keele, Staffordshire, ST5 SBG. England (Received
4 January
1977)
for comparing separations was measured using a pattern of 3 parallel lines. Line length and separation were varied within a maximum eccentricity of 3f”. Threshold was constant for lines of length 30’ arc or shorter; above 30’ performance improved with line length. Performance fell with increasing separation. When compared with an ideal processor, efficiency was much lower than has been recorded for discriminating orientation or shape in small patterns. Efficiency falls with line length, at first steeply but then gradually; the change occurs at a line length of order 30min arc. It is proposed that the positional specificity of cortical units is the effective information source for the task. and that this specificity is low. Abstract-Acuity
Andrews, Webb and Miller (1974) showed that disis more efficiently performed with continuous lines (Fig. 1A) than with the end parts The present study concerns a detail of a wider probalone (Fig. IS), although the end parts contain ail lem, how is the visual input organized into the unitary and apparently continuous visual world which we ex- the spatial information which is relevant to the task. A gap in a line appears always to depress spatial perperience? We can judge the spatial relation between formances. A similar result was observed by Andrews any two targets which stimulate the retina together. (1967b) and Andrews, Butcher and Buckley (1973) for It follows that there must be a connection (direct or orientation and curvature tasks. In these experiments, indirect) between every input to the visual pathway the effect of a gap could be described as precipitating and every other. How is this system of connections for small patterns the kind of processing losses engineered? usually found onty for large patterns. There was no Our approach to this problem is statistical: the geometrical statistics of the retina are known, so we evidence that gaps could depress performance below this base level. can calculate the best possible performance that the retina could support in a given visual task. This is These differences in efficiency can readily be underachieved by considering an ideal machine which stood in terms of a two-stage process, as proposed shares the same input limitations as the human retina, by Andrews. Webb and Miller (1974). At the first level but which integrates the positional information availare units which encode spatial parameters such as able to it without loss. For example, in judging the orientation, disparity, or shape; these units make full orientation of a line, the ideal machine would fit a use of the detailed positional information from the line of least squares’ through the positions of its reretina, but this is not available at any higher level. sponding receptors. The variance of the regression All spatial parameters not encoded at the first level coefficient can be calculated explicitly, and the result have to be derived from the positional specificity of compared with human psychophysical performance. the coding units, which is low. The ratio of these two variances is called efiiciency The primary coding process is intended to reflect (of use of positional information). the feature-specific behaviour of single units in the In applying this approach, some performances are visual cortex, and most of the specificities required found to be more efficient than others. Andrews of the coding process have been observed there. (1967a, b) showed that orientation in lines up to The present experiments extend the findings of 10min arc is perceived almost as accurately as the Andrews, Webb and Miller (1974). If estimation of retina would allow, but longer lines are perceived distance depends on secondary positional informawith progressively greater processing losses. Andrews, tion, as they suggest, we can expect that performance Butcher and Buckley (1973) showed that curvature wit1 be inde~ndent of stimulus size up to the field of Lines is just as eEcientIy processed as orientation, size of the units, just as the retina can not differentiate but the optimum efficiency is maintained for lines up signals which fall within one receptor. Likewise, patto 30min arc in length, so that for lines between terns which do extend more than one functional unit 10min and 30min arc, curvature is perceived more may be perceived more accurately than those which efficiently than orientation. The same is true for do not. detecting vernier offset. It appears therefore that The stimulus arrangement of Fig. l(C) was used to shape is coded independently of orientation. find whether our ability to equate separations is dependent on the length of the lines. The results should show whether there is integration of spatial ’ This is a simplified statement; see the section on e& information in the task, and if so, how efficiently the ciency for a fuller account. integration is accomplished. INTRODUCTION
tance comparison
615
616
D.
P.
ANVREWS and D. T. MILLJS
.
Two experienced subjects, DPA and DTM. were used. Both had normal vision, DTM after correction for slight astigmatism in the right eye. Figure l(B, C) shows the stimulus patterns used. The lines were horizontal and thus the distances to be compared were in the vertical meridian. The principal variable of interest was the length of the lines: the effect of varying the separation was also explored.
.
EXPERIMENT I The inter-line separation remained fixed at X2’ arc (except of course for the small variation required to
. A Fig. The and was
8
C
1. Stimulus con~gurations for distance comparisons. present paper is concerned primarily with line length line separation in configuration (C). A fixation spot shown at the centre of each figure for I set immediately preceding the stimulus presentation (2 set).
The procedure has been fully described by Andrews. Webb and Miller (1974) and is here only briefly summarized. The patterns (Fig. IB. C) were displayed on a DEC 338 computer-controlled display and viewed binocularly from a distance of f m. Line width and spot size were approximately 2 min arc. Screen luminance was 0.5 log !I L and stimuli were at 1.0 Iog ft L. Subjects were instructed to fixate on the fixation point (always at the centre of the pattern) during a pre-stimulus period of I set and the stimulus period of 2 sec. A blank period of 3 set completed a 6sec cycle. The method of constant stimuli was used; the subject gave a forced choice response after each presentation according to which separation appeared larger. Both separations varied between trials, the outer lines moving together and the central line remaining fixed. There were four separation values in force at any one time, with the computer selecting one of the four at random for each trial. A suitable range of stimulus values was set up by the computer from the responses during a short pre-run. The run proper consisted of 120 trials. Within the run the set of four stimuli was reset automatically from time to time in the light of the subject’s recent responses. The object of this procedure was to optimize measurement of the subject’s response error distribution; the optimum occurs when the stimulus range is centred on the response distribution, and encompasses about 2.7 standard deviations of the response error distribution. This condition was tracked as accurately as we could achieve within a simple computer program. (The statistical work to establish optimal stimulus range and optimal tracking procedure will be the subject of separate communications. Details of the working programme are available on request from the authors.) A probit analysis of the response counts (see Finney. 1952) gave estimates of the mean and SD of the response error distribution. The mean is a measure of the average or constant error when the separations appeared equal to the subject. (This value rarely corresponded to objective equality of separations.) The SD is a measure of sensitivity to separation change. Threshold for separation change is defined as the SD of the error distribution.
Ideal
\ IO spa*
50 l-one
length,
200
rn~narc
Fig. 2. Threshold for separation discrimination as a function of line length at a fixed line separation of 82’ arc. The upper line was fitted to the RMS value (4) of 6 observations. DTM (0) performed consistently better than DPA (0). Tire lower function shows the performance of an ideal machine in the same task. (Assumptions and calculation are described in the text.) The idea] machine has the same retinal input as the human observer. but makes rhe best possible integration of the available information. The spots were about 2min arc in length. The corresponding ideal performance is not shown, because it depends critically on the number of receptors covered. The central Function shows the ratio of observed to ideal performance on an efficiency scale (to the right of the figure). Efficiency is calculated as the ratio of ideal to observed variance. (Variance = threshold23
617
Acuity for spatial separation Line iength:164mmarc l -.... ... 26Zmm arc D
DPA
DTM
Ideal
/
................
..............
....... ........ .......... .” ..........
200
$0 Line
,’
SO
lb
separation,
2bO
mtn clrc
Fig. 3. Observed threshold for separation discrimination as a function of separation for each of two line lengths. The lower curves shows the performance of an ideal machine in the same tasks.
determine the threshoid). Six different line lengths between 2’ arc (i.e. spots) and 262’ arc were tested in the order 2, 33, 131,262, 65, 16 min arc. The series was then repeated in the reverse order. After Experiment II had been completed a third series was run to see if performance had changed. There was no evidence of a si~ific~t change in performance and the results of all three series are shown in Fig. 2. Performance was approximately constant up to a line length of about 30’ arc and then gradually improved. DTM’s performance was consistently better than DPA’s (by a factor of about 1.4). For both subjects the constant errors were very different from the null value of zero. The expected sampling variation of the constant error for a run of 120 trials is about 15% of the SD (see Finney, 1952); the observed magnitudes were up to fifteen times this value, when separations of 77.9 and 86.1’ appeared equal. DPA’s constant errors were consistently larger than those of DTM and the lower interval always seemed larger than the upper when objectively they were equal. For DTM, constant errors had this same polarity for lines up to about 65’ arc in length; small constant errors of the opposite polarity occurred among longer lines. Wolfe (1923) also reported large constant errors in midpoint estimation which varied between individuals. Most individuals see the lower interval as the larger. As in the present experiments, these differences were unrelated to visual acuity. Constant error in matching intervals has also been found to vary with fixation point and with exposure duration (Andrews and Butcher, in preparation).
EXPERIMENT II
Experiment II measured how performance varies with separation at a fixed line length. Four separations (20.5, 41, 82, 164’ arc) and two line lengths (16.4, 262’ arc) were used, the 8 combinations being tested in random order. The series was then repeated in the reverse order. The thresholds obtained are shown in Fig. 3. For a fixed line length, acuity for separation fell as the separation of the lines increased. There was an approximately linear relationship between SD and separation on a log-log plot. DTM’s performance was again better than DPA’s, but significantly so only for the greater separations. Constant errors again differed significantly from zero, and DPA’s were once more all in the same direction. RESULTS EXPRESSED AS EFFICIENCIES
Efficiency is the info~ation used by the human observer as a fraction of the information available. It is calculated by comparing human performance with that of an ideal machine. The ideal machine has a finite mosaic of inputs with the same statistics as the human retina. Given this limitation, the ideal machine makes a maximum likelihood estimate of the spatial parameter which the human observer was asked to judge. The resulting error of performance can be calculated. The case of separation judgement between 3 lines is particularlv simnle. If cones were uniformlv senar-
618
D. P. ANDREWS and D. T. MILLER
ated by a mean distance d. then a horizontal line containing II cones could be located vertically with a variance dz/12n. (The distribution of cone positions about the image is assumed rectangular.) For cones of unequal size, the best estimate of vertical position is given by
where the weight wi is proportional to l/d!. Intercone spacing is taken from Fig. 2 of Andrews, Butcher and Buckley (1973). The variance of separation judgements VT is given by VT = 2V, + 4V,, where V, is the variance of an outer line. V, the variance of the centre line. Various factors are not taken into account in the above calculation. During the 2sec of exposure duration. an ideal machine could reposition the stimulus pattern on its mosaic an arbitrary number of times. and so achieve a greater precision. The appropriate number should reflect the restraints shared by a human retina-finite integration time and limited variability of eye position during voluntary fixation. The burden of this is that Vr should be divided by an unknown factor n. the number of independent samples available to the ideal machine. Other factors have been considered by Andrews (1967b. p. 1007). and these also change the value of n. In drawing efficiency functions such as those of Figs. 2 and 3. n is taken as unity; the functions therefore represent relative efficiency rather than absolute efficiency. However, their shape remains the same for any value of n, and the conclusions we wish to draw mainly concern the shapes of the functions. Different perceptual tasks may be compared with each other in their efficiency of positional information use. The proviso for doinn so is that n mav be assumed constant: the present experiments may be compared with others in which there was voluntary fixation and a 2-set exposure duration. and preferably using the same subjects. (Where these conditions are not met, large differences in efficiency may still be established by allowing for variation of the relevant factor.) The best relative efficiency in estimating the separation of lines is about ly{ for closely spaced short lines. (For spots, efficiency might be still higher, but the calculation is suspect because it depends critically on the number of cones which the target is supposed to stimulate.) For lines of length 30’ and spaced 82’ arc, relative efficiency was about 0.12%, gradually falling with increasing separation or line length. In the worst case, relative efficiency was about 0.045’%. For targets of 30’ or over, overall efficiency fell by a factor of about 0.80 for each doubling of length or separation.
’ If curvature detection rested on a fixed central part of the target and the outer parts were ignored, then threshold (as defined) should rise as the square of line length. and the efficiency function should fall about four times as steeply as it does. It follows that the end parts of the curve are not ignored, though it remains possible that there is some differential weighting between retinal sites.
DISCUSSION
Figure 2 shows that threshold for separation judgement was nearly constant for short lines. For longer lines. threshold fell steadily, showing that the extra line length does benefit performance, though not greatly. For an ideal processor of course, all increase in line length would improve performance. Comparing human with ideal performance the most obvious difference is in level: relative efficiency is very low. Comparable values for discriminating orientations or detecting curvature are about 50% for small targets. (Andrews, Butcher and Buckley, 1973). The question arises whether this difference in et%ciency may be associated with retinal site rather than with perceptual task; might curvature detection for example be equally inefficient at retinal eccentricities of the order l-2”? This question can be answered by reference to data from Andrews, Butcher and Buckley (1973). Relative efficiency for curvature detection in a line 3” long was about IO?%, and falling at a rate of about xi-for each doubling of line length. Now, in this task, the central third of the pattern by itself provides less than 5% of useful information concerning curvature, so performance rests alsmot entirely on the outer parts of the pattern.’ In comparable retinal loci therefore. curvature detection is performed between 10 and 100 times more efficiently than distance comparison. The smaller patterns used in Experiment II support the same estimate. The shape of the efficiency function is also of interest. For short lines, efficiency fell sharply. Between a line length of 2’ arc (spots) and 30’ arc there was no change in human performance, though an ideal machine could reduce threshold tenfold given the same stimulus range. (Note that Fig. 2 does not show all of the corresponding efficiency drop, and the steep portion could justifiably be extended much higher on the left.) For lines over 30’ arc in length, efficiency falls much less sharply. The results do not permit the knee to be located very accurately, and a smooth curve might represent the data better, but there is no doubt that efficiency function falls first sharply and then shallowly as line length is increased. These results are consistent with the hypothesis that judgements of separation depend on a low-grade source of positional information. It is as though for the purpose of judging separations we had a retina with very few receptors. and all stimuli which fall within one ‘receptor’ are assigned the same location. Stimuli large enough to excite more than one ‘receptor’ can be positioned more accurately than by one alone. The retina is clearly not the true site of this resolution loss. However, the functional grain size of the visual pathway does appear to be about 30 times coarser for the purpose of distance estimation than for the best known visual performances. Loss of spatial resolution can be achieved by a convergence of many lines on a single output, and it seems a fair speculation that this does occur. With less confidence we would speculate on the site of the convergence; single unit studies show that cortical cells have response specificities for orientation, binocular disparity, spatial wavelength and the like. and these properties require extensive convergence of con-
Acuity for spatial separation nections. We propose that this is the convergence which precipitates the observed loss of positional specificity. This hypothesis has the virtue that no new entities need be proposed. It offers the further economy that no special class of ‘local sign bearers’ is required: all units share in the task of carrying positional information, albeit of low specificity. CONSTANT ERRORS
Separations judged equal were usually unequal, sometimes by as much as l@k of the distance to be judged. Large constant errors have been observed in judgements of parallelism, which were highly correlated between the two eyes (Andrews 1967a,b): Constant errors have been reported in other distance comparisons: patterns A and B (Fig. 1) elicit different constant errors from individual subjects (Andrew% Webb and Miller, 1974). All of these constant errors vary with location in the field and between subjects. These findings can be assimilated into the above hypothesis by supposing that primary coding units are mislabelled in the domains of orientation and location. For example, the responses of a cell which is orientation selective will be received higher in the pathway as si~~ying some particular o~en~tion #. The stimulus orientation 8 which elicits optimal response is not necessarily the same, and assignment errors are to be. expected unless the signal lines are perfectly ordered and scaled. Similarly, errors are to be expected in the scaling of stimulus location and stimulus size. SECOND-ORDER
INTEGRATION
The results show that separation comparison does improve with line length for long lines. It follows that
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the positional information available within large patterns is partially integrated, and it can be seen in Figs. 2 and 3 that the secondary integration is a fairly efficient process. Efficiency falls, but only slightly. Unfortunately, this high efficiency prevents any hy pothesis about how the secondary integration is effected: all summation processes which do not involve heavy loss would serve equally well. The way forward is to look for circumstances in which processing is less efficient, when the kind of loss observed may favour one hypothesis rather than another. Stimuli larger or briefer than ours might show up the limits of efficient secondary inte~ation. Ac~owledgements-We thank the Medical Research Council for their support, and Dr. P. Hammond for helpful discussion of the manuscript.
REFERENCES
Andrews D. P. (1967a) Perception of contour orientation in the central fovea, Part I: Short lines. Vision Res. 7, 915-991. Andrews D. P. (1967b) Perception of contour orientation in the centrat fovea. Part II: Spatial integration. Vision Res. 7, 999-1013. Andrews D. P., Butcher A. K. and Buckley B. R. (1973) Acuities for spatial a~angement in line figures: human and ideal observers cornoared. Vision Res. 13. 599-620. Andrews D. P. and Butche; A. K. (In preparation). Acuity for separation of visual targets. Andrews D. P., Webb J. M. and Miller D. T. (1974) Acuity for length comparison in continuous and broken lines. Vision Res., 14, 757-166. Finney D. J. (1952) Probit Analysis. Cambridge University Press, Cambridge. Polyak S. L. (1941) The Retina. Univ. of Chicago press, Chicago. Wolfe H. K. (1923) On the estimation of the middle of lines. Am. J. Psychol. 34, 313-358.