- Email: [email protected]
The spatial requirements for fine stereoacuity
THE SPATIAL
REQUIREMENTS STER~QA~~ITY*
FOR FINE
SUZANNE I’. MCKEE Smith-~ettiewei~ Institute of Visual Sciences. 2232 Webster Street. San Francisco. CA 94i f5. U.S.A.
Abstract-The finest human stereoacuity, which in some gifted individuals amounts to the detection of binocular disparities of less than 5 arc set, is found with isotated vertical target lines IO-15 min of arc in length, Summation aIong the vertical dimension of the lines is physiological in origin. and is not due to probability summation of disparity signals from multiple point targets. What is being summed is nor the quantity of light, but rather information about its distribution-positional signals leading to finer judgments of disparity. Increasing target length beyond 20arc min produces littie improvement in disparity thresholds because stereoacuity decreases at even small eccentricities. The threshold at 30 min away from the fixation point is nearly twice its value at the very center of the fovea. Fine stereoacuity has one additional constraint: the compared features should be disjoint. Connecting lines between test
and reference targets can increase the stereo threshold markedly. Stereoacuity
Binocular vision
Stereo&s
INTRODUff1Di%
Under ideal circumstances, the threshold for binocular disparity is less than $arc set, equalling the best thresholds for such mon~ular hyperacuities as vernier acuity, orientatjon detection, and motion dispIacement. Stereoacuity may be the most precise of the hyperacuities by some criteria. In observers with excellent stereoacuity, the disparity threshold frequently matches the best vernier threshold (Berry, 1948; Stigmar, 1970; Westheimer and McKee, 1979) and, as the disparity threshold usually represents the sum of the relative lateral displacements in the two eyes, the actual monocular components of the stereo threshold are about half the vernier threshold. This tendency to group the spatial hyperacuities together on the basis of their threshold magnitudes does obscure some important differences in their operations. For example, all of the hyperacuities are sensitive to the distance separating the compared features. but the optimum ~paration varies depending on the task. If the two lines which form a vernier target are separated by more tkan about 5 min arc, the vernier threshold rises significantly (Berry, t948: Westheimer and McKee, 1977). For a stereo threshold the best separation for target lines is 10 to 30 min arc (Hirsch and Weymouth, 1948; Westheimer and McKee, 19SOa). Stereoacuity has an additional positioning requirement: for greatest sensitivity, the target must fall on or near the fixation plane. Present-
*This work was conducted in the laboratory of Dr Gerald Westheimer at the University of California in Berkeley and was supported by the National Eye institute. U.S. Public Health Service under Grant EY-00592. and The Srn~~h-Kattl~well Eye Research Faundation.
Foveat vision
tfyperacuity
ing the target as little as 5 min arc in front or behind the fixation plane elevates the stereo threshold (Westheim~r and McKee, 1978}. Several decades ago, the most important item on a list of the conditions for optimum stereoacuity would surely have been the length of the target lines. The prevailing explanation for the fineness of a11 hyperacuity thresholds was that the spatial position signals, the “focal signs”. associated with the individual receptive elements stimulated by the long target lines were averaged to yield a localization more precise than the diameter of a single cone (Hering, 1899; Anderson and Weymouth, 1923). Moreover, there was experimental evidence showing that vernier acuity improved with increasing target length (French, 1917; Weymouth et ul., 1923). Then, in 1953, Ludvigh observed that a target composed of three points, two aligned vertically, and a third central point which could be displaced laterally right or left, produced misalignment thresholds as low as those found with vernier targets of any length. More recent work has shown that two points will suffice (Sullivan et al., 1972; Westheim~r and McKee, 19771, The apparent inconsistency of these findings hinges on the. particular con~guration used to demonstrate the effect of line length-a pair of ahuttbrg vertical lines. The thresholds found with dots will equal the thresholds for line targets only if the dots are separated by at least 2-4min arc: abutting dots or short abutting lines are extremely poor targets for vernier acuity. Anderson and Weymouth (19231 had also demonstrated that stereoacuity improved with increasing target length and here, the evidence for the rignificance of this variable is d~~~nlt to refute. Their target consisted of three vertical lines, separated horizontally by 26 min arc. a spacing which produces very good
192
S~XAUNE
stereoacuity. Yet even their results should have raised questions about the explanatory power of a “mean local sign” theory of hyperacuity. because the threshold for the shortest target length used in their study f3 arc min) was a mere 4 arc sec. Lengthening the target to a subtense of 3’ lowered the disparity threshold to 2 arc sec. While the findings of Anderson and Weymouth do not reveal the mechanism of fine stereoacuity. their results may instead show the spatial summation properties of the physiological units responsive to disparity. It is this aspect which I shall examine in some detail in this study. Stereoacuity has some other. rather unusual. configuration specifications, which I shall describe as well. METHODS
The basic target pattern used in these experiments to measure stereoscopic acuity consisted of three bright vertical lines presented against a dark background and separated laterally by 13 min arc, a separation found in earlier work to produce good stereoacuity. The subject’s task was to judge whether the central test line was in front or behind the plane defined by the outer two reference lines. For some experiments only two target lines were used; the subject then had to judge the depth of the right-hand line relative to that of the left line. In the experiments using only two lines, the lateral separation between test and reference was 20min arc. Variations of these patterns used in the experimental manipulations are diagrammed in the figures. The target pattern was drawn by computer-generated signals on the screens of two 602 Tektronix display units. cathode ray tubes equipped with a P-4 phosphor. The images on the two CRT screens were super-imposed by a beam-splitting pellicle. Orthogonally-oriented poIarizers placed in front of the CRT screens and the subject’s eyes guaranteed that only one screen was visible to each eye. Target lines were actually composed of a row of small dots. about 1 min arc in diameter and separated by 3Osec arc. This structure. however. could not be resolved by the subject. These dots could be positioned to an accuracy of 1.7 set arc. a value which limited the minimum change in symmetric binocular disparity to 3.4 set arc. Target luminance was estimated by measuring with an SE1 photometer the luminance of a small square patch also made up of dots separated by 30 set arc and generated at a “refresh rate” identical to that of the target lines (IOmsec). Except where indicated in the results section. target luminance was 6ft-L. To maintain high target contrast. the screens were shaded from room illumination by a box in front of the CRT screens \thlch housed the pellicle. Thus, the illuminimce of the screens was only 0.02 ft-L. Ambient room llluminarion. supplied by an incandescent lamp, was at a low photopic level; furniture and equipment in the room were clearly visible to the subject. The viewing distance for these experiments was always 3 m.
P. MCKEIDuring an experimental session. the stimulus pattern appeared on the CRT screens every 3 sec. Target duration was 500 msec except for the eccentricit! experiments, where the duration was shortened to 150msec to prevent voluntary eye movements to the position of the eccentrically-placed target. In the period between stimulus presentations. the subject saw a fixation pattern consisting of four brackets. each bracket made of two 4min iines connected to form a right angle. The four brackets outlined a square 314’ on a side. This pattern. which was visible for 3 set between target presentations. disappeared during the 500 msec in which the stimulus pattern was presented. The sharp outline of the brackets helped the subjects maintain optimal accommodation and convergence on the target plane. The fine stereoacuity of both subjects is evidence that they were able to maintain adequate accommodation and convergence since both defocus and inappropriate convergence degrade stereoacuity markedly (Westheimer and McKee. 1978: 198Ob). The experimental procedure used to obtain these data was the method of constant stimuli. On each stimulus presentation. the test line was shown with a different disparity chosen at random from a set of seven equaliy-spaced positions: three uncrossed disparities, three crossed disparities. and a control position of no disparity relative to the reference lines. The reference lines always appeared in the same plane as the fixation pattern. On each trial the subject indicated the apparent position of the test line. i.e. in front or behind the reference lines, by setting a switch in one of two positions. If the subject’s choice was incorrect. a brief error flash appeared on the CRT screen but no error Rashes were given for either response to the control disparity. All thresholds in this paper are based on at least 300 forced choice responses. Threshold is defined here as that stimulus disparity producing a change in response frequency from the SO”, to the 75”(, level of the psychometric function. The data which generate the psychometric function are the percentage of trials on which the subject responded “behind” for each of the seven tested disparities. Ideally. this percentage will change from 0 to loo”,, as the test target is changed from the most extreme crossed disparity to the most extreme uncrossed disparity of the tested set. To estimate the stimuli corresponding to the relevant response levels. a cumulative normal curve was fitted to the data by probit analysis. The threshold was estimated to be equal to half the distance (in disparity) between the stimuli corresponding to the 25 and 75”;; level of the psychometric function, a value which averages the minimum detectable uncrossed and crossed disparities. Probit analysis was also used to estimate the standard error of the threshold; for 3M) responses. the standard error is usually about lo’?;, of the threshold. The two subjects used for these experiments were chosen for their exceptionally fine stereoacuity. One
193
The spatial requirements for fine stereoacuity subject (SM.) was slightly hyperopic ( -0.5 D) and the other (D.M.) somewhat myopic (- 1.0 D). The sub-
jects’ resolution acuity for the 3 m viewing distance was 20/20, with appropriate optical corrections if needed. Stereoacuity of less than 5 arc set is a rare ability in the human population, but the improvement in the stereo threshold found with increasing target length has been noted in many other subjects used in this laboratory in the past whatever the quality of their stereoacuity. As the first experiment will demonstrate, this basic finding is easy to replicate and thus, my results will probably apply to the stereoacuity of less gifted human subjects. RESULTS
open squares plotted in Fig. 1 give the threshold disparities for the central test line as a function of target length, thresholds obtained through the use of the computer-driven “electronic stereoscope” described above. The continuous curve is based on the actual values of the thresholds measured by Anderson and Weymouth (1923) data obtained with three pieces of thread and an episcotister. In spite of a 60 year difference in technology, there is considerable agreement, particularly on the two main findings: (1) the improvement in the disparity threshold with line length is small, barely a factor of two, and (2) increasing the target length beyond 20 min arc has a negligible effect on the stereo threshold. Although the effect of target length is slight. these results still argue for some sort of summation. But what is being summed? The quantity of light in point targets may be inadequate for optimum stereoacuity and increasing the length may lower thresholds through the simple summation of luminance. HowThe
01 0
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Fig. 1. Threshold for horizontal binocular disparity as a function of length of target lines. Subject identified whether the center line appeared to be in front or behind the plane defined by the outer two reference lines. Target duration 500 msec. Target separation I3 arc min. Bars through symbols represent + 1 SE. Continuous curve is based on the data presented by Anderson and Weymouth (1923).
1
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Fig. 2. Threshold for horizontal binocular disparity (in arc set) of central test line relative to the position in depth of outer two reference lines. Targets on the left I arc min in length. targets on the right I5 arc min long. Vertical separation 13 arc min. Increasing target length along the horizontal axis does not improve stereoacuity.
ever, tripling the intensity of targets 1 min arc in length produced no improvement in the disparity threshold. The targets shown in Fig. 2 provide additional evidence that quantity of light is not the limiting factor for stereoacuity with point targets. If lines are lengthened horizontally, rather than vertically, there is no improvement in the stereoacuity, although clearly there is more light in the longer lines. Only the endpoints of the horizontal lines contain any information about horizontal disparity. As Anderson and Weymouth surmised. stereoacuity improves with lengthening along the vertical dimension because more receptive elements are stimulated by a positional signal related to horizontal disparity. The summation process ceases when the lines exceed a length of about 20min arc as though the additional information in the longer lines fails to produce a useful disparity signal. The next experiment shows that there is a marked decline in stereoacuity at the small eccentricities stimulated by the end portions of the lines. The thresholds in Fig. 3 were measured using short targets (4 arc min) presented for a duration (150 msec) too brief to permit a voluntary change in fixation. Subjects were instructed to fixate the center of the fixation square and on each trial the three-line target could appear at random in one of several possible positions: in the center, or above or below the center at one of the eccentricities indicated on the graph. Subjects were instructed to fixate the center and this behaviour was encouraged by presenting a disproportionate number of targets in the central position. The results show that stereoacuity has fallen to half its best value at an eccentricity of 30 arc min. a decline which exceeds the decrement observed in ordinary visual acuity at this eccentricity (Weymouth er (II.. 1928; Millodot. 1972). The small improvement in the stereo threshold found with long lines could arise from probability
receptive field of a hypothetical disparity unit. The postulated summation field is about IO-I5 min arc long. and once the two dots are separated by more than this distance. there is an abrupt decrement in sensitivity. On the other hand. stereoacuity falls off SO steeply with eccentricity that the decrement observed when the points are separated could represent the probability summation of less sensitive eccentric point detectors. but the curves do not quite fit the fall-off predicted by the eccentricity curves in Fig. 3 for a probability summation scheme. Simple probabihty summation would predict that the thresholds for the endpoint targets would fail off monotonically as OEZ or both points are positioned on eccentric regions. In fact. the continuous curves in Fig. 4 are slightly U-shaped. There are circumstances in which the presence of physiological summation would degrade stereoacuity rather than improve it. If targets at different depths are sufficiently close together to fall within a summation area. then the effective depth of the whole configuration would be an average of the target components. One such configuration is diagrammed in the upper right hand corner of Fig. 5-a vertical test target consisting of three line segments. two at the the
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Fig. 3. Threshold for horizontal binocular disparity of central test line as a function of vertical eccentricity. Targets. 4arc min in length. appeared at random above or below the center of the fixation square for 150 msec. Fine stereoacuity is a property only of the foveola.
summation among independent sensing units which are responsive only to point targets. Alternatively. the longer vertical targets may be a more adequate stimulus for a specialized disparity sensing apparatus which is optimally stimulated by Iirzes with the improved stereoacuity due to physiological summation. Physiological summation impiies summation only within some circumscribed area-the region which stimulates a particular unit. Consider a target similar to the three-line configuration used for the first experiments. but constructed only of the endpoints of the target lines (see diagram at the top of Fig. 4). If the summation process depends on probability summation of point signals. then the stereoacuity should be indifferent to the positioning of the points, provided that the points fall within the central fovea-within a region of about i5-20min in diameter. The solid curves in Fig. 4 show the thresholds for the “endpoint” targets as a function of the distance between the endpoints. For comparison, the thresholds for line targets are also plotted as a function of line length (dotted curves). These data support a physiological summation interpretation: the endpoints are nearly as effective as an entire target line until one endpoint falls outslde
01 0
t3
30
43
Fig. 4. Threshold disparity for central test elements as a function of target length (O-----O) or separation between endpoints (C----~--W. The endpoint targets are adequate substitutes for a complete line until the points are separsymbols Bars through IO- 15min arc. ated by represent & 1 SE.
The spatial
requirements
Fig. 5. Threshold for binocular disparity as a function of the length of the test segment, indicated in the diagram on the right by the two offset segments (A-A,). Test segment is equal to one third of total target length. Control data showing thresholds for isolated test targets equal in
length to only the test segment of 3-segmented (O----O).
target
depth of the fixation plane and a central segment at a different depth. If the depth of the outer line segments is summed physiologically with the depth signal from the central segment, then the minimum detectable disparity of the central test segment will rise due to the
I II I
195
for fine stereoacuit!
dilution of the disparity signal. The magnitude of the increase in the psychophysical threshold would depend on the details of the physiological arrangements, such as the degree of overlap of the receptive fields of the disparity units. Nevertheless. this loss due to summation should cease once the central segment exceeds the vertical limits of the summation region. The continuous curves in Fig. 5 show the disparity threshold for a three-segmented test line as a function of the length of the central segment. The target lines are actually three times this length but the outer segments of the test line lie in the same plane as the parallel reference lines. For comparison. data based on a test target equal in length to the central segment alone are also plotted (dotted lines). As expected the threshold for the three-segmented test line is elevated. The surprise is that the elevation persists even when the central segment is 25-30min long. a summation region which is much larger than the estimate from the previous experiment. Tyler (1975a) used a single cycle of a square wave as a stereo target, a target similar to the central segmented line shown in Fig. 5. and his threshold data also reach a minimum at a pulse width (half-cycle) of 30 min or greater. Fluctuations in the relative vertical fixation positions of the two eyes could produce faulty fusion of this target. which would also degrade stereoacuity. but it is unlikely that vertical fixation disparity would be as large as 25 or 30 arc min. The outer segments of the three-segmented test line seem to be interfering with disparity detection. Richards (1973) has described a related disparity masking effect produced by attaching target components at one disparity to the ends of a long (2 ) test target at another disparity. It is tempting to attribute the elevation in the thresholds shown in Fig. 5 to physiological inhibition, but usually such inhibitory interactions are between signals produced by stimuli falling on adjacent spatial regions. The experiment
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Fig. 6. Each panel shows threshold binocular disparity (in arc set) for test target. indicated in the drawing to have a different disparity from reference lines by adjacent continuous and dotted lines. Left panel: target length I5 arc min. Lateral separation 13 arc min. Middle panel: total target length 45 arc min. central test segment 15 arc min. Lateral separation I3 arc min. Right panel: total target length 45 arc min. Central test segment 8 arc min. Shortening the test segment to create small breaks between central and outer segments improves disparity detection.
196
SCZAKYE
P.
MCKEE
diagrammed in Fig. 6 shows that no simple spatial arrangement for inhibition can explain the increase in the disparity thresholds. Thresholds improve when the central disparate section of the three-segmented test line is shortened. leaving small breaks (holes) between the outer segments and the central portion even though the distance between center and surround segments is unchanged. Apparently. there is a fundamental difficulty associated with detecting disparity when target components at different disparities are joined in some fashion. Westheimer (1979) noted that if the test and reference lines of the traditional Howard-Dolman target were connected to form a square, thresholds rose precipitously. Figure 7 shows how connections between target lines can affect stereoacuity for targets of various lengths. The vertical lines were separated laterally by 20arc min. a distance which produces quite low thresholds in the absence of the horizontal connections. The “box” targets inadequately signal the presence of disparity even when their vertical length reaches one degree, a length which places the endpoints outside of the foveola.* There are. however. no special attributes of endpoint connections. Joining the target lines at other places along their length also raises the threshold (Fig. 8). As with the three-segmented stimulus (Fig. 6), a small break in the connections improves the stereo threshold. Fig. 7. Threshold for horizontal binocular disparity as a DISCUSSION
The finest stereoacuity requires isolated vertical target lines of at least 1Omin in length. imaged in the foveola. The small improvement in sensitivity found with increasing target length undoubtedly reflects physiological summation of disparity signals within specialized sub-units sensitive to depth. This summation property would seem to distinguish stereoacuity from the monocular hyperacuities. because two properly positioned points will produce excellent vernier acuity, equal to that found with line targets. However, the most precise detection of line orientation also depends on the summation of positional signals, and, as in the case of stereoacuity. the sum-
mation region is 10-15 min arc (Andrews cf a/.. 1973; Westheimer. 1982). The influence of vertical length on stereo thresholds raises the ghost of a “mean local sign” theory of hyperacuity, but the data offer scant support for a model in which averaging along the length of the *All subjects tested in the Westheimer taboratory show a diminished capacity to detect the disparity of two hnes when they are joined to form a “box” as in Figs 7 and 8. a result first described in a tong monograph by Werner (1937).This ‘*connectivity” effect is most spectacular in the visual system of the author. Nevertheless. there are some “box” configurations in which the disparity of the vertical contours is easily detected. even by myself, as will be shown in a subsequent paper by G. J. Mitchison and Gerald Westheimer.
function of length of two vertical lines separated laterally by 20arc min. Subject judged whether right vertical line was in front or behind the left vertical line. Target duration 500 msec. Isolated vertical lines (Cl----O); vertical lines connected by horizontal lines at ends to form box IO-----Ok
target lines can account for a spatial precision finer than a single fovea1 cone. Even point targets can be localized in the third dimension with a precision of 68 arc sec. The enhanced detection of binocular disparity found with longer line targets is likely to be a secondary consequence of another functional objective of the human visual system--the identification of local contours which share a common disparity. Certainly. the more puzzling results come from the experiments on the stereoacuity for continuous figures. Connections between features at different disparities impair the ability to detect small differences in depth. Slanted surfaces and inclined lines-stimuli in which there are continuous changes in disparityare also known to be poor targets for stereoacuity (Ogle and Ellerbrock. 1946; Gulick and Lawson. 1976; Youngs. 1976). This curious suppression of the disparity information which is embedded in a larger continuous form highlights an interesting conflict in binocular processing. The cortex must solve two problems in dealing with the inputs from the two eyes. The obvious problem is the detection of the relative disparities of features, but the visual system must also solve the “fusion” problem-determining which
197
The spatial requirements for fine stereoacuit?
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Fig. 8. Each panel gives the horizontal binocular disparity for the target configuration shown above it. Lateral separation for all targets 20 arc min. For subject SM. vertical length 60 arc min. For subject D.M. vertical length IO arc min. Any type of horizontal connection between the vertical target lines degrades stereoacuity; breaking the connection improves the disparity detection somewhat.
feature in the image in one eye belongs to a feature from the image in the other eye. Ultimately “fusing” a binocular stimulus means discarding information about the difference in the positions of the two monocular features. For example, Tyler (1975b) found that stereomovement, consisting of equal and opposite displacements of the monocular half-images of a binocularly-fused image, is undetectable at displacement amplitudes which are easily seen monocularly. The process of fusion and disparity detection may proceed in parallel, or as a “co-operative” interaction (Julesz, 1971; Nelson, 1975; Marr and Poggio. 1976). Nevertheless, there may be occasional competition between the two processes, leading to a decrement in
disparity detection. Continuous figures could constitute a powerful input to a “global” fusion mechanism which might average the disparities of the component features to assign a single depth value to the figure as a whole. Local variations in disparity which are intrinsically detectable would produce inadequate signals when pitted against the averaged depth assigned by the fusion process, the result being that only larger disparity differences could be observed in continuous contours. Ac~)IOM./edgclnellts--l would like to thank Dr Gerald Westheimer, Dr Jeremiah Nelson, and Dr G. J. Mitcluson for helpful discussions about the meaning of these results. REFERENCES Andersen E. E. and Weymouth F. W. (1923) Visual perception and the retinal mosaic: I. Retmal mean local
sign-an explanation of the fineness of binocular perception of distance. Am. J. Physiol. 64, 561-594. Andrews D. P.. Butcher A. K. and Buckley B. R. (1973) Acuities for spatial arrangement in line figures: Human and ideal observers compared. vision Res. 13, 599-620. Berry R. N. (1948) Quantitative relations among vernier. real depth and stereoscopic depth acuities. J. esp. Psj,chol. 38, 708-721. French J. W. (1917) The unaided eye; part III. Trans. Opt. Sot. 21, 127-156. &lick W. L. and Lawson R. B. (1976) Hunrat Srereopsis. a Ps#lophysical Approach, Chap. 8, pp. 201-224. Oxford Univ. Press. Hering E. (1899) Uber die Grenzen der Sehscharfe. Brr. Math-phys. Cl. d. Konigl. Sachs Natur. Wiss. Teil., pp. 1624.
Gesell.
Wiss.
Leip:ig.
Hirsch M. J. and Weymouth F. W. (1948) Distance discrimination. II. Effect on threshold of lateral separation of the test objects. Archs Ophthal. 39, 224-231. Julesz B. (1971) Foundations of CyclopeanPerception. Univ. of Chicago Press. Chicago. Ludvigh E. (1953) Direction sense of the eye. Am. J. Opht/la/. 36, 139- 142. Marr D. and Poggio T. (1976) Cooperative computation of stereo disparity. Science 194, 283-287. Millodot M. (1972) Variation of visual acuity in the central region of the retina. Br. J. Physiol. Opr. 27. 2428. Nelson J. I. (1975) Globality and stereoscopic fusion in binocular vision. J. Theor. Biol. 49, l-88. Ogle K. N. and Ellerbrock V. J. (1946) Cyclofusional movements. Archs Ophrhal. 36. 7CKk735. Richards W. (1973) Disparity masking. vision Res. 12, 1113-1124. Stigmar G. (1970) Observations on vernier and stereoacuity with special reference lo their relationship. Acta ophthal.
48. 979-997.
Sullivan G. D., Oatley K. and Sutherland N. S. (1972) Vernier acuity as affected by target length and separation. Percept.
Psychophys.
12, 438444.
REQUIREMENTS STER~QA~~ITY*
FOR FINE
SUZANNE I’. MCKEE Smith-~ettiewei~ Institute of Visual Sciences. 2232 Webster Street. San Francisco. CA 94i f5. U.S.A.
Abstract-The finest human stereoacuity, which in some gifted individuals amounts to the detection of binocular disparities of less than 5 arc set, is found with isotated vertical target lines IO-15 min of arc in length, Summation aIong the vertical dimension of the lines is physiological in origin. and is not due to probability summation of disparity signals from multiple point targets. What is being summed is nor the quantity of light, but rather information about its distribution-positional signals leading to finer judgments of disparity. Increasing target length beyond 20arc min produces littie improvement in disparity thresholds because stereoacuity decreases at even small eccentricities. The threshold at 30 min away from the fixation point is nearly twice its value at the very center of the fovea. Fine stereoacuity has one additional constraint: the compared features should be disjoint. Connecting lines between test
and reference targets can increase the stereo threshold markedly. Stereoacuity
Binocular vision
Stereo&s
INTRODUff1Di%
Under ideal circumstances, the threshold for binocular disparity is less than $arc set, equalling the best thresholds for such mon~ular hyperacuities as vernier acuity, orientatjon detection, and motion dispIacement. Stereoacuity may be the most precise of the hyperacuities by some criteria. In observers with excellent stereoacuity, the disparity threshold frequently matches the best vernier threshold (Berry, 1948; Stigmar, 1970; Westheimer and McKee, 1979) and, as the disparity threshold usually represents the sum of the relative lateral displacements in the two eyes, the actual monocular components of the stereo threshold are about half the vernier threshold. This tendency to group the spatial hyperacuities together on the basis of their threshold magnitudes does obscure some important differences in their operations. For example, all of the hyperacuities are sensitive to the distance separating the compared features. but the optimum ~paration varies depending on the task. If the two lines which form a vernier target are separated by more tkan about 5 min arc, the vernier threshold rises significantly (Berry, t948: Westheimer and McKee, 1977). For a stereo threshold the best separation for target lines is 10 to 30 min arc (Hirsch and Weymouth, 1948; Westheimer and McKee, 19SOa). Stereoacuity has an additional positioning requirement: for greatest sensitivity, the target must fall on or near the fixation plane. Present-
*This work was conducted in the laboratory of Dr Gerald Westheimer at the University of California in Berkeley and was supported by the National Eye institute. U.S. Public Health Service under Grant EY-00592. and The Srn~~h-Kattl~well Eye Research Faundation.
Foveat vision
tfyperacuity
ing the target as little as 5 min arc in front or behind the fixation plane elevates the stereo threshold (Westheim~r and McKee, 1978}. Several decades ago, the most important item on a list of the conditions for optimum stereoacuity would surely have been the length of the target lines. The prevailing explanation for the fineness of a11 hyperacuity thresholds was that the spatial position signals, the “focal signs”. associated with the individual receptive elements stimulated by the long target lines were averaged to yield a localization more precise than the diameter of a single cone (Hering, 1899; Anderson and Weymouth, 1923). Moreover, there was experimental evidence showing that vernier acuity improved with increasing target length (French, 1917; Weymouth et ul., 1923). Then, in 1953, Ludvigh observed that a target composed of three points, two aligned vertically, and a third central point which could be displaced laterally right or left, produced misalignment thresholds as low as those found with vernier targets of any length. More recent work has shown that two points will suffice (Sullivan et al., 1972; Westheim~r and McKee, 19771, The apparent inconsistency of these findings hinges on the. particular con~guration used to demonstrate the effect of line length-a pair of ahuttbrg vertical lines. The thresholds found with dots will equal the thresholds for line targets only if the dots are separated by at least 2-4min arc: abutting dots or short abutting lines are extremely poor targets for vernier acuity. Anderson and Weymouth (19231 had also demonstrated that stereoacuity improved with increasing target length and here, the evidence for the rignificance of this variable is d~~~nlt to refute. Their target consisted of three vertical lines, separated horizontally by 26 min arc. a spacing which produces very good
192
S~XAUNE
stereoacuity. Yet even their results should have raised questions about the explanatory power of a “mean local sign” theory of hyperacuity. because the threshold for the shortest target length used in their study f3 arc min) was a mere 4 arc sec. Lengthening the target to a subtense of 3’ lowered the disparity threshold to 2 arc sec. While the findings of Anderson and Weymouth do not reveal the mechanism of fine stereoacuity. their results may instead show the spatial summation properties of the physiological units responsive to disparity. It is this aspect which I shall examine in some detail in this study. Stereoacuity has some other. rather unusual. configuration specifications, which I shall describe as well. METHODS
The basic target pattern used in these experiments to measure stereoscopic acuity consisted of three bright vertical lines presented against a dark background and separated laterally by 13 min arc, a separation found in earlier work to produce good stereoacuity. The subject’s task was to judge whether the central test line was in front or behind the plane defined by the outer two reference lines. For some experiments only two target lines were used; the subject then had to judge the depth of the right-hand line relative to that of the left line. In the experiments using only two lines, the lateral separation between test and reference was 20min arc. Variations of these patterns used in the experimental manipulations are diagrammed in the figures. The target pattern was drawn by computer-generated signals on the screens of two 602 Tektronix display units. cathode ray tubes equipped with a P-4 phosphor. The images on the two CRT screens were super-imposed by a beam-splitting pellicle. Orthogonally-oriented poIarizers placed in front of the CRT screens and the subject’s eyes guaranteed that only one screen was visible to each eye. Target lines were actually composed of a row of small dots. about 1 min arc in diameter and separated by 3Osec arc. This structure. however. could not be resolved by the subject. These dots could be positioned to an accuracy of 1.7 set arc. a value which limited the minimum change in symmetric binocular disparity to 3.4 set arc. Target luminance was estimated by measuring with an SE1 photometer the luminance of a small square patch also made up of dots separated by 30 set arc and generated at a “refresh rate” identical to that of the target lines (IOmsec). Except where indicated in the results section. target luminance was 6ft-L. To maintain high target contrast. the screens were shaded from room illumination by a box in front of the CRT screens \thlch housed the pellicle. Thus, the illuminimce of the screens was only 0.02 ft-L. Ambient room llluminarion. supplied by an incandescent lamp, was at a low photopic level; furniture and equipment in the room were clearly visible to the subject. The viewing distance for these experiments was always 3 m.
P. MCKEIDuring an experimental session. the stimulus pattern appeared on the CRT screens every 3 sec. Target duration was 500 msec except for the eccentricit! experiments, where the duration was shortened to 150msec to prevent voluntary eye movements to the position of the eccentrically-placed target. In the period between stimulus presentations. the subject saw a fixation pattern consisting of four brackets. each bracket made of two 4min iines connected to form a right angle. The four brackets outlined a square 314’ on a side. This pattern. which was visible for 3 set between target presentations. disappeared during the 500 msec in which the stimulus pattern was presented. The sharp outline of the brackets helped the subjects maintain optimal accommodation and convergence on the target plane. The fine stereoacuity of both subjects is evidence that they were able to maintain adequate accommodation and convergence since both defocus and inappropriate convergence degrade stereoacuity markedly (Westheimer and McKee. 1978: 198Ob). The experimental procedure used to obtain these data was the method of constant stimuli. On each stimulus presentation. the test line was shown with a different disparity chosen at random from a set of seven equaliy-spaced positions: three uncrossed disparities, three crossed disparities. and a control position of no disparity relative to the reference lines. The reference lines always appeared in the same plane as the fixation pattern. On each trial the subject indicated the apparent position of the test line. i.e. in front or behind the reference lines, by setting a switch in one of two positions. If the subject’s choice was incorrect. a brief error flash appeared on the CRT screen but no error Rashes were given for either response to the control disparity. All thresholds in this paper are based on at least 300 forced choice responses. Threshold is defined here as that stimulus disparity producing a change in response frequency from the SO”, to the 75”(, level of the psychometric function. The data which generate the psychometric function are the percentage of trials on which the subject responded “behind” for each of the seven tested disparities. Ideally. this percentage will change from 0 to loo”,, as the test target is changed from the most extreme crossed disparity to the most extreme uncrossed disparity of the tested set. To estimate the stimuli corresponding to the relevant response levels. a cumulative normal curve was fitted to the data by probit analysis. The threshold was estimated to be equal to half the distance (in disparity) between the stimuli corresponding to the 25 and 75”;; level of the psychometric function, a value which averages the minimum detectable uncrossed and crossed disparities. Probit analysis was also used to estimate the standard error of the threshold; for 3M) responses. the standard error is usually about lo’?;, of the threshold. The two subjects used for these experiments were chosen for their exceptionally fine stereoacuity. One
193
The spatial requirements for fine stereoacuity subject (SM.) was slightly hyperopic ( -0.5 D) and the other (D.M.) somewhat myopic (- 1.0 D). The sub-
jects’ resolution acuity for the 3 m viewing distance was 20/20, with appropriate optical corrections if needed. Stereoacuity of less than 5 arc set is a rare ability in the human population, but the improvement in the stereo threshold found with increasing target length has been noted in many other subjects used in this laboratory in the past whatever the quality of their stereoacuity. As the first experiment will demonstrate, this basic finding is easy to replicate and thus, my results will probably apply to the stereoacuity of less gifted human subjects. RESULTS
open squares plotted in Fig. 1 give the threshold disparities for the central test line as a function of target length, thresholds obtained through the use of the computer-driven “electronic stereoscope” described above. The continuous curve is based on the actual values of the thresholds measured by Anderson and Weymouth (1923) data obtained with three pieces of thread and an episcotister. In spite of a 60 year difference in technology, there is considerable agreement, particularly on the two main findings: (1) the improvement in the disparity threshold with line length is small, barely a factor of two, and (2) increasing the target length beyond 20 min arc has a negligible effect on the stereo threshold. Although the effect of target length is slight. these results still argue for some sort of summation. But what is being summed? The quantity of light in point targets may be inadequate for optimum stereoacuity and increasing the length may lower thresholds through the simple summation of luminance. HowThe
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Fig. 1. Threshold for horizontal binocular disparity as a function of length of target lines. Subject identified whether the center line appeared to be in front or behind the plane defined by the outer two reference lines. Target duration 500 msec. Target separation I3 arc min. Bars through symbols represent + 1 SE. Continuous curve is based on the data presented by Anderson and Weymouth (1923).
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Fig. 2. Threshold for horizontal binocular disparity (in arc set) of central test line relative to the position in depth of outer two reference lines. Targets on the left I arc min in length. targets on the right I5 arc min long. Vertical separation 13 arc min. Increasing target length along the horizontal axis does not improve stereoacuity.
ever, tripling the intensity of targets 1 min arc in length produced no improvement in the disparity threshold. The targets shown in Fig. 2 provide additional evidence that quantity of light is not the limiting factor for stereoacuity with point targets. If lines are lengthened horizontally, rather than vertically, there is no improvement in the stereoacuity, although clearly there is more light in the longer lines. Only the endpoints of the horizontal lines contain any information about horizontal disparity. As Anderson and Weymouth surmised. stereoacuity improves with lengthening along the vertical dimension because more receptive elements are stimulated by a positional signal related to horizontal disparity. The summation process ceases when the lines exceed a length of about 20min arc as though the additional information in the longer lines fails to produce a useful disparity signal. The next experiment shows that there is a marked decline in stereoacuity at the small eccentricities stimulated by the end portions of the lines. The thresholds in Fig. 3 were measured using short targets (4 arc min) presented for a duration (150 msec) too brief to permit a voluntary change in fixation. Subjects were instructed to fixate the center of the fixation square and on each trial the three-line target could appear at random in one of several possible positions: in the center, or above or below the center at one of the eccentricities indicated on the graph. Subjects were instructed to fixate the center and this behaviour was encouraged by presenting a disproportionate number of targets in the central position. The results show that stereoacuity has fallen to half its best value at an eccentricity of 30 arc min. a decline which exceeds the decrement observed in ordinary visual acuity at this eccentricity (Weymouth er (II.. 1928; Millodot. 1972). The small improvement in the stereo threshold found with long lines could arise from probability
receptive field of a hypothetical disparity unit. The postulated summation field is about IO-I5 min arc long. and once the two dots are separated by more than this distance. there is an abrupt decrement in sensitivity. On the other hand. stereoacuity falls off SO steeply with eccentricity that the decrement observed when the points are separated could represent the probability summation of less sensitive eccentric point detectors. but the curves do not quite fit the fall-off predicted by the eccentricity curves in Fig. 3 for a probability summation scheme. Simple probabihty summation would predict that the thresholds for the endpoint targets would fail off monotonically as OEZ or both points are positioned on eccentric regions. In fact. the continuous curves in Fig. 4 are slightly U-shaped. There are circumstances in which the presence of physiological summation would degrade stereoacuity rather than improve it. If targets at different depths are sufficiently close together to fall within a summation area. then the effective depth of the whole configuration would be an average of the target components. One such configuration is diagrammed in the upper right hand corner of Fig. 5-a vertical test target consisting of three line segments. two at the the
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Fig. 3. Threshold for horizontal binocular disparity of central test line as a function of vertical eccentricity. Targets. 4arc min in length. appeared at random above or below the center of the fixation square for 150 msec. Fine stereoacuity is a property only of the foveola.
summation among independent sensing units which are responsive only to point targets. Alternatively. the longer vertical targets may be a more adequate stimulus for a specialized disparity sensing apparatus which is optimally stimulated by Iirzes with the improved stereoacuity due to physiological summation. Physiological summation impiies summation only within some circumscribed area-the region which stimulates a particular unit. Consider a target similar to the three-line configuration used for the first experiments. but constructed only of the endpoints of the target lines (see diagram at the top of Fig. 4). If the summation process depends on probability summation of point signals. then the stereoacuity should be indifferent to the positioning of the points, provided that the points fall within the central fovea-within a region of about i5-20min in diameter. The solid curves in Fig. 4 show the thresholds for the “endpoint” targets as a function of the distance between the endpoints. For comparison, the thresholds for line targets are also plotted as a function of line length (dotted curves). These data support a physiological summation interpretation: the endpoints are nearly as effective as an entire target line until one endpoint falls outslde
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Fig. 4. Threshold disparity for central test elements as a function of target length (O-----O) or separation between endpoints (C----~--W. The endpoint targets are adequate substitutes for a complete line until the points are separsymbols Bars through IO- 15min arc. ated by represent & 1 SE.
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Fig. 5. Threshold for binocular disparity as a function of the length of the test segment, indicated in the diagram on the right by the two offset segments (A-A,). Test segment is equal to one third of total target length. Control data showing thresholds for isolated test targets equal in
length to only the test segment of 3-segmented (O----O).
target
depth of the fixation plane and a central segment at a different depth. If the depth of the outer line segments is summed physiologically with the depth signal from the central segment, then the minimum detectable disparity of the central test segment will rise due to the
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dilution of the disparity signal. The magnitude of the increase in the psychophysical threshold would depend on the details of the physiological arrangements, such as the degree of overlap of the receptive fields of the disparity units. Nevertheless. this loss due to summation should cease once the central segment exceeds the vertical limits of the summation region. The continuous curves in Fig. 5 show the disparity threshold for a three-segmented test line as a function of the length of the central segment. The target lines are actually three times this length but the outer segments of the test line lie in the same plane as the parallel reference lines. For comparison. data based on a test target equal in length to the central segment alone are also plotted (dotted lines). As expected the threshold for the three-segmented test line is elevated. The surprise is that the elevation persists even when the central segment is 25-30min long. a summation region which is much larger than the estimate from the previous experiment. Tyler (1975a) used a single cycle of a square wave as a stereo target, a target similar to the central segmented line shown in Fig. 5. and his threshold data also reach a minimum at a pulse width (half-cycle) of 30 min or greater. Fluctuations in the relative vertical fixation positions of the two eyes could produce faulty fusion of this target. which would also degrade stereoacuity. but it is unlikely that vertical fixation disparity would be as large as 25 or 30 arc min. The outer segments of the three-segmented test line seem to be interfering with disparity detection. Richards (1973) has described a related disparity masking effect produced by attaching target components at one disparity to the ends of a long (2 ) test target at another disparity. It is tempting to attribute the elevation in the thresholds shown in Fig. 5 to physiological inhibition, but usually such inhibitory interactions are between signals produced by stimuli falling on adjacent spatial regions. The experiment
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Fig. 6. Each panel shows threshold binocular disparity (in arc set) for test target. indicated in the drawing to have a different disparity from reference lines by adjacent continuous and dotted lines. Left panel: target length I5 arc min. Lateral separation 13 arc min. Middle panel: total target length 45 arc min. central test segment 15 arc min. Lateral separation I3 arc min. Right panel: total target length 45 arc min. Central test segment 8 arc min. Shortening the test segment to create small breaks between central and outer segments improves disparity detection.
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diagrammed in Fig. 6 shows that no simple spatial arrangement for inhibition can explain the increase in the disparity thresholds. Thresholds improve when the central disparate section of the three-segmented test line is shortened. leaving small breaks (holes) between the outer segments and the central portion even though the distance between center and surround segments is unchanged. Apparently. there is a fundamental difficulty associated with detecting disparity when target components at different disparities are joined in some fashion. Westheimer (1979) noted that if the test and reference lines of the traditional Howard-Dolman target were connected to form a square, thresholds rose precipitously. Figure 7 shows how connections between target lines can affect stereoacuity for targets of various lengths. The vertical lines were separated laterally by 20arc min. a distance which produces quite low thresholds in the absence of the horizontal connections. The “box” targets inadequately signal the presence of disparity even when their vertical length reaches one degree, a length which places the endpoints outside of the foveola.* There are. however. no special attributes of endpoint connections. Joining the target lines at other places along their length also raises the threshold (Fig. 8). As with the three-segmented stimulus (Fig. 6), a small break in the connections improves the stereo threshold. Fig. 7. Threshold for horizontal binocular disparity as a DISCUSSION
The finest stereoacuity requires isolated vertical target lines of at least 1Omin in length. imaged in the foveola. The small improvement in sensitivity found with increasing target length undoubtedly reflects physiological summation of disparity signals within specialized sub-units sensitive to depth. This summation property would seem to distinguish stereoacuity from the monocular hyperacuities. because two properly positioned points will produce excellent vernier acuity, equal to that found with line targets. However, the most precise detection of line orientation also depends on the summation of positional signals, and, as in the case of stereoacuity. the sum-
mation region is 10-15 min arc (Andrews cf a/.. 1973; Westheimer. 1982). The influence of vertical length on stereo thresholds raises the ghost of a “mean local sign” theory of hyperacuity, but the data offer scant support for a model in which averaging along the length of the *All subjects tested in the Westheimer taboratory show a diminished capacity to detect the disparity of two hnes when they are joined to form a “box” as in Figs 7 and 8. a result first described in a tong monograph by Werner (1937).This ‘*connectivity” effect is most spectacular in the visual system of the author. Nevertheless. there are some “box” configurations in which the disparity of the vertical contours is easily detected. even by myself, as will be shown in a subsequent paper by G. J. Mitchison and Gerald Westheimer.
function of length of two vertical lines separated laterally by 20arc min. Subject judged whether right vertical line was in front or behind the left vertical line. Target duration 500 msec. Isolated vertical lines (Cl----O); vertical lines connected by horizontal lines at ends to form box IO-----Ok
target lines can account for a spatial precision finer than a single fovea1 cone. Even point targets can be localized in the third dimension with a precision of 68 arc sec. The enhanced detection of binocular disparity found with longer line targets is likely to be a secondary consequence of another functional objective of the human visual system--the identification of local contours which share a common disparity. Certainly. the more puzzling results come from the experiments on the stereoacuity for continuous figures. Connections between features at different disparities impair the ability to detect small differences in depth. Slanted surfaces and inclined lines-stimuli in which there are continuous changes in disparityare also known to be poor targets for stereoacuity (Ogle and Ellerbrock. 1946; Gulick and Lawson. 1976; Youngs. 1976). This curious suppression of the disparity information which is embedded in a larger continuous form highlights an interesting conflict in binocular processing. The cortex must solve two problems in dealing with the inputs from the two eyes. The obvious problem is the detection of the relative disparities of features, but the visual system must also solve the “fusion” problem-determining which
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The spatial requirements for fine stereoacuit?
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Fig. 8. Each panel gives the horizontal binocular disparity for the target configuration shown above it. Lateral separation for all targets 20 arc min. For subject SM. vertical length 60 arc min. For subject D.M. vertical length IO arc min. Any type of horizontal connection between the vertical target lines degrades stereoacuity; breaking the connection improves the disparity detection somewhat.
feature in the image in one eye belongs to a feature from the image in the other eye. Ultimately “fusing” a binocular stimulus means discarding information about the difference in the positions of the two monocular features. For example, Tyler (1975b) found that stereomovement, consisting of equal and opposite displacements of the monocular half-images of a binocularly-fused image, is undetectable at displacement amplitudes which are easily seen monocularly. The process of fusion and disparity detection may proceed in parallel, or as a “co-operative” interaction (Julesz, 1971; Nelson, 1975; Marr and Poggio. 1976). Nevertheless, there may be occasional competition between the two processes, leading to a decrement in
disparity detection. Continuous figures could constitute a powerful input to a “global” fusion mechanism which might average the disparities of the component features to assign a single depth value to the figure as a whole. Local variations in disparity which are intrinsically detectable would produce inadequate signals when pitted against the averaged depth assigned by the fusion process, the result being that only larger disparity differences could be observed in continuous contours. Ac~)IOM./edgclnellts--l would like to thank Dr Gerald Westheimer, Dr Jeremiah Nelson, and Dr G. J. Mitcluson for helpful discussions about the meaning of these results. REFERENCES Andersen E. E. and Weymouth F. W. (1923) Visual perception and the retinal mosaic: I. Retmal mean local
sign-an explanation of the fineness of binocular perception of distance. Am. J. Physiol. 64, 561-594. Andrews D. P.. Butcher A. K. and Buckley B. R. (1973) Acuities for spatial arrangement in line figures: Human and ideal observers compared. vision Res. 13, 599-620. Berry R. N. (1948) Quantitative relations among vernier. real depth and stereoscopic depth acuities. J. esp. Psj,chol. 38, 708-721. French J. W. (1917) The unaided eye; part III. Trans. Opt. Sot. 21, 127-156. &lick W. L. and Lawson R. B. (1976) Hunrat Srereopsis. a Ps#lophysical Approach, Chap. 8, pp. 201-224. Oxford Univ. Press. Hering E. (1899) Uber die Grenzen der Sehscharfe. Brr. Math-phys. Cl. d. Konigl. Sachs Natur. Wiss. Teil., pp. 1624.
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Hirsch M. J. and Weymouth F. W. (1948) Distance discrimination. II. Effect on threshold of lateral separation of the test objects. Archs Ophthal. 39, 224-231. Julesz B. (1971) Foundations of CyclopeanPerception. Univ. of Chicago Press. Chicago. Ludvigh E. (1953) Direction sense of the eye. Am. J. Opht/la/. 36, 139- 142. Marr D. and Poggio T. (1976) Cooperative computation of stereo disparity. Science 194, 283-287. Millodot M. (1972) Variation of visual acuity in the central region of the retina. Br. J. Physiol. Opr. 27. 2428. Nelson J. I. (1975) Globality and stereoscopic fusion in binocular vision. J. Theor. Biol. 49, l-88. Ogle K. N. and Ellerbrock V. J. (1946) Cyclofusional movements. Archs Ophrhal. 36. 7CKk735. Richards W. (1973) Disparity masking. vision Res. 12, 1113-1124. Stigmar G. (1970) Observations on vernier and stereoacuity with special reference lo their relationship. Acta ophthal.
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Sullivan G. D., Oatley K. and Sutherland N. S. (1972) Vernier acuity as affected by target length and separation. Percept.
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