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Distance in stereoscopic vision: The three-point problem
DISTANCE IN STEREOSCOPIC VISION: THE THREE-POINT PROBLEM1 J. M. FOLEY University (Received
of California,
Santa
Barbara
1I March 1969; in revisedfbrm
93106.
U.S.A.
20 May 1969)
THE STUDIESto be reported here are concerned with the relation between perceived depth and horizontal disparity. The situation is one in which points of light are presented stereoscopically in otherwise dark surroundings, secondary cues to distance being eliminated. The horizontal disparity between two points is the absolute value of the difference between the horizontal components of the visual angles subtended by the points in the two eyes. This is equal to the difference between the convergence angles of the two points. This definition is illustrated in Fig. 2(a). It has been known for 130 years that disparity is associated with the perception of depth, but relatively little work has been reported on the quantitative relation between disparity and perceived depth. There have been two basic approaches to this problem. One has been to proceed by direct judgments of distance and relative distance (i.e. relations between distances). The other approach, introduced by LUNEBURG(1947), has been to determine the intrinsic geometry of the visual space and, concluding that it is hyperbolic, to use the absolute unit of this geometry to measure distances. These intrinsic distances depend on relative distances (typically matches) and angles, and are calculated using the hyperbolic trigonometry. Any relation between relative distance in these intrinsic units and relative distance as directly judged has yet to be established. However, they must be related if an observer’s judgments of angles and relative distances are to form a coherent whole. Further, although the results of the two approaches are inconsistent in some respects, there is remarkable agreement about the basic relation between depth and disparity. HARDYet al. (1953) and BLANK(1958) working in the Luneburg tradition and FOLEY (1967a, 1967b, 1968) using direct judgments have presented evidence that the ratio of the egocentric distance of one point to the egocentric distance of another point is a function of the horizontal disparity between the points, that is: 0; -=jV-,)
(1)
Where 0: is the perceived distance to the neater point, 0; is the perceived distance to the farther point, and r,, is the horizontal disparity. Hardy et al. restricted the domain of this equation to point pairs which included the farthest point. The significance and validity of this restriction will be discussed below. Originally, equation 1 was interpreted to mean that relative distance is a function of disparity and disparity alone, that is, a given disparity always corresponds to the same distance ratio regardless ofthe configuration. Recent experiments by FOLEY(1967b) indicate that this must be qualified in one respect. He found that as ‘This research was supported by U.S. Public Health Service Grant MH 08878 from the National Institute Mental Health. The author is grateful to James Comerford for his assistance in conducting these experiments.
1SOS
of
J.XI.FOLEI
1506
the distance between the configuration and the observer decreases, more disparity is required to maintain a constant distance ratio (depth micropsia). Apart from this qualification. equation 1 is consistent with results obtained by both approaches using a variety of configurations and experimental tasks. Ifequation 1 is correct and the unspecified function,f(T,,,.), is the same for all point pairs in a configuration of three or more points at different distances, it follows that: _I LJ/
-=e
Qr"l
(2)
where Q is a constant and e is the base of the natural logarithms. The proof of this assertion is given elsewhere (See FOLEY, 1968, p. 269).2 Here it will be simply illustrated by an example. Figure 1 will aid in following the example. Place one point in the field and define its perceived distance as 16. Introduce a second point and vary its distance until it is perceived to be half the distance of the first point. Its perceived distance will therefore be 8 and D;/D’n will be 2. Call the disparity between these points r. j. Introduce a third point nearer than the first two and with a disparity of T., with respect to the second point. Its perceived distance will be half that of the second point or 4. With respect to the first point its disparity will be 2IY.j and 0; /D’” will be 4. In this way the relation given in Fig. 2 is obtained. It is described by the function: D;(T”fir.s) -=, DZ or more generally
An alternative to the logical derivation of the unknown function of equation 1 from seemingly plausible assumptions is to establish the function by experimental means. HARDY et al.(1953) did this using the methods of Luneburg and some of their own. FOLEY(1967a. 1965) used the more conventional methods of magnitude estimation, magnitude production, and intramodality matching of depth to size. These studies suggest that, instead of equation 2, the following relation holds:
where Kand Pare constants.3 Hardy et al. found this equation to apply for point pairs which included the farthest point. They took the exponent P to be 1. Foley worked with configurations in vvhich there was only one value of disparity. He found P to vary between 1 and 2 for different observers. A reanalysis of the data of St-ttPLEY(1957) on fractionated alleys is also consistent with this equation with P slightly greater than 1 (SHIPLEYand WILLLOIS, 1965). Foley made a direct comparison between the goodness of fit of equations 2 and 3 to his data. ‘There are two errors in the argument in that paper as printed. In place of r3, =r2, ‘r3, read r3,=r2iIr32.
Inplaced-m,,) .mz2)=f(rz,-rj2) readAr2,).ftr,,)=m-z,- b2). iThis is usually written
in the equivalent d;n -=kT$ 0’”
form: where+.==&-
D’,,
Distance
in Stereoscopic
Vision: The Three-Point
r .5
I507
Problem
2
2r.5
4
3r.5
8
4r.5
16
8 FIG. I. Illustration
of the relation between distance ratio and disparity which must hold if the same function applies to all point pairs in the configuration.
Equation 3, the power function, gave the better fit in all cases. This may seem like quibbling over the not-very-interesting matter of goodness of fit, with the two-parameter function not surprisingly coming out better than the one-parameter function. However, it will be seen that the choice between these two equations has some important implications, The conclusion that the function is exponential follows from the premise that the same function describes the relation between relative distance and disparity for all point pairs in the configuration. If, as the data indicate, the function is not exponential, then the premise is false and the samefimction cannot apply to allpointpairs in the configuration. Perhaps the power function (equation 3) applies to some subset of point pairs. But, if so, what subset is it‘? As has been noted, Hardy et al. proposed an answer to this question. They hypothesized that “the perceived ratio of radial distance for any point of a stimulus to that of the point of perceived greatest radial distance depends only on the difference in convergence between
1508
J. M. FOLEY
the two points. independently of the stimulus (HARDYer al.. 1953, p. 18). The “difference in convergence” is the horizontal disparity. Put in other words, the hypothesis says that all relative distances in a configuration depend on a particular subset of the disparities among the points: they depend on the disparities between the far point and each of the other points. This hypothesis will be referred to as the far point hypothesis. Working with a variety of configurations Hardy et al. confirmed this hypothesis. They found that equation 3 describes rather well the relation between relative distance (in intrinsic units) and disparity for point pairs that include the farthest point. The far point hypothesis which Hardy et al. tested and verified is only one of several hypotheses which might describe how equation 3 could apply to a configuration of three or more points. It could apply only to point pairs that include the nearest point (near-point hypothesis), pairs that include the middlemost point (middle point hypothesis), pairs adjacent in depth (depth adjacency hypothesis), pairs adjacent in direction (direction adjacency hypothesis), or possibly some other subset. Another possibility is that there is no subset of point pairs to which equation 3 applies. but rather in configurations with two or more disparities the relativ-e distance of one point to another may depend jointly on its disparity with respect to several or all the points in the configuration Qoint determination hypothesis). Although the consistency of their data with the far point hypothesis is impressive, Hardy et nl. do not report comparisons of their data with the implications of any of these other hypotheses. If the far point hypothesis is correct, it poses the problem of why the far point has this critical role, a problem to which there does not seem to be any obvious answer. Further. no test of any of these hypotheses has been made in which direct judgments of relative distances were used. In the present study these hypotheses were tested directly by means of a simple experiment. The basic procedure was as follows: Two points were presented in the eye level plane, one on either side of the median plane. The light on the right, b, was fixed; the light on the left, a, was variable in distance. The observer’s task was to piace the variable light so that the distance from himself to it equalled the distance from it to the fixed light. See Fig. 2(b). The dependent variable was the disparity required to satisfy this criterion. The independent variable was the position of a third light, c, referred to as the probe light. Light c appeared in various positions within and above the eye-level plane. The task does not involve a judgment of relative egocentric distance. In order to apply equation 3, it is assumed that the match between the two sides implies a constant ratio of the two egocentric distances. This assumption will be correct even if the geometry is nonEuclidean, unless the perceived angles of the triangle change. Evidence supporting the assumption will be given below. Suppose equation 3 holds and it applies only to some subset of point pairs. It is then possible to infer the expected outcome of this experiment for various hypotheses about which set of point pairs and associated disparities determine perceived spatial relations. When there are only three points in the configuration, one of them will belong to both pairs of the subset under the far point, middle point, near point, depth adjacency, or direction adjacency hypotheses. This point which belongs to all pairs of the subset will be referred to as the reference point. If we assume, for example, that the far point is the reference point, then when the probe point is the far point, we have: 0; 0; -= 1 AT-&+- 1 and-=KFL106 Db
Distance in Stereoscopic Vision: The Three-Point Problem
LEFT EYE
RW3M
1509
EYE
a
b
FIG. 2. (a) Illustration of the .definition of horizontal disparity and schematic diagram of the apparatus (top view). Horizontal disparity, rab=[a; -a,] =‘/,,-‘/b. The lined rectangles represent Polaroids; the fiHed circles. real lights; and the open circle, a simulated light. (b) Illustration of the experimental task. Observer varied the distance of a until oo equalled ab. Open circles indicate some of the possible positions of the probe light, c. A different subset of these was used in each experiment. Neither illustration is drawn to scale.
where a denotes the variable light, b the fixed light, and c the probe light. It follows that: 06 KI-:+l -=0’0 M-L+1 which by our interpretation of the instructions equals a constant. Denoting this constant by C and remembering that rf=yn--y, one can solve for the disparity between a and b: C- 1 ‘/P a;+ rbc (4) ( > K Since every term on the right is a constant except rbc, and since I+yb-ycr with yb being also a constant, we have r,b as a function of ye. If the middle light with respect to distance is the reference point, then when the probe light ‘is the middle light, by a similar argument we have : 1 1/P c r,,b=
+
--
mb= (
rcb (5)
)
K(KI-Lfl) K If the nearest light is the reference light, then when the probe light c is the nearest, we have: (6)
isio
J. M. FOLEY
* Under any of the three hypotheses. if the probe light is in a position where it is not the reference light or if the probe is absent, Tabwill be a constant. specifically : c-1 r&=
iP
i’
(3
‘i
K
This equation is obtained by setting the left side of equation 3 equal to C and solving for T’,,.
8
‘1,
60
)
,
50
100
‘
,
I
,
,
0
-50
0
CONVERGENCE
130 ANGLE
230
200 OF
PROBE
300
350
(MIN.)
FIG. 3.
Predicted functions relating disparity and convergence angle of the probe on th-e assumption that the probe is the reference point. Parameter C equals 1.9. -lb equals 150, and rob in no-probe condition eql;als 100.
Figure 3 illustrates the predictions for the case in which the convergence angle at b is 150 min, f;b= 100 min when the probe is absent (i.e. the convergence angle of a is 250 min), and C= l-9. The actual value of C for any observer can only be determined empirically. The value 1.9 used in this simulation is approximately the value that C would have if the geometry were Euclidean and the visual angle between n and h were veridically perceived. The form of the functions obtained, however, is the same over a wide range in values of C. Four functions for values of P between 1 and 1.6 are given. The dashed lines indicate the predicted disparity when the probe light is the reference point (i.e. the values of the function defined by equations 4, 5, and 6):The solid horizontal tine indicates the predicted disparity when the probe light is not the reference point or when it is absent (equation 7). Note that all the dashed lines intersect the solid line at the convergence
Distance in Stereoscopic Vision: The Three-Point Problem
I511
angle of 6, and the convergence angle of a when the probe is absent. According to the far point hypothesis the farthest point is always the reference point. This will be the probe light only when the probe light is farther away than light b. Thus, for the far point hypothesis the predicted relation between T,, and y, will consist of the segment of one of the dashed curves for convergence angles < 150 min. and the solid line for convergence angles > 150 min. For the middle point hypothesis the prediction will follow the solid line except between yC= 150 and y, = 250 min, where it will follow the dashed line for the appropriate value of P. For the near point hypothesis the prediction will follow the solid line for yCc250 and one of the dashed lines for ~~‘250 min. When there are only three points, the hypothesis that relative distances are determined by disparities between points adjacent in depth yields exactly the same prediction as the middle point hypothesis. The implication of the hypothesis that relative distances are determined by disparities between points adjacent in direction depends on the direction of the probe light with respect to a and 6. If the probe is between a and b, then the probe will be the reference point throughout, and the prediction will follow one of the dashed lines at all values of convergence angle. On the other hand, if c is not between a and b, then it will never be the reference point and the prediction will follow the solid horizontal line. It is logically possible that the probe light could always be the reference point regardless of its position. In this ease, the prediction will follow one of the dashed lines over the entire range of convergence angle regardless of directional adjacency. This will be called the probe reference hypothesis. If there is no reference point and the relative distances of a and b depend on all three disparities T,,, I-,,, and TbC,then it is impossible to predict what wili happen. without additional assumptions about the manner in which the three disparities combine to determine relative distance. No combination rule seems obvious to the author. The problem is complicated by the possibility that the three disparities may be differentially weighted in their influence depending on the positions of the lights. However, it does seem plausible to expect that, if relative distance is jointly determined, then I-‘,,would lie somewhere between the predicted value if determined by disparities with respect to c and the predicted constant value if it does not depend on c. That is, Teb would lie somewhere between one of the dashed curves and the solid horizontal line in Fig. 3. Finally, two ways in which the probe point might indirectly cause an effect should be considered. First, the average convergence of the eyes may depend on the presence and position of the probe (average convergence change hypothesis). FOLEY (1967b) has shown that the convergence required to fixate the lights of a two point configuration is directly related to the disparity associated with a constant perceived distance ratio. If we make the plausible assumption that the average convergence varies with the position of the probe because the observer tends to look at the probe, it follows that disparity should increase as the probe moves in toward the observer. The second indirect effect hypothesis is that the presence of the probe induces a change in the perceived absolute distances ofa.and b and the disparity depends on these distances (perceived distance change hypothesis). Although there is no evidence that the disparity associated with a constant perceived distance ratio depends on perceived absolute distances, this is at least a logical possibility. This hypothesis would be eliminated, however, if it were shown that no change in perceived absolute distance was induced by the probe.
1512
J. MM.FOLEY EXPERlMENTS
The apparatus was essentially the same as that used in previous studies by the author (FOLEY1964, 1967a, 1967b, 1968). The stimuli consisted of configurations of point sources of light {ophthalmoscope bulbs) presented in otherwise dark surroundings. The lights had an intensity of about two log units above fovea1 threshold and had the appearance of bright stars. Presentation was stereoscopic. Each variable point was simulated by a pair of horizontally separated bulbs, one member of which was visible to each eye. Light a in Fig. i(a) represents such a simulated Iight. The stereoscopic presentation &as achieved by means of properly oriented pieces of Polaroid film, one in front of each buib and one in front of each eye. The bulbs were in a frontal plane at a distance of 1.5 m. Points behind this plane were simulated by using uncrossed Polaroids; points in front of this plane were simulated by crossing the Polaroids, as shown. The simulated distance of a point was varied by changing the horizontaf separation of the lights. For one movable Iight this separation could be varied continuously by observer or experimenter. Each had a pair of buttons which controlled a reversible motor (60 rev/min). The motor drove a screw (43 threads. in.) that moved the lights together or apart while leaving their midpoint fixed. Ten observers were used in the experiments to be reported: All were university students between the ages of 18 and 26. All had uncorrected visual acuity of better than 1 min of arc and stereoacuity of better than 30 set of arc as measured by Keystone Orthoscope tests. Five experiments will be reported here which differ primarily in the set of loci occupied by the probe point. Each experiment required 8 to 10 I-hr sessions for each observer. A practice session was given at the beginning of each experiment. The conditions were presented in random order, except that when some conditions required crossed Polaroids and others, uncrossed Polaroids, all conditions of one type were presented, then all of the other, alternately starting with crossed or uncrossed conditions on successive days. Each condition came up once during a session and three settings were made at each condition. The observer’s head was held fixed by means of a bite board. but his eyes were allowed to move freely. In all experiments light b, the fixed light, was a single real light mounted on the frame of the stereoscope, as illustrated in Fig. 2(a). In four of the experiments the Polaroids of light a were crossed, con~ning it to simulated positions closer than the frame. In these experiments the observers were instructed to vary the distance of light a until the distance oa from themselves to a equalled the distance ab from a to 6. They were told to pay no attention to the probe light. Each experiment included one condition in which no probe was present as well as several conditions with the probe in different positions. In expe~ment 2 the procedure was slightly different, as will be noted. All the data presented are for individual observers. Experiment I In the first experiment ail three lights were in the horizontal, eye level plane. The probe was positioned in the median plane at various simulated distances from the observer. The fixed and variable lights were 8 deg to the right and left of the median plane, respectively. There were three observers. The results are given in Fig. 4. (In this and subsequent figures the dashed Iines are placed at the value of the disparity obtained when no probe was present.) It is clear that the disparity required for this setting is not independent of the position of the probe. In all three cases when probe convergence angle is small (i.e. the probe is far away), the disparity is greater than in the no-probe condition. As convergence angle increases, disparity decreases. The function passes through the no-probe value. reaches a
1513
Distance in Stereoscopic Vision: The Three-Point Problem
minimum, and then begins to increase. In two of the three cases (observer G and observer P) the minimum occurs when the probe is between a and b in depth. (In this and subsequent graphs the convergence angle of b and the average convergence angle of a in the no-probe 180. OBSERVER
160.
G
0
T
.
P
8
140.
;? 5
120.
> k : m
100.
ii
8s
60.
0 -50
I
.O
I
50
I
I
I
100
1so
CONVERGENCE
ANGLE
200 OF
PROSE
t
250
I
300
1
350
(MIN.)
FIG.
4. Median disparity (r,,b) between u and 6 as a function of the convergence angle of the probe, *fCProbe in median plane at eye level. The convergence angle of b, and the median convergence angle of a in the no-probe condition are indicated by short vertical lines. n = 24.
condition are indicated by short vertical lines intersecting the no-probe line.) In both of these cases the disparity increases to a value greater than the no-probe disparity. For Observer T the minimum is reached when the probe is just inside the nearer light and, although disparity then increases, it does not reach the no-probe level. The standard deviation of these points ranges from 7 to 19.5 min of arc. Since with n=24 the standard deviation of the mean will’be about l/5 of this, it is clear that these effects are statistically significant. Although the relation between variance and convergence angle of the probe is not a smooth function, the data suggest that it has a form similar to the relation between median disparity and convergence angle. That is, when the.probe is far away variance is higher than in the no-probe condition. As the probe moves in, variance decreases. It passes through the no-‘probe value, reaches a minimum, and then increases. ‘Experiment 2
The second experiment was like the first except that the Polaroids on the variable light, 6, were uncrossed, so that the simulated position of the light was farther than the frame of
151-l
J. hf. FOLEY
the stereoscope. The task was to adjust the distance of light CIuntil the distance ab squalled the distance ob. This change in the task increased the portion of the probe range in which the probe was nearer than both a and 6. As in the first experiment the probe was in the median plane at eye level and a and b were 8 deg to the left and right of the median plane. There were three observers. The results are given in Fig. 5. Here and elsevvhere in this report the same 140
OBSERVER 120
T
.
B
A
C
0
100
? 2
8o
c' E % 'u)
60
5
40
20
0
-so
,
0
I
SO
I
CONVERGENCE FIG. 5.
Median disparity
I
I
ANGLE
I
200
100 OF
PROBE
250
I
360
I
350
(MIN.)
(l-,6) between n and b as a function of the convergence -fC. Probe in median plane at eye level. n = 2-I.
angle of the probe,
letter always refers to the same observer. Again an effect of the position of the probe on disparity is evident. Here the functions for Observers T and B have a different form from those in Fig. 4. They have two minima with a maximum in between. For Observer T, this maximum occurs when the probe is between a and b in depth. For Observer B the peak occurs when the probe is closer to the observer than both a and b. The function for Observer C is similar to those of Fig. 4. The minimum occurs when the probe is closer than both a and b and there is no clear increase after the minimum. Standard deviation of these points ranged from 5.3 to 13.0 min of arc. Again there is an indication that standard deviation may vary systematically with convergence angle. The relation is not as clear as in the first experiment. However, in two of the three cases the minimum variance occurs when the probe is between 4 and b in depth and in all cases this minimum variance is substantially lower than the noprobe variance.
1515
Distance in Stereoscopic Vision: The Three-Point Problem
During the last three sessions of the experiment all three observers made verbal estimates of the distance in feet from themselves to each of the three lights. These were made at the end of the last trial for each condition. The probe light appeared to move in toward the observer as convergence angle increased. Ranges were as follows: C, 8 to 2 ft ; T, 5.5 to 1 ft ; B, 5 to 1 ft. For Observers B and T estimates of the fixed point are relatively constant and show no trend. For C perceived distance to the fixed point increased as the probe moved in. For all three observers, however, the ratio of perceived distances to a and b was relatively constant with no indication of a trend. This confirms the assumption that the task used here yields a constant ratio of egocentric distances. Median perceived distance ratios were 1.65 for C, 1.5 for T, and 1.4 for B. E.lcperiment 3 The third experiment was essentially the same as the first, except that the locus of the probe was a straight line in the median plane 5 deg above the other two lights. Here lights a and b were 4 deg to the left and right of the median plane respectively. There were three observers. The results are given in Fig. 6. The presence of the probe again produces an effect. The function of Observer R has a single minimum. That of Observer G has two minima with a maximum occurring when the probe is between a and b in depth. The function of 180
160
140
5 z 5
120
t z $
100
OBSERVER
a
00
e
0
R
0
M
0
60
0 -50
6
i0
10-O
CONVERSENCE
li0 ANSLE
20.0 OF
PROSE
2s’o
00.0
GO
(MIN.)
FIG. 6. Median disparity (rab> between a and b as a function of the convergence angle of the probe, yC.Probe in median plane 5 deg abovea and 6. Middle dashed line indicates no-probe disparity for G; lower line, no-probe disparity for R. n=24.
J. >I.
1516
FOLEY
Observer 34 is more difficult to classify. It is like the double minimum functions except that there is no real ma.ximum between a and b, and because so much disparity is required for the setting, there is no part of the range in which the probe is inside both a and b. Two other things to be noted are that the curve for G seems to be shallower here than in the first study, and the curve for Observer R decreases slightly at low values of convergence angle. Standard deviation of these points range from 9 to 23 min of arc. When the probe is far away variance is again higher than in no-probe condition. There is a tendency for variance to decrease as convergence angle increases, but here the correspondence between the variance function and the median function is again not as close as in the first experiment. Experiment 4
The fourth experiment compared the effect of the probe when it is in the median plane with its effect when it is 8 deg to the right of light 6. All three lights were in the eye level plane. Lights a and b were 8 deg to the left and right of the median plane respectively. Three observers were used. The results for one of them are shown in Fig. 7. The functions are very much like those of Fig. 4. Here the point at which the functions cross the no-probe line is very close to the convergence angle of 6, and the point at which they would, if projected, recross this line is very close to the convergence angle of a. Againthe variance function has 180
160
CENTER
M
RIGHT
C-
4
140
OBSERVER
80
V
4c
C I
-50
6
I
53
I
100
CONVERGENCE
I
150 ANQLE
I
I
210
200 (DF PRO8E
I
300
(MIN.)
Median disparity (rrrb) between a and b as a function of the convergence angle oi the probe. -fC. Probe in eye level plane and, either between a and b or to the ri_ght of both a and b. n=23.
FIG. 7.
1
350
Distance in Stereoscopic Vision: The Three-point Problem
I517
roughly the same form as the median function. Standard deviation ranged from 7.7 to 19.1 min of arc. The functions for the two positions of the probe have essentially the same form except that the curve for the probe on the right seems to be slightly shallower. This means that the probe has slightly less effect when it is to the right of both lights than when it is between them. The second observer gave very similar resufts (not shown here). Both functions were shallower than those of V, but that for the probe on the right was again the shallower of the two. The data of the third observer were highly variable, and the highly irregular function obtained is not considered to provide any information concerning perception. Experiment5 In the fifth experiment the convergence angle of the probe was held fixed and its lateral direction varied. Lights a and b were in the eye level plane 6 deg to the left and right of the median plane. The probe light was 3 deg above the eye level plane and appeared in positions from 18 deg to the left of the median plane to 18 dkg to the right of the median plane. Convergence angles of the probe for the two observers were 0 for P and 25 min of arc for T. The results are given in Fig. 8. Ill0 OSSeRVER
160
P
a
7
A
140,
;
---
----------------
? 12cJ
:: E 111 %
loo
5
80
------------------60
0
1
la
1
12
I 6
I
0
d
LEFT
I 12
I
18 RIM
AZIMUTH
OF
PROBE
(DEG.)
FIG. 8. Median disparity (Tab) between a and b as Bfunction of the azimuth angle of the probe. Probe 3 deg above eye level plane. Convergence angle 0 for P, 25 min for T. n =i 24.
ISIS
J. kl. FOLEY
The probe effect is substantial over the entire range of probe direction. For P it seems to decrease slightly as the probe moves toward the periphery. For T a minimum is reached when the probe and the variable light have the same azimuth. Differences among the means are significant at the O-01 level. Standard deviations of the points lie in the range of 3.6 to 14-4 min of arc. No trend is evident in the variability as a function of the direction of the probe. Although the data of all five experiments were obtained with free eye movement, a few sessions have been run in which the observer was instructed to fixate on point b. the fixed point. The setup and procedure were otherwise the same as in the first experiment. The presence of the probe had essentially the same effect as in the eye movement condition. Eye movement therefore is not a necessary condition for the occurrence of the probe point effect. Not enough data have been obtained to make a detailed comparison of the tvvo conditions. DlSCUSSION
The principal generalization that can be made from these results is that the presence of the probe does affect the disparity required to maintain a constant perceived distance ratio. It does so regardless of its position in depth or direction relative to the two points involved in the judgment. (The only exceptions to this statement are those points where the empirical functions cross the no-probe line.) This qualitative conclusion is sufficient to eliminate most of the hypotheses outlined in the introduction. The far point, middle point, and near point hypotheses al1 predict that the probe will have an effect only over part of its range. The same is true of the depth adjacency hypothesis. The direction adjacency hypothesis predicts that the probe will have an effect only when it is directionally adjacent to a and b. None of these hypotheses are consistent with the data. The verbal distance estimates (second experiment) indicate that the probe effect can occur in the absence of any change in the perceived distances of a and b. This.would seem to rule out the perceived distance change hypothesis according to which the effect would be mediated by a change in perceived distance. A decision in regard to the average convergence change hypothesis requires only that we examine the general trend of the data. As noted above, the most plausible interpretation of this hypothesis predicts disparity will increase with probe convergence angfe. The dominant trend in the data is just the opposite. What are left, then, are the probe reference hypothesis and the joint dete~ination hypothesis. Assuming that relative distances depend on disparities, these seem to exhaust the possibilities, Since relative distance is not determined by the disparity between a and b alone, either it is determined by disparities with respect to the probe point alone (probe reference hypothesis) or it is determined jointly by the disparities among the three points (joint determination hypothesis). Although the probe reference hypothesis must be considered as a Iogical possibility, it does not seem very likely, The probe light is the probe light only by virtue of its being designated as such by the experimenter. It is the light which moves around the most between trials. It is the one light which is not involved in the judgment and the light which the observer is told to ignore. None of these properties seem a probable basis for its being the reference point. One does not have to rely on considerations of plausibility. however, in deciding between these hypotheses. The implications of each can be compared with the data. It will be recalled that on the assumption that the power function (equation 3) is correct, the probe reference hypothesis implies that the obtained functions will have the form of one of the dashed curves in Fig. 3. The joint determination hypothesis implies that the functions
Distance
in Stereoscopic
Vision:
The Three-Point
Problem
1519
will lie between a curve of this form and the no-probe line. That is, they would have the form of the curves in Fig. 3, but be shallower. The curves of Fig. 3 are based on a particular set of parameters which obviously do not correspond to those of the empirical functions. However, sets of functions very much similar to those of Fig. 3 are obtained over the full range of parameter values which could reasonably be expected here. Qualitatively there is certainly a resemblance between the empirical curves and the theoretical ones. Both have two basic forms, the single minimum form (Fig. 4: G, P, and T; Fig. 6: R; Fig. 7: V) and the double minimum form (Fig. 5 : T and B; Fig. 6: G). Like the theoretical curves, the empirical curves generally intersect the no-probe line twice or would do so if extended. In general the empirical curves are shallower than the theoretical ones, thus favoring the joint determination. hypothesis over the probe reference hypothesis. This is especially evident at the left-hhnd end of the single minimum functions. The theoretical curves are much higher than the empirical ones. If we substitute the values for C suggested by the verbal estimates (C= 1.4, 1.5 and 1.65) the situation is improved slightly but the theoretical curves are still much higher, as they are over wide ranges of this and the other parameters. This fact provides further evidence against the far point hypothesis. Even if it were claimed that the probe effects occurring when the probe was not the far point were just noise, the data still diverge markedly from the quantitative predictions made by the far point hypothesis. Two ways in which the empirical curves differ from shallower versions of the theoretical curves should be noted. First the empirical curves do not always cross the no-probe line at the convergence angles of a and b (indicated by vertical marks). Observer T (Fig. 4) and Observer R (Fig. 6) are most deviant in this respect..Other functions cross reasonably clpse to the predicted points, given the error of the measurements. See; for example, Observer T (Fig. 5) and Observer V (Fig. 7j.The second type ofdeviation from shallower versions of the theoretical curves occurs at the high end of the convergence angle range. Here the function should be increasing and, except for those that peak between a and 6, should be above the no-probe line. This is not the case for Observer T (Fig. 4) or for Observer C (Fig. 5). The author has no explanation of these two common deviations, except to note that both are a consequence of the relative positions of the function and the no-probe line, both of which are empirically determined and therefore subject to measurement error. A slight vertical adjustment of the no-probe line (within the range of error) would considerably improve the correspondence in several cases. The interpretation of the empirical functions of Figs. 4-7 as shallower versions of theoretical curves (equations 4, 5 and 6) means that the form of the curve indicates at least roughly the exponent of the power function (equation 3). The obtained functions correspond most closely to theoretical functions for P between 1 and 1.6. At lower values the theoretical curves dip deeply below the no-probe line and rise very steeply on either side of this minimum. For values of P greater than 1.6 the first minimum becomes very pronounced and constitutes most or all of the left tail. Curiously in different experiments Observer T produced functions of different types (Figs. 4 and 5) indicating different exponents. This suggests that the exponent may not be stable. The outcome of this experiment is reminiscent of experiments in which conflicting cues are presented simultaneously and the resolution is intermediate to what it would be if one or the other cue determined the outcome. Here disparity has one value if the relative distance of two points depends only on the disparity between them (no-probe condition). It would
J. bl. FOLEY
1520
have another value if relative distance depended only on disparity with respect to tix probe light. In fact, it has a value intermediate between these two. This outcome can be interpreted as the resolution of a contlict between the disparity between a and b on the one hand and the disparities between each of these lights and the probe light on the other. It is impossible to say on the basis of these results if one of these two cues dominates and if so which one. The results do, however, suggest that relative dominance differs between observers. Given the inte~retation of these experiments as studies in cue conflict, the work of GOGEL(1956, 1963) would lead one to expect that the effect of the probe would be inversely related to the distance between the probe and the other two lights (the adjacency principle). The fourth experiment (Fig. 7) did show a small decrement in the probe effect as directional adjacency of the probe decreased. But the results of the fifth experiment (Fig. 8) indicate that this decrement is at best very small while the probe is within i 15 deg of the median plane. Gogel’s formulation would also predict a decrease in the dominance of the probe as the adjacency in depth of the probe to Q and b decreased. Such a decrease is suggested by the data of Observer R {Fig. 6) in which disparity actuahy decreases at low values ofconvergence angle. This decrease does not occur in the other functions. However, a decrease in the relative dominance of the probe cue only requires that the disparity diverge more and more from the appropriate probe reference curve as depth adjacency decreases. These data are not sufficiently precise to determine whether or not this happens. In the four experiments in which the probe moved in depth, it was noted that thevariance of the settings was not constant. At the ends of the convergence angle range, variance tended to be high relative to the no-probe condition, but it reached a minimum lower than in the no-probe condition in the middle of the range. Tests of homogeneity of variance indicate significant differences among variances in only about one-third of the cases. However, the fact that the relation between variance and probe convergence angle is roughly the same in most cases indicates that the effect is real. The fact that variance is sometimes less with three lights than with two suggests that variance is a decreasing function of the number of lights. The more lights present or the more cues to relative distance present, the more precisely determined are the relative distances in the configuration. On the other hand, if the probe is at one end of its range, particularly the far end, variance is higher than in the no-probe condition. But this is also the region in which the probe produces its greatest effect. This suggests that variance is an increasing function of cue conflict.
REFERENCES BL+.xK,A. A. (1958). Analysis of experiments in binocular space perception. f. opr. SW. .&w. d&91 l-925. FOLEY,J. ?4. (1964). Desarguesian property in vi&l space. .I. opt. Sot. Am. Y, 68ti92. FOLEY,J. hl. (1967a). Binocular disparity and perceived relative distance: An examination of two hypotheses. Vision Res. 7,655-670.
FOLEY,.I. M. (l967b). Disparity increase with convergence for constant perceptual criteria.
Percept.
fsychoph.w
2,605-608.
FOLEY,J. M. f 1968). Depth. size and distance in stereoscopic vision. Percept. Psychopiqx 3,265-2X. GOGEL,W. C. (1956). Relative visual direction as a factor in relative distance perceptions. Psr_eltoi. 70,
Monbgr.
Ii.
I-19. GOGEL,W. C. (1963).
The visual perception ofsize and distance. Vision Res. 3, 101-120. HARDY.L. H., RAND,G., RITTLER.M. C., BLANK.A. A. and BOEDER,P. (1953). The Geometr? ofBinocular Space Petceprion, Knapp Memorial taboratories, Institute of Ophtkalmoio~y, Columbia Umversity. College of Physicians and Surgeons, New York.
Distance
in Stereoscopic
Vision: The Three-point
Problem
1521
LUNEBURC. R. K. (1947). .lfnrhemuricul AnnlvGs of’Binoculur Vision. Princeton University Press. Princeton, New Jersey. SHIPLEY. T. (!957). Convergence function in binocular visual space. J. opt. Sot. rim. 47.79.~811. SHIPLEY, T. and W~LLIMS. D. (1968). The relationship between retinal disparity and relative visual distance. C’ision Rex. 8.325-332. the function relating the horizontal disparity between two points and the ratio of their perceived egocentric distances has any form other than exponential, then the distance ratio between two points will be influenced by the presence and position of a third point. Five experiments are reported in which this dependence is shown to exist and is measured as a function of the position of the third point. The functions obtained are inconsistent with several possible explanations of the effect, but are consistent with the hypothesis that the perceived distance ratio of two points is jointly determined by the positions of all the points in the configuration. Abstract-If
R&urn&-Si la fonction qui relie la disparite horizontale entre deux points et Ie rapport de leurs distances Cgocentriques percues est autre qu’une exponentielle. le rapport de distance entre deux points sera influence par la presence et lo position d’un troisiime point. On expose cinq experiences dans lesquelles on montre que cette dependence existe. et on la mesure en fonction de la position du troisieme point. Les fonctions obtenues ne sont pas en accord avec diverses explications possibles. mais en accord avec I’hypothese que le rapport de distance percue des deux points est determine conjointement par les positions de tous les points de la configuration. Zusammenfassung-Wenn die die Horizontaldisparitlt zweier Punkte und das Verhaltnis der egozentrischen Beobachtungsentfernungen verbindende Funktion anders als exponentiell verlauft, dann wird das Verhiiltnis zwischen den Beobachtungsentfernungen zwischen zwei Punkten durch die Gegenwart und Position eines dritten Punktes beeinflusst. Es werden fiinf V’ersuche mitgeteilt. in welchen dieser Einfluss bestatigt und als Funktion der Position des dritten Punktes bestimmt wird. Die Funktionen konnen mit keiner Erkhirung des Effektes in Ubereinstimmung gebracht werden. aber stimmen mit der Annahme iiberein, derzufolge das beobachtete Entfernungsverhaltnis zweier Punkte gleichgehend durch die Positionen aller Konfigurationspunkte bestimmt wird. Pexome
-
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of California,
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20 May 1969)
THE STUDIESto be reported here are concerned with the relation between perceived depth and horizontal disparity. The situation is one in which points of light are presented stereoscopically in otherwise dark surroundings, secondary cues to distance being eliminated. The horizontal disparity between two points is the absolute value of the difference between the horizontal components of the visual angles subtended by the points in the two eyes. This is equal to the difference between the convergence angles of the two points. This definition is illustrated in Fig. 2(a). It has been known for 130 years that disparity is associated with the perception of depth, but relatively little work has been reported on the quantitative relation between disparity and perceived depth. There have been two basic approaches to this problem. One has been to proceed by direct judgments of distance and relative distance (i.e. relations between distances). The other approach, introduced by LUNEBURG(1947), has been to determine the intrinsic geometry of the visual space and, concluding that it is hyperbolic, to use the absolute unit of this geometry to measure distances. These intrinsic distances depend on relative distances (typically matches) and angles, and are calculated using the hyperbolic trigonometry. Any relation between relative distance in these intrinsic units and relative distance as directly judged has yet to be established. However, they must be related if an observer’s judgments of angles and relative distances are to form a coherent whole. Further, although the results of the two approaches are inconsistent in some respects, there is remarkable agreement about the basic relation between depth and disparity. HARDYet al. (1953) and BLANK(1958) working in the Luneburg tradition and FOLEY (1967a, 1967b, 1968) using direct judgments have presented evidence that the ratio of the egocentric distance of one point to the egocentric distance of another point is a function of the horizontal disparity between the points, that is: 0; -=jV-,)
(1)
Where 0: is the perceived distance to the neater point, 0; is the perceived distance to the farther point, and r,, is the horizontal disparity. Hardy et al. restricted the domain of this equation to point pairs which included the farthest point. The significance and validity of this restriction will be discussed below. Originally, equation 1 was interpreted to mean that relative distance is a function of disparity and disparity alone, that is, a given disparity always corresponds to the same distance ratio regardless ofthe configuration. Recent experiments by FOLEY(1967b) indicate that this must be qualified in one respect. He found that as ‘This research was supported by U.S. Public Health Service Grant MH 08878 from the National Institute Mental Health. The author is grateful to James Comerford for his assistance in conducting these experiments.
1SOS
of
J.XI.FOLEI
1506
the distance between the configuration and the observer decreases, more disparity is required to maintain a constant distance ratio (depth micropsia). Apart from this qualification. equation 1 is consistent with results obtained by both approaches using a variety of configurations and experimental tasks. Ifequation 1 is correct and the unspecified function,f(T,,,.), is the same for all point pairs in a configuration of three or more points at different distances, it follows that: _I LJ/
-=e
Qr"l
(2)
where Q is a constant and e is the base of the natural logarithms. The proof of this assertion is given elsewhere (See FOLEY, 1968, p. 269).2 Here it will be simply illustrated by an example. Figure 1 will aid in following the example. Place one point in the field and define its perceived distance as 16. Introduce a second point and vary its distance until it is perceived to be half the distance of the first point. Its perceived distance will therefore be 8 and D;/D’n will be 2. Call the disparity between these points r. j. Introduce a third point nearer than the first two and with a disparity of T., with respect to the second point. Its perceived distance will be half that of the second point or 4. With respect to the first point its disparity will be 2IY.j and 0; /D’” will be 4. In this way the relation given in Fig. 2 is obtained. It is described by the function: D;(T”fir.s) -=, DZ or more generally
An alternative to the logical derivation of the unknown function of equation 1 from seemingly plausible assumptions is to establish the function by experimental means. HARDY et al.(1953) did this using the methods of Luneburg and some of their own. FOLEY(1967a. 1965) used the more conventional methods of magnitude estimation, magnitude production, and intramodality matching of depth to size. These studies suggest that, instead of equation 2, the following relation holds:
where Kand Pare constants.3 Hardy et al. found this equation to apply for point pairs which included the farthest point. They took the exponent P to be 1. Foley worked with configurations in vvhich there was only one value of disparity. He found P to vary between 1 and 2 for different observers. A reanalysis of the data of St-ttPLEY(1957) on fractionated alleys is also consistent with this equation with P slightly greater than 1 (SHIPLEYand WILLLOIS, 1965). Foley made a direct comparison between the goodness of fit of equations 2 and 3 to his data. ‘There are two errors in the argument in that paper as printed. In place of r3, =r2, ‘r3, read r3,=r2iIr32.
Inplaced-m,,) .mz2)=f(rz,-rj2) readAr2,).ftr,,)=m-z,- b2). iThis is usually written
in the equivalent d;n -=kT$ 0’”
form: where+.==&-
D’,,
Distance
in Stereoscopic
Vision: The Three-Point
r .5
I507
Problem
2
2r.5
4
3r.5
8
4r.5
16
8 FIG. I. Illustration
of the relation between distance ratio and disparity which must hold if the same function applies to all point pairs in the configuration.
Equation 3, the power function, gave the better fit in all cases. This may seem like quibbling over the not-very-interesting matter of goodness of fit, with the two-parameter function not surprisingly coming out better than the one-parameter function. However, it will be seen that the choice between these two equations has some important implications, The conclusion that the function is exponential follows from the premise that the same function describes the relation between relative distance and disparity for all point pairs in the configuration. If, as the data indicate, the function is not exponential, then the premise is false and the samefimction cannot apply to allpointpairs in the configuration. Perhaps the power function (equation 3) applies to some subset of point pairs. But, if so, what subset is it‘? As has been noted, Hardy et al. proposed an answer to this question. They hypothesized that “the perceived ratio of radial distance for any point of a stimulus to that of the point of perceived greatest radial distance depends only on the difference in convergence between
1508
J. M. FOLEY
the two points. independently of the stimulus (HARDYer al.. 1953, p. 18). The “difference in convergence” is the horizontal disparity. Put in other words, the hypothesis says that all relative distances in a configuration depend on a particular subset of the disparities among the points: they depend on the disparities between the far point and each of the other points. This hypothesis will be referred to as the far point hypothesis. Working with a variety of configurations Hardy et al. confirmed this hypothesis. They found that equation 3 describes rather well the relation between relative distance (in intrinsic units) and disparity for point pairs that include the farthest point. The far point hypothesis which Hardy et al. tested and verified is only one of several hypotheses which might describe how equation 3 could apply to a configuration of three or more points. It could apply only to point pairs that include the nearest point (near-point hypothesis), pairs that include the middlemost point (middle point hypothesis), pairs adjacent in depth (depth adjacency hypothesis), pairs adjacent in direction (direction adjacency hypothesis), or possibly some other subset. Another possibility is that there is no subset of point pairs to which equation 3 applies. but rather in configurations with two or more disparities the relativ-e distance of one point to another may depend jointly on its disparity with respect to several or all the points in the configuration Qoint determination hypothesis). Although the consistency of their data with the far point hypothesis is impressive, Hardy et nl. do not report comparisons of their data with the implications of any of these other hypotheses. If the far point hypothesis is correct, it poses the problem of why the far point has this critical role, a problem to which there does not seem to be any obvious answer. Further. no test of any of these hypotheses has been made in which direct judgments of relative distances were used. In the present study these hypotheses were tested directly by means of a simple experiment. The basic procedure was as follows: Two points were presented in the eye level plane, one on either side of the median plane. The light on the right, b, was fixed; the light on the left, a, was variable in distance. The observer’s task was to piace the variable light so that the distance from himself to it equalled the distance from it to the fixed light. See Fig. 2(b). The dependent variable was the disparity required to satisfy this criterion. The independent variable was the position of a third light, c, referred to as the probe light. Light c appeared in various positions within and above the eye-level plane. The task does not involve a judgment of relative egocentric distance. In order to apply equation 3, it is assumed that the match between the two sides implies a constant ratio of the two egocentric distances. This assumption will be correct even if the geometry is nonEuclidean, unless the perceived angles of the triangle change. Evidence supporting the assumption will be given below. Suppose equation 3 holds and it applies only to some subset of point pairs. It is then possible to infer the expected outcome of this experiment for various hypotheses about which set of point pairs and associated disparities determine perceived spatial relations. When there are only three points in the configuration, one of them will belong to both pairs of the subset under the far point, middle point, near point, depth adjacency, or direction adjacency hypotheses. This point which belongs to all pairs of the subset will be referred to as the reference point. If we assume, for example, that the far point is the reference point, then when the probe point is the far point, we have: 0; 0; -= 1 AT-&+- 1 and-=KFL106 Db
Distance in Stereoscopic Vision: The Three-Point Problem
LEFT EYE
RW3M
1509
EYE
a
b
FIG. 2. (a) Illustration of the .definition of horizontal disparity and schematic diagram of the apparatus (top view). Horizontal disparity, rab=[a; -a,] =‘/,,-‘/b. The lined rectangles represent Polaroids; the fiHed circles. real lights; and the open circle, a simulated light. (b) Illustration of the experimental task. Observer varied the distance of a until oo equalled ab. Open circles indicate some of the possible positions of the probe light, c. A different subset of these was used in each experiment. Neither illustration is drawn to scale.
where a denotes the variable light, b the fixed light, and c the probe light. It follows that: 06 KI-:+l -=0’0 M-L+1 which by our interpretation of the instructions equals a constant. Denoting this constant by C and remembering that rf=yn--y, one can solve for the disparity between a and b: C- 1 ‘/P a;+ rbc (4) ( > K Since every term on the right is a constant except rbc, and since I+yb-ycr with yb being also a constant, we have r,b as a function of ye. If the middle light with respect to distance is the reference point, then when the probe light ‘is the middle light, by a similar argument we have : 1 1/P c r,,b=
+
--
mb= (
rcb (5)
)
K(KI-Lfl) K If the nearest light is the reference light, then when the probe light c is the nearest, we have: (6)
isio
J. M. FOLEY
* Under any of the three hypotheses. if the probe light is in a position where it is not the reference light or if the probe is absent, Tabwill be a constant. specifically : c-1 r&=
iP
i’
(3
‘i
K
This equation is obtained by setting the left side of equation 3 equal to C and solving for T’,,.
8
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)
,
50
100
‘
,
I
,
,
0
-50
0
CONVERGENCE
130 ANGLE
230
200 OF
PROBE
300
350
(MIN.)
FIG. 3.
Predicted functions relating disparity and convergence angle of the probe on th-e assumption that the probe is the reference point. Parameter C equals 1.9. -lb equals 150, and rob in no-probe condition eql;als 100.
Figure 3 illustrates the predictions for the case in which the convergence angle at b is 150 min, f;b= 100 min when the probe is absent (i.e. the convergence angle of a is 250 min), and C= l-9. The actual value of C for any observer can only be determined empirically. The value 1.9 used in this simulation is approximately the value that C would have if the geometry were Euclidean and the visual angle between n and h were veridically perceived. The form of the functions obtained, however, is the same over a wide range in values of C. Four functions for values of P between 1 and 1.6 are given. The dashed lines indicate the predicted disparity when the probe light is the reference point (i.e. the values of the function defined by equations 4, 5, and 6):The solid horizontal tine indicates the predicted disparity when the probe light is not the reference point or when it is absent (equation 7). Note that all the dashed lines intersect the solid line at the convergence
Distance in Stereoscopic Vision: The Three-Point Problem
I511
angle of 6, and the convergence angle of a when the probe is absent. According to the far point hypothesis the farthest point is always the reference point. This will be the probe light only when the probe light is farther away than light b. Thus, for the far point hypothesis the predicted relation between T,, and y, will consist of the segment of one of the dashed curves for convergence angles < 150 min. and the solid line for convergence angles > 150 min. For the middle point hypothesis the prediction will follow the solid line except between yC= 150 and y, = 250 min, where it will follow the dashed line for the appropriate value of P. For the near point hypothesis the prediction will follow the solid line for yCc250 and one of the dashed lines for ~~‘250 min. When there are only three points, the hypothesis that relative distances are determined by disparities between points adjacent in depth yields exactly the same prediction as the middle point hypothesis. The implication of the hypothesis that relative distances are determined by disparities between points adjacent in direction depends on the direction of the probe light with respect to a and 6. If the probe is between a and b, then the probe will be the reference point throughout, and the prediction will follow one of the dashed lines at all values of convergence angle. On the other hand, if c is not between a and b, then it will never be the reference point and the prediction will follow the solid horizontal line. It is logically possible that the probe light could always be the reference point regardless of its position. In this ease, the prediction will follow one of the dashed lines over the entire range of convergence angle regardless of directional adjacency. This will be called the probe reference hypothesis. If there is no reference point and the relative distances of a and b depend on all three disparities T,,, I-,,, and TbC,then it is impossible to predict what wili happen. without additional assumptions about the manner in which the three disparities combine to determine relative distance. No combination rule seems obvious to the author. The problem is complicated by the possibility that the three disparities may be differentially weighted in their influence depending on the positions of the lights. However, it does seem plausible to expect that, if relative distance is jointly determined, then I-‘,,would lie somewhere between the predicted value if determined by disparities with respect to c and the predicted constant value if it does not depend on c. That is, Teb would lie somewhere between one of the dashed curves and the solid horizontal line in Fig. 3. Finally, two ways in which the probe point might indirectly cause an effect should be considered. First, the average convergence of the eyes may depend on the presence and position of the probe (average convergence change hypothesis). FOLEY (1967b) has shown that the convergence required to fixate the lights of a two point configuration is directly related to the disparity associated with a constant perceived distance ratio. If we make the plausible assumption that the average convergence varies with the position of the probe because the observer tends to look at the probe, it follows that disparity should increase as the probe moves in toward the observer. The second indirect effect hypothesis is that the presence of the probe induces a change in the perceived absolute distances ofa.and b and the disparity depends on these distances (perceived distance change hypothesis). Although there is no evidence that the disparity associated with a constant perceived distance ratio depends on perceived absolute distances, this is at least a logical possibility. This hypothesis would be eliminated, however, if it were shown that no change in perceived absolute distance was induced by the probe.
1512
J. MM.FOLEY EXPERlMENTS
The apparatus was essentially the same as that used in previous studies by the author (FOLEY1964, 1967a, 1967b, 1968). The stimuli consisted of configurations of point sources of light {ophthalmoscope bulbs) presented in otherwise dark surroundings. The lights had an intensity of about two log units above fovea1 threshold and had the appearance of bright stars. Presentation was stereoscopic. Each variable point was simulated by a pair of horizontally separated bulbs, one member of which was visible to each eye. Light a in Fig. i(a) represents such a simulated Iight. The stereoscopic presentation &as achieved by means of properly oriented pieces of Polaroid film, one in front of each buib and one in front of each eye. The bulbs were in a frontal plane at a distance of 1.5 m. Points behind this plane were simulated by using uncrossed Polaroids; points in front of this plane were simulated by crossing the Polaroids, as shown. The simulated distance of a point was varied by changing the horizontaf separation of the lights. For one movable Iight this separation could be varied continuously by observer or experimenter. Each had a pair of buttons which controlled a reversible motor (60 rev/min). The motor drove a screw (43 threads. in.) that moved the lights together or apart while leaving their midpoint fixed. Ten observers were used in the experiments to be reported: All were university students between the ages of 18 and 26. All had uncorrected visual acuity of better than 1 min of arc and stereoacuity of better than 30 set of arc as measured by Keystone Orthoscope tests. Five experiments will be reported here which differ primarily in the set of loci occupied by the probe point. Each experiment required 8 to 10 I-hr sessions for each observer. A practice session was given at the beginning of each experiment. The conditions were presented in random order, except that when some conditions required crossed Polaroids and others, uncrossed Polaroids, all conditions of one type were presented, then all of the other, alternately starting with crossed or uncrossed conditions on successive days. Each condition came up once during a session and three settings were made at each condition. The observer’s head was held fixed by means of a bite board. but his eyes were allowed to move freely. In all experiments light b, the fixed light, was a single real light mounted on the frame of the stereoscope, as illustrated in Fig. 2(a). In four of the experiments the Polaroids of light a were crossed, con~ning it to simulated positions closer than the frame. In these experiments the observers were instructed to vary the distance of light a until the distance oa from themselves to a equalled the distance ab from a to 6. They were told to pay no attention to the probe light. Each experiment included one condition in which no probe was present as well as several conditions with the probe in different positions. In expe~ment 2 the procedure was slightly different, as will be noted. All the data presented are for individual observers. Experiment I In the first experiment ail three lights were in the horizontal, eye level plane. The probe was positioned in the median plane at various simulated distances from the observer. The fixed and variable lights were 8 deg to the right and left of the median plane, respectively. There were three observers. The results are given in Fig. 4. (In this and subsequent figures the dashed Iines are placed at the value of the disparity obtained when no probe was present.) It is clear that the disparity required for this setting is not independent of the position of the probe. In all three cases when probe convergence angle is small (i.e. the probe is far away), the disparity is greater than in the no-probe condition. As convergence angle increases, disparity decreases. The function passes through the no-probe value. reaches a
1513
Distance in Stereoscopic Vision: The Three-Point Problem
minimum, and then begins to increase. In two of the three cases (observer G and observer P) the minimum occurs when the probe is between a and b in depth. (In this and subsequent graphs the convergence angle of b and the average convergence angle of a in the no-probe 180. OBSERVER
160.
G
0
T
.
P
8
140.
;? 5
120.
> k : m
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ii
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.O
I
50
I
I
I
100
1so
CONVERGENCE
ANGLE
200 OF
PROSE
t
250
I
300
1
350
(MIN.)
FIG.
4. Median disparity (r,,b) between u and 6 as a function of the convergence angle of the probe, *fCProbe in median plane at eye level. The convergence angle of b, and the median convergence angle of a in the no-probe condition are indicated by short vertical lines. n = 24.
condition are indicated by short vertical lines intersecting the no-probe line.) In both of these cases the disparity increases to a value greater than the no-probe disparity. For Observer T the minimum is reached when the probe is just inside the nearer light and, although disparity then increases, it does not reach the no-probe level. The standard deviation of these points ranges from 7 to 19.5 min of arc. Since with n=24 the standard deviation of the mean will’be about l/5 of this, it is clear that these effects are statistically significant. Although the relation between variance and convergence angle of the probe is not a smooth function, the data suggest that it has a form similar to the relation between median disparity and convergence angle. That is, when the.probe is far away variance is higher than in the no-probe condition. As the probe moves in, variance decreases. It passes through the no-‘probe value, reaches a minimum, and then increases. ‘Experiment 2
The second experiment was like the first except that the Polaroids on the variable light, 6, were uncrossed, so that the simulated position of the light was farther than the frame of
151-l
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the stereoscope. The task was to adjust the distance of light CIuntil the distance ab squalled the distance ob. This change in the task increased the portion of the probe range in which the probe was nearer than both a and 6. As in the first experiment the probe was in the median plane at eye level and a and b were 8 deg to the left and right of the median plane. There were three observers. The results are given in Fig. 5. Here and elsevvhere in this report the same 140
OBSERVER 120
T
.
B
A
C
0
100
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CONVERGENCE FIG. 5.
Median disparity
I
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ANGLE
I
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(l-,6) between n and b as a function of the convergence -fC. Probe in median plane at eye level. n = 2-I.
angle of the probe,
letter always refers to the same observer. Again an effect of the position of the probe on disparity is evident. Here the functions for Observers T and B have a different form from those in Fig. 4. They have two minima with a maximum in between. For Observer T, this maximum occurs when the probe is between a and b in depth. For Observer B the peak occurs when the probe is closer to the observer than both a and b. The function for Observer C is similar to those of Fig. 4. The minimum occurs when the probe is closer than both a and b and there is no clear increase after the minimum. Standard deviation of these points ranged from 5.3 to 13.0 min of arc. Again there is an indication that standard deviation may vary systematically with convergence angle. The relation is not as clear as in the first experiment. However, in two of the three cases the minimum variance occurs when the probe is between 4 and b in depth and in all cases this minimum variance is substantially lower than the noprobe variance.
1515
Distance in Stereoscopic Vision: The Three-Point Problem
During the last three sessions of the experiment all three observers made verbal estimates of the distance in feet from themselves to each of the three lights. These were made at the end of the last trial for each condition. The probe light appeared to move in toward the observer as convergence angle increased. Ranges were as follows: C, 8 to 2 ft ; T, 5.5 to 1 ft ; B, 5 to 1 ft. For Observers B and T estimates of the fixed point are relatively constant and show no trend. For C perceived distance to the fixed point increased as the probe moved in. For all three observers, however, the ratio of perceived distances to a and b was relatively constant with no indication of a trend. This confirms the assumption that the task used here yields a constant ratio of egocentric distances. Median perceived distance ratios were 1.65 for C, 1.5 for T, and 1.4 for B. E.lcperiment 3 The third experiment was essentially the same as the first, except that the locus of the probe was a straight line in the median plane 5 deg above the other two lights. Here lights a and b were 4 deg to the left and right of the median plane respectively. There were three observers. The results are given in Fig. 6. The presence of the probe again produces an effect. The function of Observer R has a single minimum. That of Observer G has two minima with a maximum occurring when the probe is between a and b in depth. The function of 180
160
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OBSERVER
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CONVERSENCE
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FIG. 6. Median disparity (rab> between a and b as a function of the convergence angle of the probe, yC.Probe in median plane 5 deg abovea and 6. Middle dashed line indicates no-probe disparity for G; lower line, no-probe disparity for R. n=24.
J. >I.
1516
FOLEY
Observer 34 is more difficult to classify. It is like the double minimum functions except that there is no real ma.ximum between a and b, and because so much disparity is required for the setting, there is no part of the range in which the probe is inside both a and b. Two other things to be noted are that the curve for G seems to be shallower here than in the first study, and the curve for Observer R decreases slightly at low values of convergence angle. Standard deviation of these points range from 9 to 23 min of arc. When the probe is far away variance is again higher than in no-probe condition. There is a tendency for variance to decrease as convergence angle increases, but here the correspondence between the variance function and the median function is again not as close as in the first experiment. Experiment 4
The fourth experiment compared the effect of the probe when it is in the median plane with its effect when it is 8 deg to the right of light 6. All three lights were in the eye level plane. Lights a and b were 8 deg to the left and right of the median plane respectively. Three observers were used. The results for one of them are shown in Fig. 7. The functions are very much like those of Fig. 4. Here the point at which the functions cross the no-probe line is very close to the convergence angle of 6, and the point at which they would, if projected, recross this line is very close to the convergence angle of a. Againthe variance function has 180
160
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(MIN.)
Median disparity (rrrb) between a and b as a function of the convergence angle oi the probe. -fC. Probe in eye level plane and, either between a and b or to the ri_ght of both a and b. n=23.
FIG. 7.
1
350
Distance in Stereoscopic Vision: The Three-point Problem
I517
roughly the same form as the median function. Standard deviation ranged from 7.7 to 19.1 min of arc. The functions for the two positions of the probe have essentially the same form except that the curve for the probe on the right seems to be slightly shallower. This means that the probe has slightly less effect when it is to the right of both lights than when it is between them. The second observer gave very similar resufts (not shown here). Both functions were shallower than those of V, but that for the probe on the right was again the shallower of the two. The data of the third observer were highly variable, and the highly irregular function obtained is not considered to provide any information concerning perception. Experiment5 In the fifth experiment the convergence angle of the probe was held fixed and its lateral direction varied. Lights a and b were in the eye level plane 6 deg to the left and right of the median plane. The probe light was 3 deg above the eye level plane and appeared in positions from 18 deg to the left of the median plane to 18 dkg to the right of the median plane. Convergence angles of the probe for the two observers were 0 for P and 25 min of arc for T. The results are given in Fig. 8. Ill0 OSSeRVER
160
P
a
7
A
140,
;
---
----------------
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LEFT
I 12
I
18 RIM
AZIMUTH
OF
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(DEG.)
FIG. 8. Median disparity (Tab) between a and b as Bfunction of the azimuth angle of the probe. Probe 3 deg above eye level plane. Convergence angle 0 for P, 25 min for T. n =i 24.
ISIS
J. kl. FOLEY
The probe effect is substantial over the entire range of probe direction. For P it seems to decrease slightly as the probe moves toward the periphery. For T a minimum is reached when the probe and the variable light have the same azimuth. Differences among the means are significant at the O-01 level. Standard deviations of the points lie in the range of 3.6 to 14-4 min of arc. No trend is evident in the variability as a function of the direction of the probe. Although the data of all five experiments were obtained with free eye movement, a few sessions have been run in which the observer was instructed to fixate on point b. the fixed point. The setup and procedure were otherwise the same as in the first experiment. The presence of the probe had essentially the same effect as in the eye movement condition. Eye movement therefore is not a necessary condition for the occurrence of the probe point effect. Not enough data have been obtained to make a detailed comparison of the tvvo conditions. DlSCUSSION
The principal generalization that can be made from these results is that the presence of the probe does affect the disparity required to maintain a constant perceived distance ratio. It does so regardless of its position in depth or direction relative to the two points involved in the judgment. (The only exceptions to this statement are those points where the empirical functions cross the no-probe line.) This qualitative conclusion is sufficient to eliminate most of the hypotheses outlined in the introduction. The far point, middle point, and near point hypotheses al1 predict that the probe will have an effect only over part of its range. The same is true of the depth adjacency hypothesis. The direction adjacency hypothesis predicts that the probe will have an effect only when it is directionally adjacent to a and b. None of these hypotheses are consistent with the data. The verbal distance estimates (second experiment) indicate that the probe effect can occur in the absence of any change in the perceived distances of a and b. This.would seem to rule out the perceived distance change hypothesis according to which the effect would be mediated by a change in perceived distance. A decision in regard to the average convergence change hypothesis requires only that we examine the general trend of the data. As noted above, the most plausible interpretation of this hypothesis predicts disparity will increase with probe convergence angfe. The dominant trend in the data is just the opposite. What are left, then, are the probe reference hypothesis and the joint dete~ination hypothesis. Assuming that relative distances depend on disparities, these seem to exhaust the possibilities, Since relative distance is not determined by the disparity between a and b alone, either it is determined by disparities with respect to the probe point alone (probe reference hypothesis) or it is determined jointly by the disparities among the three points (joint determination hypothesis). Although the probe reference hypothesis must be considered as a Iogical possibility, it does not seem very likely, The probe light is the probe light only by virtue of its being designated as such by the experimenter. It is the light which moves around the most between trials. It is the one light which is not involved in the judgment and the light which the observer is told to ignore. None of these properties seem a probable basis for its being the reference point. One does not have to rely on considerations of plausibility. however, in deciding between these hypotheses. The implications of each can be compared with the data. It will be recalled that on the assumption that the power function (equation 3) is correct, the probe reference hypothesis implies that the obtained functions will have the form of one of the dashed curves in Fig. 3. The joint determination hypothesis implies that the functions
Distance
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1519
will lie between a curve of this form and the no-probe line. That is, they would have the form of the curves in Fig. 3, but be shallower. The curves of Fig. 3 are based on a particular set of parameters which obviously do not correspond to those of the empirical functions. However, sets of functions very much similar to those of Fig. 3 are obtained over the full range of parameter values which could reasonably be expected here. Qualitatively there is certainly a resemblance between the empirical curves and the theoretical ones. Both have two basic forms, the single minimum form (Fig. 4: G, P, and T; Fig. 6: R; Fig. 7: V) and the double minimum form (Fig. 5 : T and B; Fig. 6: G). Like the theoretical curves, the empirical curves generally intersect the no-probe line twice or would do so if extended. In general the empirical curves are shallower than the theoretical ones, thus favoring the joint determination. hypothesis over the probe reference hypothesis. This is especially evident at the left-hhnd end of the single minimum functions. The theoretical curves are much higher than the empirical ones. If we substitute the values for C suggested by the verbal estimates (C= 1.4, 1.5 and 1.65) the situation is improved slightly but the theoretical curves are still much higher, as they are over wide ranges of this and the other parameters. This fact provides further evidence against the far point hypothesis. Even if it were claimed that the probe effects occurring when the probe was not the far point were just noise, the data still diverge markedly from the quantitative predictions made by the far point hypothesis. Two ways in which the empirical curves differ from shallower versions of the theoretical curves should be noted. First the empirical curves do not always cross the no-probe line at the convergence angles of a and b (indicated by vertical marks). Observer T (Fig. 4) and Observer R (Fig. 6) are most deviant in this respect..Other functions cross reasonably clpse to the predicted points, given the error of the measurements. See; for example, Observer T (Fig. 5) and Observer V (Fig. 7j.The second type ofdeviation from shallower versions of the theoretical curves occurs at the high end of the convergence angle range. Here the function should be increasing and, except for those that peak between a and 6, should be above the no-probe line. This is not the case for Observer T (Fig. 4) or for Observer C (Fig. 5). The author has no explanation of these two common deviations, except to note that both are a consequence of the relative positions of the function and the no-probe line, both of which are empirically determined and therefore subject to measurement error. A slight vertical adjustment of the no-probe line (within the range of error) would considerably improve the correspondence in several cases. The interpretation of the empirical functions of Figs. 4-7 as shallower versions of theoretical curves (equations 4, 5 and 6) means that the form of the curve indicates at least roughly the exponent of the power function (equation 3). The obtained functions correspond most closely to theoretical functions for P between 1 and 1.6. At lower values the theoretical curves dip deeply below the no-probe line and rise very steeply on either side of this minimum. For values of P greater than 1.6 the first minimum becomes very pronounced and constitutes most or all of the left tail. Curiously in different experiments Observer T produced functions of different types (Figs. 4 and 5) indicating different exponents. This suggests that the exponent may not be stable. The outcome of this experiment is reminiscent of experiments in which conflicting cues are presented simultaneously and the resolution is intermediate to what it would be if one or the other cue determined the outcome. Here disparity has one value if the relative distance of two points depends only on the disparity between them (no-probe condition). It would
J. bl. FOLEY
1520
have another value if relative distance depended only on disparity with respect to tix probe light. In fact, it has a value intermediate between these two. This outcome can be interpreted as the resolution of a contlict between the disparity between a and b on the one hand and the disparities between each of these lights and the probe light on the other. It is impossible to say on the basis of these results if one of these two cues dominates and if so which one. The results do, however, suggest that relative dominance differs between observers. Given the inte~retation of these experiments as studies in cue conflict, the work of GOGEL(1956, 1963) would lead one to expect that the effect of the probe would be inversely related to the distance between the probe and the other two lights (the adjacency principle). The fourth experiment (Fig. 7) did show a small decrement in the probe effect as directional adjacency of the probe decreased. But the results of the fifth experiment (Fig. 8) indicate that this decrement is at best very small while the probe is within i 15 deg of the median plane. Gogel’s formulation would also predict a decrease in the dominance of the probe as the adjacency in depth of the probe to Q and b decreased. Such a decrease is suggested by the data of Observer R {Fig. 6) in which disparity actuahy decreases at low values ofconvergence angle. This decrease does not occur in the other functions. However, a decrease in the relative dominance of the probe cue only requires that the disparity diverge more and more from the appropriate probe reference curve as depth adjacency decreases. These data are not sufficiently precise to determine whether or not this happens. In the four experiments in which the probe moved in depth, it was noted that thevariance of the settings was not constant. At the ends of the convergence angle range, variance tended to be high relative to the no-probe condition, but it reached a minimum lower than in the no-probe condition in the middle of the range. Tests of homogeneity of variance indicate significant differences among variances in only about one-third of the cases. However, the fact that the relation between variance and probe convergence angle is roughly the same in most cases indicates that the effect is real. The fact that variance is sometimes less with three lights than with two suggests that variance is a decreasing function of the number of lights. The more lights present or the more cues to relative distance present, the more precisely determined are the relative distances in the configuration. On the other hand, if the probe is at one end of its range, particularly the far end, variance is higher than in the no-probe condition. But this is also the region in which the probe produces its greatest effect. This suggests that variance is an increasing function of cue conflict.
REFERENCES BL+.xK,A. A. (1958). Analysis of experiments in binocular space perception. f. opr. SW. .&w. d&91 l-925. FOLEY,J. ?4. (1964). Desarguesian property in vi&l space. .I. opt. Sot. Am. Y, 68ti92. FOLEY,J. hl. (1967a). Binocular disparity and perceived relative distance: An examination of two hypotheses. Vision Res. 7,655-670.
FOLEY,.I. M. (l967b). Disparity increase with convergence for constant perceptual criteria.
Percept.
fsychoph.w
2,605-608.
FOLEY,J. M. f 1968). Depth. size and distance in stereoscopic vision. Percept. Psychopiqx 3,265-2X. GOGEL,W. C. (1956). Relative visual direction as a factor in relative distance perceptions. Psr_eltoi. 70,
Monbgr.
Ii.
I-19. GOGEL,W. C. (1963).
The visual perception ofsize and distance. Vision Res. 3, 101-120. HARDY.L. H., RAND,G., RITTLER.M. C., BLANK.A. A. and BOEDER,P. (1953). The Geometr? ofBinocular Space Petceprion, Knapp Memorial taboratories, Institute of Ophtkalmoio~y, Columbia Umversity. College of Physicians and Surgeons, New York.
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LUNEBURC. R. K. (1947). .lfnrhemuricul AnnlvGs of’Binoculur Vision. Princeton University Press. Princeton, New Jersey. SHIPLEY. T. (!957). Convergence function in binocular visual space. J. opt. Sot. rim. 47.79.~811. SHIPLEY, T. and W~LLIMS. D. (1968). The relationship between retinal disparity and relative visual distance. C’ision Rex. 8.325-332. the function relating the horizontal disparity between two points and the ratio of their perceived egocentric distances has any form other than exponential, then the distance ratio between two points will be influenced by the presence and position of a third point. Five experiments are reported in which this dependence is shown to exist and is measured as a function of the position of the third point. The functions obtained are inconsistent with several possible explanations of the effect, but are consistent with the hypothesis that the perceived distance ratio of two points is jointly determined by the positions of all the points in the configuration. Abstract-If
R&urn&-Si la fonction qui relie la disparite horizontale entre deux points et Ie rapport de leurs distances Cgocentriques percues est autre qu’une exponentielle. le rapport de distance entre deux points sera influence par la presence et lo position d’un troisiime point. On expose cinq experiences dans lesquelles on montre que cette dependence existe. et on la mesure en fonction de la position du troisieme point. Les fonctions obtenues ne sont pas en accord avec diverses explications possibles. mais en accord avec I’hypothese que le rapport de distance percue des deux points est determine conjointement par les positions de tous les points de la configuration. Zusammenfassung-Wenn die die Horizontaldisparitlt zweier Punkte und das Verhaltnis der egozentrischen Beobachtungsentfernungen verbindende Funktion anders als exponentiell verlauft, dann wird das Verhiiltnis zwischen den Beobachtungsentfernungen zwischen zwei Punkten durch die Gegenwart und Position eines dritten Punktes beeinflusst. Es werden fiinf V’ersuche mitgeteilt. in welchen dieser Einfluss bestatigt und als Funktion der Position des dritten Punktes bestimmt wird. Die Funktionen konnen mit keiner Erkhirung des Effektes in Ubereinstimmung gebracht werden. aber stimmen mit der Annahme iiberein, derzufolge das beobachtete Entfernungsverhaltnis zweier Punkte gleichgehend durch die Positionen aller Konfigurationspunkte bestimmt wird. Pexome
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