DEPTH CUES AND CONSTANCY SCALING IN THE HORIZONTAL-VERTICAL ILLUSION: THE BISECTION ERROR'f JOAN S. GIRGUS The City College of the City University of New York STANLEY COREN University of British Columbia ABSTRACT
Analysis of the horizontal-vertical illusion in terms of possible depth cues allows the prediction of a bisection illusion in which the length of the lower portion of the vertical is underestimated relative to the length of the upper segment. Significant variations in illusion magnitude as a function of line length and angle of inclination indicates that height in the plane is the depth cue which is evoking inappropriate size constancy scaling.
1855 OPPEL NOTICED that the apparent length of a vertical line was overestimated relative to that of a horizontal line of equal extent. This is, of course, the horizontal-vertical illusion. Forty-two years later (1897), Wundt attempted to explain this distortion on the basis of eye movement feedback. He reasoned that the effort expended in traversing a vertical line was greater than that used to view a horizontal line. He argued that this greater effort was then translated into the perception of greater extent by the higher centres. Hicks and Rivers (1908) were able to discredit the importance of eye movements by showing that the illusion was still present in tachistoscopic exposures where the presentation was too brief for eye movements to occur. Since this early attempt at explanation, only one theoretical interpretation of the horizontal-vertical illusion has gained any widespread acceptance. The major premise of this position is that the perimeter of the visual field is shaped like an ellipse with its major axis horizontally oriented (Kunnapas, 1955a,fo, 1957a,b, 1959). In terms of closeness to the perimeter then, a vertical line takes up more of the available visual field than does an equally long horizontal line. To the extent that length judgments are made relative to the proportion of the visual field covered by the stimulus, the vertical line will be perceived as longer. In dealing with other illusion configurations, a number of investigators IN
*We wish to acknowledge the assistance of Francine Shapiro in the collection of these data and Raymond Pass in the statistical analysis. fThis research was supported in part by a grant from the National Research Council of Canada (A-9783). 59 CANAD. J. PSYCHOL./REV. CANAD. PSYCHOI,., 1975, 29
(1)
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have suggested that these figures contain pictorial depth cues which may prompt observers to interpret the figure as a two-dimensional representation of a three-dimensional array. If these implicit depth cues are sufficiently strong, they could evoke the constancy scaling mechanisms for size and shape. Given the fact that such constancy adjustment is inappropriate for two-dimensional figures, the resultant percept would be distorted. Misapplied constancy scaling explanations, based upon inferred depth cues in illusion configurations, have been reasonably successful in explaining a variety of illusions of extent (Day, 1972; Gregory, 1963), direction (Gillam, 1971), and shape (Coren & Festinger, 1967). In light of the general success of such constancy explanations of visual illusions, it seems reasonable to ask if there are potential depth cues which might lead to the overestimation of a vertical line relative to a horizontal line. A horizontal line in a two-dimensional array seems to offer little in the way of suggested depth cues. A vertical line, on the other hand, frequently represents a line receding into the third dimension as it is projected onto a two-dimensional surface. Consider Figure 1A. In this figure, an eye is viewing a line lying horizontally along a surface. This line is receding away from the observer. Notice that point a along the line is farther away from the observer than is point b. Let us now look at the two-dimensional representation of this projection, as it might appear in a photograph or in the retinal image. This is shown in Figure IB. Note that here point a is represented by a point higher on the picture plane than is point b. This is the basis of the artistic convention of representing more distant points as being higher up on the picture plane. Thus in Figure 1C, we have two identical figures. Although they differ only in their relative height on the surface of the paper, most observers will tend to see the higher figure as being more distant than the lower figure. If we now translate this argument into an analysis of a vertical line, it is clear that the cue of relative height in the picture plane could conceivably serve to indicate that the top of the vertical line is more distant than the bottom of the line. If this source of depth information is encoded as such by the observer, we have a situation in which the horizontal fine and the vertical fine subtend the same visual angle, but the vertical fine appears to lie at a greater distance from the observer for most of its length than does the horizontal. In normal threedimensional perception, size constancy would cause an apparently more distant target to be perceived as larger than an apparently closer target of equal retinal size. This same mechanism may be responsible for the overestimation of the vertical fine if it is seen as receding into the distance, while the horizontal fine is not. If such a depth cue is indeed operative in this situation, there is an interesting new illusion which can be predicted. The geometry of the
1975]
HORIZONTAL-VERTICAL ILLUSION
61 • a
b
c
B
A
C FIGURE 1 (A & B). The geometric projection of a horizontal line on a two-dimensional surface, showing how equally spaced points are projected with unequal spacing as a function of distance from the surface. (C) The higher square looks farther away than the lower square.
situation is such that a constant retinal unit should represent a greater change in distance for the upper part of the line than for the lower part of the line. This is illustrated in Figure 1A where points a, b, and c are all equally spaced. In the two-dimensional projection of this configuration, however, it is clear that points a and b, representing the two more distant elements, are more closely spaced than are points b and c. Thus, if an observer is asked to bisect a vertical line and that vertical line is interpreted as receding in depth, we should find an overestimation of the upper half of the line relative to the lower half. In addition, if height in the picture plane is the major determinant, then gradually varying the inclination of a line from horizontal through intermediate orientations to vertical should lead to a corresponding increase in the amount of bisection error since, as the orientation changes in this fashion, the vertical discrepancy between the highest and lowest points on the line gradually increases. The experiment reported below attempts to demonstrate the existence of this bisec-
62
GIRGUS & COREN
[Vol. 29, No. 1
tion illusion, which is predicted from a depth processing interpretation of the horizontal-vertical illusion. METHOD
Subjects Thirty-eight undergraduate volunteers served as subjects in this experiment. Each observer was tested under all stimulus conditions. Stimuli The stimuli consisted of 1 mm wide lines that could be either 6, 8, 10, or 12 cm in length. The lines could be presented at inclinations from the horizontal of 0, 30, 60, or 90°. Each line appeared singly, centred on a sheet of white paper which was 8.5 inches square. Procedure The stimuli were presented one at a time in random order. Each stimulus was placed flat on a low table in front of the subject. The subject was asked to bend over the stimulus so that he was centred above it with his face parallel to the plane of the stimulus. He was told that his task was to bisect each individual line by placing a pencil mark on it at a point that divided the line into two perceptually equal segments. Data were scored to the closest 0.5 mm. RESULTS AND DISCUSSION
The data from this experiment are plotted in Figure 2 in terms of the difference between the apparent size of the upper and lower line segments (or between the segments to the right and left of the bisection point in the case of the horizontal line). A positive value means that the upper half of the line was overestimated relative to the lower half. (For the horizontal line, a positive value is arbitrarily assigned for the relative overestimation of the right side.) Note that, in terms of our scoring procedure, an overestimation of the upper half of the line manifests itself in terms of a bisection point that is placed higher on the line than is physically warranted. The subject thus indicates that it takes a physically longer lower line segment to match the perceived length of the upper line segment. It is quite clear that the predicted bisection error exists. We need only look at the horizontal bisections which show negligible errors as compared to the vertical bisection scores which show strong positive errors in the direction predicted by the depth cue analysis (F = 6.04; df = 1/37; p < .05). Such a bisection error is clearly predictable if the vertical line is seen as receding into space. However, if we are to believe the cogency of the height in the picture plane cue, we must look at the totality of the data in order to assess its effectiveness in evoking constancy scaling.
1975]
HORIZONTAL-VERTICAL ILLUSION
5
4
3
3
63
< 2 NOis
z o
0
30
60
90
LINE ORIENTATION (DEG) FIGUHE 2. The overestimation of the size of the upper portion of the line is plotted as a function of angle of inclination for each line length.
If height in the picture plane operates as a cue for depth and is responsible for this observed bisection error, we would expect a significant increase in this bisection error as a function of line length. Such a line length effect is clearly visible in Figure 2 and is statistically reliable (F = 4.32; df = 3/111; p < .01). There is also a statistically significant, monotonically increasing linear trend in bisection error as a function of line length (F = 12.97; df = 1 / 111; p
Analysis of the horizontal-vertical illusion in terms of possible depth cues allows the prediction of a bisection illusion in which the length of the lower portion of the vertical is underestimated relative to the length of the upper segment. Significant variations in illusion magnitude as a function of line length and angle of inclination indicates that height in the plane is the depth cue which is evoking inappropriate size constancy scaling.
1855 OPPEL NOTICED that the apparent length of a vertical line was overestimated relative to that of a horizontal line of equal extent. This is, of course, the horizontal-vertical illusion. Forty-two years later (1897), Wundt attempted to explain this distortion on the basis of eye movement feedback. He reasoned that the effort expended in traversing a vertical line was greater than that used to view a horizontal line. He argued that this greater effort was then translated into the perception of greater extent by the higher centres. Hicks and Rivers (1908) were able to discredit the importance of eye movements by showing that the illusion was still present in tachistoscopic exposures where the presentation was too brief for eye movements to occur. Since this early attempt at explanation, only one theoretical interpretation of the horizontal-vertical illusion has gained any widespread acceptance. The major premise of this position is that the perimeter of the visual field is shaped like an ellipse with its major axis horizontally oriented (Kunnapas, 1955a,fo, 1957a,b, 1959). In terms of closeness to the perimeter then, a vertical line takes up more of the available visual field than does an equally long horizontal line. To the extent that length judgments are made relative to the proportion of the visual field covered by the stimulus, the vertical line will be perceived as longer. In dealing with other illusion configurations, a number of investigators IN
*We wish to acknowledge the assistance of Francine Shapiro in the collection of these data and Raymond Pass in the statistical analysis. fThis research was supported in part by a grant from the National Research Council of Canada (A-9783). 59 CANAD. J. PSYCHOL./REV. CANAD. PSYCHOI,., 1975, 29
(1)
60
GIRGUS & COREN
[Vol. 29, No. 1
have suggested that these figures contain pictorial depth cues which may prompt observers to interpret the figure as a two-dimensional representation of a three-dimensional array. If these implicit depth cues are sufficiently strong, they could evoke the constancy scaling mechanisms for size and shape. Given the fact that such constancy adjustment is inappropriate for two-dimensional figures, the resultant percept would be distorted. Misapplied constancy scaling explanations, based upon inferred depth cues in illusion configurations, have been reasonably successful in explaining a variety of illusions of extent (Day, 1972; Gregory, 1963), direction (Gillam, 1971), and shape (Coren & Festinger, 1967). In light of the general success of such constancy explanations of visual illusions, it seems reasonable to ask if there are potential depth cues which might lead to the overestimation of a vertical line relative to a horizontal line. A horizontal line in a two-dimensional array seems to offer little in the way of suggested depth cues. A vertical line, on the other hand, frequently represents a line receding into the third dimension as it is projected onto a two-dimensional surface. Consider Figure 1A. In this figure, an eye is viewing a line lying horizontally along a surface. This line is receding away from the observer. Notice that point a along the line is farther away from the observer than is point b. Let us now look at the two-dimensional representation of this projection, as it might appear in a photograph or in the retinal image. This is shown in Figure IB. Note that here point a is represented by a point higher on the picture plane than is point b. This is the basis of the artistic convention of representing more distant points as being higher up on the picture plane. Thus in Figure 1C, we have two identical figures. Although they differ only in their relative height on the surface of the paper, most observers will tend to see the higher figure as being more distant than the lower figure. If we now translate this argument into an analysis of a vertical line, it is clear that the cue of relative height in the picture plane could conceivably serve to indicate that the top of the vertical line is more distant than the bottom of the line. If this source of depth information is encoded as such by the observer, we have a situation in which the horizontal fine and the vertical fine subtend the same visual angle, but the vertical fine appears to lie at a greater distance from the observer for most of its length than does the horizontal. In normal threedimensional perception, size constancy would cause an apparently more distant target to be perceived as larger than an apparently closer target of equal retinal size. This same mechanism may be responsible for the overestimation of the vertical fine if it is seen as receding into the distance, while the horizontal fine is not. If such a depth cue is indeed operative in this situation, there is an interesting new illusion which can be predicted. The geometry of the
1975]
HORIZONTAL-VERTICAL ILLUSION
61 • a
b
c
B
A
C FIGURE 1 (A & B). The geometric projection of a horizontal line on a two-dimensional surface, showing how equally spaced points are projected with unequal spacing as a function of distance from the surface. (C) The higher square looks farther away than the lower square.
situation is such that a constant retinal unit should represent a greater change in distance for the upper part of the line than for the lower part of the line. This is illustrated in Figure 1A where points a, b, and c are all equally spaced. In the two-dimensional projection of this configuration, however, it is clear that points a and b, representing the two more distant elements, are more closely spaced than are points b and c. Thus, if an observer is asked to bisect a vertical line and that vertical line is interpreted as receding in depth, we should find an overestimation of the upper half of the line relative to the lower half. In addition, if height in the picture plane is the major determinant, then gradually varying the inclination of a line from horizontal through intermediate orientations to vertical should lead to a corresponding increase in the amount of bisection error since, as the orientation changes in this fashion, the vertical discrepancy between the highest and lowest points on the line gradually increases. The experiment reported below attempts to demonstrate the existence of this bisec-
62
GIRGUS & COREN
[Vol. 29, No. 1
tion illusion, which is predicted from a depth processing interpretation of the horizontal-vertical illusion. METHOD
Subjects Thirty-eight undergraduate volunteers served as subjects in this experiment. Each observer was tested under all stimulus conditions. Stimuli The stimuli consisted of 1 mm wide lines that could be either 6, 8, 10, or 12 cm in length. The lines could be presented at inclinations from the horizontal of 0, 30, 60, or 90°. Each line appeared singly, centred on a sheet of white paper which was 8.5 inches square. Procedure The stimuli were presented one at a time in random order. Each stimulus was placed flat on a low table in front of the subject. The subject was asked to bend over the stimulus so that he was centred above it with his face parallel to the plane of the stimulus. He was told that his task was to bisect each individual line by placing a pencil mark on it at a point that divided the line into two perceptually equal segments. Data were scored to the closest 0.5 mm. RESULTS AND DISCUSSION
The data from this experiment are plotted in Figure 2 in terms of the difference between the apparent size of the upper and lower line segments (or between the segments to the right and left of the bisection point in the case of the horizontal line). A positive value means that the upper half of the line was overestimated relative to the lower half. (For the horizontal line, a positive value is arbitrarily assigned for the relative overestimation of the right side.) Note that, in terms of our scoring procedure, an overestimation of the upper half of the line manifests itself in terms of a bisection point that is placed higher on the line than is physically warranted. The subject thus indicates that it takes a physically longer lower line segment to match the perceived length of the upper line segment. It is quite clear that the predicted bisection error exists. We need only look at the horizontal bisections which show negligible errors as compared to the vertical bisection scores which show strong positive errors in the direction predicted by the depth cue analysis (F = 6.04; df = 1/37; p < .05). Such a bisection error is clearly predictable if the vertical line is seen as receding into space. However, if we are to believe the cogency of the height in the picture plane cue, we must look at the totality of the data in order to assess its effectiveness in evoking constancy scaling.
1975]
HORIZONTAL-VERTICAL ILLUSION
5
4
3
3
63
< 2 NOis
z o
0
30
60
90
LINE ORIENTATION (DEG) FIGUHE 2. The overestimation of the size of the upper portion of the line is plotted as a function of angle of inclination for each line length.
If height in the picture plane operates as a cue for depth and is responsible for this observed bisection error, we would expect a significant increase in this bisection error as a function of line length. Such a line length effect is clearly visible in Figure 2 and is statistically reliable (F = 4.32; df = 3/111; p < .01). There is also a statistically significant, monotonically increasing linear trend in bisection error as a function of line length (F = 12.97; df = 1 / 111; p
