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Searching for invariance: Geographical and optical slant
Vision Research 149 (2018) 30–39
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Vision Research journal homepage: www.elsevier.com/locate/visres
Searching for invariance: Geographical and optical slant ⁎
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Olivia C Cherry , Geoffrey P Bingham Indiana University, Bloomington, United States
A R T I C LE I N FO
A B S T R A C T
Number of reviewers = 2
When we move through rigid environments, surface orientations of static objects do not appear to change. Most studies have investigated the perception of optical slant which is dependent on the perspective of the observer. We investigated the perception of geographical slant, which is invariant across different viewing perspectives, and compared it to optical slant. In Experiment 1, participants viewed a 3D triangular target surface with triangular phosphorescent texture elements presented at eye level at one of 5 slants from 0° to 90°, at 0° or 40° tilt. Participants turned around to adjust a 2D line or a 3D surface to match the slant of the target surface. In Experiment 2, the difference between optical and geographical slant was increased by changing the height of the surface to be judged. In Experiment 3, target surfaces were rotated by 50° ( ± 25°) and viewed in both a dark and lighted room. In Experiment 1, the overall pattern of judgments exhibited only slight differences between response measures. In Experiment 2, slant judgments were slightly overestimated when the surface was at a low height and at 0° tilt. We compared optical slants of the surfaces to geographical slants. While sometimes inaccurate, participants’ slant judgments remained invariant across changes in viewing perspective. In Experiment 3, judgments were the same in the dark and lighted conditions. There was no effect of target motion on judgments, although variability decreased. We conclude that participants’ judgments were predicted by geographical slant, not optical slant.
Keywords: Geographical slant Slant perception 3D perception
1. Introduction Numerous investigations of the perception of slant have studied optical slant primarily (Braunstein & Payne, 1969; Koenderink & van Doorn, 1976; Norman, Todd, Norman, Clayton, & McBride, 2006, Norman, Todd, & Phillips, 1995; Perrone, 1982; Phillips, 1970; Stevens, 1983; Todd, Thaler, & Dijkstra, 2005; van Ee & Erkelens, 1996; van Ee, van Dam, & Erkelens, 2002). Optical slant is egocentric and is defined as the angle between the surface normal and the line of sight (Todd & Perotti, 1999). Any change to the viewer’s position or the surface’s position yields changes in a surface’s optical slant, even though the orientation of the surface to the surroundings does not change. A single large scale surface has multiple optical slant values at different points across the surface. For example, if an observer stands at one end of a long table, the closest end has a very different optical slant value than the middle of the table or the opposite end. However, the orientation of different portions of the table’s surface are generally perceived and judged to be the same, that is, the surface is flat and level. In addition, when navigating through a static environment, people do not perceive rigid surfaces as constantly changing in orientation. If optical slant was used to inform us about the slant of surfaces in the world, the experience would be chaotic and disorienting.
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Fortunately, perception of the slant of rigid surfaces remains stable with changes in viewer position (e.g., a flat table continues to appear flat no matter where the observer is). Without this stability, interaction with our environment would be incredibly difficult. The instability of optical slant makes it less likely that we use it to inform us about the slant of surfaces. Sedgwick and Levy (1985) suggest that it would be more efficient for the visual system to rely on geographical slant rather than continuously update itself with fluctuating optical slants. Geographical slant is the slant of a surface relative to gravity which remains stable with changes in a viewer’s position. Therefore, investigating the perception of geographical slant may be a more ecological approach to understanding human perception of surface orientation. Perception of the slant of large-scale surfaces has been studied extensively (Proffitt, Bhalla, Gossweiler, & Midgett, 1995; Proffitt, Creem, & Zosh, 2001; Stefanucci, Proffitt, Clore, & Parekh, 2008). Much of this research has concluded that the perception of the slant of large-scale surfaces (e.g., hills and ramps) depends on extraneous factors such as glucose levels (Schnall, Zadra, & Proffitt, 2010) and social support (Schnall, Harber, Stefanucci, & Proffitt, 2008). Therefore they assumed constancy across the large surfaces while suggesting that judgment accuracy reflects states of the perceiver/actor as much as the actual slant of the surface. In general, studies have focused on the relative
Corresponding author at: Department of Psychological and Brain Sciences, Indiana University, Bloomington, 1101 W. 10th St., Bloomington, IN 47405, United States. E-mail address: [email protected] (O.C. Cherry).
https://doi.org/10.1016/j.visres.2018.05.008 Received 30 January 2018; Received in revised form 15 May 2018; Accepted 17 May 2018 0042-6989/ © 2018 Elsevier Ltd. All rights reserved.
Vision Research 149 (2018) 30–39
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accuracy of slant perception or the lack thereof. Durgin, Li, and Hajnal (2010) conducted a study investigating the tendency to overestimate the slant of small-scale surfaces in near visual space. They had participants judge geographical slants by verbal estimation and a free-hand matching task. They found that there was a bias in the estimation of geographical slant: participants showed a tendency to perceive slanted surfaces to be steeper than they actually were. Durgin et al. also used a two alternative forced choice paradigm and a psychometric staircase procedure to calculate the point of bisection (i.e., the equidistant point between vertical and horizontal) perceived by their participants. On each trial, participants would be shown a single slanted surface and would indicate whether the slant was closer to vertical or horizontal. On average, the equidistant point between horizontal and vertical was 34°. The current investigation is focused on determining whether or not there is constancy of slant judgments across different viewing perspectives rather than whether those slant judgments are accurate. Smaller scale surfaces have fewer different optical slant values that are almost equivalent to each other. Optical slant can be manipulated easily using smaller surfaces by changing viewing perspective. Thus, optical slant can be compared to geographical slant at each viewing perspective. The current study uses small-scale surfaces in near space for this reason. The response measure that has been used most frequently in studies of geographical slant perception is the palm board (Coleman & Durgin, 2014). The palm board is typically placed at waist level, and is out of the participants’ sight. Participants place their hand on it to adjust the board to be parallel to the angle of the target surface they are observing. Gibson (1950) began using a palm board to provide participants with a non-verbal method to estimate slant. Proffitt et al. (1995) found that participants tended to overestimate geographical slant when making verbal estimates or adjusting an angle on a disk to match the crosssection of the inclination of the hill, but when using the palm board, they were more accurate. Witt and Proffitt (2007) conducted a study directly comparing slant judgments of large scale hills using three response measures: a palm board, a relative visual matching task, and an absolute visual matching task. They found that judgments made with the palm board were significantly more accurate than the other tasks and were not significantly different from the actual slope of the hill. Witt and Proffitt concluded that judgments made with the palm board were accurate because the measure did not require explicit perceptual awareness but instead relied on visuomotor control, which they posit to be separate processes. Durgin, Hajnal, Li, Tonge, and Stigliani (2010) conducted several experiments investigating the accuracy and reliability of palm boards. They found that judgments of slants in near space made with palm boards were consistently underestimated compared to a free-hand measure. Coleman and Durgin (2014) found that there is a systematic bias for haptic perception of surface orientation. Participants would underestimate the slant angles when the palm board was set at a lower height (aligned with the participant’s navel) and overestimate slant angles when the palm board was higher (at eye level). The primary purpose of the current study was to investigate whether perceived slant remains invariant under changes in viewing perspective. In Experiment 1, the surfaces were presented at either 0° tilt or 40° tilt. Geographical tilt is defined here as the amount of rotation around a vertical axis through the surface. For a slanted surface presented at 0° tilt, the surface would be slanted directly away from the observer. If that surface was presented at 90° tilt, the observer would see the slanted surface edge on. The optical slant for this surface would be very different in both cases, while the geographical slant would remain the same. Geographical slant is unaffected by changes in tilt while optical slant varies. Fig. 1 shows optical slant values plotted against geographical slant values. The solid thin line represents the relationship between optical and geographical slant when the target surface was at eye level and 0° tilt. In this case, optical and geographical slant are equivalent. The dashed thin line corresponds to the condition in which the target surface was presented at eye level at 40° tilt. In this case,
Fig. 1. Transformed optical slant values are plotted against corresponding geographical slant values for each condition. Optical slant values were linearly transformed so that 0° and 90° optical slant would correspond to 0° and 90° geographical slant. Optical slants were computed continuously for corresponding geographical slants from 0° to 90° presented at and below eye level at 0° and 40° tilt. Points on the functions correspond to geographical slant values presented to participants in the study. The solid lines and circles indicate surfaces presented at tilt 0° and the dashed lines and squares indicate those presented at tilt 40°. The thin lines and open shapes indicate surfaces presented at eye level (Experiment 1) and the thick lines and closed shapes indicate surfaces presented below eye level (Experiment 2).
optical slant varies with the same geographical slant values. In Experiment 2, the surfaces were presented at 0° and 40° tilt again, but at a lower eye height to amplify the difference between geographical and optical slant. The solid thick line corresponds to target surfaces presented at 0° tilt below eye level and the dashed thick line corresponds to target surfaces presented at 40° tilt below eye level. Again, optical slant varies. For the same unchanging set of geographical slants, optical slant changes dramatically across the different viewing conditions (for 5 geographical slants there are 18 different optical slants). If participants perceive optical slant, then their judgments of geographical slant should reflect these differences. Given the controversial nature of the palm board, we used two visually guided response measures: an adjustable 3D surface that participants controlled and matched to the perceived geographical slant and a 2D line viewed on a computer screen and adjusted to match the perceived geographical slant. The participant sat between the response measure and target surface so they could not be viewed simultaneously. Durgin and Li (2011) found a bias to overestimate the slant of 2D lines, which would prove problematic for using a line task as a dependent measure. A secondary purpose of Experiment 1 was to determine whether adjusting a 2D line would yield the same results as adjusting a 3D surface. 2. Experiment 1 2.1. Methods 2.1.1. Participants Ten adults ranging in age from 20 to 30 (two males) participated in this experiment. All participants had normal or corrected-to-normal vision and passed a stereo fly test (Stereo Optical Co., Inc.) measuring stereo acuity. Participants were required to identify a target circle with a disparity of 80 s of arc in order to participate in the study. All participants gave written informed consent prior to taking part in the study. Research was conducted in accordance with the Code of Ethics of the 31
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back of each surface so it could be easily attached and removed from the presentation and response apparatus. The presentation apparatus consisted of a small Plexiglas platform that was attached to the top of a metal rod extending from a wooden block. The metal rod was stationary and oriented vertically. Velcro was attached to the surface of the platform so the surfaces could be mounted on it. At the top of the rod, the platform could be rotated to set it at various slants at 0° tilt and at eye height. By keeping the center of rotation close to the surface, the surface’s slant with respect to the observer’s line of sight was similar to the geographical slant of the surface. A small piece of wood was marked to indicate the angles at which the platform could be set (i.e., 0°–90° in increments of 10) and was attached to the stationary metal rod extending from the block. An additional straight metal rod was attached to the bottom of the platform to act as a pointer so the experimenter could easily set the surface to the correct slant value. The target surface was elevated approximately 35 cm off the table (111.2 cm off the ground) and was at the participant’s eye level. The pointer, platform, and piece of wood could not be seen by the observer because they were occluded by the target surface. The observer could only see the block, the metal rod, and the surface. There were two different response measures used in this experiment: a line displayed on a computer screen and a response surface mounted on an apparatus. The line response measure was displayed on an ASUS Notebook Q500A with a screen resolution of 1366 × 768 using a program generated and run using MatLab R2016a. The response surface apparatus was constructed similarly to the target surface apparatus with the following exceptions. The response surface apparatus was attached to the table and could not be moved (i.e., the tilt of the response surface could not be altered from 0°). The metal rod attached to the platform was instead made of plastic in order to not interfere with the magnetic field generated by the MiniBird measurement system (described below). The screws attaching the plastic rod to the wooden block and the plastic rod to the platform were stainless steel (no magnetism). The plastic rod extending from the block was also not stationary. The slant of the surface was adjusted by moving this plastic rod as opposed to moving the platform, so the center of rotation was located at the top of the wooden block. This means that the optical slant of the response surface was different from the optical slant of the target surface at each geographical slant value (see Fig. 3). The response surface also was not placed at eye level (the response surface was 2 cm shorter). These differences in eye height and center of rotation between the target and response apparatuses were included to ensure that the task was not trivial. Different triangles with different sized texture elements were also used as target and response surfaces for each trial to enhance the difference between them (e.g., if the target surface had large texture elements the response surface would have small texture elements and these surfaces would be different shapes). The response and target surfaces were located on opposite sides of the participants so they could not be compared simultaneously. Participants were not simply matching image structure. A MiniBird magnetic measurement system was used to record the slant of the response surface. Two markers were attached to the front and back sides of the flat platform that the surface rested on. The magnetic field emitter was placed directly underneath the table in line with the response apparatus. The trackers’ position coordinates were recorded by a program executed on a Dell E5200 computer so that the slant could be calculated from these positions.
Fig. 2. A photograph of the triangular surfaces mounted on the presentation and response apparatus in Experiments 1, 2, and 3. The left column shows triangles with large texture elements (average area = 0.64 cm2) and the right column shows triangles with small texture elements (average area = 0.16 cm2).
World Medical Association (Declaration of Helsinki).
2.1.2. Materials and apparatus Target slant values were presented using an actual surface that was mounted on an apparatus that allowed adjustment and positioning of the surface orientation. Different surface shapes and surface textures were used throughout the experiment. As can be seen in Fig. 2, the surfaces consisted of two sets of six different black triangles made of black PVC plastic with equivalent surface areas (mean = 70.49 cm2) and an average angular size of 46.4°. Triangles were chosen because they do not provide any information about the slant of the surface (Koenderink & van Doorn, 1991). On one of these sets, small triangularshaped texture elements were placed randomly on the surfaces (see right column of Fig. 2). On the other set, larger versions of the texture elements were placed on the surfaces (see left column of Fig. 2). The texture elements were made of phosphorescent tape. Triangular-shaped texture was used so that the shape of the individual texture elements could not provide information about the slant of the surface. The shape of the texture elements matched the shape of the surface they were on. Before the texture was placed on the surfaces, the surfaces were rubbed with steel wool to reduce specularity. A piece of velcro was glued to the
2.1.3. Procedure After reading and signing the informed consent, participants were given a stereo fly test to assure that their stereo acuity was adequate. Participants then sat in a swivel chair that had been modified to include an adjustable chin rest. The chin rest could be moved closer or farther from the participant and its height could also be adjusted. See Fig. 4 for a diagram of the experimental set up. The swivel chair was placed between two tables: on one table was the target apparatus and on the 32
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were informed that the response and target apparatus were slightly different, so the only component of the apparatus that mattered was the surface and its slant. For trials during which the line task was used, the response surface was replaced with a laptop. Participants were instructed to adjust the slope of a single 2D line on the computer screen to match the slant of the target surface if it were viewed from the side. The line would be randomly oriented either vertically or horizontally at the beginning of each trial. To adjust the slope of the line, participants used the left and right arrow keys and when they were ready to move on to the next trial, they pressed the space bar. Participants could look at the target surface and the response measure as many times as they liked by rotating the chair, while still in the chin rest, to view either the target or the response surface. There was no time limit.
2.2. Results and discussion The primary purpose of Experiment 1 was to determine whether slant judgments would remain invariant over changes in viewing perspective (e.g., differences in geographical tilt). Fig. 5 plots the mean judged slant for tilt 0° (open circles) and tilt 40° (filled circles) as a function of geographical (left panel) and optical (right panel) slant. In the right panel, the dashed line indicates the geographical slant values for 0° tilt and the solid line indicates geographical slant values for 40° tilt. At tilt 0°, optical slant and geographical slant are equivalent. One can easily see that the mean slant judgments at tilt 0° are very close to the optical and geographical slant values of the target surface. At tilt 40°, however, the mean slant judgments do not align with the optical slant values but instead are very close to the corresponding geographical slant values of the target surface. If viewing perspective was affecting participants’ judgments, mean slant judgments for both tilt 0° and 40° should overlap in the right panel of Fig. 5 rather than in the left panel. This is not the case in the current experiment. Mean slant judgments were closer to the corresponding geographical slants (left panel of Fig. 5) than optical slants (right panel of Fig. 5). On average, the 90° target slant was underestimated compared to the other slant values (mean errors for 0°, 20°, 40°, 60°, and 90° were −0.96, −2.23, −6.72, −2.43, and 3.28, respectively). As can be seen in Fig. 6, the 40° and 60° slants were overestimated more when adjusting the line (filled circles) relative to adjusting the surface (open circles). Also, the 90° slant was underestimated more when using the response surface compared to the line. The change in tilt did not affect participants’ slant judgments. The pattern of slant judgments remained constant across variations in viewing perspective. A 5 (geographical slant: 0°, 20°, 40°, 60°, 90°) × 2 (tilt: 0° and 40°) × 2 (response measure: 3D surface and 2D line task) repeated measures analysis of variance (ANOVA) was conducted on the mean error of participants’ slant judgments. Error was calculated by subtracting participants’ slant judgments from the geographical slant of the target surface. These errors were averaged across the five repetitions of each geographical slant value per condition to produce the mean error. The Greenhouse-Geisser correction for violations of sphericity was applied to all results to correct for the violation of homogeneity of variance. There was no effect of
Fig. 3. Transformed optical slant values are plotted against corresponding geographical slant values for the response surface apparatus. Optical slant values were linearly transformed so that 0° and 90° optical slant would correspond to 0° and 90° geographical slant. Filled circles correspond to the five geographical slant values used in Experiments 1, 2, and 3. The dashed line indicates a perfect relationship between optical and geographical slant (optical slant = geographical slant).
other table was the response apparatus or laptop. Participants’ viewing distance from the target surface was 76.2 cm (this was also the distance from the response surface) and their eye height was in line with the middle of the target surface (111.2 cm). Participants completed 100 trials: 50 trials adjusting the response surface and 50 trials adjusting the line on the computer. Within each block of 50 trials, the target surface was oriented at zero degree tilt for 25 trials (surface was slanted away from the participants directly) and forty degree tilt for 25 trials (the apparatus was rotated by forty degrees to the left). The response surface was fixed at 0° tilt throughout the duration of the experiment. Every five trials, the surfaces were replaced with a different triangle with a different sized texture so that the response and target surface were never the same shape or texture. The surfaces were also placed on the target and response apparatuses in random orientations rotated around the surface normal. The order of response types and degree of tilt was counter-balanced across participants. For trials during which the response surface was used, participants were instructed to view the slant of the target surface and then turn around in their chair to adjust the slant of the response surface to match. If the target surface was flat/horizontal, they should also adjust the response surface to be flat/horizontal. In other words, participants were instructed to judge the geographical slant of the surfaces. They
Fig. 4. A diagram of the experimental setup when using the response surface in Experiment 1. The participant is resting his chin on the chin rest and looking at the target surface set at some angle depicted on the left side of the diagram. The response surface is directly behind the participant so that the participant loses full sight of the target surface when adjusting the response surface. The response surface has two tracking markers attached underneath it to the MiniBird measurement system and the emitter is underneath the table below the response surface. The response apparatus on the right is replaced by a laptop when performing the line task. 33
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Fig. 5. The mean slant judgments plotted as a function of geographical slant for Experiment 1 are shown in the left panel. The mean slant judgments plotted as a function of optical slant are shown in the right panel. The open circles indicate the mean slant judgments for target surfaces presented at tilt 0°. The filled circles indicate the mean slant judgments for tilt 40°. The dashed line in the left plot indicates perfect accuracy (judged slant = geographical slant). In the right panel, the continuously computed geographical slants are plotted as a function of optical slant for tilt 0° (dashed line) and tilt 40° (solid curved line). The error bars are standard error of the mean.
Fig. 6. The mean slant judgments plotted as a function of geographical slant for Experiment 1. The open circles indicate the mean slant judgments for the surface response measure. The filled circles indicate the mean slant judgments for the 2D line task. The best-fitting regression line for the surface response measure is shown by the dashed line: y = 0.91x + 3.93, R2 = 0.99. The best-fitting regression line for the 2D line task is shown by the solid line: y = 1.0x + 3.55, R2 = 0.99. The dotted line indicates perfect accuracy (judged slant = geographical slant). The error bars are standard error of the mean.
Fig. 7. The mean standard deviation for each geographical slant value in Experiment 1. The error bars are standard error of the mean.
each geographical slant value is plotted in Fig. 7. The variability in judging 0° and 90° geographical slants was far smaller than the variability in judging the middle slant values. The largest variability was found at the 40° target slant. The variability of 0° and 90° slant judgments were not significantly different from each other but were significantly different from the other slant values. 20° slant judgments were significantly different from 40° slant judgments, but not significantly different from 60° slant judgments. The variability (standard
tilt, either main effect or interaction. There was a significant main effect of geographical slant (F(2.5, 22.2) = 12.19, p < .01, η2 = 0.58). The slant by response measure interaction was significant (F(1.4, 12.7) = 6.12, p = .02, η2 = 0.41). The mean standard deviation of participants’ slant judgments for 34
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variability would suggest that participants’ judgments were primarily influenced by geographical slant values. However, the difference between optical and geographical slant values may not have been large enough to detect a difference. Geographical slant and optical slant were the same when the target surface was viewed at 0° tilt (see the thin solid line in Fig. 1). So optical slant only differed from geographical slant when the target surface was presented at 40° tilt (see the thin dashed line in Fig. 1). If optical slant influences slant judgments, then lowering and tilting the target surface should amplify this difference (see the thick lines in Fig. 1).
deviation) in these slant judgments was also subjected to a 5 (geographical slant) × 2 (tilt) × 2 (response measure) repeated measures ANOVA. There was a main effect of geographical slant (F(2.2, 19.7) = 50.09, p < .01, η2 = 0.85). Notably, there was no main effect of response measure or tilt and no significant interactions. Four simple linear regressions predicting slant judgments from geographical slants were performed for each individual participant: surface response measure and target surface at 0° tilt, line response measure and target surface at 0° tilt, surface response measure and target surface at 40° tilt, and finally, line response measure and target surface at 40° tilt. The slopes, intercepts, and R2 values were then analyzed in a 2 (response measure) × 2 (tilt) repeated measures ANOVA. No significant effects of response measure, tilt, or the interaction between response measure and tilt were found for R2 values or intercepts. A significant main effect of response measure (F(1, 9) = 51.44, p < .01, η2 = 0.85) on slope values indicated that the mean slope of judgments made in the line task (1.00) was higher than the mean slope of judgments made in the surface task (0.92). To eliminate the potential anchoring effects of 0° and 90° slant values, we also ran the previous analyses using only the middle slant values (i.e., 20°, 40°, 60°). These results were the same as those including 0° and 90° slants. Two simple linear regressions were conducted on the mean slant judgments made using the line and the surface response measure. Mean slant judgments as well as their respective regression lines are plotted in Fig. 6. Although there are significant differences between certain geographical slant values, the overall pattern of results is very similar for both response measures. The largest difference between the response measures is the difference in slope which is driven by the overestimation of 40° and 60° slant values when participants used the line. These linear regression analyses were also conducted on the middle slant values and the results from these analyses revealed very similar slopes and R2 values. The intercept values for the surface response measure increased from 3.9° to approximately 5.7° and the intercept for the line response measure decreased from 3.5° to approximately 1.9°. One goal of Experiment 1 was to determine whether there was a significant difference between adjusting a line on a computer screen and adjusting a small surface to match the slant of a target surface. Despite the significant interaction between response measure and geographical slant value, the patterns of slant judgments are very similar for the two response measures. On average, the 60° slant judgments made when adjusting the line were only 6.4° higher than those made using the surface response measure. 40° slant judgments were only 5.5° higher when adjusting the line as opposed to the surface. These are small deviations across these response measures and the overall pattern of results is still very similar across response measures as can be seen in Fig. 6. There was no main effect of response measure for the measure of variability. Also, no significant differences were found in terms of R2 or intercept values and the slopes were higher on average for the line response measure, which is unsurprising when considering the 60° and 40° slant values discussed previously. Considering the small magnitude of the differences found between judgments made in these conditions, one can conclude that both methods of estimation are similarly effective. The surface response measure may have higher face validity than the line task, but in terms of overall efficacy we conclude that the line task is a viable option.
3.1. Methods The methods in Experiment 2 were the same as in Experiment 1 except for the following. Due to the similarity between judgments made using the line and the surface response measures in the previous experiment, only the response surface was used in this experiment. The target surface was lowered by 35.7 cm. The response surface remained at the same height (2 cm below eye height). As was the case in Experiment 1, the target surface was viewed at 0° and 40° tilts while the response surface remained at 0° tilt throughout the experiment. The total number of trials was also reduced to 50 in order to keep this condition comparable to the previous study.
3.1.1. Participants The same ten participants from Experiment 1 also participated in the current experiment. All participants gave written informed consent prior to participation. Research was conducted in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki).
3.2. Results and discussion The mean error for tilt 0° at the lower surface height (−4.4) was larger than in the other conditions (eye height, tilt 0°: −0.5; eye height, tilt 40°: −0.1; low, tilt 40°: −2.5). A 5 (slant) × 2 (surface height) × 2 (tilt) repeated measures ANOVA was performed on mean errors from Experiments 1 and 2. Judgments of 90° slants were underestimated relative to all other slant values. Also, 40° target slant mean error was less than the 60° error. There was a main effect of geographical slant (F (2.3, 20.7) = 20.47, p < .01, η2 = 0.70). At the lower surface height, there was a larger difference between judgments made at 0° and 40° tilt than those made at eye height. There was primarily one condition driving the interaction between surface height and tilt, namely slant judgments at 0° tilt at the lower surface height were more underestimated than those in the other conditions by 2° (low, tilt 40°), 4° (eye height, tilt 0°), and 4.3° (eye height, tilt 40°). A significant surface height by tilt interaction was observed (F(1, 9) = 31.49, p = .02, η2 = 0.46). The variability of slant judgments from Experiments 1 and 2 were analyzed using a 5 (slant) × 2 (surface height) × 2 (tilt) repeated measures ANOVA. There was a main effect of geographical slant (F(3.4, 30.4) = 47.06, p < .01, η2 = 0.84; see Fig. 8). Pairwise comparisons revealed that the standard deviations for judgments of 0° and 90° geographical slants were not significantly different from each other but were significantly less than the other slant values. The variability in judging 60° slant values was larger than all other slant values. The variability of 20° and 40° geographical slant judgments were not different from each other, but were different than all other slant values. There was also a main effect of surface height (F(1, 9) = 8.05, p = .02, η2 = 0.47). Slant judgments made when viewing the target surface at eye level were more variable on average (SD = 4.98) than those made when the target surface was lowered (SD = 4.14).
3. Experiment 2 The purpose of Experiment 2 was to increase the difference between optical slant and geographical slant to determine whether participants’ slant judgments would be influenced by further alterations in their viewing perspective. In Experiment 1, no significant main effects of tilt were found for mean error or variability despite the fact that the optical slants were dependent on the surface tilt. This result coupled with the significant main effect of geographical slant on mean error and 35
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Fig. 9. The mean slant judgments collapsed across the motion and static conditions (Experiment 3 and the 3D surface condition of Experiment 1) are plotted against geographical slant. The open circles represent mean errors of surfaces presented at 0° tilt and the filled circles represent mean errors of surfaces presented at 40° tilt. The error bars are standard error of the mean. The dashed line indicates perfect accuracy.
Fig. 8. The mean standard deviation of slant judgments from Experiment 2 for each geographical slant value. The error bars are standard error of the mean.
4. Experiment 3 In the previous two experiments, the participants’ visual environment was not restricted. The target surface was not the only surface that could be seen. Participants could easily see walls, a table, books, the top of the block that the surface was mounted on, etc. Judgments of 0° and 90° slant were less variable than the middle slant values in Experiments 1 and 2. Did the other horizontal and vertical surfaces act as a frame of reference for target and response surfaces, thereby affecting the participants’ judgments of 0° and 90°? To investigate this issue, Experiment 3 was conducted in the same viewing condition as the first two experiments and in the dark to remove all other surfaces from view. Participants’ slant judgments were extremely variable for the middle slant values (20°, 40°, and 60°). Can the variability in these judgments be reduced? To test this question, the target surface was rotated while viewed by the participant. Previous studies (Bingham and Lind, 2008; Lee, Lind, Bingham, & Bingham, 2012; Lind, Lee, Mazanowski, Kountouriotis, & Bingham, 2014) found that a 45° continuous perspective change improved the perception of metric shape. So perhaps rotating the target surface would increase the stability of participants’ slant judgments.
accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki). 4.2. Results and discussion Participants underestimated 90° slants and overestimated 0°, 20°, 40°, and 60° slants. 40° slants were overestimated more than 60° slants. Surfaces presented at 40° tilt (mean error of −2.12) were overestimated more than those presented at 0° tilt (mean error of −1.41). There was no effect of lighting on participants’ slant judgments. A 5 (slant) × 2 (lighting) × 2 (tilt) repeated measures ANOVA was performed on the mean errors. There was a main effect of slant (F(2.2, 17.8) = 22.1, p < .01, η2 = 0.73). There was also a main effect of tilt (F(1, 8) = 22.7, p = .01, η2 = 0.57). The same pattern of mean slant judgments was observed (with slight numerical variations) when data from Experiment 3 and the 3D surface condition of Experiment 1 were combined (see Fig. 9) as was seen in Experiments 1 and 2. Motion had no effect on mean slant judgments. A 5 (slant) × 2 (motion) × 2 (tilt) repeated measures ANOVA was performed to compare the mean errors from the 3D surface condition of Experiment 1 (no motion) and the mean errors from both conditions of Experiment 3 (motion). There was a main effect of slant (F(2.3, 18.3) = 14.8, p < .01, η2 = 0.65). There was also a significant slant by tilt interaction (F(2.4, 18.8) = 3.8, p < .04, η2 = 0.32). As shown by mean standard deviations, slant judgments of 20°, 40°, and 60° slants were more variable than judgments of 0° and 90°. There was no effect of tilt or presence of light on the variability of slant judgments. A 5 (slant) × 2 (lighting) × 2 (tilt) repeated measures ANOVA was performed on the standard deviations of slant judgments from Experiment 3. There was a main effect of slant (F(2.4, 19.2) = 41.67, p < .01, η2 = 0.84). Judgments of the middle slants were still more variable than judgments of 0° and 90° when standard deviations of surfaces presented with and without motion were compared (see Fig. 10). Also, judgments of rotating surfaces were less variable overall than those of static surfaces. There was no effect of tilt or interactions. A 5 (slant) × 2 (motion) × 2 (tilt) repeated measures ANOVA was performed on the standard deviations of slant judgments
4.1. Methods The methods in Experiment 3 were the same as in Experiment 2 except for the following. The target surface was presented at eye level and was positioned on a platform. The experimenter manually rotated the platform at 1.8 Hz ± 25° for a total rotation of 50° on every trial. The target surface was tilted by either 0° or 40° prior to rotation. 50 trials were performed with the lights off and 50 were performed with the lights on. The texture elements were phosphorescent and visible in the dark. Only the texture elements on the target and response surfaces could be seen. Every 5 trials, the experimenter would shine a flashlight on the target and response surfaces to reserve the luminosity. The order of these conditions was counter-balanced. 4.1.1. Participants Nine of the ten participants from Experiments 1 and 2 also participated in the current experiment. All participants gave written informed consent prior to participation. Research was conducted in 36
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Fig. 11. Mean judged slant values are plotted against geographical slant values for each condition. The open circles indicate the mean judgments for surfaces presented at eye level at 0° tilt, the filled circles indicate surfaces presented below eye level at 0° tilt, the open squares indicate surfaces presented at eye level at 40° tilt, and the filled squares indicate surfaces presented below eye level at 40° tilt. The predicted optical slant values are also plotted against geographical slant values and these four lines correspond to the four conditions as previously described in Fig. 1.
Fig. 10. The mean standard deviations of slant judgments from Experiment 3 and the 3D surface condition of Experiment 1 are plotted against geographical slant. The filled circles represent the mean standard deviations of judgments of slants presented statically (Experiment 1) and the open circles represent the mean standard deviations of judgments of slants presented with motion (Experiment 3). The error bars are standard error of the mean.
from the 3D surface condition of Experiments 1 and 3. There were main effects of slant (F(2.3, 18.03) = 35.95, p < .01, η2 = 0.82) and motion (F(1, 8) = 6.98, p = .03, η2 = 0.47). There was no effect of lighting which suggests that participants did not use other visible surfaces in the room as a frame of reference for their slant judgments. The pattern of results for the slant judgments was similar to that observed in corresponding data in Experiment 1. The addition of motion did not affect the mean errors of participants’ slant judgments. A slant by tilt interaction was detected, but the differences driving this interaction were very small. As shown in Fig. 9, the judgments of slants presented at 40° tilt were more overestimated for slants 0°, 20°, and 40° and more underestimated for 60° and 90° slants compared to judgments of surfaces presented at 0° tilt. However, the error bars overlap at each geographical slant value and the differences between these mean errors were small (the largest difference is 2.2°). Motion did decrease the variability of slant judgments overall and, as can be seen in Fig. 10, judgments of 60° slant became more stable when motion was added.
variations in the tilt and eye level, the geographical slant of 60° corresponds to optical slants of 77.8° (below eye level, 0° tilt), 60° (at eye level, 0° tilt), 51.7° (below eye level, 40° tilt), and 41.6° (at eye level, 40° tilt). For a constant value of geographical slant, the optical slant varies by a factor of 2! However, the participants’ slant judgments are not so different in each of the conditions. The participants’ mean judgments for each slant in each condition are plotted in Fig. 11 against the predicted optical slant values shown in Fig. 1. While the mean slant judgments do not match the geographical slant of the target exactly (see the solid black line), they certainly do not vary as the values predicted by the optical slant values do. When we conducted several multiple regression analyses using optical and geographical slant values, the results confirm what is shown in Fig. 11. Slant judgments made by adjusting the response surface in each experiment were included in the analyses. Factors common to each regression analysis include eye height, tilt, and the interaction between eye height and tilt. For each multiple regression analysis, factors were removed using the least F value until only significant factors (p < .05) remained. The factors used in the analysis conducted on geographical slant included geographical slant and the following interaction terms: geographical slant x surface height, slant x tilt, surface height x slant x tilt. The final model included geographical slant (t = 119.58, p < .01, β = 0.97), tilt (t = −2.44, p < .02, β = −0.2) and surface height (t = −6.61, p < .01, β = −0.53) which resulted in a significant regression equation (F(3, 996) = 4783.2, p < .01) and an R2 value of 0.94. The analysis conducted on optical slant (the factors were the same as those used in the previous analysis with optical slant instead of geographical slant) did not require any factors to be eliminated. The regression equation was significant (F(7, 992) = 960.27, p < .01) with an R2 value of 0.87. The values for each coefficient can be seen in Table 1. The model with only geographical slant had the fewest significant factors and was thus the most parsimonious with a high R2 of 0.94. In contrast, every factor in the optical slant model was significant and contributed to a lower R2 value. The inclusion of every factor in the optical slant model points to the perspective-dependent nature of
5. General discussion The primary purpose of Experiments 1 and 2 was to determine whether participants’ slant judgments remained invariant over changes in the perspective on the surfaces. In Experiment 1, there was no effect of tilt for either mean error or variability. These results would seem to indicate that changes in viewing perspective do not influence participants’ slant judgments (see Fig. 5). In Experiment 2, there was an interaction between surface height and tilt, but it was driven primarily by the low surface height at tilt 0°, and the differences between this condition and the others were small in magnitude (largest difference was < 5°). The findings from the previous two experiments indicate that participants perceived geographical slant rather than optical slant. To compare optical slant with geographical slant, we calculated the optical slants for each geographical slant (0°, 20°, 40°, 60°, and 90°) in each condition of Experiments 1 and 2. As shown in Fig. 1, the geographical slants and optical slants are very different. For example, with 37
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errors of slant judgments. However, the presence of motion did decrease the variability in participants’ slant judgments, especially when viewing a geographical slant of 60°. We note that the motion was rotation around a vertical axis through the target surface and thus, consisted of variation in tilt that would have produced significant variations in optical slant. If judgments of slant depended on optical slant, then one would have expected this motion to have made the judgments of geographical slant less reliable. Instead, the motion yielded increased reliability.
Table 1 Multiple Regression Model for Optical Slant. Factor
β
t
p
Optical Slant Eye Height Tilt Eye Height × Tilt Eye Height × Optical Slant Tilt × Optical Slant Eye Height × Tilt × Optical Slant
1.173 0.498 −0.187 0.124 −0.337 0.584 −0.970
76.291 17.958 −6.745 4.470 −11.271 19.619 −3.225
< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 0.001 < 0.001
6. Conclusions optical slant (i.e., its lack of constancy). The model including geographical slant was a better predictor of slant judgments than the model including optical slant. However, the geographical slant model did not account for all the variance within participants’ slant judgments. To determine whether optical slant predicted the unexplained variance, we first conducted a simple linear regression analysis regressing geographical slant on participants’ mean slant judgments from Experiments 1 and 2. The regression equation was significant (F(1, 998) = 13648.2, p < .01) with an R2 value of 0.93. We then conducted a simple linear regression in which we regressed optical slant on the residuals calculated in the previous regression analysis to see how much of the remaining variance could be accounted for by optical slant. The regression equation was significant (F(1, 998) = 23.12, p < .01) with an R2 value of 0.02. So optical slant only accounted for 2% of the variance in the residuals. This analysis further confirms that optical slant does not explain the data from the current study as well as geographical slant. A secondary purpose of Experiment 1 was to determine whether the 2D line task would produce the same results as the 3D surface response measure. The results indicate that participants tended to overestimate the 40° and 60° geographical target slants more when adjusting the line as opposed to the surface. However, this difference was very small in magnitude (i.e., 5.6° for 40° target slants and 6.4° for 60° target slants) and the overall pattern of slant judgments was remarkably similar. Judgments of 90° geographical slants were underestimated more (by approximately 5°) when using the surface response measure than when using the line. The variability of participants’ slant judgments in Experiments 1 and 2 was certainly not equivalent across different geographical slant values (see Figs. 5 and 7). Judgments of 0° and 90° target surface slants were far less variable than judgments of the other three geographical slant values. This small amount of variation in judgments clearly indicates that there is something special about vertical and horizontal surfaces. Importantly, this lower variability for 0° and 90° occurred at both lower and eye level surface heights for both tilts with no difference. This means there is something unique about the perception of vertical and horizontal surfaces that is invariant to changes in viewing perspective (i.e., geographical slant). In Experiments 1 and 2, the participants’ view of surrounding surfaces in the room was not restricted for the target or response surface. Therefore, participants could see horizontal (e.g., table, floor) and vertical (e.g., wall) surfaces that they could have used as references to assist in their judgments of 0° and 90°. Experiment 3 was primarily conducted to address this potential issue. Participants judged geographical slant with the lights on and with the lights off. If they were using the other visible horizontal and vertical surfaces in the room as a frame of reference, there should have been a noticeable difference between performance in these two conditions. However, there was no effect of lighting on slant judgments. This result suggests that other visible horizontal and vertical surfaces in the room did not assist participants in their judgments of 0° and 90° geographical slants. A second purpose of Experiment 3 was to determine whether continuously rotating the target surface would improve participants’ performance in judging slant. There was no effect of motion on the mean
We conducted the current study to investigate invariance in slant judgments across different conditions. There was no significant effect of tilt found in Experiment 1, which suggests constancy in judgments across changes in viewing perspective. Experiment 2 was conducted to amplify the difference between optical and geographical slant by changing the surface height as well as tilt. In Experiment 2, the slant judgments of the lowered target surface at 0° tilt were underestimated relative to the other conditions. However, these differences were very small in magnitude (2°–4.3°) and were certainly not predicted by changes in optical slant (maximum difference predicted to be 45°). Variations in optical slant could not account for the pattern of results observed in this study. Participants were able to perceive geographical slant directly as others have found previously (Gibson and Cornsweet, 1952; Sedgwick & Levy, 1985). Given geographical slant is perceived rather than optical slant, future investigations should examine what information specifies geographical slant. 7. Declarations of interest None. References Bingham, G. P., & Lind, M. (2008). Large continuous perspective transformations are necessary and sufficient for accurate perception of metric shape. Attention, Perception, & Psychophysics, 70, 524–540. http://dx.doi.org/10.3758/PP.70.3.524. Braunstein, M. I., & Payne, J. W. (1969). Perspective and form ratio as determinants of relative slant judgments. Journal of Experimental Psychology, 81, 584–590. http://dx. doi.org/10.1037/h0027886. Coleman, A., & Durgin, F. H. (2014). Egocentric reference frame bias in the palmar haptic perception of surface orientation. Psychonomic Bulletin Review, 21, 955–960. http:// dx.doi.org/10.3758/s13423-013-0552-7. Durgin, F. H., Hajnal, A., Li, Z., Tonge, N., & Stigliani, A. (2010). Palm boards are not action measures: An alternative to the two-systems theory of geographical slant perception. Acta Psychologica, 134, 182–197. http://dx.doi.org/10.1016/j.actpsy. 2010.01.009. Durgin, F. H., & Li, Z. (2011). The perception of 2D orientation is categorically biased. Journal of Vision, 11. http://dx.doi.org/10.1167/11.8.13. Durgin, F. H., Li, Z., & Hajnal, A. (2010). Slant perception in near space is categorically biased: Evidence for a vertical tendency. Attention, Perception, & Psychophysics, 72, 1875–1889. http://dx.doi.org/10.3758/APP.72.7.1875. Gibson, J. (1950). The perception of visual surfaces. The American Journal of Psychology, 63, 367–384. http://dx.doi.org/10.2307/1418003. Gibson, J., & Cornsweet, J. (1952). The perceived slant of visual surfaces – Optical and geographical. Journal of Experimental Psychology, 44, 11–15. http://dx.doi.org/10. 1037/h0060729. Koenderink, J. J., & van Doorn, A. J. (1976). Geometry of binocular vision and a model for stereopsis. Biological Cybernetics, 21, 29–35. http://dx.doi.org/10.1007/ BF00326670. Koenderink, J. J., & van Doorn, A. J. (1991). Affine structure from motion. Journal of the Optical Society of America A, 8, 377–385. http://dx.doi.org/10.1364/JOSAA.8. 000377. Lee, Y., Lind, M., Bingham, N., & Bingham, G. P. (2012). Object recognition using metric shape. Vision Research, 69, 23–31. http://dx.doi.org/10.1016/j.visres.2012.07.013. Lind, M., Lee, Y., Mazanowski, J., Kountouriotis, G. K., & Bingham, G. P. (2014). Affine operations plus symmetry yield perception of metric shape with large perspective changes (≥45°): Data and model. Journal of Experimental Psychology: Human Perception and Performance, 40, 81–93. http://dx.doi.org/10.1037/a0033245. Norman, J. F., Todd, J. T., Norman, H. F., Clayton, A. M., & McBride, T. R. (2006). Visual discrimination of local surface structure: Slant, tilt, and curvedness. Vision Research, 46. http://dx.doi.org/10.1016/j.visres.2005.09.034. Norman, J. F., Todd, J. T., & Phillips, F. (1995). The perception of surface orientation from multiple sources of optical information. Perception & Psychophysics, 57,
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248–260. http://dx.doi.org/10.1016/S0734-189X(85)80008-6. Stefanucci, J. K., Proffitt, D. R., Clore, G. L., & Parekh, N. (2008). Skating down a steeper slope: Fear influences the perception of geographical slant. Perception, 37, 321–323. http://dx.doi.org/10.1068/p5796. Stevens, K. A. (1983). Slant-tilt: The visual encoding of surface orientation. Biological Cybernetics, 46, 183–195. http://dx.doi.org/10.1007/BF00336800. Todd, J. T., & Perotti, V. J. (1999). The visual perception of surface orientation from optical motion. Perception & Psychophysics, 61, 1577–1589. http://dx.doi.org/10. 3758/BF03213119. Todd, J. T., Thaler, L., & Dijkstra, T. M. H. (2005). The effects of field of view on the perception of 3D slant from texture. Vision Research, 45, 1501–1517. http://dx.doi. org/10.1016/j.visres.2005.01.003. van Ee, R., & Erkelens, C. J. (1996). Temporal aspects of binocular slant perception. Vision Research, 36, 43–51. http://dx.doi.org/10.1016/0042-6989(95)00078-E. van Ee, R., van Dam, L. C. J., & Erkelens, C. J. (2002). Bi-stability in perceived slant when binocular disparity and monocular perspective specify different slants. Journal of Vision, 2, 597–607. http://dx.doi.org/10.1167/2.9.2. Witt, J. K., & Proffitt, D. R. (2007). Perceived slant: A dissociation between perception and action. Perception, 36, 249–257. http://dx.doi.org/10.1068/p5449.
629–636. http://dx.doi.org/10.3758/BF03213268. Perrone, J. A. (1982). Visual slant underestimation: A general model. Perception, 11, 641–654. http://dx.doi.org/10.1068/p110641. Phillips, R. J. (1970). Stationary visual texture and the estimation of slant angle. Journal of Experimental Psychology, 22, 389–397. http://dx.doi.org/10.1080/ 14640747008401912. Proffitt, D. R., Bhalla, M., Gossweiler, R., & Midgett, J. (1995). Perceiving geographical slant. Psychonomic Bulletin & Review, 2, 409–428. http://dx.doi.org/10.3758/ BF03210980. Proffitt, D. R., Creem, S. H., & Zosh, W. D. (2001). Seeing mountains in mole hills: Geographical-slant perception. Psychological Science, 12, 418–423. http://dx.doi.org/ 10.1111/1467-9280.00377. Schnall, S., Harber, K. D., Stefanucci, J. K., & Proffitt, D. R. (2008). Social support and the perception of geographical slant. Journal of Experimental Social Psychology, 44, 1246–1255. http://dx.doi.org/10.1016/j.jesp.2008.04.011. Schnall, S., Zadra, J. R., & Proffitt, D. R. (2010). Direct evidence for the economy of action: Glucose and the perception of geographical slant. Perception, 39, 464–482. http://dx.doi.org/10.1068/p6445. Sedgwick, H. A., & Levy, S. (1985). Environment-centered and viewer-centered perception of surface orientation. Computer Vision, Graphics, and Image Processing, 31,
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Vision Research journal homepage: www.elsevier.com/locate/visres
Searching for invariance: Geographical and optical slant ⁎
T
Olivia C Cherry , Geoffrey P Bingham Indiana University, Bloomington, United States
A R T I C LE I N FO
A B S T R A C T
Number of reviewers = 2
When we move through rigid environments, surface orientations of static objects do not appear to change. Most studies have investigated the perception of optical slant which is dependent on the perspective of the observer. We investigated the perception of geographical slant, which is invariant across different viewing perspectives, and compared it to optical slant. In Experiment 1, participants viewed a 3D triangular target surface with triangular phosphorescent texture elements presented at eye level at one of 5 slants from 0° to 90°, at 0° or 40° tilt. Participants turned around to adjust a 2D line or a 3D surface to match the slant of the target surface. In Experiment 2, the difference between optical and geographical slant was increased by changing the height of the surface to be judged. In Experiment 3, target surfaces were rotated by 50° ( ± 25°) and viewed in both a dark and lighted room. In Experiment 1, the overall pattern of judgments exhibited only slight differences between response measures. In Experiment 2, slant judgments were slightly overestimated when the surface was at a low height and at 0° tilt. We compared optical slants of the surfaces to geographical slants. While sometimes inaccurate, participants’ slant judgments remained invariant across changes in viewing perspective. In Experiment 3, judgments were the same in the dark and lighted conditions. There was no effect of target motion on judgments, although variability decreased. We conclude that participants’ judgments were predicted by geographical slant, not optical slant.
Keywords: Geographical slant Slant perception 3D perception
1. Introduction Numerous investigations of the perception of slant have studied optical slant primarily (Braunstein & Payne, 1969; Koenderink & van Doorn, 1976; Norman, Todd, Norman, Clayton, & McBride, 2006, Norman, Todd, & Phillips, 1995; Perrone, 1982; Phillips, 1970; Stevens, 1983; Todd, Thaler, & Dijkstra, 2005; van Ee & Erkelens, 1996; van Ee, van Dam, & Erkelens, 2002). Optical slant is egocentric and is defined as the angle between the surface normal and the line of sight (Todd & Perotti, 1999). Any change to the viewer’s position or the surface’s position yields changes in a surface’s optical slant, even though the orientation of the surface to the surroundings does not change. A single large scale surface has multiple optical slant values at different points across the surface. For example, if an observer stands at one end of a long table, the closest end has a very different optical slant value than the middle of the table or the opposite end. However, the orientation of different portions of the table’s surface are generally perceived and judged to be the same, that is, the surface is flat and level. In addition, when navigating through a static environment, people do not perceive rigid surfaces as constantly changing in orientation. If optical slant was used to inform us about the slant of surfaces in the world, the experience would be chaotic and disorienting.
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Fortunately, perception of the slant of rigid surfaces remains stable with changes in viewer position (e.g., a flat table continues to appear flat no matter where the observer is). Without this stability, interaction with our environment would be incredibly difficult. The instability of optical slant makes it less likely that we use it to inform us about the slant of surfaces. Sedgwick and Levy (1985) suggest that it would be more efficient for the visual system to rely on geographical slant rather than continuously update itself with fluctuating optical slants. Geographical slant is the slant of a surface relative to gravity which remains stable with changes in a viewer’s position. Therefore, investigating the perception of geographical slant may be a more ecological approach to understanding human perception of surface orientation. Perception of the slant of large-scale surfaces has been studied extensively (Proffitt, Bhalla, Gossweiler, & Midgett, 1995; Proffitt, Creem, & Zosh, 2001; Stefanucci, Proffitt, Clore, & Parekh, 2008). Much of this research has concluded that the perception of the slant of large-scale surfaces (e.g., hills and ramps) depends on extraneous factors such as glucose levels (Schnall, Zadra, & Proffitt, 2010) and social support (Schnall, Harber, Stefanucci, & Proffitt, 2008). Therefore they assumed constancy across the large surfaces while suggesting that judgment accuracy reflects states of the perceiver/actor as much as the actual slant of the surface. In general, studies have focused on the relative
Corresponding author at: Department of Psychological and Brain Sciences, Indiana University, Bloomington, 1101 W. 10th St., Bloomington, IN 47405, United States. E-mail address: [email protected] (O.C. Cherry).
https://doi.org/10.1016/j.visres.2018.05.008 Received 30 January 2018; Received in revised form 15 May 2018; Accepted 17 May 2018 0042-6989/ © 2018 Elsevier Ltd. All rights reserved.
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accuracy of slant perception or the lack thereof. Durgin, Li, and Hajnal (2010) conducted a study investigating the tendency to overestimate the slant of small-scale surfaces in near visual space. They had participants judge geographical slants by verbal estimation and a free-hand matching task. They found that there was a bias in the estimation of geographical slant: participants showed a tendency to perceive slanted surfaces to be steeper than they actually were. Durgin et al. also used a two alternative forced choice paradigm and a psychometric staircase procedure to calculate the point of bisection (i.e., the equidistant point between vertical and horizontal) perceived by their participants. On each trial, participants would be shown a single slanted surface and would indicate whether the slant was closer to vertical or horizontal. On average, the equidistant point between horizontal and vertical was 34°. The current investigation is focused on determining whether or not there is constancy of slant judgments across different viewing perspectives rather than whether those slant judgments are accurate. Smaller scale surfaces have fewer different optical slant values that are almost equivalent to each other. Optical slant can be manipulated easily using smaller surfaces by changing viewing perspective. Thus, optical slant can be compared to geographical slant at each viewing perspective. The current study uses small-scale surfaces in near space for this reason. The response measure that has been used most frequently in studies of geographical slant perception is the palm board (Coleman & Durgin, 2014). The palm board is typically placed at waist level, and is out of the participants’ sight. Participants place their hand on it to adjust the board to be parallel to the angle of the target surface they are observing. Gibson (1950) began using a palm board to provide participants with a non-verbal method to estimate slant. Proffitt et al. (1995) found that participants tended to overestimate geographical slant when making verbal estimates or adjusting an angle on a disk to match the crosssection of the inclination of the hill, but when using the palm board, they were more accurate. Witt and Proffitt (2007) conducted a study directly comparing slant judgments of large scale hills using three response measures: a palm board, a relative visual matching task, and an absolute visual matching task. They found that judgments made with the palm board were significantly more accurate than the other tasks and were not significantly different from the actual slope of the hill. Witt and Proffitt concluded that judgments made with the palm board were accurate because the measure did not require explicit perceptual awareness but instead relied on visuomotor control, which they posit to be separate processes. Durgin, Hajnal, Li, Tonge, and Stigliani (2010) conducted several experiments investigating the accuracy and reliability of palm boards. They found that judgments of slants in near space made with palm boards were consistently underestimated compared to a free-hand measure. Coleman and Durgin (2014) found that there is a systematic bias for haptic perception of surface orientation. Participants would underestimate the slant angles when the palm board was set at a lower height (aligned with the participant’s navel) and overestimate slant angles when the palm board was higher (at eye level). The primary purpose of the current study was to investigate whether perceived slant remains invariant under changes in viewing perspective. In Experiment 1, the surfaces were presented at either 0° tilt or 40° tilt. Geographical tilt is defined here as the amount of rotation around a vertical axis through the surface. For a slanted surface presented at 0° tilt, the surface would be slanted directly away from the observer. If that surface was presented at 90° tilt, the observer would see the slanted surface edge on. The optical slant for this surface would be very different in both cases, while the geographical slant would remain the same. Geographical slant is unaffected by changes in tilt while optical slant varies. Fig. 1 shows optical slant values plotted against geographical slant values. The solid thin line represents the relationship between optical and geographical slant when the target surface was at eye level and 0° tilt. In this case, optical and geographical slant are equivalent. The dashed thin line corresponds to the condition in which the target surface was presented at eye level at 40° tilt. In this case,
Fig. 1. Transformed optical slant values are plotted against corresponding geographical slant values for each condition. Optical slant values were linearly transformed so that 0° and 90° optical slant would correspond to 0° and 90° geographical slant. Optical slants were computed continuously for corresponding geographical slants from 0° to 90° presented at and below eye level at 0° and 40° tilt. Points on the functions correspond to geographical slant values presented to participants in the study. The solid lines and circles indicate surfaces presented at tilt 0° and the dashed lines and squares indicate those presented at tilt 40°. The thin lines and open shapes indicate surfaces presented at eye level (Experiment 1) and the thick lines and closed shapes indicate surfaces presented below eye level (Experiment 2).
optical slant varies with the same geographical slant values. In Experiment 2, the surfaces were presented at 0° and 40° tilt again, but at a lower eye height to amplify the difference between geographical and optical slant. The solid thick line corresponds to target surfaces presented at 0° tilt below eye level and the dashed thick line corresponds to target surfaces presented at 40° tilt below eye level. Again, optical slant varies. For the same unchanging set of geographical slants, optical slant changes dramatically across the different viewing conditions (for 5 geographical slants there are 18 different optical slants). If participants perceive optical slant, then their judgments of geographical slant should reflect these differences. Given the controversial nature of the palm board, we used two visually guided response measures: an adjustable 3D surface that participants controlled and matched to the perceived geographical slant and a 2D line viewed on a computer screen and adjusted to match the perceived geographical slant. The participant sat between the response measure and target surface so they could not be viewed simultaneously. Durgin and Li (2011) found a bias to overestimate the slant of 2D lines, which would prove problematic for using a line task as a dependent measure. A secondary purpose of Experiment 1 was to determine whether adjusting a 2D line would yield the same results as adjusting a 3D surface. 2. Experiment 1 2.1. Methods 2.1.1. Participants Ten adults ranging in age from 20 to 30 (two males) participated in this experiment. All participants had normal or corrected-to-normal vision and passed a stereo fly test (Stereo Optical Co., Inc.) measuring stereo acuity. Participants were required to identify a target circle with a disparity of 80 s of arc in order to participate in the study. All participants gave written informed consent prior to taking part in the study. Research was conducted in accordance with the Code of Ethics of the 31
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back of each surface so it could be easily attached and removed from the presentation and response apparatus. The presentation apparatus consisted of a small Plexiglas platform that was attached to the top of a metal rod extending from a wooden block. The metal rod was stationary and oriented vertically. Velcro was attached to the surface of the platform so the surfaces could be mounted on it. At the top of the rod, the platform could be rotated to set it at various slants at 0° tilt and at eye height. By keeping the center of rotation close to the surface, the surface’s slant with respect to the observer’s line of sight was similar to the geographical slant of the surface. A small piece of wood was marked to indicate the angles at which the platform could be set (i.e., 0°–90° in increments of 10) and was attached to the stationary metal rod extending from the block. An additional straight metal rod was attached to the bottom of the platform to act as a pointer so the experimenter could easily set the surface to the correct slant value. The target surface was elevated approximately 35 cm off the table (111.2 cm off the ground) and was at the participant’s eye level. The pointer, platform, and piece of wood could not be seen by the observer because they were occluded by the target surface. The observer could only see the block, the metal rod, and the surface. There were two different response measures used in this experiment: a line displayed on a computer screen and a response surface mounted on an apparatus. The line response measure was displayed on an ASUS Notebook Q500A with a screen resolution of 1366 × 768 using a program generated and run using MatLab R2016a. The response surface apparatus was constructed similarly to the target surface apparatus with the following exceptions. The response surface apparatus was attached to the table and could not be moved (i.e., the tilt of the response surface could not be altered from 0°). The metal rod attached to the platform was instead made of plastic in order to not interfere with the magnetic field generated by the MiniBird measurement system (described below). The screws attaching the plastic rod to the wooden block and the plastic rod to the platform were stainless steel (no magnetism). The plastic rod extending from the block was also not stationary. The slant of the surface was adjusted by moving this plastic rod as opposed to moving the platform, so the center of rotation was located at the top of the wooden block. This means that the optical slant of the response surface was different from the optical slant of the target surface at each geographical slant value (see Fig. 3). The response surface also was not placed at eye level (the response surface was 2 cm shorter). These differences in eye height and center of rotation between the target and response apparatuses were included to ensure that the task was not trivial. Different triangles with different sized texture elements were also used as target and response surfaces for each trial to enhance the difference between them (e.g., if the target surface had large texture elements the response surface would have small texture elements and these surfaces would be different shapes). The response and target surfaces were located on opposite sides of the participants so they could not be compared simultaneously. Participants were not simply matching image structure. A MiniBird magnetic measurement system was used to record the slant of the response surface. Two markers were attached to the front and back sides of the flat platform that the surface rested on. The magnetic field emitter was placed directly underneath the table in line with the response apparatus. The trackers’ position coordinates were recorded by a program executed on a Dell E5200 computer so that the slant could be calculated from these positions.
Fig. 2. A photograph of the triangular surfaces mounted on the presentation and response apparatus in Experiments 1, 2, and 3. The left column shows triangles with large texture elements (average area = 0.64 cm2) and the right column shows triangles with small texture elements (average area = 0.16 cm2).
World Medical Association (Declaration of Helsinki).
2.1.2. Materials and apparatus Target slant values were presented using an actual surface that was mounted on an apparatus that allowed adjustment and positioning of the surface orientation. Different surface shapes and surface textures were used throughout the experiment. As can be seen in Fig. 2, the surfaces consisted of two sets of six different black triangles made of black PVC plastic with equivalent surface areas (mean = 70.49 cm2) and an average angular size of 46.4°. Triangles were chosen because they do not provide any information about the slant of the surface (Koenderink & van Doorn, 1991). On one of these sets, small triangularshaped texture elements were placed randomly on the surfaces (see right column of Fig. 2). On the other set, larger versions of the texture elements were placed on the surfaces (see left column of Fig. 2). The texture elements were made of phosphorescent tape. Triangular-shaped texture was used so that the shape of the individual texture elements could not provide information about the slant of the surface. The shape of the texture elements matched the shape of the surface they were on. Before the texture was placed on the surfaces, the surfaces were rubbed with steel wool to reduce specularity. A piece of velcro was glued to the
2.1.3. Procedure After reading and signing the informed consent, participants were given a stereo fly test to assure that their stereo acuity was adequate. Participants then sat in a swivel chair that had been modified to include an adjustable chin rest. The chin rest could be moved closer or farther from the participant and its height could also be adjusted. See Fig. 4 for a diagram of the experimental set up. The swivel chair was placed between two tables: on one table was the target apparatus and on the 32
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were informed that the response and target apparatus were slightly different, so the only component of the apparatus that mattered was the surface and its slant. For trials during which the line task was used, the response surface was replaced with a laptop. Participants were instructed to adjust the slope of a single 2D line on the computer screen to match the slant of the target surface if it were viewed from the side. The line would be randomly oriented either vertically or horizontally at the beginning of each trial. To adjust the slope of the line, participants used the left and right arrow keys and when they were ready to move on to the next trial, they pressed the space bar. Participants could look at the target surface and the response measure as many times as they liked by rotating the chair, while still in the chin rest, to view either the target or the response surface. There was no time limit.
2.2. Results and discussion The primary purpose of Experiment 1 was to determine whether slant judgments would remain invariant over changes in viewing perspective (e.g., differences in geographical tilt). Fig. 5 plots the mean judged slant for tilt 0° (open circles) and tilt 40° (filled circles) as a function of geographical (left panel) and optical (right panel) slant. In the right panel, the dashed line indicates the geographical slant values for 0° tilt and the solid line indicates geographical slant values for 40° tilt. At tilt 0°, optical slant and geographical slant are equivalent. One can easily see that the mean slant judgments at tilt 0° are very close to the optical and geographical slant values of the target surface. At tilt 40°, however, the mean slant judgments do not align with the optical slant values but instead are very close to the corresponding geographical slant values of the target surface. If viewing perspective was affecting participants’ judgments, mean slant judgments for both tilt 0° and 40° should overlap in the right panel of Fig. 5 rather than in the left panel. This is not the case in the current experiment. Mean slant judgments were closer to the corresponding geographical slants (left panel of Fig. 5) than optical slants (right panel of Fig. 5). On average, the 90° target slant was underestimated compared to the other slant values (mean errors for 0°, 20°, 40°, 60°, and 90° were −0.96, −2.23, −6.72, −2.43, and 3.28, respectively). As can be seen in Fig. 6, the 40° and 60° slants were overestimated more when adjusting the line (filled circles) relative to adjusting the surface (open circles). Also, the 90° slant was underestimated more when using the response surface compared to the line. The change in tilt did not affect participants’ slant judgments. The pattern of slant judgments remained constant across variations in viewing perspective. A 5 (geographical slant: 0°, 20°, 40°, 60°, 90°) × 2 (tilt: 0° and 40°) × 2 (response measure: 3D surface and 2D line task) repeated measures analysis of variance (ANOVA) was conducted on the mean error of participants’ slant judgments. Error was calculated by subtracting participants’ slant judgments from the geographical slant of the target surface. These errors were averaged across the five repetitions of each geographical slant value per condition to produce the mean error. The Greenhouse-Geisser correction for violations of sphericity was applied to all results to correct for the violation of homogeneity of variance. There was no effect of
Fig. 3. Transformed optical slant values are plotted against corresponding geographical slant values for the response surface apparatus. Optical slant values were linearly transformed so that 0° and 90° optical slant would correspond to 0° and 90° geographical slant. Filled circles correspond to the five geographical slant values used in Experiments 1, 2, and 3. The dashed line indicates a perfect relationship between optical and geographical slant (optical slant = geographical slant).
other table was the response apparatus or laptop. Participants’ viewing distance from the target surface was 76.2 cm (this was also the distance from the response surface) and their eye height was in line with the middle of the target surface (111.2 cm). Participants completed 100 trials: 50 trials adjusting the response surface and 50 trials adjusting the line on the computer. Within each block of 50 trials, the target surface was oriented at zero degree tilt for 25 trials (surface was slanted away from the participants directly) and forty degree tilt for 25 trials (the apparatus was rotated by forty degrees to the left). The response surface was fixed at 0° tilt throughout the duration of the experiment. Every five trials, the surfaces were replaced with a different triangle with a different sized texture so that the response and target surface were never the same shape or texture. The surfaces were also placed on the target and response apparatuses in random orientations rotated around the surface normal. The order of response types and degree of tilt was counter-balanced across participants. For trials during which the response surface was used, participants were instructed to view the slant of the target surface and then turn around in their chair to adjust the slant of the response surface to match. If the target surface was flat/horizontal, they should also adjust the response surface to be flat/horizontal. In other words, participants were instructed to judge the geographical slant of the surfaces. They
Fig. 4. A diagram of the experimental setup when using the response surface in Experiment 1. The participant is resting his chin on the chin rest and looking at the target surface set at some angle depicted on the left side of the diagram. The response surface is directly behind the participant so that the participant loses full sight of the target surface when adjusting the response surface. The response surface has two tracking markers attached underneath it to the MiniBird measurement system and the emitter is underneath the table below the response surface. The response apparatus on the right is replaced by a laptop when performing the line task. 33
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Fig. 5. The mean slant judgments plotted as a function of geographical slant for Experiment 1 are shown in the left panel. The mean slant judgments plotted as a function of optical slant are shown in the right panel. The open circles indicate the mean slant judgments for target surfaces presented at tilt 0°. The filled circles indicate the mean slant judgments for tilt 40°. The dashed line in the left plot indicates perfect accuracy (judged slant = geographical slant). In the right panel, the continuously computed geographical slants are plotted as a function of optical slant for tilt 0° (dashed line) and tilt 40° (solid curved line). The error bars are standard error of the mean.
Fig. 6. The mean slant judgments plotted as a function of geographical slant for Experiment 1. The open circles indicate the mean slant judgments for the surface response measure. The filled circles indicate the mean slant judgments for the 2D line task. The best-fitting regression line for the surface response measure is shown by the dashed line: y = 0.91x + 3.93, R2 = 0.99. The best-fitting regression line for the 2D line task is shown by the solid line: y = 1.0x + 3.55, R2 = 0.99. The dotted line indicates perfect accuracy (judged slant = geographical slant). The error bars are standard error of the mean.
Fig. 7. The mean standard deviation for each geographical slant value in Experiment 1. The error bars are standard error of the mean.
each geographical slant value is plotted in Fig. 7. The variability in judging 0° and 90° geographical slants was far smaller than the variability in judging the middle slant values. The largest variability was found at the 40° target slant. The variability of 0° and 90° slant judgments were not significantly different from each other but were significantly different from the other slant values. 20° slant judgments were significantly different from 40° slant judgments, but not significantly different from 60° slant judgments. The variability (standard
tilt, either main effect or interaction. There was a significant main effect of geographical slant (F(2.5, 22.2) = 12.19, p < .01, η2 = 0.58). The slant by response measure interaction was significant (F(1.4, 12.7) = 6.12, p = .02, η2 = 0.41). The mean standard deviation of participants’ slant judgments for 34
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variability would suggest that participants’ judgments were primarily influenced by geographical slant values. However, the difference between optical and geographical slant values may not have been large enough to detect a difference. Geographical slant and optical slant were the same when the target surface was viewed at 0° tilt (see the thin solid line in Fig. 1). So optical slant only differed from geographical slant when the target surface was presented at 40° tilt (see the thin dashed line in Fig. 1). If optical slant influences slant judgments, then lowering and tilting the target surface should amplify this difference (see the thick lines in Fig. 1).
deviation) in these slant judgments was also subjected to a 5 (geographical slant) × 2 (tilt) × 2 (response measure) repeated measures ANOVA. There was a main effect of geographical slant (F(2.2, 19.7) = 50.09, p < .01, η2 = 0.85). Notably, there was no main effect of response measure or tilt and no significant interactions. Four simple linear regressions predicting slant judgments from geographical slants were performed for each individual participant: surface response measure and target surface at 0° tilt, line response measure and target surface at 0° tilt, surface response measure and target surface at 40° tilt, and finally, line response measure and target surface at 40° tilt. The slopes, intercepts, and R2 values were then analyzed in a 2 (response measure) × 2 (tilt) repeated measures ANOVA. No significant effects of response measure, tilt, or the interaction between response measure and tilt were found for R2 values or intercepts. A significant main effect of response measure (F(1, 9) = 51.44, p < .01, η2 = 0.85) on slope values indicated that the mean slope of judgments made in the line task (1.00) was higher than the mean slope of judgments made in the surface task (0.92). To eliminate the potential anchoring effects of 0° and 90° slant values, we also ran the previous analyses using only the middle slant values (i.e., 20°, 40°, 60°). These results were the same as those including 0° and 90° slants. Two simple linear regressions were conducted on the mean slant judgments made using the line and the surface response measure. Mean slant judgments as well as their respective regression lines are plotted in Fig. 6. Although there are significant differences between certain geographical slant values, the overall pattern of results is very similar for both response measures. The largest difference between the response measures is the difference in slope which is driven by the overestimation of 40° and 60° slant values when participants used the line. These linear regression analyses were also conducted on the middle slant values and the results from these analyses revealed very similar slopes and R2 values. The intercept values for the surface response measure increased from 3.9° to approximately 5.7° and the intercept for the line response measure decreased from 3.5° to approximately 1.9°. One goal of Experiment 1 was to determine whether there was a significant difference between adjusting a line on a computer screen and adjusting a small surface to match the slant of a target surface. Despite the significant interaction between response measure and geographical slant value, the patterns of slant judgments are very similar for the two response measures. On average, the 60° slant judgments made when adjusting the line were only 6.4° higher than those made using the surface response measure. 40° slant judgments were only 5.5° higher when adjusting the line as opposed to the surface. These are small deviations across these response measures and the overall pattern of results is still very similar across response measures as can be seen in Fig. 6. There was no main effect of response measure for the measure of variability. Also, no significant differences were found in terms of R2 or intercept values and the slopes were higher on average for the line response measure, which is unsurprising when considering the 60° and 40° slant values discussed previously. Considering the small magnitude of the differences found between judgments made in these conditions, one can conclude that both methods of estimation are similarly effective. The surface response measure may have higher face validity than the line task, but in terms of overall efficacy we conclude that the line task is a viable option.
3.1. Methods The methods in Experiment 2 were the same as in Experiment 1 except for the following. Due to the similarity between judgments made using the line and the surface response measures in the previous experiment, only the response surface was used in this experiment. The target surface was lowered by 35.7 cm. The response surface remained at the same height (2 cm below eye height). As was the case in Experiment 1, the target surface was viewed at 0° and 40° tilts while the response surface remained at 0° tilt throughout the experiment. The total number of trials was also reduced to 50 in order to keep this condition comparable to the previous study.
3.1.1. Participants The same ten participants from Experiment 1 also participated in the current experiment. All participants gave written informed consent prior to participation. Research was conducted in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki).
3.2. Results and discussion The mean error for tilt 0° at the lower surface height (−4.4) was larger than in the other conditions (eye height, tilt 0°: −0.5; eye height, tilt 40°: −0.1; low, tilt 40°: −2.5). A 5 (slant) × 2 (surface height) × 2 (tilt) repeated measures ANOVA was performed on mean errors from Experiments 1 and 2. Judgments of 90° slants were underestimated relative to all other slant values. Also, 40° target slant mean error was less than the 60° error. There was a main effect of geographical slant (F (2.3, 20.7) = 20.47, p < .01, η2 = 0.70). At the lower surface height, there was a larger difference between judgments made at 0° and 40° tilt than those made at eye height. There was primarily one condition driving the interaction between surface height and tilt, namely slant judgments at 0° tilt at the lower surface height were more underestimated than those in the other conditions by 2° (low, tilt 40°), 4° (eye height, tilt 0°), and 4.3° (eye height, tilt 40°). A significant surface height by tilt interaction was observed (F(1, 9) = 31.49, p = .02, η2 = 0.46). The variability of slant judgments from Experiments 1 and 2 were analyzed using a 5 (slant) × 2 (surface height) × 2 (tilt) repeated measures ANOVA. There was a main effect of geographical slant (F(3.4, 30.4) = 47.06, p < .01, η2 = 0.84; see Fig. 8). Pairwise comparisons revealed that the standard deviations for judgments of 0° and 90° geographical slants were not significantly different from each other but were significantly less than the other slant values. The variability in judging 60° slant values was larger than all other slant values. The variability of 20° and 40° geographical slant judgments were not different from each other, but were different than all other slant values. There was also a main effect of surface height (F(1, 9) = 8.05, p = .02, η2 = 0.47). Slant judgments made when viewing the target surface at eye level were more variable on average (SD = 4.98) than those made when the target surface was lowered (SD = 4.14).
3. Experiment 2 The purpose of Experiment 2 was to increase the difference between optical slant and geographical slant to determine whether participants’ slant judgments would be influenced by further alterations in their viewing perspective. In Experiment 1, no significant main effects of tilt were found for mean error or variability despite the fact that the optical slants were dependent on the surface tilt. This result coupled with the significant main effect of geographical slant on mean error and 35
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Fig. 9. The mean slant judgments collapsed across the motion and static conditions (Experiment 3 and the 3D surface condition of Experiment 1) are plotted against geographical slant. The open circles represent mean errors of surfaces presented at 0° tilt and the filled circles represent mean errors of surfaces presented at 40° tilt. The error bars are standard error of the mean. The dashed line indicates perfect accuracy.
Fig. 8. The mean standard deviation of slant judgments from Experiment 2 for each geographical slant value. The error bars are standard error of the mean.
4. Experiment 3 In the previous two experiments, the participants’ visual environment was not restricted. The target surface was not the only surface that could be seen. Participants could easily see walls, a table, books, the top of the block that the surface was mounted on, etc. Judgments of 0° and 90° slant were less variable than the middle slant values in Experiments 1 and 2. Did the other horizontal and vertical surfaces act as a frame of reference for target and response surfaces, thereby affecting the participants’ judgments of 0° and 90°? To investigate this issue, Experiment 3 was conducted in the same viewing condition as the first two experiments and in the dark to remove all other surfaces from view. Participants’ slant judgments were extremely variable for the middle slant values (20°, 40°, and 60°). Can the variability in these judgments be reduced? To test this question, the target surface was rotated while viewed by the participant. Previous studies (Bingham and Lind, 2008; Lee, Lind, Bingham, & Bingham, 2012; Lind, Lee, Mazanowski, Kountouriotis, & Bingham, 2014) found that a 45° continuous perspective change improved the perception of metric shape. So perhaps rotating the target surface would increase the stability of participants’ slant judgments.
accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki). 4.2. Results and discussion Participants underestimated 90° slants and overestimated 0°, 20°, 40°, and 60° slants. 40° slants were overestimated more than 60° slants. Surfaces presented at 40° tilt (mean error of −2.12) were overestimated more than those presented at 0° tilt (mean error of −1.41). There was no effect of lighting on participants’ slant judgments. A 5 (slant) × 2 (lighting) × 2 (tilt) repeated measures ANOVA was performed on the mean errors. There was a main effect of slant (F(2.2, 17.8) = 22.1, p < .01, η2 = 0.73). There was also a main effect of tilt (F(1, 8) = 22.7, p = .01, η2 = 0.57). The same pattern of mean slant judgments was observed (with slight numerical variations) when data from Experiment 3 and the 3D surface condition of Experiment 1 were combined (see Fig. 9) as was seen in Experiments 1 and 2. Motion had no effect on mean slant judgments. A 5 (slant) × 2 (motion) × 2 (tilt) repeated measures ANOVA was performed to compare the mean errors from the 3D surface condition of Experiment 1 (no motion) and the mean errors from both conditions of Experiment 3 (motion). There was a main effect of slant (F(2.3, 18.3) = 14.8, p < .01, η2 = 0.65). There was also a significant slant by tilt interaction (F(2.4, 18.8) = 3.8, p < .04, η2 = 0.32). As shown by mean standard deviations, slant judgments of 20°, 40°, and 60° slants were more variable than judgments of 0° and 90°. There was no effect of tilt or presence of light on the variability of slant judgments. A 5 (slant) × 2 (lighting) × 2 (tilt) repeated measures ANOVA was performed on the standard deviations of slant judgments from Experiment 3. There was a main effect of slant (F(2.4, 19.2) = 41.67, p < .01, η2 = 0.84). Judgments of the middle slants were still more variable than judgments of 0° and 90° when standard deviations of surfaces presented with and without motion were compared (see Fig. 10). Also, judgments of rotating surfaces were less variable overall than those of static surfaces. There was no effect of tilt or interactions. A 5 (slant) × 2 (motion) × 2 (tilt) repeated measures ANOVA was performed on the standard deviations of slant judgments
4.1. Methods The methods in Experiment 3 were the same as in Experiment 2 except for the following. The target surface was presented at eye level and was positioned on a platform. The experimenter manually rotated the platform at 1.8 Hz ± 25° for a total rotation of 50° on every trial. The target surface was tilted by either 0° or 40° prior to rotation. 50 trials were performed with the lights off and 50 were performed with the lights on. The texture elements were phosphorescent and visible in the dark. Only the texture elements on the target and response surfaces could be seen. Every 5 trials, the experimenter would shine a flashlight on the target and response surfaces to reserve the luminosity. The order of these conditions was counter-balanced. 4.1.1. Participants Nine of the ten participants from Experiments 1 and 2 also participated in the current experiment. All participants gave written informed consent prior to participation. Research was conducted in 36
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Fig. 11. Mean judged slant values are plotted against geographical slant values for each condition. The open circles indicate the mean judgments for surfaces presented at eye level at 0° tilt, the filled circles indicate surfaces presented below eye level at 0° tilt, the open squares indicate surfaces presented at eye level at 40° tilt, and the filled squares indicate surfaces presented below eye level at 40° tilt. The predicted optical slant values are also plotted against geographical slant values and these four lines correspond to the four conditions as previously described in Fig. 1.
Fig. 10. The mean standard deviations of slant judgments from Experiment 3 and the 3D surface condition of Experiment 1 are plotted against geographical slant. The filled circles represent the mean standard deviations of judgments of slants presented statically (Experiment 1) and the open circles represent the mean standard deviations of judgments of slants presented with motion (Experiment 3). The error bars are standard error of the mean.
from the 3D surface condition of Experiments 1 and 3. There were main effects of slant (F(2.3, 18.03) = 35.95, p < .01, η2 = 0.82) and motion (F(1, 8) = 6.98, p = .03, η2 = 0.47). There was no effect of lighting which suggests that participants did not use other visible surfaces in the room as a frame of reference for their slant judgments. The pattern of results for the slant judgments was similar to that observed in corresponding data in Experiment 1. The addition of motion did not affect the mean errors of participants’ slant judgments. A slant by tilt interaction was detected, but the differences driving this interaction were very small. As shown in Fig. 9, the judgments of slants presented at 40° tilt were more overestimated for slants 0°, 20°, and 40° and more underestimated for 60° and 90° slants compared to judgments of surfaces presented at 0° tilt. However, the error bars overlap at each geographical slant value and the differences between these mean errors were small (the largest difference is 2.2°). Motion did decrease the variability of slant judgments overall and, as can be seen in Fig. 10, judgments of 60° slant became more stable when motion was added.
variations in the tilt and eye level, the geographical slant of 60° corresponds to optical slants of 77.8° (below eye level, 0° tilt), 60° (at eye level, 0° tilt), 51.7° (below eye level, 40° tilt), and 41.6° (at eye level, 40° tilt). For a constant value of geographical slant, the optical slant varies by a factor of 2! However, the participants’ slant judgments are not so different in each of the conditions. The participants’ mean judgments for each slant in each condition are plotted in Fig. 11 against the predicted optical slant values shown in Fig. 1. While the mean slant judgments do not match the geographical slant of the target exactly (see the solid black line), they certainly do not vary as the values predicted by the optical slant values do. When we conducted several multiple regression analyses using optical and geographical slant values, the results confirm what is shown in Fig. 11. Slant judgments made by adjusting the response surface in each experiment were included in the analyses. Factors common to each regression analysis include eye height, tilt, and the interaction between eye height and tilt. For each multiple regression analysis, factors were removed using the least F value until only significant factors (p < .05) remained. The factors used in the analysis conducted on geographical slant included geographical slant and the following interaction terms: geographical slant x surface height, slant x tilt, surface height x slant x tilt. The final model included geographical slant (t = 119.58, p < .01, β = 0.97), tilt (t = −2.44, p < .02, β = −0.2) and surface height (t = −6.61, p < .01, β = −0.53) which resulted in a significant regression equation (F(3, 996) = 4783.2, p < .01) and an R2 value of 0.94. The analysis conducted on optical slant (the factors were the same as those used in the previous analysis with optical slant instead of geographical slant) did not require any factors to be eliminated. The regression equation was significant (F(7, 992) = 960.27, p < .01) with an R2 value of 0.87. The values for each coefficient can be seen in Table 1. The model with only geographical slant had the fewest significant factors and was thus the most parsimonious with a high R2 of 0.94. In contrast, every factor in the optical slant model was significant and contributed to a lower R2 value. The inclusion of every factor in the optical slant model points to the perspective-dependent nature of
5. General discussion The primary purpose of Experiments 1 and 2 was to determine whether participants’ slant judgments remained invariant over changes in the perspective on the surfaces. In Experiment 1, there was no effect of tilt for either mean error or variability. These results would seem to indicate that changes in viewing perspective do not influence participants’ slant judgments (see Fig. 5). In Experiment 2, there was an interaction between surface height and tilt, but it was driven primarily by the low surface height at tilt 0°, and the differences between this condition and the others were small in magnitude (largest difference was < 5°). The findings from the previous two experiments indicate that participants perceived geographical slant rather than optical slant. To compare optical slant with geographical slant, we calculated the optical slants for each geographical slant (0°, 20°, 40°, 60°, and 90°) in each condition of Experiments 1 and 2. As shown in Fig. 1, the geographical slants and optical slants are very different. For example, with 37
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errors of slant judgments. However, the presence of motion did decrease the variability in participants’ slant judgments, especially when viewing a geographical slant of 60°. We note that the motion was rotation around a vertical axis through the target surface and thus, consisted of variation in tilt that would have produced significant variations in optical slant. If judgments of slant depended on optical slant, then one would have expected this motion to have made the judgments of geographical slant less reliable. Instead, the motion yielded increased reliability.
Table 1 Multiple Regression Model for Optical Slant. Factor
β
t
p
Optical Slant Eye Height Tilt Eye Height × Tilt Eye Height × Optical Slant Tilt × Optical Slant Eye Height × Tilt × Optical Slant
1.173 0.498 −0.187 0.124 −0.337 0.584 −0.970
76.291 17.958 −6.745 4.470 −11.271 19.619 −3.225
< 0.001 < 0.001 < 0.001 < 0.001 < 0.001 0.001 < 0.001
6. Conclusions optical slant (i.e., its lack of constancy). The model including geographical slant was a better predictor of slant judgments than the model including optical slant. However, the geographical slant model did not account for all the variance within participants’ slant judgments. To determine whether optical slant predicted the unexplained variance, we first conducted a simple linear regression analysis regressing geographical slant on participants’ mean slant judgments from Experiments 1 and 2. The regression equation was significant (F(1, 998) = 13648.2, p < .01) with an R2 value of 0.93. We then conducted a simple linear regression in which we regressed optical slant on the residuals calculated in the previous regression analysis to see how much of the remaining variance could be accounted for by optical slant. The regression equation was significant (F(1, 998) = 23.12, p < .01) with an R2 value of 0.02. So optical slant only accounted for 2% of the variance in the residuals. This analysis further confirms that optical slant does not explain the data from the current study as well as geographical slant. A secondary purpose of Experiment 1 was to determine whether the 2D line task would produce the same results as the 3D surface response measure. The results indicate that participants tended to overestimate the 40° and 60° geographical target slants more when adjusting the line as opposed to the surface. However, this difference was very small in magnitude (i.e., 5.6° for 40° target slants and 6.4° for 60° target slants) and the overall pattern of slant judgments was remarkably similar. Judgments of 90° geographical slants were underestimated more (by approximately 5°) when using the surface response measure than when using the line. The variability of participants’ slant judgments in Experiments 1 and 2 was certainly not equivalent across different geographical slant values (see Figs. 5 and 7). Judgments of 0° and 90° target surface slants were far less variable than judgments of the other three geographical slant values. This small amount of variation in judgments clearly indicates that there is something special about vertical and horizontal surfaces. Importantly, this lower variability for 0° and 90° occurred at both lower and eye level surface heights for both tilts with no difference. This means there is something unique about the perception of vertical and horizontal surfaces that is invariant to changes in viewing perspective (i.e., geographical slant). In Experiments 1 and 2, the participants’ view of surrounding surfaces in the room was not restricted for the target or response surface. Therefore, participants could see horizontal (e.g., table, floor) and vertical (e.g., wall) surfaces that they could have used as references to assist in their judgments of 0° and 90°. Experiment 3 was primarily conducted to address this potential issue. Participants judged geographical slant with the lights on and with the lights off. If they were using the other visible horizontal and vertical surfaces in the room as a frame of reference, there should have been a noticeable difference between performance in these two conditions. However, there was no effect of lighting on slant judgments. This result suggests that other visible horizontal and vertical surfaces in the room did not assist participants in their judgments of 0° and 90° geographical slants. A second purpose of Experiment 3 was to determine whether continuously rotating the target surface would improve participants’ performance in judging slant. There was no effect of motion on the mean
We conducted the current study to investigate invariance in slant judgments across different conditions. There was no significant effect of tilt found in Experiment 1, which suggests constancy in judgments across changes in viewing perspective. Experiment 2 was conducted to amplify the difference between optical and geographical slant by changing the surface height as well as tilt. In Experiment 2, the slant judgments of the lowered target surface at 0° tilt were underestimated relative to the other conditions. However, these differences were very small in magnitude (2°–4.3°) and were certainly not predicted by changes in optical slant (maximum difference predicted to be 45°). Variations in optical slant could not account for the pattern of results observed in this study. Participants were able to perceive geographical slant directly as others have found previously (Gibson and Cornsweet, 1952; Sedgwick & Levy, 1985). Given geographical slant is perceived rather than optical slant, future investigations should examine what information specifies geographical slant. 7. Declarations of interest None. References Bingham, G. P., & Lind, M. (2008). Large continuous perspective transformations are necessary and sufficient for accurate perception of metric shape. Attention, Perception, & Psychophysics, 70, 524–540. http://dx.doi.org/10.3758/PP.70.3.524. Braunstein, M. I., & Payne, J. W. (1969). Perspective and form ratio as determinants of relative slant judgments. Journal of Experimental Psychology, 81, 584–590. http://dx. doi.org/10.1037/h0027886. Coleman, A., & Durgin, F. H. (2014). Egocentric reference frame bias in the palmar haptic perception of surface orientation. Psychonomic Bulletin Review, 21, 955–960. http:// dx.doi.org/10.3758/s13423-013-0552-7. Durgin, F. H., Hajnal, A., Li, Z., Tonge, N., & Stigliani, A. (2010). Palm boards are not action measures: An alternative to the two-systems theory of geographical slant perception. Acta Psychologica, 134, 182–197. http://dx.doi.org/10.1016/j.actpsy. 2010.01.009. Durgin, F. H., & Li, Z. (2011). The perception of 2D orientation is categorically biased. Journal of Vision, 11. http://dx.doi.org/10.1167/11.8.13. Durgin, F. H., Li, Z., & Hajnal, A. (2010). Slant perception in near space is categorically biased: Evidence for a vertical tendency. Attention, Perception, & Psychophysics, 72, 1875–1889. http://dx.doi.org/10.3758/APP.72.7.1875. Gibson, J. (1950). The perception of visual surfaces. The American Journal of Psychology, 63, 367–384. http://dx.doi.org/10.2307/1418003. Gibson, J., & Cornsweet, J. (1952). The perceived slant of visual surfaces – Optical and geographical. Journal of Experimental Psychology, 44, 11–15. http://dx.doi.org/10. 1037/h0060729. Koenderink, J. J., & van Doorn, A. J. (1976). Geometry of binocular vision and a model for stereopsis. Biological Cybernetics, 21, 29–35. http://dx.doi.org/10.1007/ BF00326670. Koenderink, J. J., & van Doorn, A. J. (1991). Affine structure from motion. Journal of the Optical Society of America A, 8, 377–385. http://dx.doi.org/10.1364/JOSAA.8. 000377. Lee, Y., Lind, M., Bingham, N., & Bingham, G. P. (2012). Object recognition using metric shape. Vision Research, 69, 23–31. http://dx.doi.org/10.1016/j.visres.2012.07.013. Lind, M., Lee, Y., Mazanowski, J., Kountouriotis, G. K., & Bingham, G. P. (2014). Affine operations plus symmetry yield perception of metric shape with large perspective changes (≥45°): Data and model. Journal of Experimental Psychology: Human Perception and Performance, 40, 81–93. http://dx.doi.org/10.1037/a0033245. Norman, J. F., Todd, J. T., Norman, H. F., Clayton, A. M., & McBride, T. R. (2006). Visual discrimination of local surface structure: Slant, tilt, and curvedness. Vision Research, 46. http://dx.doi.org/10.1016/j.visres.2005.09.034. Norman, J. F., Todd, J. T., & Phillips, F. (1995). The perception of surface orientation from multiple sources of optical information. Perception & Psychophysics, 57,
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