Perception of the physical stability of asymmetrical three ...

In press, Journal of Vision

Perception of the physical stability of asymmetricalthree-dimensional objects

Steven A. Cholewiak1, Roland W. Fleming2, Manish Singh1

1Department of Psychology and Center for Cognitive Science, Rutgers University, New Brunswick2Department of Experimental Psychology, University of Giessen, Germany

Visual estimation of object stability is an ecologically important judgment that allows ob-servers to predict the physical behavior of objects. A natural method that has been used inprevious work to measure perceived object stability is the estimation of perceived “criticalangle” – the angle at which an object appears equally likely to fall over versus return to itsupright stable position. For an asymmetric object, however, the critical angle is not a singlevalue, but varies with the direction in which the object is tilted. The current study addressedtwo questions: (1) Can observers reliably track the change in critical angle as a function oftilt direction? (2) How do they visually estimate the overall stability of an object, given thedifferent critical angles in various directions? To address these questions, we employed twoexperimental tasks using simple asymmetric 3D objects (skewed conical frustums): settings ofcritical angle in different directions relative to the intrinsic skew of the 3D object (Experiment1), and stability matching across 3D objects with different shapes (Experiments 2 & 3). Ourresults showed that (1) Observers can perceptually track the varying critical angle in differentdirections quite well; and (2) Their estimates of overall object stability are strongly biased to-ward the minimum critical angle (i.e., the critical angle in the least stable direction). Moreover,the fact that observers can reliably match perceived object stability across different 3D shapessuggests that perceived stability is likely to be represented along a single dimension.

Keywords: 3D Shape, Perceived Object Stability, Critical Angle, Stability Matching

Introduction

In addition to estimating the current properties of objectsand surfaces based on visual inputs, human observers arevery good at predicting how objects are likely to behave inthe near future. For example, observers can visually extrapo-late the trajectory of a moving object (Becker & Fuchs, 1985;Pavel, Cunningham, & Stone, 1992; Verghese & McKee,2002), and predict where an object that disappears behindan occluder is likely to re-emerge from (Scholl & Pylyshyn,1999; Graf, Warren, & Maloney, 1995; Shah, Fulvio, &Singh, 2013). Even more impressive are cases where ob-servers can make visual predictions about object behaviorbased on the inference of unseen forces, such as momentum(Kim, Feldman, & Singh, 2013; Newman, Choi, Wynn, &Scholl, 2008; Todd & Warren Jr., 1982), gravity, and supportrelations (Barnett-Cowan, Fleming, Singh, & Bülthoff, 2011;Hamrick, Battaglia, & Tenenbaum, 2011; Samuel & Kerzel,2011). Indeed, infants as young as 8 months of age have been

This work was supported by NSF Grants CCF-0541185 andDGE-0549115 (IGERT: Interdisciplinary Training in PerceptualScience) and by NIH Grant EY021494.

shown to be visually sensitive to support relations and grav-ity; and are surprised when shown a scene in which an in-adequately supported object appears to maintain its positionin space (Baillargeon & Hanko-Summers, 1990; Baillargeon,Needham, & DeVos, 1992). Predictions such as these rely ona “causal understanding” of the scene based on visual infor-mation (Cooper, Birnbaum, & Brand, 1995).

Traditional work on naïve physics has documented vari-ous ways in which people’s intuitions are often inconsistentwith Newtonian mechanics (e.g., McCloskey, Caramazza, &Green, 1980). However, when shown dynamic “real-time”displays simulating physical behavior, observers can be quiteaccurate at detecting deviations from Newtonian mechanics(Kaiser, Proffitt, & Anderson, 1985; Kaiser, Proffitt, Whelan,& Hecht, 1992; Proffitt & Gilden, 1989). More recent workhas also demonstrated that observers correctly take into ac-count acceleration due to gravity – consistent with Newton’slaws of motion – when timing their hand movements in or-der to catch a falling ball (McIntyre, Zago, Berthoz, & Lac-quaniti, 2001; Zago & Lacquaniti, 2005). These results arealso consistent with our day-to-day experiences of manipu-lating and interacting with objects, where we are generallyquite good at predicting the physical behavior of objects andusing these predictions to guide our motor actions.

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2 CHOLEWIAK, FLEMING, & SINGH

ba

Figure 1. Two objects sitting near the edge of a table, onethat is very perceptually stable (a) and one that is very per-ceptually unstable (b).

In a recent study on the perception of object stability,Samuel and Kerzel (2011) used planar polygonal objects,shown sitting on a supporting edge/base, with varying de-grees of “imbalance” – measured in terms of where the COMof the object lies relative to the center of the supporting edge(including the possibility of being outside this supportingbase – in which case the object would not physically stayupright). Subjects indicated whether a given object (shownresting on its supporting edge) would stay upright or fallover. Their responses exhibited a conservative (or anticipa-tory) bias – i.e., in the direction of perceiving an object tobe unstable, even though physically it would maintain its up-right posture. These results were interpreted in terms of aconservative tendency to keep judgments of object stabilitystay “on the safe side.”

In the current paper, our interest is in the visual estimationof the degree of physical stability of 3D objects which are instable equilibrium (in the sense that, if left unperturbed, theywould maintain their current position). For example, the twoobjects in Fig. 1 are both in stable equilibrium (in each case,the COM of each object is directly above the center of a cir-cular base). However, it is visually apparent that the objectin Fig. 1a is physically more stable than the one in Fig. 1b. Inother words, based on vision alone, one would naturally ex-pect that the object in Fig. 1a is more resistant to the actionsof perturbing forces than is the object in Fig. 1b.

One natural way of capturing object stability, therefore,is in terms of the maximal extent to which an object can betilted away from its “upright” stable position and still returnto that upright position when released. We refer to this an-gle of tilt as the critical angle (see Fig. 2). At this angle oftilt, the center of mass (COM) of the object is directly ver-tically above the point of contact on the base, around whichthe object is being rotated (see Fig. 2c). By definition, anytilt greater in magnitude than this critical angle will result inthe object toppling over, rather than returning to its uprightposition. With this definition in mind, it is easy to appreciatethat the object in Fig. 1a is physically more stable because it

a

CriticalAngle

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Figure 2. Coffee cup with COMs – represented by the bluecircle and blue line representing gravity vector – verticallyabove the base (green) (a), directly over the contact pointwhen the object is at its critical angle (b), and outside thebase (c).

has a larger critical angle than the one in Fig. 1b.

In previous work, we have used the visual estimationof critical angle to measure the perception of stability forthree-dimensional objects that were rotationally symmetric(Barnett-Cowan et al., 2011; Cholewiak, Singh, Fleming, &Pastakia, 2010). While critical angle provides a natural mea-sure of overall physical stability for such objects, the situa-tion is more complicated for asymmetric 3D objects. Specif-ically, for an asymmetric object, the critical angle is not asingle value, but depends on the direction in which the objectis tilted. Since the object can be tilted in any radial directionvarying from 0◦ to 360◦, the critical angle is really a functiondefined on [0◦, 360◦). However, as we will see, observers canreliably judge the overall stability of such an asymmetric 3Dobject. This raises the question of how people combine theinformation about different critical angles in different possi-ble tilt directions into a single overall estimate of the physicalstability of an object? To address this question, we use twoexperimental methods: visual estimation of critical angle indifferent directions (Experiment 1) and perceptual matchingof overall stability across objects with different shapes (Ex-periment 2 & Experiment 3).

In Experiment 1, we explicitly test how well people canestimate the critical angle of an object as a function of thedirection in which the object is tilted relative to its directionof intrinsic skew. In Experiments 2 and 3, we ask how thesedifferent critical angles in different tilt directions are com-bined into a single estimate of overall stability. Specifically,we compare two natural combination rules: (i) that the over-all perceived stability is determined by the average criticalangle across all possible tilt directions; and (ii) overall per-ceived stability is determined by the minimum critical angle(i.e., the critical angle in the least stable direction).

PERCEIVED STABILITY OF ASYMMETRIC 3D OBJECTS 3

2:2 3:2 4:2

Figure 3. Cropped examples for Exp. 1 showing the three as-pect ratios of the asymmetric conical frustums (2:2, 3:2, 4:2)with a constant skew. Note that 3D volumes were equatedand all of the objects were skewed by 25◦ from the vertical.

Experiment 1: Measuring Critical Angle in DifferentDirections

In the first experiment, we investigated whether observerscould accurately track the critical angle of asymmetric 3Dobjects as a function of the direction in which they are tilted.The objects used in these experiments were skewed conicalfrustums (see Fig. 3). In what follows, it will be importantto distinguish between: (i) the direction in which the objectis tilted (the magnitude of tilt is adjusted interactively by theobserver); and (ii) the intrinsic direction of skew of the con-ical frustum objects. The angle of relevance to us is the di-rection in which the object was tilted relative to the intrinsicdirection of skew of the object. We refer to this angle as α.In the experiments, the object was always tilted directly overthe precipitous edge of a table, but the direction of intrin-sic skew of the object was varied from trial to trial, therebymanipulating α (see Fig. 4).

Methods

Observers. 14 Rutgers University undergraduate stu-dents participated for course credit. All reported normal orcorrected-to-normal vision.

Apparatus. The stimuli were generated in MATLAB2012a using Psychotoolbox-3 (Brainard, 1997; Pelli, 1997;Kleiner, Brainard, & Pelli, 2007) running on an HP desk-top computer – with an Intel Core i7 870 processor (8MBcache, 4-cores running at 2.93GHz) and 4 GB of RAM –and presented on a Sony Trinitron 20 inch CRT with a 1024x 768 pixel resolution at a refresh rate of 140Hz. WithinPsychtoolbox-3, experimental scenes were rendered usingthe Matlab OpenGL (MOGL) toolbox and were presentedstereoscopically using an NVIDIA Quadro 4000 graphicscard and 3D Vision 2 LCD shutter glasses1. Observers werecomfortably seated with a chin-rest supporting their head 80cm from the screen.

The individual object meshes and all relevant quantities(volumes, COM locations, critical angles, etc.) were calcu-lated using Mathematica 8. The experimental scenes were

120°

240° 300°180°

0° 60°

Figure 4. Examples for Exp. 1 of the 6 possible skew stim-uli (α = 0◦, 60◦, 120◦, 180◦, 240◦, 300◦) for a frustum with aconstant aspect ratio (4:2). Note that the aspect ratio or skewangle may appear to change in these illustrative figures; how-ever the stimuli were presented stereoscopically, so a skewdirection of α = 120◦ was facing into the screen/scene andα = 300◦ was facing out toward the observer.

illuminated with OpenGL’s per-vertex lighting model usinga fixed function pipeline with specular highlights applied totextured objects, and with the specular reflection angles cal-culated assuming a constant view direction parallel to thedirection of the -z axis. There were 2 light sources, bothwith white ambient, diffuse, and specular components andthe objects’ surfaces had ambient, diffuse, and specular re-flectances. Objects were textured with 2D planar texturesmapped to their surfaces.

Stimuli and Design. The experiment used the methodof adjustment to measure the perceived critical angle forobjects in a 3 (aspect ratios) × 6 (skew directions) facto-rial design. The objects were conical frustums2, that wereplaced close to the edge of a rendered table. The objectshad 3 possible aspect ratios (Height:Base diameter; 2:2,3:2, 4:2) and subtended approximately 6.4-10.7 DVA (seeFig. 3). The frustums were skewed by 25◦ from the verti-cal in one of 6 directions relative to the edge of the table(α = 0◦, 60◦, 120◦, 180◦, 240◦, 300◦) (see Fig. 4). At 0◦ theobject’s skew was directed towards the precipice, in the range0◦ − 180◦ the object was facing away from the observer, at180◦ the object was skewed in the opposite direction of theprecipice, and in the range 180◦−360◦ the object was skewedtoward the observer.

Objects were always tilted directly toward the edge of thetable, so the skew direction determines the tilt direction rela-tive to the object’s intrinsic skew. The observers’ viewpointin the scene was fixed. The objects had a woodgrain tex-

1The binocular views were rendered using a fixed camera sepa-ration of 6 cm.

2A conical frustum is a cone with the top portion removed. Theconical frustums were then skewed to produce asymmetric shapes.The magnitude of skew was such that their central axis was orientedat 25◦ from the vertical.


0° 60° 120° 180° 240° 300°

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Figure 5. Average group critical angle performance for the 3object aspect ratios (red = 2:2, green = 3:2, blue = 4:2) and 6skew directions (α = 0◦, 60◦, 120◦, 180◦, 240◦, 300◦). Dottedlines show the veridical critical angle curves for the 3 aspectratios. Error bars are standard error. On average, observerstracked the physical critical angle quite well.

ture to reinforce the percept of solid, uniform density objects.They were described to observers as solid blocks of wood (asif cut out from the trunk of a tree). Visual cues – includingshading and a metallic reflection of the objects on the table– were added to aid in the realism of the scene and, in ad-dition, scenes were presented stereoscopically. The volumesof objects were equated across shape manipulations.

Procedure. The observers’ task was to adjust the tilt ofthe object until it was perceived to be equally likely to fall off

the table versus return to its upright position on the table (theperceived critical angle). As noted previously, the object’smotion was constrained to move orthogonal to the edge ofthe table, with the axis of rotation at the point on the base ofthe object closest to the table’s edge.

On each trial, the conical frustum was shown with an ini-tial tilt angle of either 0◦ (upright with base fully on the table)or 90◦ (tilted so that the entire object was over the precipice).The initial tilt angle was counterbalanced to later analyze forpossible hysteresis (which would manifest itself as a reliabledifference in settings between trials with an initial angle of0◦ vs. 90◦).

Each observer performed a total of 144 adjustments: 8 ad-justments for each of the 3 × 6 combinations of aspect ratioand skew directions. Half of these had an initial tilt angle of0◦, and the other half had an initial tilt angle of 90◦.

Results

Observer performance was evaluated by examining theircritical angle settings as a function of the skew direction (α)and aspect ratio. A repeated measures ANOVA was con-ducted on the pooled observer data, showing highly signif-icant effects for aspect ratio (F(2, 26) = 86.57, p < 0.01),α (F(5, 65) = 280.31, p < 0.01) and initial tilt angle(F(1, 13) = 11.95, p < 0.01). There was a small but sig-nificant effect of initial tilt angle on the observers’ responses.

ba

Figure 6. Example scenes illustrating situations where acylinder is perceptually more stable than a skewed conicalfrustum (a) and where a cylinder is perceptually less stablethan the skewed frustum (b).

Fig. 5 shows the average data plotted as a function of αalong with the predicted critical angles for all three aspectratios. Observers were, on average, quite good at estimatingthe critical angles when the α was changed. The only no-table exception was for α = 0◦ (especially at aspect ratiosof 3:2 and 4:2), where observers overestimated the criticalangle (and hence the stability) of the object – which wouldhave caused the object to fall off the table in the real world.Observers’ settings showed that they tracked the critical an-gle as a function of the skew remarkably well.

Discussion

In the first experiment, we found that not only were ob-servers able to perform the task, they were also very good attracking the critical angle as a function of the skew direction.For all three aspect ratios, observers adjustments followedthe physical prediction as a function of α.

Interestingly, on average, observers’ perceptual judgmentswere very close to the physical predictions, indicating thatthere was no systematic conservative bias for judgments “onthe safe side” (Samuel & Kerzel, 2011). And for α = 0◦,observers actually made liberal critical angle estimates, sug-gesting that there may be a more complex interaction be-tween shape and the observers’ biases to over or underes-timate the critical angle.

Experiment 2: Matching Overall Stability

Motivation

The results of the first experiment showed that observerstracked the critical angle as a function of the tilt directionrelative to the intrinsic skew of the object (α). However, inaddition to judging the critical angle in any given direction,observers can also estimate overall object stability. For ex-ample, in Fig. 6a it is visually apparent that the cylindricalobject on the right is more stable than the skewed frustum;whereas in Fig. 6b the cylindrical object is less stable.


This raises the question of how observers combine infor-mation about the various critical angles into a unitary perceptof object stability. Because the critical angle task inherentlymeasures the perceived stability in a single direction of tilt,we can only answer this question by using a different taskthat assesses the overall stability of an object, not just stabil-ity in a given direction. To do this, we use an overall stabilitymatching task.

Observers were asked to compare the overall stabilities oftwo objects – a standard object (corresponding to one of theskewed conical frustums from the previous experiment) anda simple cylindrical comparison object – and to adjust theaspect ratio of the comparison object until it was perceivedto be equally stable as the standard. When the stabilities ofthe two objects were perceptually matched, the two shapescould be considered stability metamers3.

It was not obvious a priori whether observers would beable to perform such matches. A pilot study revealed, how-ever, that observers find this a natural task, and their settingsare quite precise. This is consistent with the idea that physi-cal stability is natural perceptual dimension.

How do observers combine the various critical angles intoa single stability judgment? Two natural combination rulesthat could be used are: (1) to take the average of all the crit-ical angles; and (2) to simply use the minimal critical angle.If observers assume that an object could have a force im-pulse applied from any direction (uniformly sampled fromevery radial direction around the object), then it would makesense to integrate across all possible tilt directions, whichwould result in the average critical angle strategy (Hamricket al., 2011). This average critical angle model incorporatesinformation from every potential force vector direction, so ittakes into account the minimum and maximum critical angle,as well as every critical angle in-between, when judging theoverall stability.

Alternatively, observers could use the object’s minimumcritical angle as a proxy for the perceived overall stability,which may be a more salient feature if individuals are look-ing for the critical angle that is most informative about a po-tential change in the object’s state of equilibrium. A naturalway to think of physical stability is in terms of the minimumforce required to change the equilibrium state of the object,so if we consider the minimum of the forces in all directions,then the absolute minimum would be in the least stable di-rection. That is, when a force is applied in the direction ofthe minimum critical angle, it has the highest likelihood ofchanging the equilibrium state.

The fits of the average and minimum critical angle mod-els were compared to see which model better described theobservers’ settings.

2:2 3:2 4:2

Figure 7. Cropped examples for Exp. 2 of the 3 possibleaspect ratios (2:2, 3:2, 4:2) for a constant skew direction(α = 180◦), with the comparison cylinder at it’s lowest aspectratio (1:2) on the right.

Methods

Observers. 13 Rutgers University undergraduate stu-dents, with normal or corrected-to-normal visual acuity, par-ticipated in the experiment.

Stimuli and Design. The scene was composed of twoobjects, a standard and a comparison object, sitting next toeach other on a virtual table. The standard objects – skewedconical frustums as in the first experiment – had 3 possibleaspect ratios (2:2, 3:2, 4:2) and subtended approximately 4.3-6.8 DVA (see Fig. 7). The frustums were skewed by 25◦

from the vertical and were skewed towards the left or right(see left panes in Fig. 8). In addition, the standard objectswere placed on either the left or the right of the comparisonobject (see center panes in Fig. 8). The comparison objects –cylinders – had one of 2 initial aspect ratios (1:2 or 16:2), thetwo opposite ends of the aspect ratio spectrum: short-and-wide and narrow-and-tall, that subtended approximately 3.0and 12.3 DVA, respectively (see right panes in Fig. 8). Theyhad the same wood grain texture and material properties asthe conical frustums and had a variable aspect ratio, to beadjusted by the observers.

The experiment used the method of adjustment for stabil-ity matching using a 3 (standard aspect ratios) × 2 (skew di-rections) × 2 (presentation locations) × 2 (initial comparisonobject aspect ratios) factorial design.

Procedure. On each trial, a standard conical frustumwas shown alongside a comparison cylinder. The observers’task was to adjust the aspect ratio of the comparison objectuntil it was judged to have the same overall stability as thestandard object. The comparison object’s volume was heldconstant across changes in aspect ratio, hence the observer’sadjustments affected both height and radius. Since the twoobjects had the same surface properties and there were no

3This is analogous to color metamerism, where two colors withdifferent spectral compositions are perceived to be the same. In thecurrent context, two 3D objects that are physically very different areperceived to be the same in their stability.


Standard α = 0°

Standard α = 180°

Standard on Left

Standard on Right

Initial ComparisonAspect Ratio

= 16:2

Initial ComparisonAspect Ratio

= 1:2

Figure 8. Examples of the two standard skew directions(α = 0◦and180◦), the two standard object locations (leftand right), and the 2 initial comparison aspect ratios (1:2 and16:2).

other stability cues than the visual ones, observers had to relyon shape alone in making their judgments.

When the observer finished their aspect ratio adjustment(i.e., when the perceived stabilities of the comparison andstandard were subjectively the same), the two objects couldbe considered stability metamers (similar to color metamers,where two colors are perceived to be the same even thoughthey have differing wavelength compositions).

Each observer performed a total of 192 adjustments: 8adjustments for each of the 3 × 2 × 2 × 2 combinations ofaspect ratio, skew direction, presentation location, and initialcomparison aspect ratio.

Results

A repeated measures ANOVA was conducted on thepooled observer data (see Fig. 9), showing significant effectsfor the standard aspect ratio (F(2, 24) = 50.80, p < 0.01) andinitial comparison aspect ratio (F(1, 12) = 21.12, p < 0.01).There was no effect of skew direction (F(1, 12) = 3.67, p >0.05) or presentation location (F(1, 12) = 1.23, p > 0.05).There was a small but significant effect of initial aspect ratioon the observers’ responses.

Observers’ settings were compared against predictions de-rived from (i) the average critical angle model and (ii) theminimum critical angle model. If observers used the lowestcritical angle to make their judgments, then we would expecttheir data to be close to the minimum critical angle predic-tion (blue dashed curve in Fig. 9). Conversely, if observersused an estimate that uniformly took into account the object’scritical angle in all possible directions, then their judgmentsshould be closer to the average critical angle prediction (reddashed curve in Fig. 9). On average, the observers’ settingswere closer to the minimum critical angle model’s predic-tions than the average critical angle model.

In order to evaluate which model (average critical angleor minimum critical angle) better described observers’ per-formance, we calculated the likelihood ratios for each ob-

0.5 1 1.5 2 2.5

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Figure 9. Average group performance for the 3 object aspectratios (2:2, 3:2, 4:2) collapsed across skew direction and pre-sentation location. The green lines represent the observers’responses, separated by initial comparison aspect ratio (solid= 16:2, dashed = 1:2). The blue line represents the predictedcomparison aspect ratio if observers used the minimum criti-cal angle and the red line represents the average critical angleprediction. Error bars represent 95% confidence intervals.

server’s data. The likelihood ratio test allows us to comparethe two stability estimation models and judge which modelbetter explains the observed data.

Normally, we would compare performance using theBayes factor (K), which is the ratio of the probability of thedata given the first model (Pr(D|Mavg)) over the probabilityof the data given the second model (Pr(D|Mmin))(see Eq. 1).

K =Pr(D|Mavg)Pr(D|Mmin)

(1)

However, in this case the two models we are comparinghave no free parameters, which means the Bayes Factors sim-ply reduces to the ratio of the two model likelihoods. Thislikelihood ratio can now be used to judge which model betterexplains the data, with likelihood ratio values greater than 1supporting Mavg and values less than 1 supporting Mmin. Theratios of the average and minimum likelihoods were calcu-lated for each observer (see Eq. 2).

Λobserver =

∏ni=1 Pr(Di|Mavg)∏ni=1 Pr(Di|Mmin)

(2)

Log likelihoods were calculated for the individual modelsand for the likelihood ratios – positive log likelihoods sup-porting Mavg and negative log likelihoods supporting Mmin.For 10 of the 13 observers, the log likelihood ratios favoredthe minimum model over the average model (see Table 1),indicating that the minimum critical angle provides a bettermodel of the observers’ stability matches than the averagecritical angle.

Discussion

Observers’ judgments were better explained by the mini-mum critical angle, rather than the mean critical angle. Thus


Table 1Log likelihoods for each model and the log likelihood ratios(Log(Likelihoodavg/Likelihoodmin)) for each observer. Notethat for 10 of the 13 observers, the log likelihood ratio fa-vored the minimum model – that is, the likelihood ratio wasnegative.

Observer Log(Likelihoodavg) Log(Likelihoodmin) Log(Λ) Model Favored1 −507.33 −154.81 −352.51 Minimum2 −1920.61 −374.93 −1545.69 Minimum3 −561.97 −218.72 −343.25 Minimum4 −1094.45 −258.80 −835.66 Minimum5 −328.60 −213.95 −114.65 Minimum6 −1659.79 −254.89 −1404.89 Minimum7 −367.33 −243.18 −124.15 Minimum8 −585.69 −246.68 −339.01 Minimum9 −1054.09 −116.44 −937.65 Minimum10 −683.06 −218.44 −464.62 Minimum11 −8317.27 −29086.70 20769.40 Average12 −270.65 −3203.57 2932.92 Average13 −231.59 −702.86 471.27 Average

the least-stable direction has a disproportionately large influ-ence in visually estimating the overall stability of an object.As noted earlier, the direction of least stability is the mostinformative one about a potential change in the object’s stateof equilibrium. And, indeed, it tends to dominate observers’judgments.

Experiment 3: Matching Overall Stability in DifferentDirections

Motivation

The results of the stability matching task in Experiment 2indicated that observers’ overall stability judgments were in-formed by the minimum critical angle of the asymmetric ob-jects. The idea behind the stability matching task presentedin Exp. 2 was that, unlike the critical angle task, the stabilitymatches for a given object should be constant as the directionof tilt varies relative to the intrinsic skew of the object. Herewe test if this is indeed the case.

Dependencies on the viewing direction are common in 3Dshape perception – and are particularly acute when the axisof elongation of an object is foreshortened (Marr & Nishi-hara, 1978; Biederman, 1987; Humphrey & Jolicoeur, 1993;Lawson & Humphreys, 1998). Because of this, one may ex-pect some effect of viewing direction on visual judgmentsof object stability (if the 3D shape appears different, its per-ceived stability will, of course, also be affected). However,we would expect that any such effects are much smaller thanthe large influence of skew direction on perceived critical an-gles, observed in Experiment 1.

In this experiment, observers compared the overall stabil-ities of two objects – a standard and a comparison – and ad-justed the comparison object until it was perceived to be thesame stability as the standard, just as in Exp. 2. However,this experiment differed from Exp. 2 in that the direction of

skew was also manipulated (analogous to the manipulationof α in the first experiment).

Methods

Observers. 13 Rutgers University undergraduate stu-dents, with normal or corrected-to-normal visual acuity, par-ticipated in the experiment.

Stimuli and Design. The experiment used the same sta-bility matching task as Exp. 2. It used a 3 (standard aspectratios) × 6 (skew directions) × 2 (presentation locations) × 2(initial comparison object aspect ratios) factorial design.

As in the second experiment, the rendered scene containedtwo objects, a standard and a comparison object, sitting nextto each other on top of a table. The standard objects had 3possible aspect ratios (2:2, 3:2, 4:2), were skewed by 25◦

from the vertical, and were skewed in 1 of 6 skew direc-tions (0◦, 60◦, 120◦, 180◦, 240◦, 300◦). In addition, the stan-dard objects were placed on either the left or the right of thecomparison object and the comparison objects had one oftwo initial aspect ratios (1:2 or 16:2). See Figure 8 for exam-ples of the scene layout.

Procedure. The testing procedure was identical to Ex-periment 2. Each observer performed a total of 216 adjust-ments: 6 adjustments for each of the 3 × 6 × 2 combina-tions of aspect ratio, skew direction, and presentation loca-tion. Half of these used an initial aspect ratio of 1:2 for thecomparison object; the other half used an initial aspect ratioof 16:2.

Results

Observers’ performance was evaluated by examining theirstability matches (aspect-ratio settings for the comparisonobject) as a function of the skew direction. A repeated mea-sures ANOVA was conducted on the pooled data, show-ing a highly significant effect of aspect ratio (F(2, 24) =

153.21, p < 0.01) and a significant effect of initial compari-son aspect ratio (F(1, 12) = 7.53, p < 0.05). There was alsoa significant effect of skew direction (F(5, 60) = 39.94, p <0.01), but a Post-Hoc Tukey Test confirmed that the skewdirection effect was due to the dip in performance at 60◦ and120◦ and that there were no other significant differences forany other slant directions.

As shown in Fig. 10, on average, judgments were affectedby the change in the standard aspect ratio and the directionof the skew, which had a much smaller effect. There was aslight dip in the comparison aspect ratios at 60◦ and 120◦. Itis likely this may have been due to perceived foreshorteningwhen the slant was directed into the screen.

Discussion

The results from the third experiment support the hypoth-esis that observers are able to match the overall stabilities of


0° 60° 120° 180° 240° 300°0

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Figure 10. Average group performance in Exp. 3 for the 3object aspect ratios (red = 2:2, green = 3:2, blue = 4:2) col-lapsed across the other independent variables (standard loca-tion and initial comparison aspect ratio) as a function of theskew direction. Error bars represent standard error.

objects in a consistent manner, even when the asymmetricobjects are rotated relative to the observer’s viewpoint.

Although the effect of skew direction was statistically re-liable, it was quite small when compared to the influence ofskew direction on the perceived critical angle observed inExp. 1. Given the well-known effects of foreshortening of theaxis on shape perception, it seems likely that the observed in-fluence of skew direction on perceived stability is most likelydue to the misperception of 3D shape from those viewpoints(Marr & Nishihara, 1978; Biederman, 1987; Humphrey &Jolicoeur, 1993; Lawson & Humphreys, 1998). Neverthe-less, the stability matches are surprisingly flat, especiallywhen compared with the large influence of skew directionperceived critical observed in Exp. 1.

Conclusion

The critical angle is a good measure of stability forthree-dimensional objects, incorporating information aboutmass distribution and shape to define an ecologically im-portant property of real-world objects. We have previouslyfound that people are good at visually estimating the criti-cal angles of symmetric objects (Barnett-Cowan et al., 2011;Cholewiak et al., 2010); however, it was not clear how theseresults would extend to more general, asymmetric objects.Since the critical angle is not a single value for asymmetricobjects, but depends on the direction in which the object istilted, we investigated how well observers could visually es-timate the critical angles in different tilt directions for asym-metric objects.

Observers were clearly able to track the critical angle asasymmetric conical frustums were tilted in different direc-tions relative to their intrinsic skews in Exp. 1, indicating thattheir judgments were appropriately informed by the shape’sasymmetry. These results extend our previous results for ro-tationally symmetric objects to asymmetric objects. This is

reassuring because we have few issues making guided ac-tions that are informed by our percepts of real-world objectstability in our daily lives. In contrast to the findings ofSamuel and Kerzel (2011) with 2D polygonal objects, how-ever, our results show no systematic conservative bias – i.e.,for stability settings to be judged “on the safe side”.

Surprisingly, observers were able to reliably match theperceived stability of two objects with different shapes inExp. 2. These results suggest that stability may be repre-sented along a unitary dimension. The observers were able toinfer the force of gravity acting upon the shapes and comparetheir overall stabilities even though the shapes were quite dif-ferent. Observers’ stability matches were better explainedby the minimum model than the average model, suggestingthat observers’ estimates were disproportionately influencedby the direction of least stability. Since physical stability isdefined as the minimum force required to change the equi-librium state of the object, the direction with the smallestcritical angle defines where the object has the highest likeli-hood of changing its equilibrium state. Although the averagecritical angle is a measure of the central tendency of the sta-bility for the object – taking into account the maximum andminimum stability, and all of the critical angles in-between– on average, people appeared to use the minimum criticalangle as a proxy for the perceived overall stability.

Finally, Exp. 3 demonstrated that the matches were a func-tion of the aspect ratio of the 3D shapes – with observed dataexhibiting only a slight dependence on viewpoint (likely dueto foreshortening).

As a whole, these results provide evidence that visual es-timates of overall object stability are strongly influenced bythe minimum critical angle and that perceived shape plays animportant role to inform the people of the objects’ physicalproperties.

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