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Perception & Psychophysics1992, 51(1), 86-102

Planar motion permits perception of metricstructure in stereopsis

JOSEPH S. LAPPIN and STEVEN R. LOVEVanderbilt University, Nashville, Tennessee

A fundamental problem in the study of spatial perception concerns whether and how visionmight acquire information about the metric structure of surfaces inthree-dimensionaLspacefrommotion and from stereopsis. Theoretical analyses have indicated that stereoscopic perceptionsof metric relations in depth require additional information about egocentric viewing distance;and recent experiments by James Todd and his colleagues have indicated that vision acquiresonly afline but notmetric structure from motion—that is, spatial relations ambiguous with regardto scale in depth. The purpose of the present study was to determine whether the metric shapeofplanar stereoscopic forms might be perceived from congruence underplanar rotation. In Experi-ment 1, observers discriminated between similar planar shapes (ellipses) rotating in a plane withvarying slant from the frontal-parallel plane. Experimental conditions varied the presence versusabsence of binocular disparities, magnification of the disparity scale, and moving versus station-ary patterns. Shape discriminations were accurate inall conditions withmovingpatterns and werenear chance in conditions with stationary patterns; neither-the- presence nor the magnification ofbinocular disparities had any reliable effect. In Experiment 2, accuracy decreased as the range ofrotation decreased from 800 to 100. In Experiment 3, small deviations from planarity of the mo-tion produced large decrements in accuracy. In contrast with the critical role of motion in shapediscrimination, motion hindereddiscriminations of the binocular disparity scaieirrExperiment 4.In general, planar motion provides an intrinsic metric scale that is independent of slant in depthand of the scale of binocular disparities. Vision is sensitive to this intrinsic optical metric.

Everyday human experience and performance in per-ceiving constant object shapes from changing perspectivesinvarying contexts indicate that vision acquires informa-tion about the spatial structure of environmental objectswith impressive speed, reliability, precision, and gener-ality. Despite the familiarity and functional importanceof these visual achievements, we lack a theoretical under-standing ofthe optical information and visual mechanismson which they are based.

Indeed, the specific geometric relations that might beused by vision to perceive shape have seldombeen directlyevaluated by psychophysical techniques. Sperling, Landy,Dosher, and Perkins (1989) recently criticized the litera-ture on perceiving three-dimensional (3-D) structure frommotion for having failed to provide such evidence. Theapparent 3-D shape constancy of perceived environmen-tal objects, however, suggests that vision usually providesinformation about the metric structure of environmental

This research was supported in part by NIH Grants EY05926 andP30EY08126. Experiments 1-3 were conducted by S. R. Love as partof the honors program in psychology at Vanderbilt. We are grateful toFarley Norman and John Kuster for their help in conducting Experiment 4,to Warren Craft and Rebecca Fisher for help in preparing several figures,to Randolph Blake, Myron Braunstein, James Todd, and an anonymousreviewerfor comments on an earlier version of this paper, and to JamesTodd and Farley Norman for numerous discussions of the theoretical is-sues. Correspondenceshould be addressed to Joseph S. Lappin, Depart-ment of Psychology, Vanderbilt University, Nashville, TN 37240.

surfaces; and the precision of visual-motor coordinationexhibited in athletics seems to indicate that vision also pro-vides metric information about the locations and trajec-tories of environmental objects relative to the observer’sposition.’

Recent experiments on this issue, however, have shownthat metric relations in 3-D are not perceived in manycases. Todd and Reichel (1989) found that depth relations(perpendicular to the frontal-parallel plane) among neigh-boring points on curved surfaces, seen in 3-D space byvirtue of shading and texture gradients, were discrimi-nated as if the perceived depth relations in empty Euclid-ean 3-D space (E3) were only ordinal and not metric. Inmore recent experiments on the kinetic depth effect(KDE), Todd and Bressan (1990) and Todd and Norman(1991) have found that observers are unable to discrim-nate relative lengths of line segments or to discriminatedifferences in the depth of smooth surfaces. They haveconcluded that vision probably obtains information onlyabout affine and not about metric structure from KDEpatterns—that is, that the relative scale of the depth axisis visually indeterminate in KDE patterns.2

Evidence suggesting that observers can accurately dis-criminate the curvature and depth of rotating surfaces ofrevolution was obtained by Todd (1984) in experimentsdone with cylindrical surfaces with varying radii of cur-vature rotating around their central axes in the frontal-parallel plane. As Todd pointed out, however, these dis-cnminations could have been based on differences in ye-

Copyright 1992 Psychonomic Society, Inc. 86

METRIC STRUCTURE FROM MOTION 87

locity rather than differences in depth or curvature per Se.Indeed, in subsequent experiments, Todd (personal com-munication, November 1990) found that these surfacescannot be discriminated if the rotational velocities are ran-domly varied. Thus, the metric structure of these surfacesseems to be indiscriminable.

Can vision discriminate metric structure from motionunder any conditions? In contrast with the results of Toddand his colleagues (Todd & Bressan, 1990; Todd & Nor-man, 1991), Lappin and Fuqua (1983) found accurate bi-section discriminations of 3-D distances in kinetic depthdisplays of three collinear points rotating in a slantedplane. The perceived spatial relations in these experimentswere not merely affine, because (1) the patterns in mostconditions were displayed with exaggerated perspectiverather than by orthographic projection, so that the rela-tive spacing among the points in the image plane differedfrom that in E3 and changed with the angle of rotation;(2) the three collinear points appeared subjectively tomove rigidly in depth, with the distances between pointsappearing to remain constant under changing directionsin space; and (3) observers were inaccurate in bisectingprojected lengths in the two-dimensional (2-D) imageplane. The discrepancy between these results and thoseof Todd and Bressan (1990) and Todd and Norman (1991)may well have derived from differences in differentialstructure of the retinal image motions in the two setsof studies: The projected velocity fields in Lappin andFuqua’s (1983) displays remained constant over time, cor-responding to rotation within a slanted plane, whereasthose of Todd and Bressan (1990) and Todd and Norman(1991) changed over time at any given position in the im-age, corresponding to depth rotation of volumetric andsolid objects. Thus, Lappin and Fuqua’s (1983) patternssatisfied the “planarity constraint” that Hoffman andFlinchbaugh (1982) haveshown is theoretically sufficientfor metric structure to be determined from two successiveviews, whereas those of Todd and Bressan (1990) andTodd and Norman (1991) did not satisfy this constraint.

The perceptual utility of this planarity constraint wasstudied more extensively in the present experiments. Amain objective was to determinewhether such planar ro-tation in depth was sufficient to provide accurate percep-tion of metric structure in stereoscopic patterns. To an-ticipate, we found that the metric shape of a planar formrotating in a slanted plane could be accurately discrimi-nated, even when the binocular disparities were exagger-ated and unreliable! Without such rotation, however,stereopsis permitted only poor discrimination of the shapesof stationary forms in depth.

Stereoscopic Depth ConstancyThis investigation was motivated in part by the geomet-

ric fact that the depth separation between two environ-mental points depends not only on the binocular disparitybetween their two monocular half-images butalso on theiregocentric distance from the observer. Small changes inviewing distance yield large changes in depth. The prob-

lem of stereoscopic depth constancy is the problem of howstereopsis might provide constant information about spa-tial relations in depth, independently of changes in view-ing distance (Cormack, 1984; Fox, Cormack, & Norman,1987; Mauk, Crews, & Fox, 1987; Ono & Comerford,1977). (A schematic illustration of this relationship be-tween binocular disparity, viewing distance, and depthis given in Figure 1.)

When two points lie within a few degrees of the per-pendicular bisector of the line between the nodal pointsof the two eyes, the depth separation, z, between the twopoints is given by

z = (1/2){tan[tan~(2D/I)+ ô12]} — D, (1)

where I is the interocular distance, D is the distance tothe nearerpoint, and ô is the angularbinocular disparity.If D is the viewing distance to the farther of the two points,~5and z are negative. A more general equation for the casein which the two points are not necessarily centered be-tween the eyes is given by Cormack and Fox (1985).

Figure 1. An illustration of the relationship between viewing dis-tance, D, and depth, Z, for a constant binocular dIsparity 6. [From“Perceiving the metric structure of environmental objects from mo-tion, self-motion, and stereopais,” by J. S. Lappin, 1990, in K. War-ren & A. H. Wertheiin (Eds.), Perception and controlof self-motion(pp. 541-578). Hillsdale, NJ: Erlbaum. Copyright 1990 by LawrenceErlbaum Associates. Reprinted by permission.]

1

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T

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88 LAPPIN AND LOVE

Equation 1 describes a distinctly nonlinear relation: Whenthe depth is small relative to the viewing distance, then,for a constant binocular disparity, the depth separationis roughly proportional to the square of the viewingdistance.

If the relationships illustrated in Figure 1 adequatelydescribe the geometry of stereopsis, uncertainties aboutviewing distance should have the following effect on thestereoscopically perceived shapes of solid objects: A va-riety of different potential shapes might be perceived, de-pending on theestimated viewing distance. A convex spher-ical surface, for example, would appear spherical onlywhen its distance was correctly estimated; if the estimateddistance was too small, the curvature of the surface wouldappear reduced, yieldingan effipsoid flattenedin the direc-tion of gaze; and if the estimated distance was too great,the surface would appear elongated like a football in thedirection of gaze. Whenever the estimated viewing dis-tance was incorrect, the perceived length of a contourwould vary with its orientation, stretching and compress-ing as the object rotated in depth.

One might guess that the viewing distance and thus thedepth could be obtained by triangulation, using the ver-gence angle between the two eyes to the point of fixa-tion. Although psychophysical evidence indicates that vari-ations in vergence angle do influence perceived depths(Foley, 1967), changes in viewing distance produce onlysmall changes in vergence angle but large changes in depth.That is, the partial derivative of depth relative to vergenceangle is very large and nonlinear. Therefore, vergenceangle is a poor source of information with which to scalestereoscopic depth. Moreover, for viewing distances be-yond about 2 m, the vergence angle is essentially constant,but the apparent depth of stereoscopic afterimages (i.e.,with fixed disparity) increases with viewing distances onthe order of thousands of meters (Cormack, 1984).

Thus, the binocular disparities in a single pair of mon-ocular half-images of a stationary object do notdeterminethe metric structure of the object (cf. Foley, 1980; Koen-derink & van Doom, 1991); the scaling of distances indepth relative to distances in the frontal-parallel plane isambiguous. The KDE information provided by two suc-cessive views of a moving object is no less ambiguous.Moreover, the results of Todd and Bressan (1990) andTodd and Norman (1991) suggest that vision is insensi-tive to the added geometric constraints associated withthree or more views and that only affine spatial structurecan be perceived from motion.

Because of this inherent ambiguity about the scaling ofdistances in depth, many investigators (e.g., Longuet-Higgins, 1986; Man, 1982; Mayhew& Longuet-Higgins,1982) have concluded that the retinal optical informationfrom stereopsis and KDE must be supplemented by non-retinal information. The present study, however, demon-strates that retinal patterns ofplanar rotation canbe suffi-cient for the perception of metric structure.

Spatial Structure fromCongruence under Rotation

Two successive (monocular) views of a rotating objector two simultaneous stereoscopic views of a stationaryobject pennit a one-parameter family of perceived struc-tures (Bennett, Hoffman, Nicola, & Prakash, 1989). Withorthographic projections, these potential alternative struc-tures differ from one another by affine transformationsof the scale of depth relative to the frontal-parallel plane(Koenderink &van Doom, 1991; Todd &Bressan, 1990);and with perspective projections, when the viewing dis-tance is small or when viewing is stereoscopic, the fam-ily of potential alternative structures includes additionalvariations in shape associated with variations in viewingdistance. Additional views of a rotating object produceadditional image transformations, and the scaling param-eters derived from these multiple transformations mustagree if the object remains isometric under these motions.Thus, the rigidity or congruence of moving surfaces andcontours might be used as the basis for scaling distancesin E3.3

The logic of this idea was cogently expressed by Kill-ing (1892). Two spaces are congruent if and only if theycan be covered by the same object; two objects are con-gruent if and only if they can coverthe same space. Thus,objects and spaces constitute mutually constraining rela-tional structures. The structure of both may be derivedfrom congruence under motion.

Underwhat conditions might vision be sensitive to thispotential geometric information? Different groups of mo-tions produce projected images that differ in the poten-tial visibility of information about 3-D spatial structure.One noteworthy distinction involves translation versus ro-tation in depth. Although translations and rotations mayproduce image motions that are practically indistinguish-able locally, rotations are globally simpler because theycan be characterized by a single free parameter cor-responding to the angular extent of rotation, whereastranslations require more parameters. Moreover, “mo-tion parallax” patterns arising from lateral translationoften appear to be rotations (Braunstein & Andersen,1981; Lappin, 1991). Thus, vision seems to representdif-ferential local image motions by 3-D rotations.

Other potentially important distinctions concern (1) thecomplexity of the moving surface structure (e.g., a sin-gle smooth surface vs. multiple separate surfaces or dis-connected features), (2) opaque versus transparent sur-faces, (3) the complexity of the surface’s trajectory inspace-time (e.g., a surface of revolution vs. a volume ofspace-time), (4) the angle between the surface and the axisof rotation, and (5) the anglebetween the axis of rotationand the projected image. All of these characteristics af-fect the complexity of the differential structure of theprojected image motions and thus affect the potential visi-bility ofthe underlying spatial structure. The quantitativeeffects of these conditions on the visibility of 3-D spatial


structure havenot yet been clearly established by psycho-physical research, however.

The present experiments examined a particularly sim-ple case, in which planar shapes were rotated in a slantedplane. In this case, the sequential positions of the mov-ing object formed a planar surface in space-time—a 2-Dmanifold. Additionally, the monocular half-images wereeffectively orthographic projections, with negligible per-spective. Thus, the projective mapping ofthe moving formonto its monocular images could be described by a singleaffine coordinate transform. Likewise, the metric tensorparameters for embedding the monocular images of theform into E3 were also constant over retinal positions andconstant over time (see the Appendix). This conditionsatisfies what Hoffman and Flinchbaugh (1982) termedthe planaruy constraint. They provedthat, under this con-dition, two views of three or more points (which needbe onlypiecewise rigidly connected) are sufficient to de-termine the 3-D positions and motions of the points upto a reflection about the frontal-parallel plane. A differ-ent proof of the sufficiency of this condition for recover-ing metric structure and motion is given in the Appen-dix. The hypothesis that human vision can exploit thisplanarity constraint, however, has not previously beentested experimentally. The present study provides sucha test, and it demonstrates that metric structure can in-deed be accurately perceived in this case.

The hypothesis that stereoscopicperception of 3-D spa-tial structure may be derived from congruence undermo-tion has not been tested directly, but indirect evidence isprovided by several experiments by Wallach and his col-leagues (Wallach & Karsh, l963a, 1963b; Wallach,Moore, & Davidson, 1963) and by Fisher and Ebenholtz(1986). Using a telestereoscope that magnified binoculardisparities by increasing the optical distance between theeyes, observers viewed wire forms rotating arounda ver-tical axis. Because of the exaggerated disparities that ex-panded the depth axis, these forms appeared to deformas they rotated. Wallach & Karsh (1963a, l963b) andWallach et al. (1963) found that after 10 mm of passiveobservation of such rotating forms there was a reductionin the apparently expanded depth of a stationary formjudged after the 10 mm “training” period. Fisher andEbenholtz (1986), however, concluded that this depthaftereffect did not involve adaptive recalibration ofstereopsis by motion, since the aftereffects could be ob-tained after exposure to either stereopsis alone with nomotion or to motion alone with no stereoscopic disparity.Moreover, artificial pupils eliminated these aftereffects,suggesting that they derived from changes in perceiveddistance associated withaccommodation rather than fromrecalibration of either stereopsis or KDE. It should benoted that the differential structure of the images in theseexperiments was bothcomplex and continually changing,because the sequential positions of the forms occupied avolume rather than a surface in E3. Constant spatial struc-ture in depth should havebeen difficult to perceiveunderthese conditions, and the results of Fisher and Ebenholtzsuggest that such constancy was not perceived.

The present study investigated real-time visual scalingof stereoscopic shapes rotating in a slanted plane. In Experi-ment 1, form discrimination was compared in conditionsthat varied in the presence or absence of (1) stereoscopicdisparities, (2) distorted magnifications of the disparities,and (3) motion. The effects of the range of rotation andthe complexity of the differential structure of the projectedimage motions were evaluated in Experiments 2 and 3.The opposite effects of motion on discriminations of formand on discriminations of the stereoscopic scale of depthwere contrasted in Experiment 4.

EXPERIMENT 1

The aim of Experiment 1 was to determinewhether ro-tation of a form in depth could provide sufficient visualinformation about its metric structure to correct for spa-tial distortions produced by abnormal increases in binocu-lar disparity. If stereoscopic perception of spatial structurerequires extraretinal information about viewing distance,form perception should be seriously impaired by stereo-scopic displays in which the disparities are magnified.Such stereoscopic distortion should produce an elonga-tion of the form in depth relative to its extension in thefrontal-parallel plane. On the other hand, if stereoscopicspace canbe scaled by congruence under planar rotations,such magnifications of the disparities may have little orno effect on the perceived shapes of the rotating forms.

The spatial forms employed in all the experiments wereellipses—approximately circular plane shapes differingonly slightly in the length of one axis. In most experimen-tal conditions, these forms were tilted in depth from thefrontal-parallel plane by rotation around the horizontalaxis. Of course, the projected shapes of these forms inthe plane of the display screen were also ellipses con-tracted in the vertical axis of the screen, with the degreeof contraction varying with the magnitude of slant andmagnification of the stereoscopic disparities. The valuesof both the slant and stereoscopic disparities were ran-domly varied between trials, so that the projected shapesvaried as a conjoint function of the slant, disparity, androtational position in addition to the actual shape of theellipse in E3. The projected image shapes depended lesson the shape of the ellipse in E3 than on the other threevariables, and the image velocities also depended moreon slant than on shape. Therefore, discriminations of theseshapes required reliable information about metric struc-ture in depth.

MethodDesign. Eight experimental conditions were obtained from the

orthogonal combinations ofthree variables—moving versus station-ary, stereoscopic versusnonstereoscopic (in which the patterns onthe two monitors were identical), andasimulated viewingdistancethat was either correct or reduced and variable. (Reductionof thesimulated viewing distance magnified the binoculardisparities; theperspectivity ofthe monocular images wasnegligible.) In all theseconditions, thestimulus forms were displayed as if slanted in depthby an amount that varied randomly between trials. In 2 additionalconditions, the forms were presented only in the frontal-parallel

90 LAPPIN AND LOVE

plane, either moving or stationary. The accuracy ofform discrimi-nation under each ofthese 10 conditions was evaluated in aseparateblock of trials.

Apparatus and optIcalpatterns. Theoptical patterns were dis-played on two cathode-ray tube (CRT) monitors (Tektronix 608,with P-3 1 phosphor) controlled by a computer (PDP- 11/73) witha 12-bit digital-to-analog (D/A) interface (DataTranslation DT2771).The two CRT displays were haploscopically combined by prismsand mirrors adjustedfor each observer tomatch individual vergenceand interocular separation. These displays were viewed from a dis-tance of about 114.6 cm, with the observer’s head position con-strained by the two prisms in front of each eye. The binoculardis-parity of the two monocular patterns was scaled for an interocularseparation of 6.3 cm, regardless of the observer’s actual interocularseparation. Each CRT screen was seenthrough acircular aperture10 cm (50 of visual arc) in diameter in a black baffle mounted im-mediately in front of the monitor.

The CRTs were viewed in a dimly lit room that providedamplevisual cuesabout the distance of the displays. The background lu-minance on the CRT screens was about 0.06 cd/rn2, and the lu-minance of the individual points was about 6 cd/rn2, as measuredby a Pritchard spot photometer from the observer’s eye position.The diameter of an individual point was about 0.25 mm (45” ofarc). The spatial resolution provided by the 12-bit D/A interfacewas .021 mm (3.78” ofarc). Corresponding points on the two CRTswere alternately refreshed with an interval of approximately 16 ,usecbetween successivepulses from the D/A interface. With 10 pointson each CRT, the interval between successive refreshes of thesamepoint was about 320 psec (= 16 psec x 2 x 10). When these pat-terns were moved, their positions were changed every 18 msec.The forms were rotated through an angle of 90°,±45°from thevertical midline ofthe display, aroundapoint midway betweenthecenter and the bottom edge of the ellipse. The formsrotated backand forth three times with an angular velocityof 135°/secfor 2 sec.occupying 38 separate positions over the 90°range of rotation.

The shapes to be discriminatedwere threeellipses definedby 10dots equally spaced around the perimeter. When the forms weredisplayed in the frontal-parallel plane, the vertical axis was .94°,1.0°,or 1.060 of arc, and the horizontal axis was always 1.0°ofarc. Eachofthese three shapesoccurred equally often in each blockof trials.

Conditions. In 8 of the 10 experimental conditions, the ellipseswere presented in aplane slanted in depth from the frontal-parallelplane by 40°,50°,or 60°aroundthe horizontal axis. Each ofthesethree alternative slants appeared equally often in arandom sequencewithin each block of trials. In the other two conditions, the formwas always in thefrontal-parallel plane. Figure 2 shows themonocu-lar projections ofthe threeellipses in the frontal-parallel plane andat the three values of slant.

In two conditions, the binocular disparities were displayed as ifthe observer’s viewing distance was much closer than the actual114.6 cmby an amount that varied randomly between trials—l2.6,18.9, or 25.2 cm. The width of the forms on the screen was heldconstant, independently ofthe simulated viewing distance. Thesevarying viewing distances magnified the stereoscopic depth by anamount equal to the ratio of actual to simulated viewing distance.Theamount of stereoscopic magnificationalso depended on theslantof theform, the predicted elongation being greatest for the60°slantand 12.6-cm simulated viewing distance. This stereoscopic mag-nification would have lengthened the vertical axis of the circularshape from an undistorted value of 2.0 cm (measured in the slantedplane ofthe form) to a minimum value of 6.04 cm with a 25.2-cmsimulated viewing distance and 40°slant, and a maximum valueof 15.78 cm with the 12.6-cm viewing distance and 60°slant. Aschematic illustration ofthis viewing situation is shownin Figure 3.

These two distorted-disparity conditions differed according towhether the forms were moving or stationary. In two correspond-

ing conditions with no stereoscopic distortion, the simulated view-ing distance was constant at thecorrectvalue of 114.6 cm. In fourcorresponding nonstereoscopic conditions, the monocular patternson each eye were identical—that is, they had no binocular dispar-ity. In two ofthese nonstereoscopic conditions, thesimulated view-ing distances were also shortened and variable, just as above,although these variations in viewing distance had only a negligibleeffect on the projected monocular patterns.

Thus, the principal experimental comparisons involved the ef-fects of stereoscopic distortion and motion on form discriminations.Ifvisual informationabout spatial structure in depthcan be obtainedfrom the congruence of moving forms, independently ofthe specificdisparities, then discriminations of the moving forms should be un-affected by thevarying distortions of stereoscopic space. Discrimi-nations among the stationary forms, however, should be seriouslyhindered by the stereoscopic distortions. Comparisons involvingthe other four nonstereoscopic conditions were included for con-trol purposes. Two additional frontal-parallel control conditionsprovided evaluations of shape discriminations of 2-D forms thatwere either moving or stationary.

Procedure. The fiveobservers includedthe 2 authors plus 3 paidvolunteers who were undergraduate students in psychology at Van-derbilt. All 5 had normal or corrected-to-normal visual acuity; allhad some understanding ofthe purpose ofthe experiment; all hadparticipated in at least two practice sessions prior to theexperimentaldatacollection—until form discrimination had reached asymptoticaccuracy under the condition in which the forms were moving,slanted in depth, and stereoscopic with correct and constant dis-parities.

Eachofthe observers participated in eight experimental sessions.Eachexperimental session consisted of six blocks of27 trials. Thefirst block in each session was a practice block in which the formswere moving, stereoscopic with correct and constant disparities,andslanted by 40°,50°,or 60°on each trial. In thenext fourblocksof each session, the forms were always either moving or station-ary, with order of these two conditions counterbalanced over ses-sions andoverobservers. Theconditions differed among these fourtrial blocks, involving the4 conditions with and without binoculardisparity and with simulated viewing distances either correct andconstant or shortened and variable (12.6, 18.9, or 25.2 cm, ran-domly varying between trials). The ordering of these 4 conditionswithin sessions was also counterbalanced over sessions and ob-servers. In thefinal blockof trials in each session, the forms werealwayspresented in the frontal-parallel plane. In these frontal-parallelcontrol conditions, the formswere eithermoving or stationary, con-sistent with the preceding four blocks of trials. Thus, each of the5 observers contributed data from 108 trials for each of the 10 ex-perimental conditions.

Theobserver’s task in all these conditions was the same—to iden-tilS’ whichoneof the three alternative shapes waspresented on eachtrial. Eachtrial was initiated by the observerby depressing a switchon a keypad. The pattern was displayed for 2 sec. The observerrespondedby pressing one of three alternative keys. The responsewasfollowed by visual feedback indicating which ofthe three shapeshadappeared on that trial. A fixation point reappeared on the screenwhen the next trial was ready.

Results and DiscussionThe principal results are shown inFigure 4, which gives

the percentage of correct shape-discrimination responsesfor each condition and each observer as well as the com-bined accuracy totaled over all 5 observers. These resultsare easily summarized: Discriminations among movingshapes were very accurate, but discriminations were notreliably above chance when the forms were stationary and


slanted in depth. Variations in either the correctness oreven the presence or absence of binocular disparities hadno reliableeffect on performance. Surprisingly, the mov-ing forms were more accurately discriminated even in thefrontal-parallel plane.

Subjectively, the shapes were more definite and distinctwhen they were moving. Since the orientations of the twoaxes of the ellipses rotated through an angle of 900, therelative lengths of these two axes were visually scaled in-dependently of the space in which they were located atany time—in E3, on the display screen, or on the retina.

0

HORIZONTAL

Such intrinsic optical information about the metric scaleof the form and the space added to the visibility of theshape even when the forms were confined to the frontal-parallel plane. The observer’s knowledge that the formwas not slanted may not have fully resolved inherent visualambiguity about its slant. A metric scale of the retinalcoordinates may be less perceivable than the intrinsic ge-ometry of the optical stimulation.

The failure of stereopsis toprovide accurate shape dis-crimination even when the stationary slanted forms weredisplayed with the correct binocular disparities is interest-

SHAPES

CIRCLE VERTICAL

SLANT

40

50

60

Figure 2. Photographic reproductions of the stationary images of the threeshapes at each of four differentmagnitudes of slant.[From “The perceptionof geometrical structure from congruence,” by J. S. Lappin & T. D. Wasson, 1991, in S. R. Ellis, M. K.Kaiser, & A. J. Grunwald (Eds.), Pictorial communication in virtual and real environments (pp. 425-448). London: Taylor &Francis. Copyright 1991 by Taylor & Francis Ltd. Reprinted by permission.]

92 LAPPIN AND LOVE

Display Screen

114.6cmViewingDistance

Display Screen

25.2cmViewing ~Distance

Figure 3. Schematic illustrations of the apparent shapes of a tilted circle as seenunder two different conditions of stereoscopic projection. (A) The perceived shapeof a circle slanted 40°as seen with the correct disparities at a distance of 114.6 cm.(B) The perceived shape of a circle slanted 400 as seen from a distance of 114.6 cmwith the disparities appropriate for a viewing distance of 25.2 cm.

ing and was not anticipated. Unfortunately, the poor per-formance in this condition violates part of the rationalefor this experiment: Since correctbinocular disparities didnot yield reliable discrimination of the stationary forms,accurate discrimination of the stereoscopically distortedmoving forms cannot be attributed to stereopsis per Se;the role of stereopsis in this shape-discrimination task isambiguous.

Even though the obtained discrimination accuracies donot demonstrate a role of stereopsis in this task, the sub-jective impression was that both stationary and movingstereoscopic forms appeared to be enriched in depth.When the stationary forms were displayed with magni-fied disparities, they did in fact appear abnormally elon-gated in depth. One observer (J.S.L.) did discriminate thecorrectly disparate stationary slanted shapes with an ac-

curacy well abovechance, but the other observers did notsignificantly exceed chanceaccuracy in this baseline con-dition. Even withthe correctbinocular disparities, the pre-cise slant and shape in depth were surprisingly ambigu-ous. (In Experiment 4, a different method was used todemonstrate that variations in binocular disparities do af-fect perceived shape and depth in these patterns.)

The specific geometric properties that observers dis-criminated in this task were also described by a more de-tailed analysis of the stimulus-response correlations. Theoptical patterns in four experimental conditions consistedof the 27 combinations of three shapes, three slants, andthree viewing distances. Although the observers were ex-plicitly instructed to identify the shapes, their responsesmay have been guided by any combination of these threevariables. To quantify the relative influences of these three

A

B

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CR1 Screens

METRIC STRUCTURE FROM MOTION

100 HCS

No Fronto-Stereo Stereo parallel

Control

JSL

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— -Chance

DLM

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40333

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No Frorto- No Fronto- No Fronto-Stereo Stereo parallel

c~,trotStereo Stereo parallel

ControlStereo Stereo parallel

Control

Figure 4. Accuracies of shape discrimination under each of 10 experimental conditions in Experiment 1 for each of 5 observers, andthe corresponding average accuracies combined over the 5 observers.

variables, we calculated the information transmitted (inbits) about each of the variables by each observer’s re-sponses in each of the four relevant conditions. Thesestimulus-response correlations reflect the degree to whichobservers were able to diStinguish variations in shape fromvanationS in slant and disparity.

The results of this analysis of the combined responsesof all 5 observers are shown in Figure 5. As can be seen,when the shapes were moving, the responses were cor-related almost exclusively with shape. (These stimulus—response correlations were very similaramong all the ob-servers except H.C.S., who was less accurate than theother observers and whose discriminations of the mov-ing shapes were influenced by slant about as much as byshape.) When the shapes were stationary, however, theobservers’ responses were influenced more by slant andby disparity than by shape—that is, observers were unableto discriminate variations in stationary shape from vari-ations in slant and disparity. Thus, the optical informa-tion provided by motion was necessary for perceiving theintrinsic geometric shape independently of its slant andthe scale of its stereoscopic disparities.

To summarize, the principal result was the clear andconsistent superiority in discriminating moving as opposedto stationary shapes. Variations in binocular disparitieshad virtually no effect on discrimination of the movingshapes. Rotation in depth was necessary and sufficient foraccurately perceiving metric shape in E3.

EXPERIMENT 2

Reductions in the range of rotation might be expectedto produce concomitant reductions in visible informationabout the relative scaling of distances in the two perpen-dicular directions in the image plane. These effects wereexamined in Experiment 2.

MethodWith stimulus patterns and displays identical to those in the previ-

ous experiment, the range of angular rotation in this experimentwas restricted to either 80°,40°,20°,or 10°.As before, theserotational motions occurred in a plane that was slanted in deptharound the horizontal axis by 40°,50°,or 60°,randomly varyingbetween trials. Therotations were centered at the vertical midlineof the display. The velocity of rotation was constant at 135°/sec.so that the number ofcycles occurring during the 2-sec display du-ration increasedas the rangeof rotation decreased. For each of thesevalues of angular rotation, we also compared conditions in whichstereoscopic forms were displayed with the correct or with incor-rect and variable disparities, using the same values of simulatedviewing distances as in the previous experiment. Thus, there wereeight experimental conditions—four ranges of rotation and two typesof stereoscopic disparity, orthogonally combined.

Three well-trained volunteers served as observers. All 3 hadserved in the previous experiment, and all were familiar with thepurpose ofthe experiment. Each observerparticipated in four ses-sions, each of whichcomprised four blocks of 54 trials. Two differ-ent ranges of rotation were examined in each session, with two ofthe trial blocks devoted to the correct-disparity conditions and twoto the distorted-disparity conditions.

COMBINED

C00C

E(20,

a(2a)

0C-)

93

0

a)0Ca)C)

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SRL100

80

60

41

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DEE

- -CF,onc. -Char,c.

94 LAPPIN AND LOVE

(50 500500’ 0”

Stereo No Stereo

Stat ionany

C

0. ~

:: ~Stereo No Stereo

Figure 5. Amount of information transmitted by the shape-discrimination responses about the three independent stimulusvariables—slant, disparity (as specified by the viewing distanceparameter), and shape—for each of the four conditions in Experi-ment 1 in which these properties were varied. (Under the two non-stereoscopic conditions, the variations in viewing distance had anegligible effect on the projected patterns, and therefore theresponses would not be expected to correlate with this variable.)

ResultsThe results are shown in Figure 6, which gives the

shape-discrimination accuracies for each observer andcondition. As can be seen, discrimination accuracy de-clined systematically with reductions in the range of ro-tation. Performance was similar for both the correct-disparity and the distorted-disparity conditions at all rangesof rotation. Rotations of only 10°were sufficient to pro-duce above-chance shape discriminations, even when thestereoscopic disparities were seriously distorted.

EXPERIMENT 3

Theoretical analyseshave indicated that the metric struc-ture of a rotating form may be determined by two suc-cessive views only when both the form and motion liewithin a single plane (Hoffman & Flinchbaugh, 1982; Ap-pendix). Although Ullman (1979) and others have shownthat full metric information about a rigidly rotating formand the E3 Euclidean space in which it is moving mightinprinciple be determined from three or more views, theresults of Todd and Bressan (1990) and Todd and Nor-man (1991) indicate that vision does not integrate such3-D structure over three or more views. In general, thesuccessive positions of a moving surface may occupy avolume rather than a surface in E3—a 3-D manifold. Ac-

cordingly, the E3 embedding of the images of such mov-ing forms must change over time to preserve the metricstructure of the form. In contrast, the rotating forms inExperiments 1 and 2 occupied only a plane, and the em-bedding of their images into E3 remained constant overtime. Clearly, recovery of the metric shape of a movingobject is computationally much more difficult in the gen-eral case of arbitrary motions of solid objects. In Experi-ment 3, these ideas about the visual information aboutmetric shape were tested by varying the departure of thetrajectory of the rotating ellipse from a plane.

MethodTheindependentvariable in Experiment 3 was the angle between

the ellipse and the axis of rotation. In Experiments 1 and 2, thisangle was always 90°,so that the ellipse rotated in a single planethroughout its trajectory. In Experiment 3, this angle was reduced,yielding a curvilinear trajectory in which the successive positionsof theeffipse were tangent to the surface of an imaginary cone (wherethe vertexcoincidedwith theplane ofthe display screen at the centerof rotation, the base of the cone wasbehind the display screen, andthe axis of rotation, which was the central axis of the cone, wastilted around the horizontal axis ofthe display screen). As the an-gle between the ellipse and the axis of rotation decreased, the cur-vature of the trajectory increased and the variability of the tilt andslant of the ellipse also increased.

Four alternative valuesof this angle were used: 90°(replicatingthe preceding experiments), 80°,70°,and 60°.The value of thisangle parameter was constant throughout each block of trials andwas varied between the four blocks of 27 trials which composedeach experimental session. Within each trial block, the slant of theaxis of rotation was randomly varied among the three alternativevalues, which were 40°,50°,or 600, as in Experiments 1 and 2.The same three alternative ellipses as those employed in Experi-ments 1 and2 were viewed stereoscopicallywith the correctview-ing distance parameter (114.6 cm). As in Experiment 1, the rangeof motion was constant at 900. Other conditions and procedureswere the same as in Experiments 1 and 2.

Three experienced observers who had served in Experiment 1each served for four sessions in Experiment 3. (Experiment 3 wasconductedbeforewhat is described as Experiment 2 in the presentstudy.) There were four blocks of 27 trials in each session, for atotal of 108 trials for each observer in each condition.

ResultsThe results are displayed in Figure 7, which shows the

effect on shape discrimination of the angle between theellipse and axis of rotation. As canbe seen, small amountsof curvature of the surface of rotation produced substan-tial decrements in accuracy. Merely changing the anglebetween the ellipse and axis of rotation from 90°to 80°more than tripled the percentage of errors (from about10% to 36%) by the most accurate observer, D.L.M.,and doubled the total errors of all 3 observers (from about21 % to 42%). Angles of 70°and 60°yielded accuraciesonly slightly above chance (44% and 48%, respectively,for the 3 observers combined). Subjectively, the increasedcurvature of the surface ofrotation simply rendered differ-ences in the three shapes indiscriminable; the shapes andmotions did not appear ambiguous or confusing.

The obtained decrements in shape discrimination pro-ducedby relatively small deviations from planarity of thesurface of rotation indicate that the resulting variability

Comb i

Shapa Discrimination

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Degree of Motion Degree of Motion

00 SRL 100 HCS0’

80 ~80

‘7 /0: ~~oç33.3 Chance 33.3 Chance

20 2(I I I I I I

0° 20° ‘~0° 80° 10° 20° 40° 80°Degree of Motion Degree of Motion

Figure 6. The accuracy of shape discrimination in Experiment 2 as afunctionof the rangeof motion and stereoscopicdistortion for each of the 3 observers and averaged over observers.

100 - SRL . RCS . DLM

80*Ua,

~-— 60-0~

-

CU

333Chance

20~ __-___

0 90 8070 60 90807060 908070 60

Angle Between Axis of Rotation andEllipse (deg)

Figure 7. Theaccuracyof shape discrimination as afunction of the angle be-tweenthe axis of rotation and the planar forms for each of the three observersin Experiment 3. The deviation from planarity of the rotating form increasedas the angle decreased from 90°.

96 LAPPIN AND LOVE

in the metric tensors of the image motions significantlyreduced the visual information about the intrinsic struc-ture of the moving shapes. Obviously, all motions are notequal as sources of information about the invariant shapeof the moving object.

EXPERIMENT 4

A surprising result of Experiments 1 and 2 was that thecorrectness, variability, or evenpresence of stereoscopicdisparity had no reliable effect on discriminations ofeithermoving or stationary shapes. When the shapes were sta-tionary, discriminations were usually near chance evenwith the correctbinocular disparities. One explanation forthis poverty of shape from stereopsis may be an inherentambiguity of stereoscopic information about the scale ofdepth, as many investigators have suggested. Even thoughstereoacuity for detecting a difference in depth exhibitsexquisite sensitivity, the scale of extension in depth maybe poorly resolved. In any case, the discrimination per-formance in Experiments 1-3 provided no direct evidenceabout the role of stereopsis in shape discrimination, Con-sequently, the role of motion in stereoscopic depth con-stancy remains ambiguous.

To demonstrate that binocular disparity did have a visi-ble effect on perceived depths and shapes of these pat-terns, we evaluated discriminations of the viewing dis-tance parameter for which the disparities were calculated.If these differences in disparity scaledo alter the perceivedshapes, they should be easily discriminated, at least whenthe forms are stationary. If the planar forms are rotatedwithin the plane, however, their congruence under mo-tion may calibrate the stereoscopic embedding into E3,thereby destroying information about the disparity scaleper se.

If these assumptions about the roles of stereopsis andmotion in shape perception are correct, motion shouldhave opposite effects on discriminations of shape and dis-parity. As found in the preceding experiments, movingshapes should be much more easily discriminated than sta-tionary shapes, but variations indisparity may be discrimi-nated better when the patterns are stationary. These fourconditions—discriminations of shape or disparity, withmoving or stationary patterns—constituted the main partof Experiment 4. As an additional control, the same fourconditions were replicated under nonstereoscopic condi-tions. (The simulated viewing distance parameter couldbe manipulated independently of whether there was anydisparity between the two monocular displays, and mightconceivably be discriminated by very slight differencesin perspective.)

MethodSeveral minor changes were made in the displays for Experi-

ment 4. The number of alternative shapes, disparities, and slantswas reduced to two from the three alternatives used in the preced-ing experiments. The purpose was to improve both the spatial dis-criminations and experimental efficiency.

The two alternative shapeswere ellipses, whose vertical axes wereeither 3% greater than or 3% less than the horizontal axis. Two

alternative values of the stereoscopic disparity parameter corre-sponded to viewing distances of either one halfor one fourth theactual 114.6-cm distance, thereby multiplying the disparities by afactor of two or four times the correct values. The two alternativeslants were 50°and 60°.Thus, eight equally likely alternative pat-terns were composed from these three binary variables. A schematicillustration ofthe monocular projectionsof the two ellipses at eachof the two slants is shown in Figure 8.

An additional minor change was to construct the ellipses from20 points equally spaced around theperimeter, thereby describingthe contours more accurately than in the preceding experiments.These forms rotated in the plane through an angle of 90°,as inExperiments 1 and 3. In other respects, the optical patterns werelike those in the preceding experiments.

Four well-practiced observers each served for six experimentalsessions. All had good stereoacuity as verified by other measures.All of the observers were knowledgeable about the purpose of theexperiment. Because observer J.P.K. was less accurate in dis-criminating the two alternativeshapes, the difference in the verti-cal axes of the two ellipses was increased to 8% (±4%)rather thanthe 6% difference used for the other observers.

Each session was devoted to discriminations of either shape ordisparity. Each session consisted of four blocks of 48 trials, witheach block containing40 experimental trials preceded by 8 prac-tice trials. The eight alternative forms occurred equally often ina random sequence in each block. The four conditions defined bythe moving versus stationary and stereoscopic versus nonstereo-scopic conditions were varied betweenthe four trial blocks in eachsession in counterbalanced order for the 4 observers. This provideda total of 120 trials for each observer in each of the eight conditions.

In other respects, the methods and procedures were the same asin the preceding experiments.

ResultsThe average accuracy of shape discriminations in each

of the eight conditions and the corresponding results for

A B

D

Figure 8. Schematic but accurately scaled illustrations of theprojected optical images of the two effipses at the two slants usedin Experiment 4. Each section shows the same ellipse in two differ-ent positions on the same plane—in the central position, and rotatedcounterclockwise by 45°. (A) The larger ellipse (1.03 vertical!horizontal aspect ratio) at 50°slant. (B) The larger ellipse at 60°slant. (C) The smaller ellipse (0.97 vertical/horizontal aspect ratio)at 500 slant. (D) The smaller ellipse at 60°slant. The forms actu-ally used in Experiment 4 were defined by 20 dots equally spacedaround the perimeter; the central point and vertical line were ad-ded to these illustrations simply to indicate the positions of the twoforms.


Combined100 Stereo

CombinedNo Stereo

90

80

70

60~

50

40Disparity’ Shape

Discrimination

JPKStereo

GDLStereo

Disparity ShapeDiscrimination

JSLStereo


JSLNo Stereo


JFNStereo


JFNNo Stereo

Figure 9. Discrimination accuracies for each observer in each condition of Experiment 4, and the average accuracies in each conditioncombined over observers.

each of the 4 observers are shown in Figure 9. The pri-mary results are given in the upper half of Figure 9, show-ing the opposite effects of motion on discriminations ofshape and disparity: As in the preceding experiments,moving shapes were accurately discriminated (87.5% cor-rect) despite variations in slant and disparity, but they wereessentially indiscriminable (52.9% correct) whenstation-ary. In the disparity discrimination task, however, mo-tion had the oppositeeffect, with the disparities of movingforms less accurately discriminated than those of station-ary forms—68.8% versus 80.0%, a statistically reliabledifference by chi-square test [x2(l) = 15.35, p < .0011.Disparity discriminations of the moving forms were wellabove chance, however.

Although disparities of the moving forms were discrimi-nated less accurately by all 4 observers, there was con-siderable variability between the observers in both ac-curacy of disparity discrimination and in the detrimentaleffect of motion, as can be seenin Figure 9. The clearesteffect was obtained for J.S.L., whose accuracy droppedfrom 96.7% in the stationary condition to 70.0% in themoving condition. The difference between these two con-ditions was statistically significant only for J.S.L., though

the chi-square value totaled over the 4 observers wasstatistically reliable [x2(4) = 33.86, p < .001J. Evi-dently, observers differ in their sensitivities to the mag-nitude of stereoscopic disparity.

Results for the nonstereoscopic conditions show thatmotion yielded shape-discrimination accuracies very simi-lar to those for the stereoscopic conditions. Stationaryshapes, however, were more accurately discriminated inthe nonstereoscopic than in the stereoscopic conditions—62.3% versus 52.9%—a difference that was statisticallyreliable by chi-square test [f(l) = 8.26,p < .011, basedon the combined responses totaled over observers. Ac-curacy in discriminating the stationary stereoscopic shapeswas not significantly above chance, however [x2(i) =

1.52, p > .2]. When the same tests were applied to thedata of each individual observer, only J.P.K. was signifi-cantly more accurate in the nonstereoscopic than in thestereoscopic discriminations, and the chi-square value to-taled over the 4 observers was not quite significant at a =

.05 [x2(4) = 8.93, p < .10]. Three observers were sig-nificantly above chance in discriminating the non-stereoscopic stationary forms, however, and the chi-square totaled over the 4 observers was highly signifi-

Stationary

ShapeDiscrimination

5,

0c°)5,

as

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C-I5,I-I-0L)5,

as05,5,

5,

JPKNo Stereo

GDLNo Stereo

Discrimination

98 LAPPIN AND LOVE

cant [~(4)= 33.17, p < .001]; but none of the observerswas significantly above chance in discriminating thestereoscopic stationary shapes, and the total chi-squaredid not approach significance. Thus, the varying magnifi-cations of the binocular disparities seem to have producedvarying distortions of the perceived shapes. This resultcomplements the obtained discriminations of disparitiesof both moving and stationary stereoscopic fonns inshow-ing that the binocular disparities affected the perceivedshapes. Evidently, the ambiguities inherent in this stereo-scopic information were bypassed by the intrinsic metricstructure of the moving patterns.

GENERAL DISCUSSION

The principal result of these experiments is that planarmotion provided an intrinsic metric scalefor retinal im-ages ofplanarfor,ns in 3-D space, independently of theform’s slant in depth and binocular disparities—that is,independently of the extrinsic scales of the display screenor retinae. Evidently, the perceived metric structure wasvisually definedby the congruence of the spaces and fonnsunder motion.

The precision of shape discrimination obtained in theseexperiments, with a Weber fraction in Experiment 4 ofless than 3 % (in the relative lengths of the two major axesof the ellipses), is similar to the precision obtained in otherrecent experiments on discriminations of relative positionsin stationary line segments in the frontal-parallel plane(DeValois, Lakshminarayanan, Nygaard, Schlussel, &Sladky, 1990; Levi, Klein, & Yap, 1988). The precisionof these discriminations of relative length is also similarto that obtained by Lappin and Fuqua (1983) for the rela-tive positions of three points along a single line segmentrotating in depth. The similarity of these relative lengthdiscriminations in 3-D and 2-D is remarkable. Evidently,vision is sensitive to spatial relations in the 3-D space ofthe environment rather than the 2-D space of the retina—spatial relations that remain invariant under motion.

The perception of metric structurewas achieved in theseexperiments through the “planarity constraint” (Hoffman& Flinchbaugh, 1982). As shown by Hoffman and Flinch-baugh and in the present Appendix, metric structure istheoretically recoverable from two views of a KD pat-tern when a planarform rotates within the plane. The pres-ent experiments demonstrate that human observers are infact sensitive to this optical information. In contrast, Toddand Bressan (1990), Todd and Norman (1991), and thepresent Experiment 3 demonstrate that metric structureis not perceived by human observers under a variety ofconditions that violate the planarity constraint. Whetherplanarity is a strictly necessary condition for perceivingmetric structure is uncertain. Ullman’s (1979) “structure-from-motion theorem” shows that metric structure mightin principle be recovered under much more general con-ditions involving three or more views (see also Bennettet al., 1989; Huang & Lee, 1989), buthuman observershave not yet been shown to be sensitive to this theoreti-cally available information.

Planarity of the structure and motion imposes a strongand simple constraint on the optical patterns: Orthographicprojection of the slanted plane merely alters the relativescales of two orthogonal axes of the projective plane. Aslant of 600~ for example, simply reduces the scale per-pendicular to the axis of rotation to one half that of theaxis of rotation—an affine transformation. When a pla-nar form is rotated within this rescaled plane, it is read-ily perceived as rotating rigidly in a slanted plane ratherthan as deforming in the image plane. This phenomenonis easily demonstrated ina computer-graphics displaybyfirst changing the relative scales of the horizontal and ver-tical axes of the display and then rotating a form in thisrescaled plane.4 Stereoscopic displays like those used inthe present experiments are unnecessary.

Stereopsis proved a surprisingly unreliable source ofinformation about the geometric shapes in these experi-ments. In Experiment 1, only 2 of the 5 observers weresignificantly better than chance at discriminating the cor-rectly disparate stationary ellipses, and the average forthese 2 observers was only 49% correct (33% was ex-pected by chance). Compared with stereoacuity for de-tecting binocular disparities of 10” of arc or less, thisimprecision in measuring depth seems surprising.

Such imprecision of stereopsis in measuring depth hasalso been found in two other recent studies: Nawrot (1991)recently found that observers with excellent stereoacuitywere very poor in discriminating which one of two KDpatterns (rotating random-dot spheres) was displayedstereoscopically, even after training withfeedback. Theyattribute this failure of stereoscopic detection to commo-nality of the neural mechanisms for detecting depth fromeither stereopsis or motion parallax, a hypothesis sup-ported by other recent experiments on the visual integra-tion of these two sources of depth information (Nawrot& Blake, 1989, 1991). McKee, Levi, and Bowne (1990)have found that Weber fractions for detecting incrementsto a standing disparity (of 1 ‘-20’ of arc) were about twoto five times greater than monocular acuities for detect-ing increased horizontal separations in the same patterns.McKee et al. attribute this imprecision to an absence ofbinocular neural mechanisms sensitive to the separationbetween nonoverlapping distributions of neural excitation.But this hypothesis would not seem toexplain the absenceof stereoscopic sensitivity either in the experiment ofNawrot and Blake or in the present Experiment 1, inwhich the optical patterns were dense distributions of dotsportraying a more nearlycontinuous gradient indepth thanin the two-feature patterns of McKee et al. Perhaps muchof the imprecision of stereoscopic depth results from thefact that two disparate views, from either stereopsis ormotion, provide information about spatial structure onlyup to an affine transformation indepth, as shown by Koen-derink and van Doorn (1991), Todd and Bressan (1990),and Todd and Norman (1991).

Compared to stereopsis, motion was a much more ef-fective source of information about metric structure indepth. Three different characteristics ofthese moving pat-terns may have provided this advantage: (1) planarity of


the forms and motions, (2) a larger range of angular ro-tation, and (3) a larger number of views. All threeof thesecharacteristics were probably at least indirectly impor-tant to the perception of metric structure. First, as shownby Hoffman and Flinchbaugh (1982) and in the Appen-dix, two orthographic views of a planar form rotatedwithin that plane differ simply by an affine transforma-tion in the image plane—a two-parameter transformation(corresponding to the slant of the plane and the angle ofrotation) determined by congruence of the form under ro-tation. Two stereoscopic half-images, however, differ bya perspective rather than an affine transformation, de-scribed by additional free parameters associated with the3-D location of the fixation point. Thus, stereopsis ad-mits a family of potential structures that differ from eachother by nonaffine transformations of the images.

The planar motions of forms in these experiments alsoprovided more optical information than did the station-ary stereoscopic patterns, because both the range of ro-tation and the number of views were larger. At a view-ing distance of 114.6 cm, the interocular separation of6.3 cm corresponded to a rotation of only 3.2°aroundthe center of the display screen. In contrast, the range ofrotation was 90°for most of the moving patterns. Theresults of Experiment 2 indicate that a rotation of only30 would have provided insufficient visual informationabout metric shape.

Evidently, vision integrated spatial information oversubstantially more than two successive views of these pla-nar forms and motions. This result contrasts with thoseof Todd and Bressan (1990) and Todd and Norman(1991), who found that vision was insensitive to spatialrelations over more than two successive views. In Toddand Bressan (1990) and Todd and Norman’s (1991) KDEpatterns, the depths at given image positions changed overtime, but the image-to-depth embedding remained con-stant in the present experiments. The temporal integra-tion of spatial structure in the present experiments prob-ably resulted from the temporal consistency and spatialsimplicity of the planarity constraint.

The present results might suggest that stereopsis wassimply insensitive to moving patterns in which the dis-parities of given features changed over time. Indeed, ex-perimental evidence demonstrates that vision is very slowin detecting changes in the binocular disparity of a givenfeature moving indepth only in the direction of view (e.g.,Tyler, 1975; White & Odom, 1985). But for motion inother directions, where temporal variations in disparityare correlated with spatial directions in the cyclopean im-age, stereopsis is exquisitely sensitive to changing dis-parities (Lappin, Craft, & Payne, 1986; Lappin, Love,Cook, & Norman, 1992), with no apparent decrement insensitivity produced by such motion in depth. There isno evidence and little reason to believe that stereopsis wassilenced by the motion of these forms.

One of the questions raised by the results of this studyis whether the perception of metric structure is restrictedto image motions that are congruent under affine trans-

formations, or whether metric structure can be perceivedas well from congruence under perspective transforma-tions. The theoretical results of Hoffman and Flinchbaugh(1982) and in the present Appendix show that metric struc-ture is obtainable from affine transformations associatedwith orthographic views of planar motion; and the presentexperimental results demonstrate that such motions in themonocularhalf-images are visually sufficient to perceivemetric structure. But the stereoscopic relation between thetwo half-images involves a perspective transformation,geometrically and computationally more complex than af-fine, requiring estimation of additional parameters toob-tain congruence. Insofar as the perceived spatial struc-ture of these patterns may have involved the binocularlyintegrated stereoscopic images rather than the monocu-lar half-images alone, stereopsis must have detected theperspective congruence of these two sets of half-images.The exact contribution of stereopsis indiscriminating thesemoving shapes is ambiguous, but the obtained discrimi-nations of shapes with magnified binocular disparities sug-gest that the congruent metric structure of these patternswas stereoscopically visible. In fact, the moving patternswith magnified disparities in Experiments 1 and 4 seldomappeared distorted. One of the observers mentioned thatdistortions were sometimes seen in patterns with thegreatest disparity magnification, but typically the patternsappeared quite normal. Similarly, the results of Lappinand Fuqua (1983) and recent results of Norman (1990)also indicate that vision is adept at detecting congruentstructure of moving patterns with exaggerated perspec-tive projections. More experimental and theoretical workis needed on this issue, but vision appears to detect met-ric structure from congruence under perspective transfor-mations. Theoretical analysis in the Appendix indicatesthat this is computationally tractable.

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NOTES

1. The term metric is used in its conventional mathematical sense.A relation m(a,b) is said to be metric if it satisfies the following axiomsfor all points a, b, c: positivity, m(a,b) ~ 0; symmetry, m(a,b) =

m(b,a); reflexivity, m(a,a) 0; and triangle inequality, m(a,c) m(a,b)+ m(b,c). The term metric structure is used in this paper to refer toan object or connected set of points within which the spatial relationsamong points satisfy the metric axioms and remain invariant (isomet-ric, congruent) under arbitrary motions within the space. For spacesof two, three, or any odd number of dimensions, such isometry impliesthat the space is Eudidean, hyperbolic, orspherical (cf. Suppes, Krantz,Luce, & Tversky, 1989, chap. 12).

2. Affine spatial relations are those which remain invariant under ar-bitrary linear transformations of the Cartesian coordinates. An affmetransformation t(x) of a vector x in an n-dimensional vector space hasthe form t(x) = A(x) + b, where A is a nonsingular n X n matrix andb is an n-dimensional vector. Todd and Bressan (1990) and Todd andNorman (1991) have proposed that visually perceived structure frommotion is invariant under a specific subgroup of affine transformations—underarbitrary scalar transformations ofdistances in depth in the directionof gaze. The term affine structure in the present paper also usually ap-plies to this special case of invariance up to arbitrary scalar transfor-mations ofdistances in depth relative to distances in the frontal-parallelplane.


3. Congruence is synonymous with isometty, referring to a mappingof onesetof points onto another that preserves metric relations amongall pairs of points. The groupof congruencies is slightly more generalthan the group of rigid motions. In Euclidean space, congruent trans-formations include both rigid motions and bendings. Whereas “rigidmotions” usually imply Eucidean space, congruent transformations arealso definable in hyperbolic andspherical geometric spaces. Obviously,the projected images of objects rotating in E3 are not themselves con-gruent; thecongruence is implicit, dependent on an embedding of theimages into a space in which the congruence holds.

4. Symmetric forms such as the ellipses used in these experimentsmust be rotated arounda point that is not the center of theform. Dotsor other discrete position markers must also be used instead of smoothcontours, to avoid ambiguities about the local direction of motion.

APPENDIX

The purpose of this Appendix is twofold. First, we reviewbriefly a theoretical framework and notational system for describ-ing the optical information and perception of the moving spa-tial patterns in these experiments. This theoretical frameworkhas been presented in more detail elsewhere (Lappin, 1990,1991; Lappin & Wason, 1991). Second, we prove that the met-ric structure of planar forms rotating ina plane is theoreticallydetermined by two views. This result and the required planar-ity ofthe form and motion were not given in the previously pub-lished presentations of this theory. This result is similar to thatgiven earlier by Hoffman and Flinchbaugh (1982), although thepresent result is a bit more general—applying to arbitrary reti-nal coordinates as well as Cartesiancoordinates, and applicableto perspective as well as orthographic projections—and the proofis different and in some respects simpler.

PR0POSrrION 1. Optical patterns on the retina may be regardedas images of smooth surfaces.

The insight that the geometry of vision may be simplified bytreating retinal patterns as images of environmental surfaces hasbeen recognized and developed most clearly by Koenderink andvan Doom (e.g., 1975, 1976a, 1976b, 1976c, 1977; see alsoKoenderink, 1986). Related ideas were also discussed by Gib-son (1950, 1966).

“Smooth” surfaces are differentiable almost everywhere—except at isolated local discontinuities corresponding to sharppeaks and corners and at self-occluding and boundarycontours.Patternsof discrete features (e.g., dots) that can be simply con-nected by a smooth surface may also be regarded as constitut-ing a smooth surface (cf. Pollard, Porrill, Mayhew, & Frisby,1985).

Both environmental surfaces and their images constitute smooth2-D manifolds. Moreover, there is a close correspondence be-tweenthe differential structures of these two manifolds: Imagesdefined by motion parallax, binoculardisparity, and texture arediffeomorphic with the visible regions of the corresponding en-vironmental surfaces. That is, there is a smoothly changing one-to-one mapping of the spatial derivatives of the visible regionsof an environmental surface (its gradients, curvatures, singular-ities, and critical points) onto the spatial derivatives of the im-age of thesurface; andthe inverse mapping from the image ontothe surface is also smooth, one-to-one, and onto. (This dif-feomeorphismdoes not hold for images defmed by illuminanceand shading, but a systematic correspondence between thedifferential structures of the two manifolds still exists; cf. Koen-derink & van Doom, 1980.)

It follows from Proposition 1 that the optical projection froma surfaceonto its image may be locally approximated by alinearcoordinatetransformation. Thus, let the 2 X 1 column vector [dO]

= [do1

, do2lT represent an infmitesimal displacement within alocal surfacepatch (sufficiently small that it contains negligiblecurvature) on the object surface; and let [dR] = [dr’, drhlr bethe corresponding image of this vector in the retinal image ofthe objectsurface. Then the map of any such displacement withinthat surface patch onto its retinal image may be described bythe following linear transformation:

[dR] = V[dO],

where V is a 2 x 2 Jacobian matrix of partial derivatives,

ar1/8o

1i9rVc3o

2

V=ar

2/ao

1ar

2/3o

2

(Al)

V simply describes the retinal image of a surface patch; itrepresents the local structure of the image itself, and need notbe computed or estimated. The four entries in this matrix varyas a smooth function of the position on the surface, changingwith the orientation and distance of the object surface relativeto the observer’s retina. Of course, the inverse map from theimage onto the surface is also described by a linear coordinatetransformation:

[dO] = V-1[dR].

An important consequence of this representation of the opti-cal images is that the metric structure of a local surface patchis also given by a simple quadratic function of these local im-age data. The metric structure of the local surface patch is de-termined by just threeunknown parameters that are coefficientsof the quadratic terms. This equation for the metric structureof asurface patch is givenby a basic formula in differential ge-ometry known as the firstfundamental form of the surface. Inthe present application, this formula may be expressed in termsof the retinal image dataas follows. Let [dR] be the retinal im-age coordinates for an infinitesimal displacement on the objectsurface; and let [dX} = [dx

1, dx

2, dxi be the description of

this same vector in the three orthogonalcoordinate axes of E3.Then

[dXl = P[dR], (A2)

where P = [3x”fôr”] is a 3 x 2 Jacobian matrix that specifiestheembedding ofthe retinal imageof a local surface patch intothe three orthogonal coordinate axes of E3. The metric struc-ture of this image of the surface patch is then obtained fromthe Pythagorean formula for distance:

ds2

=

= [dJ~]TpTp[cjJ~]

= [cIR1Tp*[~jR]

= [dO]TVTP*V[dO], (A3)

where P* = pTp is a s~mmetric 2 x2 matrix with entries P’th

= ~k(ax~~/ora)((9xtvar ). P” constitutes the metric tensor forthe retinal coordinates of the local surface patch. Since PIS =

Psi, this matrix contains three independent parameters.It should be emphasized that the metric tensor is independent

of theorientation of the surface patch; the metric tensor is speci-fied by just three parameters, ~*, whereas six parameters, theentries of P, are required to describe the orientation and depthat each position on the surface. Theoretical analyses of theproblem of perceiving 3-D structure haveusually assumed that

102 LAPPIN AND LOVE

surface structure must be derived from a depth map, but theproblem is actually simpler. Accurate perception of surface shapedoes not require accurate perception of orientation or depth.

PsoPosmoN 2. The metric structure of visually perceivedsur-faces and spaces derives from congruence under motion.

As given in the text, the logic of this idea was nicely statedby Killing (1892). The idea is related to the “rigidity constraint”employed by Uliman (1979) and others to infer the 3-D depthcoordinates of a rotating object from the 2-Dcoordinate valuesfrom three or more 2-D images of at least four noncoplanarpoints, though the present approach does not assume that suchcoordinate systems with an implicit metric structure have ana priori definition in either the 3-D space of the environmentor the 2-D space of the retina. The presentdevelopment derivesthe metric structure ofthe objects, images, and spaces from theircongruence under motion. A similar approach with generalizednonmetric coordinates was also discussed by Koenderink andvan Doom (1977).

The requirement that the metric structure of the image of aform remains constant over time may be formulated as an equal-ity of the metric tensor of a surface patch under motion of theobject in E3. Thus, let V represent the retinal image of a sur-face patch in one temporal frame, and let U represent the reti-nal image of the same surface patch at a subsequent momentin time, after the object and its image have moved; let ~* bethe metric tensor ofthe first image, and let Q* be the correspond-ing metric tensor for the second image. The requirement thatthe object remain congruent under motion in E3 implies that

VTP*V = UTQ*U = G, (A4)

where G is a symmetric 2 X 2 matrix containing the three met-ric tensor coefficients of the first fundamental form of the sur-face patch. In general, this equation does not yield solutions forthe unknown metric tensor parameters in the matrices ~* andQ*, since there are six unknowns but only three independentequations. Under the planarity constraint, however, the situa-tion is much simpler, because the projective mapping from theobject’s plane of motion in E3 onto the image plane remains con-stant over time. Hence, the image coordinates may be constructedin such a way that the metric embedding, ~*, of the image ofa given surface patch remains constant as it moves from oneposition to another in E3 and in the image. For example, if theplane in which the form is moving is described with rectilinearCartesian coordinates, the retinal image coordinates could bejust the projection of this Cartesian coordinate system. In thiscase, aspatially invariant Pythagorean formula for distance canapply to any position in the object plane or to any position inthe image plane. Equation A4 then simplifies to

VTP*V = UTP*U. (A5)

Although the construction of such a retinal coordinate systemwith auniform metric embedding may pose a significant com-putational problem in the general case, this turns outto be veryeasy for the case of orthographic projections of a plane. We willreturn below to the case of perspective projections of a plane.

Matrix Equation AS may also be written in the form

— I lab\1_ I labg

11— a b[P°’l~i~’i)j— a btP~~~lUilki

where v7 = ar’~/ao’and u~= 3s°/äo’,Pab are thequadratic metrictensor coefficients given above for the retinal coordinates ofthesurface patch, and the terms r” and

5a represent two different

magnitudes of displacement on retinal coordinate axis a. Sinceg12

= g21

, there are three independent equations in three un-knowns, Pu, P22, and p

12= P21. For the general case in which

Pab are three independentcoefficients, these equations have onlythe trivial solution Pu = P22 = P12 = 0, but for images of aplane p12 is a function ofp11 and P22. Without loss of general-ity, we may define r

1and r

2 as two orthogonal coordinate axes,so that for orthographic projections of a planep

12= 0. We then

obtain the following three solutions for the aspect ratio of theseparameters that embed the two retinal coordinate axes into E3:

Pu1/P22 = [(vi)2_(ui)2] I [(u~—(v~)~]

= [(v1)2 — (uD2] / [(ui)2 — (vl)2]

= [v~vi—u~ui]/ [ulul—v~v1]. (A6)

The terms on the right are ratios of differences in second-degreemagnitudes of two successive images ofa given local vectoronthe object surface, the numerators are differences in the projectedmagnitudes on retinal coordinate axis 2 (v~v~—u~u),and thedenominators are corresponding differences on coordinate axis 1(uuJ—vvJ). Thus, the aspect ratio of the metric tensor coeffi-cients for the twoorthogonal retinal coordinates, P1u/P22, is sim-

ply scaled to maintain invariant measures of object structure andmotion over varying directions and positions on the retinal coor-dinates. Under the constraints of planarity and orthographicprojection, both the image motion and this aspect ratio of theretinal coordinates constitute affine transformations, and the com-position of these transformations is also merely an affinetrans-formation.

The critical step in this proof is the assumption that the met-ric tensor for the image of a given object surface patch, P”,remains invariant under motion. Although such image coor-dinates are easily constructed for orthographic projection of aplane, this requirement is less easily satisfied for perspectiveprojections, in which the image motions involve nonaffine trans-formations. Additional parameters are required to specify theperspectiveprojection of a plane, corresponding to the locationof a vanishing point in E3. Obviously, additional image pointsand/or views would be needed to determine these perspectiveparameters. In fact, four points, no three of which are collinear,are necessary and sufficient to specify the mapping from oneperspective image of a planar form onto another, according tothe “fundamental theoremof plane perspectivity” (cf. Delone,1963). Although we cannot at present write down equations withwhich to compute the parameters for these perspective trans-formations, such perspective transformations are clearly com-putable from images of planar motions of planar forms. More-over, the evidence of Lappin and Fuqua (1983) and Norman(1990) indicates that vision achieves such computations at leastfrom patterns with rotational motions of about 90°.

(Manuscript received April 1, 1991;revision accepted for publication September 3, 1991.)= 0,

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