Picture surface illusion in parallel perspective

In the picture-surface illusion, 2D features on the picture’s surface are seen biased towards their 3D referent e.g. an angle of 200 depicting a 900 corner of a cube is seen as 300. We tested linear and parallel perspective drawings of cubes with cube drawings subtending 50 to 500. The picture-surface illusion occurs for both parallel and linear perspective drawings, at about equal strength. We argue that the 3D information at work may be the aspect ratio of a quadrilateral depicting the tilt of the cube’s top face.

In Figure 1, a 3D cube is shown, top tilted towards the viewer at angles from 5° to 85° to the picture surface, 9 drawings on the left in linear perspective and 9 on the right in parallel perspective.In each case, three receding sides are shown, around a central Y vertex.In linear perspective, pairs of lines on the picture surface converge, but depict parallels in the 3D cube.That is, the lines on the 2D picture surface take this shape / \ but they stand for parallels in the 3D object.In parallel perspective, parallels like "I I" on the 2D surface mean parallels in 3D.
In the 9 drawings on the left side of Figure 1, the top face of a cube is being shown in linear perspective by quadrilaterals that are symmetrical about the vertical, but the lines for the sides converge to left and right, up the page.The flanks of the depicted 3D cube are shown by lines that converge downwards.In the 9 parallel perspective renderings, Figure 1 right, the top face is being shown by a parallelogram, all its sides equal.Lines for the flanks remain parallel.Either way, the cube is shown correctly if it subtends a particular angle.Figure 1's linear perspective drawing top left is correct for a cube subtending approximately 5 0 .The parallel perspective drawings are correct for a cube subtending an angle close to zero.Not only are the drawings geometrically correct for 0° but, further, the drawings often appear accurate to viewers if they subtend a small angle such as 5 0 (Nicholls & Kennedy, 1993).
A picture of a cube can be viewed some distance from the observer, subtending a small angle, and then can be brought closer to the viewer with the effect that its subtense increases.In principle the angle subtended can increase from close to zero to close to 180 0 .If the angle subtended increases and becomes greater than is geometrically correct, the optic information being provided is for a block that has far edges that are larger than the near edges.However, for both parallel and linear perspective, everyday vision is quite tolerant of some degree of increase or decrease of angle subtended beyond what is strictly correct (Kubovy, 1986;Bertamini & Parks, 2005).Eventually, if the subtense increases substantially, the percept is often that the far parts are larger than the near, with the flanking sides diverging with distance.This is how vision reacts once the angle subtended by a parallel-perspective drawing increases beyond about 20 0 .A linear perspective drawing correct at 5 0 likely appears to show a cube up to about 30 0 .However, for both linear and parallel perspective, if the 5°-subtense cube-picture is viewed at a 50 0 subtended angle, apparent divergence may result, and many of the depicted angles, such as the ones on either side of the central Y vertex, may appear expanded above 90 0 , and as say 100 0 .If the Figure 1 drawings are seen at large subtenses, the parallel perspective drawing, correct at 0°, should give a more divergent appearance than the linear perspective one correct at 5°.Consider an angle observers judged in our experiment -the "target angle."This is the left angle of the 2D top-face quadrilateral and top-face parallelogram in Figure 1.Geometrically-accurate, both stand for a 90 0 angle at small or vanishingly-small subtended angles, but if the subtended angle increases to 50 0 the depicted 3D angle is greater than 90 0 .Further, the parallel perspective drawing should depict a larger angle than the linear perspective drawing.It has departed further from its correct subtense of 0 0 than the 5 0 -subtense linear-perspective drawing.
In previous studies, we asked viewers to judge 2D target angles in stimuli like Figure 1.They made errors in the direction of the depicted 3D angle, such as deeming a 2D angle of say 25 0 to be 45 0 , an error that we called the picturesurface illusion.The viewer was at the centre of projection (Hammad, Kennedy, Juricevic, & Rajani, 2008a, b;Kennedy & Hammad, in press).The explanation offered by Hammad et al. (2008a, b) is that the participant's use of optical information for any 2D features on the picture surface such as angles, parallel lines and widths between lines suffers cross-talk from the information for a scene in depth that the picture provides (Treisman, 1983;Sedgwick & Nicholls, 1993;Pizlo, 2008;Maniatis, 2009a, b).In particular, the interfering crosstalk comes from information for the tilt in depth of the top of the depicted block.The tilt could be indicated by converging lines and by the aspect ratios of the faces e.g. the more the top surface deviates from being parallel to the picture surface the more foreshortening results in the projected quadrilateral being compressed.It departs from square, which has an aspect ratio of 1.The aspect ratio, given by minor axis divided by major axis, moves towards zero.One might say the "eccentricity" of the form depicting the top surface increases.
At issue in the present paper is whether perception would continue to make this error if it is shown parallel-perspective depictions of cubes.There are three main possibilities: The parallel perspective drawing could offer (1) the same picture-surface illusion as the linear-perspective drawing, (2) little or no error, or (3) increased error.Geometrical features of the cube pictures can be used to argue in favour of each of the possibilities.
Advocating for "same" errors, Maniatis (2009a, in Perception) proposed parallel-perspective depictions (see her Figure 3, a drawing of an object in parallel perspective) would generate notable picture-surface illusions.In her drawing, the lines for the sides of the object, joining the top and bottom of the object, were parallel on the picture surface.Maniatis (2009b, in Spatial Vision) noted that Arnheim (1954, page 267) described an isometric drawing of a cube, that is a cube drawn with sides depicted by parallel lines, appears to the viewer as a "coherent, faithfully portrayed cube."She argues (page 556) "To the degree, then, that the observed bias in angle perception is contingent on the 3-D result, we should expect the same outcomes whether or not convergence is employed in the construction of the stimuli, and regardless of the degree of convergence resulting from a particular projection.Without performing the same tests on forms drawn in isometric perspective, it is not possible to say that they would differ in their outcomes from the experiments reported here [by Hammad et al., 2008a,b] (I think they would not)." An argument for "little or no" error follows from a Hammad et al. (2008a) proposal that the converging lines showing the flanks of the cube in linear perspective might be vital in creating the picture surface illusion.The middle column of the Figure 1 nine left cubes shows convergence quite clearly.The illusion Hammad et al. (2008a, b) reported was strongest for 3D objects displayed at an intermediate tilt, much like these.If the key influence on the illusion emerges from convergence in lines for the flanks, since the parallel perspective drawing lacks converging lines, the illusion should be absent.
However, a reason for rejecting a "no illusion" hypothesis and supporting the "same" illusion conjecture is that key information may have to do with the tilt of the top surface.Tilt results in foreshortening, and the more tilt the more the form depicting the top surface has an aspect ratio far from 1, in both linear and parallel projection.The role of the lines for the flanks may simply be to indicate the object is thick and, crucially, that the uppermost form depicts a top surface of an object.These conditions might allow aspect-ratio information for tilt to become effective.If so, the parallel-perspective rendering could induce the picture surface illusion at full strength.
An additional reason for the "same" illusion hypothesis is that the linear and parallel perspective drawings in Figure 1 are very similar in 2D features on the picture surface, and their differences easily overlooked on casual inspection.A parallel perspective drawing and a drawing correct at 5 0 have many features that are only slightly different physically.They have central Y junctions that offer closely matched obtuse angles.Their other angles are similar.Most notably, their aspect ratios are closely matched, in all three of the quadrilaterals that make them up.Further, the convergence of the flanking lines in the 3 linearperspective cubes on the extreme left row in Figure 1 is minor, but Hammad et al. (2008a) found they produce sizable picture-surface illusions.Convergence of the flanks likely becomes most obvious in the linear perspective drawing in the third row from the left, topmost drawing, but the converging lines are relatively short.Shortness of the lines for the flanks may reduce their influence on the illusion.Hammad et al. (2008a) noticed the illusion was reduced in strength for the three linear-perspective cube drawings on the right in Figure 1.This could be because the aspect ratio of the form depicting the top face in these three drawings is close to 1, but alternatively, the convergence in these drawings of one pair of lines to the top left, and the other pair to the top right, is quite minor, difficult to discern, and perhaps unlikely to have a sizable effect on vision.
Another reason for expecting parallel perspective cubes to offer the illusion would be that observers treat many forms as depicting cubes.Many departures from correct geometrical rendering of a cube may be accepted, as if perception had broad criteria for cubes.That is, if a drawing specifies a vertex with a 65 0 corner, slight departures from the correct subtense may lead to percepts of 70 0 corners, or 60 0 corners, but similar-sized departures from subtenses specifying 90 0 angles may be treated as insignificant by vision.The familiarity of the 90 0 form, or its Gestalt quality, or its use as a norm, may all lead to this effect.In sum, several lines of thought suggest parallel perspective drawings of cubes should generate the picture-surface illusion.Hammad et al. (2008a) tested linear-perspective pictures of cubes at their proper subtense, about 5 0 .But, unlike real objects, pictures can be viewed at incorrect subtenses.Of interest, since parallel perspective is only correct for small subtense objects, and parallel perspective pictures are not robustly consistent in appearance as cubes, beyond 20 0 , compared to linear perspective drawings, vision might not suffer much if any cross-talk from parallel-perspective depth information if a drawing is viewed at a very sizable subtense.If so, there would be no errors in judging 2D angles for some parallel-perspective drawings.On this account, viewed at 5 0 subtense there may be errors as large as those for linear-perspective pictures, but viewed at 20 0 or 50 0 , the errors might vanish.More radically, it might be that even a 5 0 viewing angle makes the parallelperspective drawing fragile, distorted looking, showing a block that appears to diverge with distance, and ineffective at generating a picture surface illusion.The moon subtends .5 0 , and 5 0 is fully ten times that.Many illustrations of cubes in books are likely viewed at 1 0 or less usually.Letters subtending the angular size of the moon, at normal reading distance, are large print.Times Roman 12-point characters on a monitor at normal reading distance often subtend about .2 0.In short, even at 5 0 , the parallel-perspective cube might be anomalous, and if vision rejects it as a cube, instead seeing the picture as little more than a flat surface, this could result in few picture-surface errors.
Finally, we turn to a third possibility -increased error.If a 2D target angle of 25 0 is viewed in a parallel-perspective picture of a cube observed at an unduly large subtense such as 50 0 then the 2D target angle should be reported as especially large, say 55 0 , because the 25 0 2D target angle depicts a larger angle than 90 0 , say a 110 0 angle in the depicted scene.If so, the bias from crosstalk would push the error towards 110 0 rather than just towards 90 0 , increasing the picture-surface error.Viewed at 50 0 , a similar increase in error should occur for linear perspective pictures of cubes that are correct at 5 0 .Their 2D angles should depict angles larger than 90 0 , such as 100 0 , if subtending unduly large angles, and bias in reporting them should increase, though perhaps modestly, given the increase is only to 100 0 , not 110 0 .
Our purpose here primarily is to test whether the picture-surface illusion occurs in parallel perspective renderings.Are the reported errors greater than zero and in the direction of the depicted angle?Our experiment has sufficient power to detect these errors if they are there.Looking beyond the main hypothesis being tested, a caution is in order.Our focus was linear versus parallel effects, and the number of participants needed to look for their differences.Other considerations need further studies, with larger numbers of participants, but we will consider what our study suggests for viewing our stimuli at different subtenses.

Method Participants
The participants were 12 undergraduate volunteers (4 male, 8 female; mean age 19; SD 1.6) from an Introductory Psychology course at the University of Toronto Scarborough.

Stimuli
Stimuli were generated using the "Java 3D API" to render cubes with white outline on a black, 6' x 6' projection screen.They were presented to participants as in Figure 1, at nine different orientations with the top face shown as tilted vertically around a horizontal azimuth in 10 0 increments between 5 0 and 85 0 .The cube drawings were in either linear or parallel perspective (perspective condition) and at 5 0 , 20 0 and 50 0 subtended angles (subtense condition).As a result of depicting the cubes tilted from the horizontal, the target angle -the left-most angle of the 2D quadrilateral or parallelogram --varied from 9 0 to 89 0 (angle condition).In addition to 2D drawings of cubes, the 2-dimensional quadrilateral and the parallelogram tops of the drawings were presented on their own (tops-alone condition).

Procedure
As a pretest, participants viewed 10 white V-shaped lines on a black background and were asked to estimate the internal V angles for each.The angles ranged from 5 0 to 85 0 .All participants were able to estimate angles within a +/-10 0 mean error rate.
Participants were randomly assigned to a "perspective and subtense" condition.In each test trial participants were presented with the cube or top-face ("tops-alone") stimulus on the screen for 4 seconds after which they estimated the 2D target angle.The estimates were given verbally.The angle shown for each trial was taken from those in Figure 1.After 10 presentations of each tilt angle, offered in a randomized order, the participants were informed that the experiment was concluded.They were debriefed as the purpose of the experiment and excused with thanks.

Results
The dependent measure was the estimation of 2D target angle size, and the central concern is illusions in viewing linear versus parallel projections.As the current experiment is concerned with deviations from accuracy, the reported angles were converted to estimation errors by subtracting the presented angle from the participants' estimations.This produced values about 0 0 , with underestimations being negative, and overestimations being positive.Analyses were conducted to an alpha level of p <0.05.In Figure 2, the steps along the horizontal axis are not equal because they show the angle presented, and the angles were created by varying tilt in equal steps of 10 0 .The cube pictures generated higher errors than the tops-alone from 26 0 to 87 0 presented-angle.The maximum error was 19.2 0 at a 54 0 presented-angle.The shape of the curve for errors -bowed upwards -is the shape reported in Hammad and Kennedy (2008a,b), who also found maximum errors at the intermediate presented angles.If the illusion is a bias towards the represented angle, 90 0 , there must be a ceiling effect for presented-angles close to 90 0 .In absolute numbers, only small increases in the smallest angles, 9 0 , and 26 0 , may be expected if the bias is to some extent a percentage of the presented-angle.Of interest, the tops-alone also generated errors, since all of their values are above zero: t(8) = 6.79, p <0.05.Mean errors in angle estimations were submitted to a 2 (cube vs. tops-alone) x 9 (9 0 , 26 0 , 41 0 , 54 0 , 65 0 , 75 0 , 80 0 , 87 0 , 89 0 ) within-subjects ANOVA.The data was collapsed across the perspective (linear vs. parallel) and subtended (5 0 , 20 0 and 50 0 ) conditions.A main effect was observed, F(1, 10) = 35.8,MSE = 1572.1,p <0.05.The 2D target angles were estimated significantly closer to 90 0 when presented in a drawing of a 3D cube than when presented in 2D tops-alone conditions.A significant interaction was evident, F(8, 80) = 6.4,MSE = 220.7,p <0.05, since the difference between the cube conditions and tops-alone was present only in the mid-range of presented angles.The range of estimation errors was greater when cubes were shown (range 17.4 0 ), than tops-alone (range 6.5 0 ). Figure 3 reveals that both linear and parallel perspective drawings of cubes provide the picture-surface illusion.At all presented angles, both the linear and parallel perspective cubes resulted in positive errors (sign test, p =.002 for both, two tailed).Also, they had similar maxima (20.0 0 at 41 0 presented-angle for the linear condition, and 21.1 0 at 54 0 for the parallel condition), at presentedangles just one step apart.A 2 (linear vs. parallel) x 9 (9 0 , 26 0 , 41 0 , 54 0 , 65 0 , 75 0 , 80 0 , 87 0 , 89 0 ) between-subjects ANOVA showed no significant main effect of perspective condition on the size of the mean estimation errors between cube conditions.Evidently, the parallel perspective drawings performed as Maniatis predicted, not differing in their outcomes from the linear conditions.Although the chief concern here is comparing linear and parallel projections, it is worth noting similar estimation errors in viewing cube angles at 5 0 , 20 0 and 50 0 subtended angle (Figure 4).There are few observers per condition, but inspection indicates the curves are similar, bowed upwards, maxima at intermediate angles 41°, 54° or 65°, and overlapping.Again, it might be worth noting, despite the few participants per condition, that parallel-perspective drawings may cause similar errors at different subtenses.On inspection, Figure 5 reveals similar bowed-upwards, overlapping curves, with maxima at 41° or 54°, suggesting little difference between subtense conditions.

Discussion
The picture-surface illusion was present at strength for the parallelperspective picture of a cube, equal in magnitude to the illusion for the linear perspective drawing that is correct at 5 0 subtense.Further, the error was present at all three subtenses tested -5 0 , 20 0 and 50 0 -at similar strength in both kinds of drawings.Positive errors were present to a modest extent in viewing topsalone, since the error in judging these shapes was about 5 0 .(It may be that participants take the tops-alone as weakly suggesting tilt in 3D.We expect that any apparent tilt, and any illusion, would reduce to zero if only the two lines forming the target angle were on display.)However, the mean tops-alone errors were in the 9 0 to 3 0 range throughout the angles-presented range, the means did not increase --bow upwards --in the mid-range of angles presented.Hence the tops-alone errors were perhaps a positive bias in judging angles, rather than a picture-surface illusion.
Increasing the angular subtense did not increase the illusion.Vision is tolerant of improper subtense, so far as drawing of cubes is concerned.The 5 0 to 50 0 range tested here may be too restricted to fully evaluate the especiallylarge-error-at-large-subtenses hypothesis.The range from 50 0 to close to 180 0 needs to be added to complete the set of possibilities.A parallel perspective drawing of a cube has many of a linear-perspective 5 0 subtense drawing's features.To distinguish the two kinds of drawings it might be wise to provide a linear-perspective drawing cube drawing correct at a large subtense, such as 50 0 or 100 0 or more.If the picture-surface illusion drives judgments of 2D angles towards their 3D referents, the parallel perspective drawing would be driven to angles much larger than 90 0 at large subtenses.Conversely, a percept of a linear perspective picture correct at 5° but viewed at 100° should be driven to angles less than 90 0 .(So far as cube drawings are concerned, we should point out these large-subtense viewing conditions would be quite unusual.Real objects such as mountains and big buildings are viewed at the full range of subtenses, but pictures of single cubes are not.) In some respects, the picture-surface illusion can only have been in existence since the invention of pictures in the Paleolithic era.Vision evolved to cope with information for depth and slant that arose from sources in natural landscapes.The information was for one depth and slant, not for two, unlike the case in pictures, in which information is provided simultaneously for a pictured scene and a picture surface.The depth information from a landscape is singular, and it is dual from a picture.Recently, Parks (2013) has proposed that two distinct pathways operate in processing depth information: an evolutionarily older, "unconscious vision-for-action dorsal stream" for guiding motor behavior, and a newer, "conscious vision-for-perception ventral stream" producing the actual visual experience of an object in the environment.In reference to the Ponzo illusion, Parks claims that a depicted depth is processed in the dorsal stream, regardless of the consciously perceived flatness of an actual depiction's surface: "to the primitive vision-for-action stream (which presumably lacks the picture-perception ability to respond to a picture as a picture-of-some-thing) that line does not represent a longer line; it simply is a longer line" (Parks, 2013).This explanation may account for the depicting feature being seen incorrectly.This proposal holds that the depicting feature is seen as what it depicts.But in the picture-surface illusion the depicting feature is merely biased in perception, biased towards what it depicts.
In the terms used by Parks (2013) interference occurs between the unconscious and conscious streams, and in our current account the dual information presented in 3D pictures crosstalks to allow two effects.The depicted scene appears flatter than it should be geometrically, and the 2D picture-surface features are seen biased towards what they depict.If the pictured scene is diminished in depth, the blocks depicted in Figure 1 should look slightly "flattened," as Maniatis (2009) observed, and the angles represented by the 2D vertices judged by our observers should appear more acute in 3D than if they depict cube vertices.That is, the depicted-cube tops should seem slightly too shallow.If observers were asked to judge the depicted 3D angles (and the aspect ratio of the 3D top-surface forms depicted by 2D quadrilaterals and parallelograms) they should report the angles less extreme than they truly are (and the aspect ratios less than 1).
Two cases in nature involve judging shapes and depths with an extra surface intervening, but their properties are not those of pictures.Shadows fall on a surface but they provide information for another.The other surface is not behind the surface bearing the shadow, in the fashion of a depicted world behind the picture surface.Nevertheless, if a V shape in the 2D shadow with a 30 0 bend indicates a sharp angle such as an elbow bent at 60 0 in the 3D person casting the shadow, the shape of the V in the shadow itself may suffer the picture surface illusion and be seen as say 45 0 .That is, it could be seen biased towards the shape of the 3D referent of the 2D shadow.The second case in nature occurs with water.Observers look through 2D water surfaces to detect 3D underwater scenes.However, there is no 2D image on the surface of the water to be judged.The crosstalk available in this case might be from highly textured water surfaces.If there are many ripples on the 2D water surface, the depth to the 3D underwater objects may be seen reduced, the more the ripples the more the reduction.Parks (2013) notwithstanding, precisely how the dual sources of information for depth and slant interact in perceptual processing to produce a particular size of illusion --a particular size of bias --remains a mystery.Gregory (1963Gregory ( , 2009) ) argued the case should be treated as inappropriate constancy scaling.But this puts emphasis on one kind of information (for depth, and its constancy), and our claim here is that two kinds of information interact.The information (such as aspect ratio) for the top-face's slant in depth interacts with information for the 2D feature, the angle.Precisely why crosstalk between the two sources should produce a particular value of a perceived 2D angle (or 2D aspect ratio) or 3D flattening is unknown.Still more mysterious is that the two kinds of information are not distinct.The angle being judged is part of an eccentric shape, a four-sided figure with aspect ratio less than 1, a quadrilateral or parallelogram, and it is eccentric precisely because it has acute and obtuse angles.The aspect ratio itself is misjudged, in the picture-surface illusion (Hammad et al., 2008b).Further, the aspect ratio and acute and obtuse angles --a collection of interdependent features --are all present in the top-faces condition, shown without the lines showing the flanks of the cube.It may be the information for slant and depth present in the features is latent, present but inactive as indicators of 3D beyond the picture surface.The flanking-side lines help make the latent indicators explicit.The flank lines act as catalysts.They help trigger an analysis that otherwise does not start.They promote processes that can engage the angles and aspect ratios as depth information.They make perception undergo the mental chemistry reaction, but they take no part in the decision making about how much slant is relevant, how much crosstalk is to occur and how much bias should result.

Figure 1 .
Figure 1.Nine renditions of a cube are shown in linear perspective on the left, and a further 9 are in parallel perspective on the right.They tilt in pitch from 5° to 85°.The linear perspective cube is correct at about 5 0 angular subtense.Observers judged the acute angle on the left in the quadrilateral showing the top of the cube.To make the illusory effect obvious, isolate lines for cube-top quadrilaterals.Use the figures central to the 9 left and right cubes.Covering all but the quadrilaterals makes the angles in the quadrilaterals look more obtuse and acute.