CHAPTER 9

Depth Perception

ELI BRENNER AND JEROEN B. J. SMEETS

People use sensory information to guide theirinteractions with the environment. One thingthat people, like most animals, almost alwayswant to know about objects in their environ-ment is where those objects are. If someoneis touching an object, he or she can feelwhere it is by combining where the objecttouches their skin with knowledge abouttheir own posture. In such haptic localizationthere is no fundamental difference betweenjudgments of distance and judgments ofazimuth or elevation (Figure 9.1). Similarly,there is no fundamental difference betweenhaptic judgments of an object’s dimension indepth and of its lateral or vertical dimension.However, people often want to know anobject’s position and dimensions before theytouch it. In that case they usually rely onvisual information. Occasionally they mightrely on auditory information, such as whenthey try to find a friend who they hear talking.In auditory localization, azimuth is judgedfrom differences between signals’ arrivaltimes and between their intensities in the twoears, elevation is judged from how the shapeof the outer ear affects the sounds’ spectralcontent and echoes sounds, and distance isprimarily judged from the intensity of thesound (although reverberation and spectralcontent probably also provide some infor-mation). In visual localization, azimuth andelevation (relative to the head) are judged

by combining the position of the object’sretinal image with information about theorientation of the eyes (Figure 9.2). Judgingthe distance is much more complicated. Thischapter deals with how people visually judgedistances in depth.

We consider three different aspects ofjudging distances in depth, which we referto as judging distance, depth, and depthordering. We use distance to refer to thedistance from the observer expressed insome metric such as meters, number of stepsrequired to reach it, number of eye-heightsaway, or any other measure that completelyspecifies the position. We use depth to referto distances between structures. This couldrefer to distances between objects, but also todistances within an object, as when referringto a single object’s extent in the viewingdirection. Depth could be estimated by com-paring two judged distances, but there arealso ways to directly judge depth. Directmeasures of depth are generally more precisethan judgments of distance, but they do notdirectly provide metric information (as isexplained). They need to be scaled by judg-ments of distance before they can provideinformation about the actual separation indepth, as one might need for judging whetheran object is too big to be grasped. We usedepth to refer to the scaled judgments. Insome cases, such as when judging whether

Stevens’ Handbook of Experimental Psychology and Cognitive Neuroscience, Fourth Edition, edited by John T. Wixted.Copyright © 2018 John Wiley & Sons, Inc.DOI: 10.1002/9781119170174.epcn209

1

2 Depth Perception

elevation

azimuth

distance

Figure 9.1 Positions relative to the head can beexpressed in terms of a distance and direction fromthe head. The direction can be expressed as anazimuth (left or right of straight ahead) and an ele-vation (up or down with respect to straight ahead).

a structure is drawn on a surface or is anobject lying on the surface, or when judgingwhether a wasp is in your room or safelybehind the glass of the window, it is enoughto know whether a structure is closer to youthan another structure, without knowing howmuch closer. The direct measures of depth

13 2

gazeangle

retinaleccentricity

(A) (B)

Figure 9.2 Two balls’ positions relative to the head (A) and their images’ positions on the retina (B).The position on the retina that is stimulated by light reflected by an object (such as the dark red ball)depends on the object’s azimuth and elevation (only the elevation is visible in (A) and on where oneis looking (here, at the blue ball). Conversely, where one is looking (gaze angle in A) and the retinaleccentricity of the object in question (arrow 2 in B) can be combined to retrieve the object’s azimuthand elevation. When looking directly at an object of interest, the retinal eccentricity is negligible (theobject’s image falls on the fovea, indicated by arrow 1 in B, which is the part of the retina with thehighest density of photoreceptors), so the azimuth and elevation correspond with the direction of gaze.Arrow 3 in B indicates the optic nerve (blind spot).

do provide information about such depthordering.

Vision is based on two-dimensional retinalimages of the three-dimensional world. Thesetwo-dimensional images represent what youcan see at a certain moment in any direction.The azimuth and elevation of all visiblestructures in the environment are representedby the positions of their images in each ofthe two eyes. Differences between structures’azimuths and elevations are directly evi-dent from the positions of their images on theretina. Distance would appear to be lost in thetransformation from the three-dimensionalworld to the two-dimensional retinal images.

The fact that humans have two eyes withlargely overlapping fields of view gives themthe possibility to recover distance: Givena structure’s direction with respect to botheyes, one can theoretically determine itsdistance through triangulation. Theoretically,such triangulation is enough to determine thedistance to all objects that are simultaneouslyvisible to both eyes, but the presence ofthis chapter specifically devoted to depthperception is an indication that in practice,judging distances is more complicated.

Depth Perception 3

The main reason for this complication is thatthe precision with which one can estimate thevisual direction with respect to each eye islimited. Therefore, humans do not only relyon differences between the visual directionswith respect to the two eyes, but also onother sources of visual information aboutdistance and depth. The different sources ofinformation are known as depth cues.

A fundamental difference between recov-ering distances by triangulation and doingso with the aid of other depth cues is thattriangulation does not require any priorknowledge or assumptions about the world.As long as one can identify a structure’simage in both eyes, one can judge the struc-ture’s distance. Other cues depend criticallyon regularities in the world. After presentinga short overview of the cues that are availablefor judging distances and depths, subsequentsections discuss these cues and the corre-sponding assumptions in greater depth. Wethen discuss how the cues’ resolutions andthe assumptions on which they rest influencethe way in which they are combined for var-ious kinds of judgments. When we mentionassumptions, we conceive of them as beingused unconsciously and of being the result ofregularities in our everyday experience. How-ever, people are certainly sometimes aware ofthe assumptions being violated, although thisdoes not necessarily decrease the extent towhich they rely on them (Muller, Brenner, &Smeets, 2008). Moreover, because reliabledepth perception is not only advantageousfor humans, but undoubtedly enhances thechances of survival in many species, theremight be an innate component to the use ofsome assumptions.

Roughly speaking, there are three kindsof cues. The first kind consists of cues thatare present in a single retinal image. Theseare sometimes also known as pictorial cues,because they can be captured and reproducedin pictures. The only direct information about

distance that is present in a single image isthat if you can see something in a certaindirection, this thing must be the nearestobject in that direction (except when lookingthrough transparent surfaces). Thus, you donot know its distance, but you know that itis closer in depth than any other structurein that direction. If the object hides part ofanother object from view, it must be nearerthan that object. If the other object occludespart of the object in question, the other objectmust be nearer. Thus, the fact that somethingis visible in a single image provides someinformation about the depth order: It tells youthat it is the nearest object in that direction.

Other pictorial cues include various prop-erties that are related to perspective, includingimage size, texture gradients, and height inthe visual field, as well as contrast and blur(reviewed extensively in Sedgwick, 1986).These cues rely on regularities that normallyexist in the world around us. Examples ofregularities are that textures are isotropic,that shapes are symmetrical, that objects reston surfaces rather than hovering in mid-air,and so on. Cues that rely on such regular-ities provide incorrect information whenthe regularities are violated. An obviousexample of such a violation is when we takea photograph of a slanted textured surface,in which case the structure of the texture inthe picture is consistent with a surface that isslanted with respect to the actual surface ofthe photograph.

The second kind of cues consists of cuesthat rely on using two eyes (reviewed exten-sively in Howard & Rogers, 1995). Thereare a number of ways to obtain informationabout distance by making use of the fact thatwe have two eyes, including extracting suchinformation from the differences betweenthe images in the two eyes (binocular dis-parities) and from the differences betweenthe directions in which the two eyes are ori-ented (ocular convergence). People who are

4 Depth Perception

unable to use binocular depth cues are oftenconsidered to be unable to see depth, but, ofcourse, this is not true. If you are not sucha person, and you close one eye, the worlddoes not suddenly look flat. However, withone eye closed you may become less precisein judging distances, and therefore have moretrouble pouring yourself a cup of tea.

The third kind of cues consists of activecues (Gibson, Gibson, Smith, & Flock,1959; Rogers & Graham, 1979). The mostprominent active cue arises when an observermoves in a static environment. When heor she does so, the directions to structuresaround him or her change in a manner thatdepends on the structures’ distances. Individ-ual changes in direction and relative directionare known as motion parallax. The combina-tion of all such changes is known as the opticflow. When moving in a static environment,combining the changes in the directionsto surrounding structures with informationabout how one is moving can provide infor-mation about the object’s distance. Even ifone does not know how much one has moved,as might be the case when looking out of atrain window, one can still obtain informationabout the depth order. Another active cue fordistance is changing the curvature of the lens

of the eye (accommodation), and detectinghow changes in such curvature influencesblur in the image.

PICTORIAL DEPTH CUES

Occlusion

Occlusion provides very reliable informationas to which of two opaque surfaces that bothpartly occupy the same direction from theobserver are closer (Figure 9.3A). It tellsus nothing about how much closer, but thedifference can be very small without anyreduction in our certainty as to which iscloser. Occlusion is therefore a very reliablecue, but it only provides information aboutwhich of two surfaces is closer, not howclose they are. Moreover, it only providesinformation about surfaces that occupypositions in space that overlap in terms oftheir direction from us. Occlusion also onlyprovides information about the depth orderif one knows how to segment the image intoobjects. In Figure 9.3A the observer sees animage that would probably be interpretedas three overlapping rectangular surfaces.It could be that the central surface is arectangle with a section removed. However,if so, one would have to assume that the

occlusion(A) (B)

Figure 9.3 Nearby objects occlude ones that are farther away if they are in the same direction. If theimage can be segmented into occluding and occluded surfaces in a straightforward manner, for instanceby assuming that the surfaces have certain shapes, the depth order is evident (A). If not, even the depthorder is ambiguous (B).

Pictorial Depth Cues 5

section that was removed from the centralrectangle is exactly aligned with the edges ofthe leftmost rectangle. Similarly, the edges ofthe central and rightmost surfaces might bealigned rather than the central one occludingpart of the rightmost one. If the surfaces wereat different distances these alignments wouldonly hold for a specially selected viewpoint,so it would be very unlikely. Thus, the inter-pretation in terms of overlapping rectanglesis reasonable. It is less evident how the imageobserved in Figure 3B should be interpreted.The small rectangle within the larger onecould be a distant surface seen through ahole in the nearer surface, a part of the samesurface that was painted a different color,or a small rectangle in front of the largerone. Thus, in this case, although occlusionstill tells you that the small rectangle is thenearest surface in that particular direction,occlusion does not tell you anything abouteven the depth order with respect to thelarge rectangle.

Height in the Visual Field

In general, there is a correlation betweenan object’s height in the visual field and itsdistance. Looking downward we usually seethings that are close to us, whereas lookingup toward the horizon we usually see thingsthat are farther away. This is because mostthings in our environment rest on surfaces.The relationship between depth order andheight in the visual field is quite straightfor-ward for small objects on a single horizontalsurface. It is not as straightforward for largeobjects or when objects are not resting onthe same horizontal surface or are not restingon surfaces at all. For large objects, it isimportant to realize that it is not the heightof the center of the object that is relevant, butthe height of its base, where the object makescontact with the surface, because it is theposition on the surface that matters. Thus, for

Figure 9.4 Trees are taller than flowers, so theywill often occupy higher positions in our visualfield, as they do in this photograph. However,despite the indicated tree extending much higherin the image, the indicated tree and the indicatedflower are judged to be at about the same distancebecause they rise from the ground at about thesame height in the image.

instance, a tree and a flower next to the treecan be considered to have the same heightin the visual field for the purpose of judgingdistance (Figure 9.4). Unlike for occlusion,the objects do not need to overlap in terms oftheir direction from the eyes for their depthorder to be determined.

There is ample evidence that people makeuse of height in the visual field to judgeobjects’ distances (Ooi, Wu, & He, 2001).When considering patterns on a surface, orwhen considering where objects make con-tact with a surface, the height in the visualfield provides direct information about thedepth order. If you can be certain that thesurface is horizontal, and you know your eyeheight with respect to this surface, heightin the visual field can provide estimates of

6 Depth Perception

height in thevisual field

hf

Δh

hn

dn

df

Δh

v

Figure 9.5 If the surface is horizontal, your own eye height is v, and your gaze angle with respect to thehorizontal is h, the distance to a small object on the surface is d = v∕ tan(h). If there are two objects onthe surface, and the distance to the farther object is known (df), the distance to the nearer object (dn) can

be judged from the difference between their heights in the visual field (Δh)∶ dn = v∕ tan(Δh + atan v

df

).

If the farther object is very far away (df ≈ ∞; hf ≈ 0; Δh ≈ hn), doing so is equivalent to relying on thegaze angle with respect to the horizontal: dn = v/tan(hn).

the actual distances of objects located on thesurface (Figure 9.5). This might for instancebe the case for a person standing in an office.He or she can be assumed to be familiar withhis or her eye height (for support for the ideaof relying on eye height see Bridgeman &Cooke, 2015; Daum & Hecht, 2009). More-over, the floor of the room can be assumedto be horizontal. Besides knowing his or herown eye height, the person would have tobe able to judge the vertical position of thehorizon. Judging this visually could be aproblem in an enclosed space, in which caseone might have to rely on a vestibular esti-mate of the horizontal eye level (Li, Dallal, &Matin, 2001), or on visual estimates based onobjects in the scene (such as other people’seye heights) or on the optic flow if one ismoving (the vertical position of the focusof expansion if one is moving through theoffice).

Considering a retinal resolution of about1 minute of arc (1/60th of a degree), theresolution of judging two structures’ depthorder from their heights in the visual field isvery good. It varies with distance, with sep-arations of less than 1 mm being discernablefor objects about 1 m away, and separationsof about 1 cm being discernable for objectsthat are several meters away (horizontalsurface curves in Figure 9.6). Of course, thisis an estimate of the resolution for detectingthat there is a depth difference between theobjects. It is difficult to estimate the resolu-tion for judging the actual distance of eitherof the objects, or of the separation betweenthem, because this requires knowledge of thegaze angle or of the visual angle with respectto the horizontal. Although the former mayor may not depend on the magnitude of theangle, the latter probably increases with thevertical separation in the image, possibly

Pictorial Depth Cues 7

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horizontal surfacefloor table

object size 5 cm

10 cm 20 cm

Figure 9.6 Resolution of height in the visualfield and image size as depth cues. The figureshows how the resolution for detecting a separa-tion in depth depends on the distance. The reso-lution is based on a threshold for detecting reti-nal separations of 1′ arc. The separation corre-sponding with this threshold is presented on aninverted logarithmic scale so that a high resolu-tion is high in the figure. Red curves: differencesin distance corresponding with a vertical retinalseparation of 1′ arc for a horizontal surface 1.5 mbelow eye height (floor) or 70 cm below eye height(table). Green curves: differences in distance cor-responding with a 1′ arc difference in imagesize for objects of three different sizes (5, 10,and 20 cm).

counteracting the increase in resolution fornearby objects to some extent. In either case,it is evident that the resolution for judging anactual distance is lower than what is shownin Figure 9.6.

If objects are not resting on the same sur-face, one can follow how objects rest on eachother to use height in the visual field to judgetheir depth order to some extent (Figure 9.7;Meng & Sedgwick, 2001). A somewhatrelated way to judge objects’ relationshipswith a ground plane when it is not evidentthat the objects are lying on the ground planeis to consider cast shadows. An object’sshadow can give an indication of its distance

from a surface: A shadow close to the objectsuggests that the object is close to the surface,whereas a larger separation suggests a largerdistance from the surface. A change in theassumed distance to the surface will modifythe relation between height in the visual fieldand perceived distance: If an object is farabove a surface, it will be perceived to benearer than a similar object for which theshadow suggests that it is lying on the samesurface (Figure 9.7; Allen, 1999; Kersten,Mamassian, & Knill, 1997).

What if the surface is not horizontal ornot flat? Height in the visual field obviouslybecomes much less reliable if the surface istilted to the side and the objects of interest areseparated laterally (i.e., in azimuth). For sur-faces that are slanted upward or downward,height in the visual field still provides reliableinformation about the depth order. For largesurfaces with small slants, even judgments ofdistances and depths could be quite reliableas long as the judgments are based on thevertical separation with respect to the visiblehorizon rather than on the gaze angle withrespect to the horizontal (for some indicationthat this is what people use see Gardner,Austerweil, & Palmer, 2010). The error thatarises from the distance of the observer’s eyesfrom the surface no longer being the heightin the direction of gravity is negligible formodest slopes. Judgments based on gazeangle relative to gravity would obviouslyprovide quite wrong estimates of the dis-tance: When walking up a slope one wouldoverestimate the distance and when walkingdown a slope one would underestimate thedistance. Even relying on height in the visualfield to judge depth order is obviously unre-liable if the objects are on a surface that isnot flat, if they are on different surfaces thatare not connected by simple visible supports,if the positions at which the objects makecontact with the surface are not visible, or ifthe objects are not on surfaces at all.

8 Depth Perception

support andshadow

Figure 9.7 Objects’ shadows and the way they rest on other objects can help determine how to interpretthe height in the visual field. Whereas the red ball’s shadow indicates that it is resting on the green surface,the blue ball’s shadow indicates that it is hovering in the air above the green surface, and is thereforenearer than its height in the visual field might suggest. Similarly, the fact that the yellow ball is evidentlyresting on the purple cube suggests that its distance corresponds with the height in the visual field of thecenter of the cube’s bottom surface, rather than with the height of the ball itself, making it appear to becloser than the red ball although they are at the same height in the visual field.

Image Size

If an observer is certain about an object’ssize, its retina image size can reveal itsdistance (Figure 9.8). Indeed, image size isused in the perception of distance (Gillam,1995; McIntosh & Lashley, 2008). Althoughone might expect image size to only influ-ence judgments of distance when the trueobject size is known, this is not the case.When judging the distance of an unknownobject, people judge a large object to becloser than a smaller object that is presentedat the same location (Collett, Schwarz, &Sobel, 1991; Lugtigheid & Welchman, 2010;Sousa, Brenner, & Smeets, 2011, 2012).People apparently use image size as a cuefor distance even when they have no directinformation about the actual object size. Anexplanation could be that people considercertain sizes to be more likely than others,

probably based on experience with similar-looking objects. The resolution for judgmentsof distance from retinal image size, assumingthat the true object size is known extremelyprecisely, and again given a retinal resolutionof 1′ arc, is shown by the object size curvesin Figure 9.6. The resolution depends onthe object’s size and decreases rapidly withdistance.

Texture

Besides considering the sizes of individualobjects’ retinal images, an obvious cue fordetermining relative distances is to considerthe gradient in the sizes of similar objects’retinal images, such as the changes in theimage sizes of stones or tiles across a surfacethat extends in depth. In this case the assump-tion is not that you know the actual size, butthat the size is constant (isotropic) across

Pictorial Depth Cues 9

image size

α

d

d = r / tan(α)

r2α

Figure 9.8 If an object is far away, its image on the retina is smaller than if it is nearby. An object’sretinal image size (represented here by the diameter of the ball’s image, 2𝛼) is determined by the ratiobetween the size of the ball (represented by its radius, r) and its distance (d). For any retinal image size,knowing the true object size could tell you its distance.

surface texture

vβ

α

d s

s

d (m)

visu

al a

ngle

0°

3°

6°

20100

β

α

v = 1.7 ms = 15 cm

α

Figure 9.9 For surface texture such as floor tiles, the image size changes with the distance from theobserver (d). The graph shows how the lateral angle (𝛼) and the angle along the depth direction (𝛽)depend on the distance when a 15 cm tile is examined from an eye-height of 1.7 m. The two curves show

𝛼 = 2 tan−1

(12

s√d2+v2

)and 𝛽 = tan−1

(d+s

v

)− tan−1

(d

v

).

space. Along a surface there are variouscomponents to the change in image size withdistance. The angle filled by structures thatare oriented orthogonal to the line of sightchanges almost linearly with the inverse of

the distance (angle 𝛼 in Figure 9.9). What istrue for such image sizes is also true for thedensity of regularly or randomly distributedtexture elements: the density of the texturein the retinal image increases in accordance

10 Depth Perception

with the decreases in single objects’ sizes.If the actual texture elements are all identicalor if their sizes vary at random across the sur-face, the distribution of the texture elements’retinal image sizes provides equivalent infor-mation. Thus, the depth order and even someindication of relative depths along the surfacecould be judged from texture gradients inthe retinal image. However, the resolutionfor detecting a difference in distance onthe basis of the local texture alone is quitepoor, irrespective of the distance (see lateraltexture density curves in Figure 9.10).

For structures that recede in depth along aground surface, the angle between the surface

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lateral texture densitydensity in depth

aspect ratio

floor table

Figure 9.10 Changes in texture density andaspect ratio with distance. Red curves: differencesin distance corresponding with a change in texturedensity of 8% (lowest value reported in Anobile,Cicchini, & Burr, 2014). Solid: lateral density (hor-izontal on the retina). Dashed: density in depth(vertical on the retina). Green curves: differencesin distance corresponding with a 5% change inaspect ratio (Nachmias, 2008). Both measures areshown for a surface 1.5 m below eye height (floor)and 70 cm below eye height (table). Other detailsas in Figure 9.6.

and the line of sight depends on the distance,so the structure’s angular extent changes ina more complicated manner with distance(angle β in Figure 9.9). A comparison of thehorizontal and vertical extents of objects’retinal images can provide information aboutthe objects’ slants if the objects’ shapes areknown. Again, what is true for the imagesize of a regularly shaped object is alsotrue for the density and sizes of regularlyor randomly distributed texture elements.For judging slant, one could also rely onother gradients than the density or size oftexture elements in the retinal image. Forinstance, if the texture consists of orientedelements, one could rely on gradients inthe distribution of orientations in the retinalimage (Warren & Mamassian, 2010). Forjudging slant it is important to realize that fortexture that is not flat on the surface, such aspebbles, matters may be more complicatedthan we have sketched above, because theslant will also determine to what extent partsof the surface, such as pebbles of many sizes,occlude each other. Slant on its own doesnot provide any information about distanceor depth, but knowing the slant could beimportant for interpreting other cues such asheight in the visual field or texture density.Moreover, if one is certain about the trueslant, one could use the gradient along theground surface (density in depth curves inFigure 9.10) or the difference between thegradients in the two directions (aspect ratiocurves in Figure 9.10) to determine the depthorder. Although the latter measures providea slightly better resolution than relying ongradients orthogonal to the line of sight, theyalso depend on more assumptions, so it isnot evident that any of these measures couldplay a major direct role in judging distancesor depths.

The use of texture gradients relies onassumptions about the elements on the sur-face, such as that the elements are identical

Pictorial Depth Cues 11

or similar, and that they are regularly or ran-domly distributed. Any systematic orderingcould be misinterpreted as depth. An extremecase of assuming a certain ordering is whenlines converge toward a single point. This isusually interpreted by the visual system asthe lines being parallel but receding in depth,so that the constant separation between themleads to a smaller separation in the imageas the distance increases (see top left ofFigure 9.9). Similarly, horizontally elongatedellipses are readily seen as slanted circles.Such cues can be very strong, especiallywhen judging slant (Muller et al., 2009). Assurface slant needs to be considered whenheight in the visual field is interpreted interms of distance, the most important influ-ence of texture cues on judged distance maybe mediated by estimates of slant in combina-tion with height in the visual field, rather thanthrough direct judgments of distance. Thisillustrates that the depth cues that we haveat our disposal might not be independent.We return to this issue in the section aboutcombining cues.

Image Quality

Image quality can be informative about dis-tance. Light reflected by objects is slightlydiffused by particulates in the air on its wayto our eyes. Consequently, contrast decreaseswith distance. Reduced contrast is thereforeindicative of a large distance (aerial perspec-tive). To use this cue to estimate distance,one must make assumptions about the localatmosphere and about the objects in question.Except under very foggy or rainy conditions,the changes in contrast with distance areso small that this cue is only effective fordetecting large separations in depth. This cuecan therefore be useful for judging which oftwo distant buildings is farther away, but itwill seldom be useful for nearby objects. Itsresolution is obviously very poor.

At small distances, blur can provide someinformation about distance. If you are look-ing at a surface at a certain distance, andaccommodation is adjusted to that viewingdistance, the images of things (edges orchanges in surface reflectance) that are at thatdistance will be sharp, whereas the imagesof things at other distances will be blurred.Consequently, if an edge between the objectthat you are looking at and another object issharp, the edge belongs to the object that youare looking at, so this object is probably infront of the other object. On the other hand,if the edge is blurred, the edge belongs to theother object, so the other object is probablyoccluding the object that you are lookingat (Marshall, Burbeck, Ariely, Rolland, &Martin, 1996). This use of blur to judge thedepth order assumes that the border of theobject itself is sharp.

Both contrast and blur appear to contributeto judgments of distance (Held, Cooper, &Banks, 2012; O’Shea, Govan, & Sekuler,1997). In order for contrast to provide infor-mation about more than the depth orderone would have to consider the weatherconditions. Similarly, in order for the instan-taneous blur to provide information aboutmore than the depth order one would haveto consider the size of the pupil and thestate of accommodation of the eye. An alter-native mechanism for using blur to obtaininformation about distances or depths isby minimizing blur at the position of inter-est through accommodation, and using therequired accommodation to judge the dis-tance. People probably use this mechanismto some extent because accommodation hasbeen shown to contribute to judgments ofdepth (Watt, Akeley, Ernst, & Banks, 2005),and distance judgments are slightly betterwhen looking normally than when lookingthrough a pinhole (in which case the image issharp irrespective of the accommodation ofthe lens; Frisby, Buckley, & Horsman, 1995).

12 Depth Perception

BINOCULAR DEPTH CUES

When thinking about judging distance ordepth, the first cues that come to mind areusually the binocular cues. It is thereforenot surprising that the many contributionsof binocular vision to depth perception havebeen studied very extensively (for reviewssee Foley, 1980; Howard & Rogers, 1995).In principle, the distance to any structurethat is being fixated could be determinedon the basis of the viewing directions ofthe two eyes, through triangulation. Beforediscussing how people make use of the smalldifferences between the images in the twoeyes (retinal disparities) to judge distancesand depths, we therefore consider judgmentsof the orientations of the eyes.

Eye Orientation

The precision with which we know the ori-entation of each eye is obviously limited.Determining how well we normally know theorientation of our eyes is difficult, becausewhen doing so subjects are necessarily keptin the dark to remove any other cues aboutdistance or direction of gaze. As the eyes driftwhen one is placed in the dark, determiningthe eye orientation in this manner will leadto it appearing to be poorer than it actuallyis. To circumvent this, one can examine howwell distances and directions can be judgedwhen comparing them across a single sac-cade (Brenner & van Damme, 1998; Enright,1991, 1996). The standard deviation in judg-ments of the orientation of each eye, whenestimated in this manner, is slightly more than6 minutes of arc (Brenner & Smeets, 2000).Converting this to 95% confidence intervalsof the perceived location shows that fixatedstructures’ positions, including their dis-tances, could be judged quite reliably on thebasis of estimates of the eyes’ orientations fornearby structures, but that for distances thatare beyond reach the judgments of distance

1 m

Figure 9.11 Estimated precision of judgingpositions on the basis of eye orientation informa-tion alone. The shaded regions are 95% confidenceintervals around the positions indicated by thedots (at distances of 30, 50, 80, 150, and 350 cmfrom the observer). These values are based onindependent errors with standard deviations ofabout 6 minutes of arc per eye.

become quite poor (Figure 9.11). Becausethese estimates are based on comparisonsacross a single saccade, they should probablybe considered to represent the best possibleprecision of judging positions on the basis ofeye orientation alone. The vergence curvesin Figure 9.12 show how the resolution forjudging depth order from information aboutthe orientation of the eyes rapidly decreaseswith distance.

A Choice of Coordinates

To describe binocular vision conveniently,we assume that the two eyes are both orientedtoward the same structure, and consider theplane including the two eyes and the struc-ture that is being fixated as the plane withinwhich the azimuth and distance of gaze isdetermined. We consider the rotation of theplane around the axis through the two eyes asthe elevation of gaze. When considering twostructures, if the angle at the eye between thedirections to the two structures is the same

Binocular Depth Cues 13

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45° tothe side

Figure 9.12 Resolution of binocular depth cuesfor structures that are straight in front of theobserver or 45∘ to the side. Values are based ona 6.5 cm separation between the eyes, and either aretinal resolution of 1′ arc (horizontal and verticaldisparity) or an oculomotor resolution of 12′ arc(vergence). The vertical disparity does not dependon the distance for objects that are straight ahead.Details as in Figure 9.6.

for both eyes, the structures are consideredto have no relative binocular disparity. Ifthe angles at the eye differ in magnitudealong the direction of the axis through theeyes, the structures are considered to have adifferent horizontal disparity. If the angles atthe eye differ in magnitude in the orthogonaldirection, the structures are considered tohave a different vertical disparity. The dis-tinction between these directions also appliesto the special case of a structure’s disparitywith respect to the structure that is fixated,in which case we refer to the differencesbetween the angles as horizontal or verticalretinal disparity.

Horizontal Disparity

For understanding the relationship betweendistance and horizontal disparities, it is con-venient to start with a description of locations

for which the aforementioned angles betweenthe directions to two structures are the samefor both eyes (i.e., points with no horizontaldisparity). These locations fall on circlesthrough the two eyes. One such circle isthe circle for which the horizontal retinaldisparity is zero: the circle through the twoeyes and the fixation point (Figure 9.13A).It is easy to see that shifting one’s gazebetween structures with the same horizontaldisparity, so that both eyes rotate by the sameamount, will not change the angle betweenthe lines of sights of the two eyes (knownas the vergence angle; Figure 9.13B). Thatpoints for which the vergence angle is thesame lie on Vieth-Müller circles is explainedin Figure 9.13C. Of course, points with thesame horizontal disparity lie on such circles,irrespective of the orientation of the eyes.

It is tempting to interpret Figure 9.13 asshowing that structures on a Vieth-Müller cir-cle have the same retinal eccentricities in botheyes. Such an interpretation would justify ourchoice of coordinate system because the reti-nal images are the basis of visual perception.However, the step from angles in the afore-mentioned coordinate system to positions onthe retina assumes a certain orientation of theeyes around the line of sight (torsion). Theorientation of the eyes around the line of sightis more or less fixed for each gaze direction(Donders’ law), and is indeed more or lessappropriate for aligning the images in the twoeyes (Cooper, Burge, & Banks, 2011). Thus,the images of structures on the Vieth-Müllercircle in Figure 9.13A can be consideredto fall on corresponding retinal positions inthe two eyes. The images of structures onother such circles, not passing through thepoint of regard, do not fall on correspondingretinal positions, but the retinal disparity isthe same for all such structures because theyhave no relative disparity with respect toeach other.

Horizontal disparity is an important sourceof depth information for everyone with

14 Depth Perception

α

β

γ

δ δ

α'

α' = α

Vieth-Müller circle

a

b

b

c

(A) (B)

(C)

d

norelative

disparity

L R

SM

Figure 9.13 The Vieth-Müller circle: a circle through both eyes (A). If the angle between two structuresis the same for both eyes (δ in B), the angle that the lines of sight would make if one were to fixate eitherof the two structures is also the same: For both triangles the angles sum to π, so 𝛼 + 𝛽 + 𝛾 = 𝜋 (redtriangle) and 𝛼′ + (𝛽 − 𝛿) + (𝛾 + 𝛿) = 𝜋 (green triangle), which means that 𝛼 = 𝛼′. To see that the pointsfor which this holds fall on the Vieth-Müller circle consider any position on that circle (e.g., that of thestructure S in C). Because the two angles indicated in red are equal, and the sum of the angles in a triangleis π, b + c = a + d. The lines MR and MS have the same length (the radius of the circle), so the anglesthat they make with the line RS are also equal d = a + b (yellowisosceles triangle RSM). Substituting a+ b for d in b + c = a + d we can see that c = 2a, showing that for any position S on the circle the anglea will be the same.

normal binocular vision, but just knowingthe disparity is not enough to determinedistances or depths, because the horizontalretinal disparity only tells you the distance ifyou know the fixation distance. Differencesbetween the retinal positions of the imagesof a structure in the two eyes provide infor-mation about the structure’s relative distance(with respect to the structure that is fixated),but the magnitude of the difference thatcorresponds with a given difference in retinaldisparity increases with the fixation distance(Figure 9.14). The sign of the differenceglobally indicates whether the structure isnearer or farther than the structure that is fix-ated, but the relationship between horizontal

retinal disparities and positions in space isnot simple. The same is true for relativedisparities between two structures. First,positions with the same horizontal disparityare on circles through the eyes, so they arenot at the same distance from the observer;neither in terms of radial distance (as definedin Figure 9.1) nor in terms of a Cartesiandistance (forward-backward, as opposed toleft-right and up-down). Secondly, equalchanges in disparity do not correspond withequal separations in distance (see the 1∘differences between consecutive circles inFigure 9.15). The same horizontal disparitycorresponds with a much larger distancewhen fixating a more distant structure.

Binocular Depth Cues 15

(A) (B)

Figure 9.14 Retinal disparity. Example showingdifferent positions of the retinal images in the righteye for objects that are aligned for the left eye,when fixating either the blue (A) or the green (B)ball. The fixated ball obviously has zero retinal dis-parity. The retinal separation for an equivalent sep-aration in space decreases with distance. The reti-nal positions alone provide no information aboutdistance because they depend on which object isfixated.

Because the angles between structuresdo not change when we rotate our eyes, butpositions of the objects’ retinal images dochange (Figure 9.14), relying on relativedisparities between visible structures ratherthan on each structure’s retinal disparity tojudge separations in depth means that we donot have to worry about the orientations ofthe eyes. Not using gaze as a reference meansthat the resolution is determined by the retinalresolution at two structures’ images’ posi-tions rather than by the resolution of judgingthe vergence angle of the eyes and the retinalresolution at a single structure’s images’positions. In order to accurately judge therelative disparity the eyes must be directedat about the same distance as the structuresof interest, because if the retinal disparity istoo large the two retinal images will not beattributed to the same structure (resulting in

1 m

1°

2°

3°

Figure 9.15 Circles of positions with the samehorizontal disparity, in steps of 1∘ relative to posi-tions at the horizon (distance = ∞). Note that thechange in disparity with distance is approximatelyinversely proportional to the distance, as can beinferred from the fact that c = 2a in Figure 9.13C.

double vision: diplopia). Moreover, althoughknowing the relative disparity provides infor-mation about the depth order, in order tointerpret a horizontal relative disparity interms of an actual depth one must also knowthe overall distance. For retinal disparities itwould be logical to obtain such knowledgeby judging where one is fixating, both interms of distance and lateral position. Con-sidering that we usually direct our head moreor less toward where we are looking, one cansee from Figure 9.15 that the adjustmentsto the lateral position are not very critical:the distance from the head does not changemuch with small angles from straight aheadwhen following Vieth-Müller circles ratherthan circles centred on a point between theeyes (the origin of our measure of distance).However, the changes in distance betweenthe circles with identical changes in disparityare very different at different distances, soknowing the relative disparity between twostructures only provides reliable informationabout their separation in depth if one knowsthe distance of one of them.

16 Depth Perception

Relative binocular disparity is the mostsensitive source of information about dis-tance, with some people being able to reliablydetect depth differences of as little as 5 sec-onds of arc for foveal targets under certainconditions (McKee, 1983). The precisionwith which people can judge relative dispar-ity obviously decreases with distance fromthe fovea, both in depth and in the frontalplane (Schor & Badcock, 1985; Siderov &Harwerth, 1995; Siderov, Harwerth, &Bedell, 1999), but the decline in precision isnot dramatic. The precision is hardly poorer ifthe target is moving (Ramamurthy, Bedell, &Patel, 2005; Westheimer & McKee, 1975,1978). That it is the relative disparity that iscritical is evident from studies showing howdisparity is judged with respect to a slantedplane (Glennerster, McKee, & Birch, 2002;Mitchison & Westheimer, 1984). Comparingrelative disparities is therefore very precise,but interpreting relative disparities in termsof actual distances is complicated.

Vertical Disparity

Until now, we have only considered binocularseparations between objects that are on theplane through the eyes and the fixation point.The elevation of this plane is irrelevant forthe cues that we are interested in, so lookingup or down will not change anything. How-ever, a large part of the image on the retinais obviously not concerned with objects thatlie within this plane. In terms of horizontaldisparity this does not matter (if our assump-tions about the relevant coordinate systemare correct), but for structures that are not onthis plane, the angle with respect to the plane(and therefore the vertical retinal eccentricityof the images in the eyes) can differ betweenthe eyes. This is easiest to understand by con-sidering the difference in size between theimages in the two eyes (Gillam & Lawergren,1983): if a vertical rod’s distance differs for

the two eyes, the vertical image sizes willdiffer in accordance with the differences indistance. The sign of the difference dependson the position: if the rod is to the left, itsimage in the left eye will be larger than that inthe right eye. A difference in vertical imageposition is referred to as vertical disparity.From the aforementioned, it should be clearthat the vertical disparity increases with theazimuth and decreases with the distance(with respect to the head). Considering thatthe same uncertainty about the eyes’ orienta-tions leads to a much larger uncertainty aboutthe distance than about the lateral position(Figure 9.11), vertical disparities might beuseful for judging the viewing distance eventhough their dependence on both distanceand azimuth means that this requires anestimate of the direction of gaze. Moreover,knowing the direction could be circumventedby relying on the gradient of vertical dispar-ity throughout the fusible part of the scene(Brenner, Smeets, & Landy, 2001).

The curves in Figure 9.16 show theazimuths and distances at which structuresthat are at three vertical retinal eccentricitieshave three different values of vertical dispar-ity. Of course, if there is vertical disparity,the vertical retinal eccentricities are not thesame in both eyes. We therefore consider themean vertical retinal eccentricities of the twoeyes to be the overall vertical retinal eccen-tricity. For nearby targets at large verticaleccentricities, vertical disparity can be quitelarge. The vertical disparity depends on theazimuth with respect to the head, the verticalretinal eccentricity, and the distance. Thus, ifyou know the direction of gaze, the verticaldisparity at a given vertical retinal eccen-tricity could provide information about thedistance (as long as the structure of interestis not straight in front of you). However,vertical disparities are quite small, unless thestructure of interest is extremely nearby, sotheir resolution (see Figure 9.12) is probably

Active Depth Cues 17

1 m

Figure 9.16 Vertical disparity. The curves showpositions for which the difference in vertical reti-nal eccentricity (vertical disparity) is 10, 20, or30 minutes of arc (indicated by increasing linewidth), while the mean vertical eccentricity is 5,12, or 30∘( red, green, and blue, respectively).On the left, the eccentricity with respect to theleft eye is larger. On the right, the eccentricitywith respect to the right eye is larger. Eccentric-ity is expressed as the vertical angle at the eyewith respect to the plane through the eyes and thefixation point.

normally insufficient for judging the distanceof individual objects. The vertical disparityincreases with increasing vertical retinaleccentricity, so vertical disparities are largerat large retinal eccentricities, but this advan-tage is probably alleviated by the decreasein resolution with retinal eccentricity. Never-theless, despite vertical disparities thereforeprobably not being useful for directly judgingan object’s distance (Cumming, Johnston, &Parker, 1991; Sobel & Collett, 1991), theoverall pattern of vertical disparities mightbe used to obtain estimates of the viewingdistance with which to judge distances ordirectly scale horizontal disparities (Adamset al., 1996; Backus, Banks, van Ee, &Crowell, 1999; Brenner et al., 2001; Duke,Oruç, Qi, Backus, 2006).

ACTIVE DEPTH CUES

We already mentioned that people mightactively accommodate to remove blur,and use the amount of accommodation tojudge the distance (Watt et al., 2005). A moreimportant active depth cue is motion parallax.Motion parallax is similar to binocular dis-parity in that it is based on having differentviews of the same scene (Figure 9.17).In motion parallax the different views areobtained at different moments. As a result,interpreting the changing image of the sceneas being caused by a change of viewpoint,and using this to derive structures’ distances,is only straightforward if the scene is sta-tionary and one knows one’s own movementquite reliably. In binocular vision, the scenebeing stationary is not an issue as the twoviews are obtained simultaneously. More-over, the distance between the eyes is fixedso we can consider it to be known with highaccuracy. For motion parallax, if the scene isindeed stationary, the resolution for detectinga difference in depth, or for detecting thedepth order, might depend on the extent ofself-motion (Figure 9.18), although the veloc-ity of self-motion is probably also important.

In order to interpret motion parallax interms of actual distances or depths one has toalso judge one’s own movement or scale themotion information in some other manner.Thus, motion parallax requires scaling ofthe retinal motion just as binocular visionrequires scaling of the horizontal dispar-ity. There are more similarities betweenmotion parallax and horizontal disparity. Forinstance, in both cases differences in depthcould be judged from the directions withrespect to the eye(s), but they could also bejudged from changes in relative positionswithin the retinal image(s) of the scene: thethree objects being aligned for the left eyebut not for the right eye in Figure 9.14 cor-respond with the three objects being aligned

18 Depth Perception

Figure 9.17 The analogy between motion paral-lax and horizontal disparity. Both rely on differ-ences in the direction to the structure of interestfrom different positions. For horizontal binocu-lar disparity the different positions are the posi-tions of the two eyes. For motion parallax thedifferent positions are positions of the same eye atdifferent moments. In this example, the observeraligns the three objects with respect to the right eyeby moving to the left, in analogy with the align-ment in the left eye in Figure 9.14. The fartherthe object from the eye, the smaller the changein angle with respect to the eye when one moves.This is evident when looking out of a train window:Nearby objects pass quickly, while distant objectspass slowly.

after but not before the head moved to theleft in Figure 9.17. Despite the similarities,there are also some fundamental differences,

86420 10

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Distance (m)

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th r

esol

utio

n (c

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lateral headmovement

0.5 cm1 cm2 cm4 cm8 cm

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Figure 9.18 Resolution of motion parallax as adepth cue for various extents of lateral motion ofthe head. Values based on a retinal resolution of 1′

arc, assuming that the scene is static. Details as inFigure 9.6.

which is probably why performance is notidentical when based on matched versionsof the two cues (Bradshaw, Hibbard, Parton,Rose, Langley, 2006).

When discussing motion parallax theemphasis is often on modest lateral self-motion (i.e., self-motion in a directionorthogonal to the distance). However, onecan also obtain information about an object’sdistance from changes in its image size asone moves toward it (Peh, Panerai, Droulez,Cornilleau-Pérès, & Cheong, 2002). Forextensive self-motion one must obviouslyconsider that most structures’ distanceswill constantly be changing. Although theobserver’s movements do not influence theactual separations between static objects, ortheir sizes, it does influence the extent towhich the separations between structures arein depth, so when it is important to isolatethe depth component, such as when judginga surface’s slant, one also has to consider thecontinuous changes. If it is safe to assume thatthe whole scene is static, it is theoreticallypossible to judge the instantaneous relative

Combining Depth Cues 19

depths within the whole scene from the opticflow (Gibson, 1979; Koenderink, 1986). Thedepths could be scaled by information aboutone’s own motion, or by any known distance,to obtain judgments of the actual distancesand depths. Such interpretation of the opticflow is presumably responsible for some ofthe ways we consider distances during every-day tasks such as locomotion (Duchon &Warren, 2002).

To know whether it is safe to interpret spe-cific image motion in terms of distance, onemust verify that the image motion is not theresult of the object in question itself movingrelative to other objects in the scene. Due tothe regularities in the retinal image motionthat is caused by self-motion, it is usuallypossible to reliably determine the direction ofself-motion (van den Berg, 1992; Warren &Hannon, 1988) and separate the influencesof self-motion from ones of object motion(Brenner & van den Berg, 1996; Warren &Rushton, 2008, 2009). When there is limitedvisual information, not all motion parallax isinterpreted as depth. In an extensive series

of experiments, Gogel and colleagues haveshown that a static object appears to movein response to lateral self-motion when itsdistance is misjudged (Gogel, 1990; Gogel &Tietz, 1973), rather than the judgment ofdistance being adjusted to conform to theobject being static. This is not the case whenthere is more visual information (Glenner-ster, Tcheang, Gilson, Fitzgibbon, & Parker,2006), so apparently people only assume thatthe scene is static if there is support for thisfrom within the image.

COMBINING DEPTH CUES

Figure 9.19 looks strange because the bananamust be closer to us than the apple, becauseit occludes part of the apple, but a numberof cues suggest that it is not. On the left sideof the picture, the image of the banana is abit small in relation to the apple. Of course,any object could give rise to an image ofany size, because image size scales with theobject’s distance (the larger the distance,

Figure 9.19 Why does this picture look strange? The banana occludes the apple, but its position on thetable indicates that it is behind the apple. The image sizes also suggest that the banana is slightly fartheraway. The shadows confirm that the apple and the right side of the banana are resting on the table.

20 Depth Perception

the smaller the image size), but the fact thatthe banana occludes the apple constrains thepossible distances. This must therefore be anexceptionally large apple, or an exceptionallysmall banana. Another cue that suggests thatthe banana is farther away than the apple isthat its base is higher in the visual field. Thatcould just mean that the banana is suspendedin mid-air, but besides that being unlikely fora banana, the banana’s shadow on the rightside of the picture confirms that it is lyingon the table. The conflict between the cuesmakes the picture look strange.

Until now, we have considered all thedifferent depth cues in isolation. We saw thatonly knowing the orientations of the eyescould directly provide an estimate of the dis-tance, and this cue’s resolution is quite poor,except perhaps at very short distances. Mostother cues can provide information about thedepth order, but require scaling to provideinformation about actual distances or depths.Most of them are also based on assumptionsthat may or may not be correct. Some canonly be used in certain circumstances (if thesurface is textured; if one is free to move). Inall cases the resolution varies with distance,usually decreasing monotonically as thedistance increases. Since the assumptionsunderlying different cues are not the same,and the required scaling and change in sen-sitivity with distance and other parametersare also different for different cues, it shouldnot surprise us to see conflicts between theestimates of distance or depth provided bydifferent cues, although the conflicts willnormally not be as evident as in Figure 9.19.

The abundance of depth cues means thateither one cue has to be selected, or they haveto be combined in some manner (Cutting &Vishton, 1995). Because some cues provideinformation faster than others (van Mierlo,Louw, Smeets, & Brenner, 2009), one mayeven have to switch between cues or adjustthe way they are combined as time passes.

Figure 9.20 The fact that the blue ball is onlyvisible to the right eye indicates that it must bewithin the range indicated by the five blue balls.

In a few cases, combining information fromwhat we have been considering different cuescould even provide additional information.For instance, the fact that a visible structureoccludes the image of a second structurein one eye limits the possible positions ofthe partly occluded structure in terms ofrelative disparities (Figure 9.20; Harris &Wilcox, 2009). The most obvious exampleof combining cues is combining the ori-entation of the eyes (ocular convergence)with horizontal disparities to judge depths.Another straightforward example is directlycombining horizontal and vertical disparities(Read, 2010; Rogers & Bradshaw, 1995),for instance to determine the most likely

Combining Depth Cues 21

structure and distance given the combinationof disparities (Bülthoff, 1991).

It seems obvious that one must somehowconsider how reliable the diverse distancecues are when combining or selectingbetween them. One must also consider whatone is trying to judge. For instance, occlusioncan provide very reliable information aboutwhich of two objects is closer than the other,but not about how much closer it is. Similarly,some cues may be more suitable for judgingdistances, whereas others may be more suit-able for judging depths. Which cues are mostsuitable also depends on the circumstances.In all cases, it would make sense to combinethe cues to obtain the most likely value of thejudgment of interest, rather than only relyingon the “best” cue. A relatively simple way toachieve this is by averaging them in the waythat maximizes the overall precision (Ernst &Banks, 2002; Hillis, Watt, Landy, & Banks,2004; Jacobs, 1999; Muller et al., 2008;van Beers, Sittig, & Denier van der Gon,1999). If the cues all provide estimates withindependent normally distributed precisions(which is not necessarily always the case;Bradshaw & Rogers, 1996; Oruç, Maloney, &Landy, 2003), the combination that gives thebest overall estimate of distance is a weightedaverage, where the weights are inversely pro-portional to the cues’ precisions. Presumably,the estimates are converted into commonunits before being combined in this manner(Landy, Maloney, Johnston, & Young, 1995).This kind of weighted averaging is oftenreferred to as optimal cue combination.

Optimizing precision is only the best wayto combine cues if the differences betweenthe estimates are really due to measurementerrors. For depth perception, most cues arebased on assumptions, so one or more of theassumptions being violated could also causediscrepancies between the cues. If peopleconsider the likelihood of assumptions beingviolated, they should reduce the weight given

to a cue when faced with evidence that anassumption that is required for using thatcue is probably not justified (Knill, 2007;Mamassian & Landy, 2001; Muller et al.,2009). In general, large cue conflicts couldindicate that an assumption must be violated.Binocular cues do not depend on assumptionsthat can be violated (although one may havefailed to correctly match the correspondingstructures in the two eyes). Pictorial cuesdo rely on assumptions that can be violated.Nevertheless, when judging slant, peopledid not increase the weight given to binoc-ular cues with respect to pictorial cues asthe cue conflict increased, except when thecue conflict was extremely large (van Ee,Adams, & Mamassian, 2003; van Ee, vanDam, & Erkelens, 2002). People did rely lesson retinal image size for judging distancewhen object size varied more on previoustrials (Seydell, Knill, & Trommershäuser,2010; Sousa et al., 2013), indicating that insome cases people do consider whether theassumption underlying the use of the cue inquestion is likely to be correct (here, that theapproximate object size is known). That thereis some flexibility in assigning weights tovarious cues is evident from a study in whichfeedback was provided about the accuracy ofthe judgment. In that case, more weight wasgiven to the more reliable cue than it wouldget based on its precision alone (van Beerset al., 2011).

One may be surprised to notice when com-paring the resolution of the cues described inFigures 9.7, 9.10, 9.12, and 9.18 that heightin the visual field is often the cue with thehighest resolution. In particular, one may besurprised that its resolution is better than thatof binocular vision. The reported resolutionfor height in the visual field is the resolutionfor determining the depth order of structuresthat are both directly on a horizontal surface.Binocular vision is more flexible in terms ofthe layout of the items in the scene. The main

22 Depth Perception

reason for height in the visual field havinga higher resolution is that it is based onvertical retinal separations in each eye ratherthan on differences between the horizontalseparations in the two eyes (or possibly inthe elevation of gaze rather than the con-vergence of the eyes). We recently found(Brenner, Driesen, & Smeets, 2014) that anequivalent difference between the resolutionof changing elevation (whereby the surface isvertical rather than horizontal) and changingbinocular information results in hitting afalling ball primarily being determined bythe changing elevation, rather than by thechanging binocular information or changingimage size (that potentially provides directinformation about the time to contact; Tresil-ian, 1993). This suggests that in daily life wemay often also rely quite strongly on heightin the visual field for judging distance.

CONSISTENCY

If people optimize the way they combinethe available cues to obtain the best possibleestimate of the attribute of interest, boththe cues and their weights will differ fordifferent judgments. This could give rise toinconsistencies between judgments. Besidesinterpreting the available information differ-ently, people may even gather informationdifferently when making different judg-ments by scanning the scene differentlywith their eyes. Thus, we should not be toosurprised by inconsistencies between errorsin, for instance, judging size and distance(Kilpatrick & Ittelson, 1953). Such inconsis-tencies are very clear in studies of perceivedmotion in depth, where the perceived dis-placement can be quite inconsistent with theperceived speed (Brenner, van den Berg, &van Damme, 1996). It should be noted thatif inconsistencies arise from combining cuesin different ways for different judgments,there should still be a reasonable correla-tion between such judgments. Indeed, such

correlations have been found, even when thejudgments themselves are far from veridical(Brenner & van Damme, 1999).

People make systematic errors whenasked to compare distances in depth withlateral or frontal separations (Kudoh, 2005;Loomis, Da Silva, Fujita, & Fukusima, 1992)and when performing exocentric pointingtasks (Cuijpers, Kappers, & Koenderink,2000; Kelly, Loomis, & Beall, 2004). Suchinconsistencies between judgments fromdifferent positions and in different direc-tions have been contrasted with the abilityto reliably walk to previously seen targetswith one’s eyes closed (Kudoh, 2005), andto the path that one takes when walking to apreviously seen target with one’s eyes closednot influencing where one ends up, evenunder conditions in which one does makeconsiderable errors (Philbeck, Loomis, &Beall, 1997). Combining different cues oreven the same cues with different weightsfor different judgments could be responsiblefor the lack of consistency between the judg-ments. The cue combinations could thereforebe influenced by subtle details of the way inwhich the response is measured or the com-parison made. For instance, asking for equaldistances might encourage people to partlyrely on directly comparing retinal imagesizes, which, of course, is not a veridical cuefor judging separations in the world if one ofthe lines is receding in depth. This cue cannotbe used to judge an object’s distance fromoneself, so it will not influence blind walking.

PERCEIVED MOTION IN DEPTH

A clear example of inconsistencies in depthperception is the comparison of perceivedmotion in depth with the perceived changesin distance due to the same motion stimulus(Brenner et al., 1996). Retinal image size isnot the main cue for judging distance, butchanging image size plays a very prominentrole when judging motion in depth (Brenner

The Special Role of Distant Structures 23

et al., 1996; Gray & Regan, 1996; Regan,1997; Regan, Kaufman, & Lincoln, 1986).A constant image size can completely over-rule changing binocular information aboutmotion in depth if the image is large (Erke-lens & Collewijn, 1985; Glennerster et al.,2006). The reason that image size plays aso much more prominent role in judgmentsof motion in depth than in judgments ofdistance is probably that rather than havingto assume that the structure of interest has acertain size, of which one cannot usually bevery certain, one only has to assume that thesize is not changing.

The motion of a small dot in a sceneconsisting of static dots is easier to detect ifthe small dot is moving in the same directionin both eyes than if it is moving in oppositedirections in the two eyes (Sumnall & Harris,2000, 2002). Perhaps the binocular cue iscombined with other depth cues, such as thenot-changing size and required accommoda-tion, despite the very small size of the target.This would be consistent with judgmentsof other attributes being determined by theresolution of the combined cues, rather thanby the resolution of the individual cues (Hilliset al., 2004; Lugtigheid, Brenner, & Welch-man, 2011; Sousa et al., 2009). However,it is also plausible that specialized motiondetectors detect lateral motion, while motionin depth is detected on the basis of changesin disparity rather than on the basis of dif-ferences between the motion in the two eyes(Harris & Rushton, 2003).

THE SPECIAL ROLE OF DISTANTSTRUCTURES

When a single object is presented in thedark, people tend to misjudge its distance(e.g., Gogel & Tietz, 1973). Different studiesreport different systematic errors, but allagree that the range of distances is under-estimated: Close objects seem to be fartheraway than they are, and far objects seem to

be nearer than they are. This underestimationof the range of distances can be interpretedas subjects considering certain distances tobe more likely than others (a distance prior).For instance, they might assume that thefarthest an object can be within the room inwhich the experiment is conducted is about2 m, or that the farthest they can see whenlooking downward is about 1.5 m belowtheir eyes (the normal distance to the groundbeneath their feet). They may even assumeboth of the above, so that the prior about anobject’s distance depends on circumstancessuch as the direction in which one is looking,possibly contributing to the systematic ten-dency to overestimate vertical with respectto horizontal distances (Higashiyama &Ueyama, 1988). However, for understandingthe underestimation of the range of distancesof isolated objects, the absence of a struc-tured background might also be relevant,because there are numerous examples ofdistant structures playing a role that theyobviously cannot play if they are not visible.

The most obvious example of distantstructures playing a special role is the role ofthe horizon when judging height in the visualfield with respect to the horizon (Gardneret al., 2010), but distant structures also appearto play a role in interpreting binocular dispar-ities and optic flow. In binocular vision thereis a minimal angle of zero degrees betweenthe directions with respect to the two eyeswhen a structure is very far away so that thelines of sight of the two eyes would be almostparallel if one were to try to fixate it. Thisminimal angle can be used to limit the rangeof possible distances for an object of interest,because no structure in the scene could giverise to a horizontal disparity correspondingwith a vergence angle of less than 0∘ (Sousaet al., 2010). In a similar manner, because thedirection to distant structures hardly changeswhen we move, it is reasonable to relatethe changes in the directions to objects ofinterest to the direction to distant structures

24 Depth Perception

(Brenner & van den Berg, 1996; van denBerg & Brenner, 1994). Relating distancesto the most distant structure will obviouslyresult in an underestimation of the range ofdistances if the farthest structure is the onlystructure. Thus, some of the systematic errorsthat are found when isolating cues might beartifacts of removing the distant structuresthat are normally used to help scale the cuesthemselves.

SIZE AND SHAPE

We have to judge distances from ourselvesfor knowing where things are. We also needjudgments of distance to interpret retinaldepth cues such as horizontal disparity interms of separations between structures indepth, to determine how far objects are fromeach other, or how far they extend along theline of sight. Without any scaling, retinaldepth cues can only tell us the depth order.Judgments of distance are therefore notonly essential for judging distance, but alsofor judging size and shape. Because thischapter is about depth perception, we haveemphasized the use of retinal image size tojudge distance, assuming or knowing that theobject has a certain size. The retinal imagesize is obviously also needed to judge anobject’s size (dimensions in the directionsof azimuth and elevation), given an estimateof the distance (r = d∕ tan 𝛼 in Figure 9.8).Using object size to estimate distance andjudged distance to estimate size providesan obvious problem if they are estimatedsequentially. It would therefore make senseto estimate both together rather than sequen-tially by finding the combination of both thatis most consistent with all the available infor-mation (the Bayesian approach proposed inBülthoff, 1991). In that case, it may appearthat only misjudging the retinal image sizecould lead to the inconsistencies betweenthe two judgments that have often been

reported (e.g., Kilpatrick & Ittelson, 1953).However, this is not necessarily true, becauseif the reported judgments were not madecompletely simultaneously, both the activeacquisition of information (for instance byeye movements) and the way the cues werecombined could have been optimized for theinstantaneous judgment of interest, even ifthe other judgment was also estimated atthat time. If so, the inconsistency would beacross moments in time, rather than betweenthe attributes (size and distance), with thejudgment that one is going to use (to performan action or report about) at each momentdetermining how the cue combination isoptimized at that moment.

Size and distance have a very straightfor-ward relationship. The relationship betweenshape and distance is more complex. Shapecould be derived from separate judgments ofsize and depth, but judgments of shape couldbe made without estimating either the actualsize or the actual depth, because only therelationship between the extents in differentdirections needs to be known. For instance,if an object is rotating, the motion in itsimage provides information about the shapewithout first requiring any scaling by dis-tance (structure from motion; Figure 9.21A;Todd, 1985). Similarly, texture cues to slantcan inform you about surface orientationeven when the distance remains unknown(Figure 9.21B; Rosenholtz & Malik, 1997).Such information could be used to recognizeobjects by their shape. It is less evident that itcould be useful for knowing where an objectis and whether one can grasp it, but knowingthe shape could theoretically contribute tojudging the distance and size because theextents in different directions (what we havebeen calling size and depth) scale differentlywith distance, so knowing the shape couldinfluence what is considered to be the mostlikely distance given the combination ofdisparities and retinal extents.

Conclusion 25

structure from motion

slant from texture

(A)

(B)

Figure 9.21 Shape and slant cues that are independent of distance. (A) Rotating a cube leads to thesame image motion if the cube is both twice as far and twice as large. (B) Judgments of local slant fromthe separation of texture elements in the retinal image are also independent of the viewing distance.

As previously mentioned, people mis-judge the distance of isolated objects in thedark. It is therefore not surprising that theyalso misjudge the shape of simulated isolatedobjects: the horizontal disparities and lateralextents are scaled by incorrect estimatesof distance (Johnston, 1991). Perhaps sur-prisingly, rotating such simulated objects tomake people see the shape correctly does notaffect the judged distance (as measured bypointing distance and judged size; Brenner &van Damme, 1999). Thus, people tolerateinconsistencies between the size and depththat are derived from scaled retinal cuesand the shape that is derived from unscaledretinal cues when judging shape and distance,rather than searching for the combination ofdistance and shape that is most consistentwith all the cues. Note that this is a differentcase than when misjudging distance givesrise to perceived motion, and vice versa(Ono & Ujike, 2005), because in that case theconflict is attributed to another percept. In the

case of rotating an object providing reliableinformation about its depth, the conflict isnot attributed to the judged size or perceiveddistance. A difficult task for future researchwill be to explain why conflicts betweenthe ways in which cues are interpreted aretolerated for some combinations of attributes,but not for others.

CONCLUSION

We have seen that many cues contribute tojudgments of distance. They do so to varyingextents depending on the circumstances. Forinstance, height in the visual field is a veryuseful cue as long as you are interested inobjects resting on flat surfaces at a knownheight. Similarly, binocular cues are veryreliable as long as the object of interest isnearby. Our description of the many cuesand their limitations is obviously far fromcomplete. Although the main cues have been

26 Depth Perception

known for many decades, research on theways in which they are combined and on howthey influence each other and are influencedby specific aspects of the surrounding isrelatively new.

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