Selective biasing of stereo correspondence in an ambiguous stereogram

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In spite of numerous studies in stereoscopic perception, it is still not clear how the visual system matches features between the two eyes. One reason is that these previous studies used stimuli that presented little perceptual ambiguity, so the
  Selective biasing of stereo correspondence in an ambiguousstereogram Ross Goutcher  * , Pascal Mamassian Department of Psychology, University of Glasgow, 58 Hillhead Street, Glasgow G12 8QB, UK  Received 27 November 2003; received in revised form 26 July 2004 Abstract In spite of numerous studies in stereoscopic perception, it is still not clear how the visual system matches features between the twoeyes. One reason is that these previous studies used stimuli that presented little perceptual ambiguity, so the correspondence prob-lem had only one solution. We present here a novel stimulus that presents a more complex correspondence problem. This stimu-lus is inspired by ‘‘wallpaper’’ stimuli and was specifically designed to put into conflict two possible constraints underlying stereocorrespondence matching. These constraints are the nearest neighbour matching rule––that biases surfaces towards the horo-pter––and the nearest disparity rule––that biases surfaces to be smooth. By varying the contrast of adjacent image features in thisstimulus, we were able to reveal and quantify a preference for nearest disparity matching. The magnitude of this preference isdependent upon the magnitude of possible disparities in the scene and is consistent with the idea that the visual system seeks tominimise local differences in disparity. We discuss these results with regard to the use of prior constraints in models of stereomatching.   2004 Elsevier Ltd. All rights reserved. Keywords:  Stereopsis; Correspondence problem; Prior constraints; Wallpaper illusion 1. Introduction The perception of depth from binocular disparitydepends upon the correct matching of correspondingfeatures between the left and right eyes   images. In com-plex scenes the visual system may be confronted withmultiple candidate features for matching and must re-duce the number of possible correspondences in orderto attain a stable, unified representation of the scene.The resolution of this correspondence problem forstereo vision has been a topic of near constant interestfor researchers in the 40 years since Julesz   popularisa-tion of the random dot stereogram (Julesz, 1964). Manycomputational models of the correspondence matchingprocess have been proposed (e.g. Jones & Malik, 1992;Marr & Poggio, 1976, 1979; Pollard, Mayhew, & Frisby,1985; Prazdny, 1985; Prince & Eagle, 2000; Qian & Zhu,1997; Read, 2002a, 2002b; Sato & Yano, 2000; Tsai &Victor, 2003). To resolve the correspondence problem,such models must limit possible matches with a seriesof constraints or rules. Models often differ in the con-straints they use and the extent to which these are em-ployed in an explicit (e.g. Marr & Poggio, 1976, 1979;Pollard et al., 1985) or implicit (e.g. Prince & Eagle,2000; Qian & Zhu, 1997; Read, 2002a, 2002b) manner.Constraints on matching include feature similarity,matching to the nearest neighbour or nearest disparity,and considering only epipolar matches (for an extensivereview of proposed matching rules, see Howard &Rogers, 2002).In this paper, we concentrate on the visual system  sadherence to the solutions provided by nearest neigh-bour, nearest disparity and contrast similarity matchingrules.  Nearest neighbour  matches (Arditi, Kaufman, & 0042-6989/$ - see front matter    2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.visres.2004.08.025 * Corresponding author. E-mail address: (R. Goutcher) Research 45 (2005) 469–483  Movshon, 1981) minimise the absolute disparity of im-age features. That is, they select the correspondencesolution that places the image feature closest to the hor-opter. In contrast, the  nearest disparity  rule (Marr &Poggio, 1976, 1979; McKee & Mitchison, 1988; Mitchi-son & McKee, 1987a, 1987b) minimises the relative dis-parity of image features, giving the correspondencesolution that minimises the difference in disparity be-tween nearby points. As such, the nearest disparity rulehas been thought of as a   smoothness   constraint and isoften referred to as a continuity or cohesiveness con-straint. The contrast similarity rule is one of a series of constraints––including also, orientation similarity––con-cerned with feature similarities. Under this constraint,matches are made between features of maximally similarcontrast (Smallman & McKee, 1995).Despite the suggestion of so many constraints in theliterature, very little research has been conducted toexamine the competition between matching rules. Thereis precious little empirical data showing which solutionthe visual system adheres to when confronted with mul-tiple plausible matches (i.e. multiple matches that satisfyone or more matching constraint). Zhang, Edwards, andSchor (2001) recently investigated this issue. Using aperiodic stimulus consisting of a one-dimensional lumi-nance Gabor flanked, above and below, by two similarGabors, they found that matching tends towards thesolution that minimises the disparity between adjacentsurfaces; that is, the solution that minimises relative dis-parity. This finding was particularly interesting sincetheir stimulus put nearest neighbour and nearest dispar-ity matching rules into conflict. Their experiments thussuggest that the process of correspondence matching isconcerned more with finding solutions that satisfy thenearest disparity constraint than those that satisfy thenearest neighbour constraint.One important characteristic of the study of  Zhanget al. (2001) is that their stimuli consisted of three iso-lated objects rather than a single surface, so their resultcan be interpreted as a contextual effect. Furthermore,the stimuli used by Zhang et al. (2001) contain a poten-tial confound between local changes in disparity and thetotal change in disparity across the scene. We here definethis maximum change in disparity across the scene as the  global relative disparity . This distinction between localand global relative disparity is clearer if one considersthe relative disparities that arise, at a global and locallevel, with different stimuli. Consider the stimuli de-picted in Fig. 1. Fig. 1a illustrates a single, fronto-paral- lel surface in depth, with two local areas––  x  and  y  ––highlighted. At area  x  relative disparity is zero, sinceall points are at the same depth. However, at area  y , rel-ative disparity is determined by the difference in dispar-ity between the stimulus and a zero disparity surround.Thus, although relative disparity is zero across much of the image, the relative disparity across the entire im-age––the  global relative disparity  ––is determined bythose few areas containing a difference in disparity be-tween stimulus and surround. Figs. 1b–d illustrateincreasingly complex stimuli, where the presence of localareas with zero relative disparity is increasingly scarce.In such stimuli the global relative disparity is determinedby the largest change in disparity across the entire scene.For example, in the case of the squarewave illustrated inFig. 1b, the global relative disparity is the peak-to-trough disparity of the waveform.Readers should note that a stimulus with a small glo-bal relative disparity may, locally, have a great deal of variation in disparity. Consider the transparent surfacesdepicted in Fig. 1d. There are no local areas containingzero relative disparity in such a stimulus since, over alocal area, both surfaces are visible. However, the globalrelative disparity––determined by the disparity betweenfront and back surfaces––may be small if the separation yxzxxxy SquarewaveFronto-parallelTransparencySawtooth (a) (b)(c) (d) Fig. 1. Distinction between locally and globally defined relativedisparities. All four figures (a–d) have identical global relativedisparity––defined as the largest change in disparity across the entirescene––but different local relative disparity structure. (a) Illustration of a fronto-parallel surface located behind a frame. Highlighted areas  x and  y  show points where relative disparity is zero ( x ) and non-zero (  y ).Global relative disparity is determined by the disparity between thesurface and the surround (  y ). (b) Illustration of a squarewavemodulation in depth. The squarewave contains many areas whererelative disparity is zero ( x  and  y ), though fewer than (a). Globalrelative disparity is determined by the peak-to-trough relative disparityof the waveform ( z ). (c) Illustration of a sawtooth modulation indepth, which contains no areas with a relative disparity of zero (e.g.area  x ). Global relative disparity is determined by the disparity at thesharp depth transitions. (d) Illustration of two overlapping transparentsurfaces in depth. Local relative disparity is never zero since bothsurfaces are present within any local area ( x ). Global relative disparityis determined by the disparity between front and back surfaces.470  R. Goutcher, P. Mamassian / Vision Research 45 (2005) 469–483  between the surfaces is small. Conversely, the global rel-ative disparity of a single fronto-parallel surface may belarge if the disparity between stimulus and surround islarge.In the stimulus used by Zhang et al. (2001) partici-pants   responses could have been driven by a preferenceto minimise either local or global relative disparity. Inthis paper, we detail a novel ambiguous stimulus thatputs nearest neighbour and nearest disparity constraintsinto conflict. By varying the disparity of surfaces arisingfrom the nearest disparity or nearest neighbour match,the global relative disparity of both matches may beequated. This allows us to investigate whether the near-est disparity bias reported by Zhang et al. (2001) is dueto a locally or globally implemented nearest disparityconstraint.In addition to the manipulation of disparity, manip-ulating the contrast similarity of image features in thestimulus can selectively bias one of the two percepts(see also Anderson & Nakayama, 1994 and Smallman & McKee, 1995 for similar uses of contrast to biasmatching). Such selective biasing may be used to exam-ine the relative strength of the solutions provided by dis-tinct matching constraints. To pre-empt our results, weconfirm and quantify the preference for nearest disparitymatching found by Zhang et al. (2001) and show thatthe tendency to match to the nearest disparity is affectedby the depth resulting from nearest neighbour and near-est disparity solutions. These results demonstrate a lo-cally implemented bias for the nearest disparity, ratherthan a bias concerned with minimising the global rela-tive disparity. We further suggest that the stimulus andmethod detailed here could be used to reveal the impactof numerous factors on stereo matching.Our stimulus makes use of the stereo   wallpaper illu-sion   (Brewster, 1844) and takes the form of a modifiedperiodic   wallpaper   dot stereogram. Wallpaper stimulirepeat an identical pattern multiple times from left toright, thereby increasing the ambiguity of the corre-spondence between features of the left and right eyes.Assuming no strong idiosyncratic preference for crossedor uncrossed disparities, the typical wallpaper stimulusis resolved to elicit the perception of a single plane eitherin front of, or behind fixation. Such percepts correspondto the solutions given by both the nearest disparityand nearest neighbour constraints. By adding small, sys-tematic offsets to image features we may put these con-straints into conflict. When in conflict, two qualitativelydifferent percepts result. The first percept consists of asingle opaque fronto-parallel surface and the secondconsists of two semi-transparent fronto-parallel sur-faces. The perception of a single, fronto-parallel surfaceindicates adherence to a nearest disparity constraint,since, in such a stimulus, relative disparity is zero acrossalmost the entire image. Conversely, the perception of stereo transparency corresponds to the nearest neigh-bour match, since, in the stimulus, such surfaces lie clo-ser to the point of fixation than the single surface match.Note that the global relative disparity of these two per-cepts may be equated if the relative disparity betweenthe single surface and fixation is the same as the relativedisparity between front and rear transparent surfaces.Fig. 2a and b provide examples of similar stimuli thatlead to such different percepts.The dominance of either the nearest disparity or thenearest neighbour constraint can thus be studied bymonitoring the perceptual preference for a single opaqueor two transparent surfaces. In an attempt to quantifythis dominance further, we also manipulate the similar-ity of the features across eyes in order to create a stim-ulus that will equally likely be perceived as opaque ortransparent. In experiment 1, we describe the construc-tion of our stimuli in detail, and provide results fromthe manipulation of the contrast similarity of the imagefeatures for two stimulus durations. In experiments 2and 3, we detail the effects of varying disparity on theresolution of our stimulus and discuss the implicationsof these findings for models of stereo correspondencematching. 2. Experiment 1 In this experiment, we seek to assess the bias fornearest disparity matching found by Zhang et al.(2001). By varying the contrast between image featuresin an ambiguous stereogram we bias matching betweenconflicting nearest neighbour and nearest disparitysolutions. We assess the bias for nearest disparitymatching for two presentation durations. In experiment1a stimuli were presented for 2 s in a raised temporalcosine window. In experiment 1b stimuli were presen-ted for 80 ms in a square-wave temporal window.Short presentation durations were used to discountany potentially confounding effects of vergence eyemovements.  2.1. Stimuli  2.1.1. Structure of the stimulus We first describe the wallpaper pattern that forms thebasis of our stimulus, and then the specific way in whichthis pattern is modified. The wallpaper pattern consistedof 16 rows of about 14 dots within an area of 7.48  · 7.48  , at a viewing distance of 80 cm (i.e. dot densitywas 4.0 dots per degree squared). Each dot was a smallcircular Gaussian blob measuring 10.5 0 . The verticalseparation between two rows was 27.1 0 and the horizon-tal distance between two consecutive dots was  a  = 30.1 0 (see Fig. 3a). In order to avoid a regular grid, the firstdot of each row was randomly shifted horizontally (byan amount  a 0  between 0 and  a ). The vertical edges of  R. Goutcher, P. Mamassian / Vision Research 45 (2005) 469–483  471  this pattern were then smoothed to diminish the visibil-ity of unmatched dots between left and right half-images. The smoothing was obtained by multiplyingthe pattern with a rectangular window attached to twohalf-Gaussian distributions on either side (the centresof the Gaussians were at 1.87   and 5.61   and the stand-ard deviation was 0.63  ). Note that this smoothing oper-ation had an effect only on the left and right sides of thestimulus, leaving the luminance constant across rows of dots. When fused, this wallpaper stimulus would lead toa single fronto-parallel plane either in front of or behindfixation.The basic wallpaper stimulus can now be modified toalso allow transparent percepts. Within each row, dotswere shifted alternately by an amount + d  or   d  in onehalf-image, and by   d  or + d  in the other half-image(here,  d  was set to 6.02 0 : see Fig. 3b). Finally, the lefthalf-image was shifted by  e  and the right half-imageby   e  to give the stimulus a pedestal disparity of    2 e ( e  was set to 3.01 0 : see Fig. 3c).  2.1.2. Dominant percepts As with any stereogram, there are multiple possibleinterpretations of the adapted wallpaper stimulus shownin Fig. 2. However, only two percepts are dominanthere. The first of these corresponds to the nearest neigh-bour solution while the second corresponds to the near-est disparity solution. We describe these two types of percept in turn.When the dots are matched according to the nearestneighbour constraint, dot A in Fig. 3c will matchwith dot C, and dot B with dot D. From that figure,one can readily see that these two pairs of dots willlead to two different disparities, consistent with two Fig. 2. An example of the ambiguous stereogram stimuli used in the experiment. The upper stereogram (a) shows the experimental stimulus withcontrast ambiguity biased towards the transparent––nearest neighbour––percept. The lower stereogram (b) shows the stimulus with a singlesurface––nearest disparity––biased contrast ambiguity level. Both examples show the stimulus with the same dot spacing as in the experiment, butwith half the number of dots (i.e. stimulus is half the size of that shown to participants, with the same dot density).472  R. Goutcher, P. Mamassian / Vision Research 45 (2005) 469–483  fronto-parallel surfaces overlaid in depth. These two dis-parities  d  1  and  d  2  are given by d  1  ¼ ð a 0  þ  d    e Þ  ð a 0    d  þ  e Þ ¼  2 d    2 e d  2  ¼ ð a 0  þ  a    d    e Þ  ð a 0  þ  a  þ  d  þ  e Þ ¼  2 d    2 e  ð 1 Þ In other words, the transparent percept will be such thatthe rear surface has an uncrossed disparity  d  1  (equal to+6.02 0 ) and the front surface has a crossed disparity  d  2 (equal to   18.06 0 ).The nearest disparity solution is found by making anext-to-nearest neighbour match. This solution resultsin the perception of a single fronto-parallel plane whosedisparity is either crossed or uncrossed. Uncrossed dis-parity will be obtained if dot A in Fig. 3c is matchedwith dot D, while crossed disparity will be obtainedwhen dot B is matched with dot C. These two disparities d  3  and  d  4  are given by d  3  ¼ ð a 0  þ  a    d    e Þ  ð a 0    d  þ  e Þ ¼  a   2 e d  4  ¼ ð a 0  þ  d    e Þ  ð a 0  þ  a  þ  d  þ  e Þ ¼  a    2 e   ð 2 Þ In absolute values, disparity  d  3  (equal to +24.08 0 ) ismuch smaller than disparity  d  4  (equal to   36.12 0 ), sothe uncrossed disparity solution will be preferred andthe single surface will be perceived behind the fixationplane.Note that the values of   a ,  d  and  e  were chosen suchthat the global relative disparity between the front andback surfaces in the transparent percept (i.e. 4 d  =24.08 0 ) equals the disparity of the surface in the singlesurface percept (i.e.  d  3  =  a    2 e ). Thus, only a matchingprocedure concerned with local relative disparities–– rather than the global relative disparity in the scene–– may underlie any observed nearest disparity bias.  2.1.3. Manipulation of contrast So far, because all the dots are identical, the stimuluswe have generated is truly ambiguous and will lead to atransparent percept if the nearest neighbour constraint isadopted, and to a single surface if the nearest disparityconstraint is adopted instead. In this paper, we look be-yond such classification of perceptions and attempt toselectively bias matching towards the percept given byone or the other constraint. The biasing of ambiguousstereograms has previously been used to demonstratethe importance of half-occlusions (Anderson & Nakay-ama, 1994) and fixation depth (McKee & Mitchison,1988) on disparity computation. In general, correspond-ence biasing is used to manipulate the perception of anambiguous stereogram and thus demonstrate the impor-tance of the biasing factor. However, in this paper suchbiasing is used as a probe to measure the relative prefer-ence for the matching solutions underlying two distinctpercepts. Here, biasing was obtained by manipulatingthe contrast of dot pairs as detailed below. Inter-ocularcontrast is known to greatly affect stereopsis both interms of stereoacuity (e.g. Halpern & Blake, 1988; Legge& Gu, 1989) and correspondence matching (Anderson &Nakayama, 1994; Smallman & McKee, 1995).To illustrate the contrast manipulation procedure, letus assume that dots A, B, C and D in Fig. 3c have lumi-nances  L A ,  L B ,  L C  and  L D  that can vary between 0 and 1(the luminances will then be scaled by the maximumluminance of the display, i.e. 19.2 cd/m 2 ). If we wantto selectively impair the nearest disparity constraint,we can simply increase the luminance difference between L A  and  L D  while preserving the same luminances for Aand C. Similarly, if we want to selectively impair thenearest neighbour constraint, we can increase the differ-ence between  L A  and  L C  while preserving the same lumi-nances for A and D. These two manipulations can begrouped together by creating a continuum that we callthe  contrast ambiguity level   and denote  / . Negative val-ues (between   1 and 0) of this parameter correspond toan impairment of the nearest disparity constraint, andpositive values (between 0 and +1) an impairment of the nearest neighbour constraint. The absolute valueof   /  corresponds to the difference in luminance betweenA and D (or A and C). Therefore, values of   /  close to  1 will strongly favour the nearest neighbour constraint Left Half-ImageRight Half-Image (a)(b)(c) ABCD Fig. 3. Illustration of the steps involved in the construction of thestimulus. (a) The basis of our stimulus is a wallpaper pattern wheredots along a particular row are equally spaced. (b) Small positive andnegative displacements are then introduced in alternate dots, and in theopposite direction in the other half-image. (c) Finally, a pedestaldisparity is introduced, shared between the left and right images. DotA in the left image can be matched to multiple dots in the right image,including dots C and D. R. Goutcher, P. Mamassian / Vision Research 45 (2005) 469–483  473
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