Is there an interaction between perceived direction and perceived aspect ratio in stereoscopic vision?

Abstract

In monocular vision, the horizontal/vertical aspect ratio (shape) of a frontoparallel rectangle can be based on the comparison
of the perceived directions of the rectangle’s edges. In binocular vision of a typical three-dimensional scene (when occlusions
are present), this is not the case: Frontoparallel rectangles would be perceived in a distorted fashion if an observer were
to base perceived aspect ratio on the perceived directions of the rectangle’s edges. We psychophysical^ investigated stereoscopically
perceived aspect ratios of frontoparallel occluding and occluded rectangles for various distances and fixation depths. We
found that observers did not perceive the distortions that would be predicted on the basis of the above-mentioned comparison
of the perceived visual directions of the edges of the rectangle. Our results strongly suggest that the mechanism that determines
perceived aspect ratio is dis-sociated from the mechanism that determines perceived direction. The consequences of the findings
for the Kanizsa, Poggendorff, and horizontal/vertical illusions are discussed.

Full text

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Perception & Psychophysics, 62, 910-926, 2000 Stereoscopic aspect ratio of occluders and occlusions
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Is there an interaction between perceived direction and perceived aspect
ratio in stereoscopic vision?
Raymond van Ee (1,2) & Casper J. Erkelens (2)
1 University of California, Berkeley
2 Helmholtz Institute, Princetonplein 5. 3584 CC, Utrecht, The Netherlands
email: r.vanee@phys.uu.nl
Abstract. In monocular vision, the horizontal/vertical aspect ratio (shape) of a fronto-parallel
rectangle can be based upon the comparison of the perceived directions of the rectangle's edges. In
binocular vision of a typical three-dimensional scene (when occlusions are present) this is not the
case: fronto-parallel rectangles would be perceived in a distorted fashion if an observer were to
base perceived aspect ratio on the perceived directions of the rectangle's edges. We
psychophysically investigated stereoscopically perceived aspect ratios of fronto-parallel occluding
and occluded rectangles for various distances and fixation depths. We found that observers did
not perceive the distortions as predicted on the basis of the above-mentioned comparison of the
perceived visual direction of the edges of the rectangle. Our results strongly suggest that the
mechanism that determines perceived aspect ratio is dissociated from the mechanism that
determines perceived direction. The consequences of the findings for the Kanizsa, Poggendorff,
and horizontal/vertical illusions are discussed.
1
Introduction
In stereoscopic vision research we are interested in how the three-dimensional (3D) lay-out of a
scene is perceived. Both perceived direction and perceived horizontal/vertical aspect ratio (or
shape) play an important role in the recovery of the 3D lay-out. Recently, Erkelens, Muijs & van Ee
(1996) studied the perceived direction of two stimuli relative to each other in the neighborhood of
occluders. The authors' results explicitly indicated that the interaction between perceived direction
and perceived aspect ratio is an unresolved topic because the existing standard model of
stereoscopic vision does not relate the two in a satisfying fashion.
Erkelens et al. (1996) instructed observers to align a stereoscopically visible slider with the rim
of a stereoscopically visible fronto-parallel circular disk (fig. 1) so that the perceived visual
directions of both the slider and the rim were identical. The disk was hovering in front of a
background and the slider was located in the background. At the leftmost part of the rim
observers positioned a vertical slider as if the slider was only viewed by the left eye. At the
rightmost part of the rim a vertical slider was aligned as if it was only viewed by the right eye.
Figure 1(B) provides an impression of the finding of Erkelens et al. At the top and bottom parts of
the rim, observers positioned a horizontal slider as if the slider was viewed from a vantage point
at eye level. In other words, the aspect ratio of the circular occluder and the aspect ratio of the
occluded region (as enclosed by the set of slider alignments) were not identical. Nevertheless, the
observers were of the opinion that they perceived an undeformed circular occluder and that the
occluded region was also circular. Thus, there appears to be a discrepancy between the
subjectively perceived aspect ratio of the occluded region and the objective aspect ratio of the
occluded region deduced from the alignment of the slider. We will refer to the latter objectively
deduced horizontal/vertical aspect ratio based on alignment as the alignment-enclosed aspect ratio.

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Occluder
Alignment-
enclosed
Region
Left
Eye
Right
Eye
Visual
Directions
Occluder
Slider
(A) Task
(B) 3D View of Result
Task and Result of
Erkelens, Muijs & van Ee (1996)
Fig. 1: Task and result according to
subject's perception of the scene in the experiment of Erkelens et al. An occluding disk was hovering
in front of a background. In fact, the occluder was monocularly invisible because it was presented as a
random-dot pattern relative to an identical background random-dot pattern (Julesz stereogram). A
binocularly visible slider was presented in the background and the task of the subject was to align the
slider with the rightmost (or leftmost) part of the disk's rim. One slider was visible at a time. The
slider consisted of two parts that were segments of an (invisible) line. The slider was oriented either
vertically (shown at the right side of the rim) or horizontally (not shown). (B) An artist's conception of
the results of Erkelens et al. The figure shows a slider when it was perceived to be aligned with the
rightmost part of the disk's rim. Erkelens et al. found that the horizontal dimension of the occlusion
(in fact, the dark area in the background comprised by the slider alignments at leftmost and rightmost
parts of the occluder's rim) is smaller than the vertical dimension of the occlusion.
Erkelens, Muijs & van Ee (1996). (A) Schematic view of the
How large is the discrepancy between the (subjectively) perceived aspect ratio of the occluded
region and the (objective) alignment-enclosed aspect ratio? Using figure 2 it is straightforward to
predict the alignment-enclosed horizontal dimension of the occlusion for fronto-parallel planes. In
this figure, line AB is occluded by line A'B'. A and B are only visible monocularly. A is visible only
with the left eye. Alignment of A with another target is performed as if both targets are viewed
with only the left eye (Erkelens et al., 1996). B is visible only with the right eye. Alignment of B
with another target is performed as if both targets are viewed with only the right eye (Erkelens et
al., 1996). For the moment, let us assume that the visual system determines perceived aspect ratio
based on low level visual information that is detected by the retinas. All retinal information that is
available to measure horizontal and vertical dimensions consists of visual angles. The visual angle
between the fovea and the retinal projection of a target is called local sign. If an observer were to
use the retinal local signs (angles α and β) directly to estimate the horizontal dimension that is
occluded in the background (γ), s/he would overestimate this dimension by an amount equal to
the disparity (δ) between the fixation plane and the background:
A' B' = α +β = γ +δ
γ = α +β −δ

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Disparity δ δ
A
B
α α
β β
γ γ
B'
A'
F
Horizontal Dimension Difference
between two Depth Planes
Left
Eye
Cyclopean
Eye
Right
Eye
γ/2 + δ/2 γ/2 + δ/2
δ/2δ/2
Fig. 2:The perceived horizontal dimension difference between two distinct depth planes. Schematic
top view of an observer fixating point F in the plane of line A'B'. This line occludes line AB. A and B
are only visible monocularly. For the observer, A looks as if it is aligned with A' and B looks as if it is
aligned with B' (Erkelens et al., 1996). If the observer were to use the retinal local signs (α and β)
directly to estimate the dimension that is occluded in the background (γ), s/he would overestimate
this dimension by an amount that is equal to the disparity between the fixation plane and the
background. Note that this statement is true for fixation in any depth plane. If fixation is behind the
plane of interest then disparity reverses its sign. From an operational point of view it is convenient to
use a single vantage point (the so-called cyclopean eye) from which the stereoscopic visual world is
viewed instead of using the two eye's views. This cyclopean eye is located midway on the line that
connects the nodal points of the eyes. But note that, mathematically speaking, the location of the
single vantage point is essentially irrelevant for the above-derived statements.
In words, the difference between the horizontal angular dimension of a frontal occluder and the
alignment-enclosed horizontal angular dimension of its occlusion is equal to the difference in the
disparity between the fixation plane and the background. If the difference in disparity is δ and the

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horizontal dimension of the foreground plane is S, the horizontal angular dimension ratio
background/foreground equals (S Ð δ)/S. For example, if the difference in disparity is 0.5 deg and
the horizontal dimension of the foreground plane is 5.2 deg (as it will be in our experiments) the
horizontal angular dimension ratio background/foreground = 4.7/5.2 = 0.9. Because the eyes are
oriented horizontally next to each other no such misestimation is predicted in the vertical
direction. If the perceived dimension of the occluder in this example were identical to the
alignment-enclosed dimension of its occlusion (the perceived visual directions along the rim), one
would expect a rectangle whose vertical dimension is 9.6% smaller than its horizontal dimension
to be perceived as a square1. Note that it is not only distortion that is predicted, but also
discontinuity in the visual direction of the background at the transition from monocular to
binocular regions (van Ee, Banks & Backus, 1999).
Figure 3 demonstrates, in the form of stereograms2, the discrepancy between the subjectively
perceived aspect ratio of the occluded region and the objective alignment-enclosed aspect ratio.
The two stereograms in figure 3 consist of a background chess-board pattern and a foreground
feature that is hovering in front of the background. Figure 3(A) addresses the perceived aspect
ratio of the foreground. In figure 3(A) observers perceive the sliders to be aligned with the rim of
the occluding disk; that is, the perceived visual directions of both the slider and the part of the rim
nearest to the slider are identical. Closer examination of the half-images of the stereogram shows
that, in fact, the left slider is physically aligned with the leftmost part of the rim only in the left
eye's half-image of the stereogram; the right slider is physically aligned with the rightmost part of
the rim only in the right eye's half-image. Observers perceive the number of chess-board elements
subtended by the occluding disk to be eight in the horizontal direction and nine in the vertical
direction, which means that the number of elements in the vertical direction is 12.5% larger than in
the horizontal direction. This significant difference is not reflected in the perceived aspect ratio
because both the aspect ratio of the occluder and the aspect ratio of the occlusion are perceived as
being equal to unity (circular). Figure 3(B) addresses the perceived aspect ratio of the background.
Most observers report that the shape of the background feature in figure 3(B) is a square or a
`standing' rectangle (with the vertical dimension slightly larger than the horizontal dimension).
However, observers perceive the number of chess-board elements in the horizontal direction to be
larger than in the vertical direction; they perceive six elements on either side of the bar. Given that
the bar is three elements wide, the number of elements in the horizontal direction is now 7% larger
than in the vertical direction. This significant difference is not reflected in the perceived aspect
ratio of the chess-board pattern in the background.

1 Note that this analysis is essentially independent of both the presence and the location of the cyclopean eye
(fig. 2). After completion of this manuscript, Mapp & Ono (1999) stated that the cyclopean eye "is both a
logical and a functional necessity for judging the direction of one object with respect to another". In addition,
they suggested that the findings about binocular alignment in different depth planes (Erkelens et al., 1996)
and about capture of visual directions (Erkelens & van Ee, 1997a, 1997b) were based on "a wandering
cyclopean eye". Their stetement and suggestions are not justified: our analyses were based on visual lines
and eye orientations which means that the location of the cyclopean eye is irrelevant for the validity of the
analyses (for the latter claim see also Banks, van Ee & Backus, 1997).
2 Stereograms in this paper consist of three half-images. Observers who have best fusion when their eyes are
crossed (which means that the half-image on the right side is seen by the left eye) should fuse the two half-
images on the right side; uncrossed fusers (or observers who use a stereoscope) should fuse the two images
on the left. Independent of the eyes' cross-mode, the half-image that is seen by the left eye is called the left
eye's half-image; the half-image seen by the right eye, the right eye's half-image.

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(A) Occluder Shape
Perceived Direction and Perceived Aspect Ratio
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1 2 3
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Fig. 3: Discrepancy between perceived alignment and perceived aspect ratio. (A) This demonstration
of the results obtained by Erkelens et al. (1996) shows that there is a discrepancy between perceived
alignment and perceived horizontal/vertical aspect ratio of the occluder. In each of the half-images
four sliders are presented: two vertical and two horizontal sliders. In this particular example
observers perceive the sliders to be aligned with the corresponding rim of the occluding disk (after
they have stable fusion of the two half-images). Closer examination of the half-images of the
stereogram shows that, in fact, the left slider is physically aligned with the leftmost part of the rim in
the left eye's half-image of the stereogram; the right slider is physically aligned with the rightmost
part of the rim in the right eye's half-image. When observers are asked to count the number of chess-
board elements subtended by the occluding disk (again after they have fused the two half-images)
they count eight elements in the horizontal direction and nine in the vertical direction, which means
that the number of elements in the vertical direction is 12.5% larger than in the horizontal direction.

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This significant difference is not reflected in the perceived shape because the shape of both the
occluder and the occlusion is perceived as being circular or only slightly elongated in the vertical
direction. In the study by Erkelens et al. the occluding disk was monocularly indistinguishable from
the background pattern [see caption of fig. 1(A)]. (B) Most observers report that the shape of the
background feature is a square or a `standing' rectangle (with the vertical dimension slightly larger
than the horizontal dimension). However, when observers count the number of chess-board elements
(after they have stable fusion of the two half-images) they find out that the number of elements in the
horizontal direction is larger than in the vertical direction; they perceive six elements on either side of
the bar. Given that the bar is three elements wide, the number of elements in the horizontal direction
(15) is now 7% larger than in the vertical direction (14). Again, this significant difference is not
reflected in the perceived shape.
The results mentioned so far can be understood if one assumes that alignment in a monocular
region is equivalent to equating the local signs (visual directions) in the eye that can see the
monocular region caused by the occluder (Erkelens & van de Grind, 1994)3.
Support for the findings of Erkelens et al. comes from reports of Lillakas, Ono & Grove (1998),
Ohtsuka & Ono (1998), and Suzuki, Segal, Lillakas & Ono (1998). Suzuki et al. (1998) and Lillakas
et al. (1998) examined background distortions by having subjects read text on a computer screen
which was partly occluded by a vertical rod. They found that the monocular areas were readable
but that visual directions were perceptually displaced. However, they reported that the perceived
aspect ratio of the letters and words were not deformed. Ohtsuka & Ono (1998) investigated both
Kanizsa's compressed square illusion and the Poggendorff illusion in stereoscopic viewing. Figure 4
shows two-dimensional (2D) examples of these illusions. In both illusions Ohtsuka & Ono (1998)
confirmed that alignment near the monocular areas caused by the occluders was based on visual
information provided by the eye to which the monocular area was visible. In the Discussion
section we will address these illusions in more detail.
The above-mentioned observations are interesting because they are not described by the
existing standard model of binocular direction perception. The standard model, formulated
originally by Alhazen, Wells, and Hering, and consolidated by Ono and others (for a review see
Howard & Rogers, 1995; Schor, 1999), states that the visual directions derived from the two eyes
images will be perceived as if the observer were viewing the scene from a single vantage point
midway between the two eyes; this point is called the cyclopean eye (fig. 2). Modern,
comprehensive and extended versions of the standard model are presented by Banks et al. (1997),
Ono & Mapp (1995), and van de Grind, Erkelens & Laan (1995)4. The cyclopean eye (like either real
eye) has a 2D structure on which each location represents a visual direction. According to the
standard model, all visual targets viewed by either eye are included in the cyclopean eye and,
thus, are perceived in binocular vision.
It is impossible to fit the complete set of visual directions into a 2D cyclopean eye when there
are occluders in the visual field because one eye will contain a number of projections of visual
targets that are not visible to the other eye (Anderson & Nakayama, 1994; Erkelens & van de
Grind, 1994). Similarly it is impossible to paint a 3D picture on canvas which has been known
since the time of Leonardo da Vinci who was one of the first to make this explicit. The cyclopean
eye has no room for more projected targets than either of the eyes. Thus, the above-mentioned
observations, in which both the background and the foreground are (at least subjectively)
perceived without distortion, are not compatible with the standard model of binocular direction
perception.

3 These results, in turn, can be understood within the larger framework of a mechanism that has been called
`capture of perceived direction' (Erkelens & van Ee, 1997a, 1997b). Van Ee et al. (1999) showed that such a
mechanism is at work even if monocular features are not due to a stimulus situation that can be interpreted
by the visual system as being caused by occlusion geometry. In this paper we will disregard mechanisms
underlying the perceived direction of targets.
4 There are slightly different versions of the standard model but all of them are based on the concept of a 2D
cyclopean eye midway between the two real eyes.

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(A) Kanizsa's compressed Square Illusion (B) The Poggendorff Illusion
The Kanizsa and Poggendorff Illusions in 2D Drawings
Fig. 4: The Kanisza and Poggendorff illusion in 2D drawings. (A) Kanizsa's compressed square
illusion. The occluded square in the top-left corner looks more compressed in the horizontal direction
than the non-occluded squares of the same size. The top-left configuration is the one that is generally
known as Kanizsa's square. The bottom-right configuration shows that after 90 deg of rotation the
illusion is just as strong, but now in the vertical direction. (B) Poggendorff's shifted line segment
illusion. The two lines that are present on either side of the rectangle are in fact segments of one line.
But, in the top-left square, it looks as if the right segment is shifted upward relative to the left
segment. The top-left configuration is the one that is generally known as the Poggendorff illusion. The
bottom-right configuration shows that after 90 deg of rotation the illusion is also strong, but now in
the horizontal direction.
2
Experiment
2.1
Motivation for new experiments
From existing experimental work it is impossible to derive systematic characteristics of
objectively perceived aspect ratio of both occluders and occluded objects in stereoscopic vision.
We will describe the results of four experiments. In the first experiment we psychophysically
investigate the impression, as reported by the observers of Erkelens et al., that one perceives an
undeformed occluder. Figure 2 shows that the fixation depth has theoretically a profound
influence on perceived aspect ratio if this ratio is based upon local sign information (see also
Banks et al., 1997). The magnitude of this influence is significant because it is of the same order as
the disparity. Therefore, in experiment 2 we investigate the influence of fixation depth on the
perceived aspect ratio of an occluder. The observers in the study of Erkelens et al. (1996) reported
that they perceived an undeformed occluded object. Therefore, in the third and fourth
experiments we systematically investigate whether an occluder distorts perceptual space behind
it.

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2.2
Subjects
Fifteen subjects participated in the four experiments. They were checked for normal stereo
vision by means of partially decorrelated Julesz random-dot test images. They were naive with
respect to the purposes of the experiment. Five subjects were experienced in stereoscopic
experiments. The other subjects were inexperienced. Nine subjects showed refraction anomalies
which were corrected by their own contact lenses or glasses. Eight subjects participated in each of
the four experiments.
2.3
Apparatus
The apparatus has been described previously (van Ee & Erkelens, 1995). The subject viewed
stereograms that were generated by an HP 750 graphics computer. Subsequently, the stimuli were
back-projected onto a large fronto-parallel translucent screen by a projection TV (Barco Data 800).
The size of the screen was 70 * 70 deg. Subjects were seated in front of the screen at a distance of
1.5 m. Head movements were restricted by a chinrest and a skullrest. The interocular axis was
both horizontal and parallel to the screen. The left and right eye's half-images of the stereogram
were shown in red and green, respectively, and both were presented in each trial afresh at a
frequency of 70 Hz. Subjects wore anaglyph glasses. The left and right eye viewed the screen
through filters matched to the emission spectra of the red and green phosphors of the TV,
respectively; no crosstalk was observed. The relative brightness of both the red and green half-
images was adjusted to look equally bright when viewed through the glasses.
Because the stimuli were large random-dot patterns, which consequently illuminated the
surroundings, it was impossible to work in a completely dark room. In order to make
uncontrolled features in the room, like furniture, impossible to see we used black cloths which
were attached between the side-edges of the screen and the headrest. There was black carpet on
the floor. In order to make the lighting conditions in the room more similar across trials (which
were relatively bright when the large background was presented compared to trials where only
the rectangle was shown) we dimly illuminated the ceiling of the room.
Rectangle
background
5.2 deg
60
deg
60 deg
86%
to
114%
of
5.2
deg
0.3 deg
1.8 deg
Window
5.2 deg
86%
to
114%
of
5.2
deg
Rectangle
(A) Experiments 1a and 1b(B) Experiment 2
1.6 deg
60
deg
60 deg
Fore-
ground
Bar
30
deg
(C) Experiment 3
background
Stimuli of Experiments 1 to 3
Fig. 5: Julesz random-dot stimuli used in experiments 1 to 3. We presented fronto-parallel rectangular
stimuli in the center of the screen. In the experiment, the background (A, C), the rectangle (A, B, C)
and the foreground bar (C) contained identical random-dot textures, so they were monocularly
indistinguishable. All dot sizes were 3 * 3 arcmin. The monocularly visible contours (width 6 arcmin,
drawn in black) around the rectangle (A, B, C) were only visible in particular conditions. The
background (60 * 60 degrees) was also only visible in particular conditions; its disparity was always 0
deg. (A) Schematic drawing of the stimulus used in experiment 1. The disparity of the rectangle was

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either 0.5, 1.0 or 1.5 deg. (B) Rectangular stimulus of experiment 2. The rectangle was always
presented with a disparity of 0.5 deg. There was a circular window (diameter 1.8 deg) in the center.
(In order to make the window clearly visible the rectangle is drawn not to scale.) In the window a
fixation dot was presented with three possible disparities. The disparity of the fixation dot was either
0, 0.5 or 1 deg. The diameter of the fixation dot was 0.3 deg. (C) Stimulus used in experiment 3.
Whenever the occluding foreground bar in front of the rectangle was presented its horizontal size was
1.6 deg and its vertical size 30 deg. The disparity of the foreground bar was always 1.2 deg. The
disparity of the rectangle was either 0.2, 0.6 or 1.0 deg. The foreground bar always contained the
monocularly visible contour around it.
2.4
Stimuli
In all of the experiments we presented fronto-parallel rectangular stimuli (figs. 5 and 10) of
which the vertical dimension relative to the horizontal dimension had to be judged both in the
presence and in the absence of an occluding bar in front of the rectangle. All dot sizes were 3.0 *
3.0 arcmin (1 pixel).
The shift between the red and the green half-images of both the rectangles and the foreground
bar were modified independently so that the rectangles were perceived at different depths in front
of the background. We will call this shift on the screen disparity. The background was always
presented in the plane of the screen so that there was no disparity between its half-images. All
disparities in this paper will be crossed disparities. Whenever there was disparity present, the left
eye's half-image was shifted to the right over half the disparity; the right eye's half-image was
shifted to the left by the same magnitude. This means that the eccentricity of the center of the
rectangle relative to the head was unaffected by the disparity.
The display duration was fixed (except for experiment 1b) at 2.0 sec. The period was kept short
so that it was impossible to perform the task by counting both the number of random dots and the
gaps between them. Except for experiment 2, the subjects were free to make eye movements while
viewing the stereogram. The blanking period between stimuli was 300 msec.
2.5
Task, procedure and data analysis
In all of the following experiments the subjects judged the horizontal/vertical aspect ratio of a
rectangle. The aspect ratio of the presented rectangles differed between 86% and 114%. There were
11 different aspect ratios presented which were taken from the set: (0.86, 0.92, 0.96, 0.98, 0.99, 1.00,
1.01, 1.02, 1.04, 1.08 or 1.14). Each trial was repeated 12 times randomly intermixed with the other
aspect ratios. Thus, a particular condition consisted of 132 (11 * 12) trials. No feedback was given
about the results.
Psychometric functions were fitted to the data in order to determine the aspect ratio where both
alternative answers were given equally often (known as the 50% point). For three subjects the
standard deviation in the 50% point across trials was determined by repeating the experiment
three times; for the rest of the subjects it was determined by a standard Monte-Carlo simulation.
These standard deviations across trials within a subject were small: about 0.3%. Subjects showed
identical trends in their data. Therefore we will present the averaged data across subjects. In all of
the graphs error bars represent the standard deviation of the data across eight subjects5.
3
Experiment 1: Perceived aspect ratio of rectangular stimuli
In this introductory experiment we investigated the influence of the disparity of the rectangle
(experiment 1a) and the display duration (experiment 1b) on the perceived aspect ratio.

5 We also determined the slopes of the psychometric curves in all of our experiments. In general, the slopes
were between 0.5% ± 0.1% and 5.5% ± 0.5%, which are similar to the slopes found by Regan & Hamstra
(1994) in a discrimination experiment in which subjects judged the aspect ratios of disparity-defined
rectangles. In general the slopes did not differ markedly between conditions for a particular subject (see also
Regan & Hamstra, 1992). In this paper we do not further analyze these slopes.

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3.1
Methods
Figure 5(A) shows a schematic drawing of the stimulus. The stimuli were essentially traditional
Julesz random-dot stereograms, which means that the different depth planes were monocularly
indistinguishable. The rectangular stimuli were always presented in the center of the screen. In
particular trials the rectangles were surrounded by a contour (to make them monocularly visible)
and/or accompanied by a planar fronto-parallel background (so that the rectangle acted as an
occluder). Whenever the background was visible its size was 60 * 60 degrees. Whenever the
contour was visible it was 6 arcmin wide.
Subjects were instructed to decide whether the vertical dimension of the rectangle was larger
than the horizontal dimension. The horizontal dimension of the rectangle was always 5.2 deg6; its
vertical dimension varied across trials. The vertical/horizontal aspect ratio of the rectangular
stimulus differed between 86% and 114%. Experiment 1 was divided into two subexperiments
denoted by 1a and 1b.
In experiment 1a we were interested in the influence of the disparity on the discrimination task;
the disparity between the half-images was either 0.5, 1.0 or as large as 1.5 deg. Pilot experiments
showed that subjects had no difficulty in judging the aspect ratios of rectangles with a disparity of
1.5 deg. Regan & Hamstra (1994) showed systematically that aspect ratio discrimination
thresholds were not negatively affected by large disparities of up to 2 deg. The larger the
disparity, the clearer the predicted aspect ratio distortion effects. In the Introduction we explained
that the predicted distortion effect equals (S Ð δ)/S where S is the horizontal dimension of the
rectangle and δ is the relative disparity between the fixation depth and the background; if the
alignment-enclosed dimension determines the perceived horizontal dimension of the rectangle, a
disparity of 0.5 deg would cause a distortion effect of 9.6%. According to a similar calculation,
disparities of 1 deg and 1.5 deg would cause distortion effects as large as 19.2% and 28.8%,
respectively. In experiment 1a (and in the other experiments reported in this paper) the display
duration was 2.0 sec.
In experiment 1b we did a control experiment in which we checked for the influence of the
display duration. In experiment 1b the display durations were 0.5, 2.0 or 3.5 sec. Eye movements
were free and subjects were encouraged not to perform the task by strict fixation. For a display
duration of 0.5 sec the subject is unable to scan the outline of the complete rectangle. But for a
display duration of 3.5 sec the subject is able to scan the complete outline. Four subjects
participated in experiment 1b. The disparity of the rectangle was 1 deg.
In both experiment 1a and experiment 1b, there were three different stimulus conditions: (1)
with background and with monocularly visible contour around the rectangle; (2) with background
and without monocularly visible contour; (3) without background.
We selected these conditions in order to investigate the difference in performance between
purely stereoscopic perception (when the rectangle is invisible monocularly) and perception with
monocularly visible information. The 3 conditions were presented in 3 individual experimental
sessions.
In experiment 1a the subject performed 1188 aspect ratio judgments deriving from:
3 experimental sessions;
3 conditions (3 disparities in experiment 1a; 3 display durations in experiment 1b);
11 aspect ratios;

6 In a control experiment we repeated one condition of the experiment while presenting rectangles with two
horizontal dimensions---5.2 deg and 8.0 deg---intermixed within one series in order to control for the
possibility that the subjects were able to perform the task by comparing the vertical dimension of the
rectangle with some internal reference (after a couple of trial blocks the range of stimuli, and consequently
the vertical dimension that is identical to the horizontal dimension, could be known to the subject and
stored in memory) instead of comparing the vertical dimension with the horizontal dimension. Results were
not significantly affected by this size variation. Regan & Hamstra (1991, 1992, 1994) explicitly disconfounded
perceived aspect ratio from size perception by requiring subjects to discriminate aspect ratios of rectangles
whose areas were altered independently of aspect ratio. They found near perfect behavior though the
subjects could not use the dimension of the rectangle sides.

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11
12 repetitions per trial.
In experiment 1b a total of 264 trials less than in experiment 1a because we did not do the
control experiment for all conditions. We fitted psychometric functions to the data in order to
determine the points of perceptual equality (µ), i.e. the aspect ratios which the subject judged as
corresponding to squares.
0.511.5
Rectangle Disparity [deg]
0.5
Display Duration [sec]
2 3.5
0 0.51
Fixation Dot Disparity [deg]
Perceived Aspect Ratio [%]
Results of Experiments 1 and 2
95
100
90
80
85
75
70
(A) Experiment 1a(B) Experiment 1b(C) Experiment 2
Contour
Contour
Background, With
Background, Without
Background
With
With
Without
Prediction based on Alignment-Enclosed Aspect Ratio
Fig. 6: Results of experiment 1 and 2. The abcissa shows the perceived horizontal/vertical aspect ratio;
the ordinate shows the various conditions. For example, a perceived aspect ratio of 95% means that
the subject perceived a rectangle, the vertical dimension of which was smaller (95% of 5.2 deg) than
the horizontal dimension, as a square. The gray lines depict predictions based on the alignment-
enclosed aspect ratio (see equation 2 and the corresponding text for details). Apart from the fusion
problem (see text) in experiment 1a [panel (A), black diamond at 0.5 deg] the rectangle disparity had
no significant effect on subject's performance. In experiments 1b [panel (B)] and experiment 2 [panel
(C)] there was no significant effect of the display duration and the fixation depth, respectively. In
experiment 1b no data have been collected for the without-background condition at a display
duration of 0.5 and 3.5 sec. Note that, in experiment 2, in the case where both background and
contour were present, there was no horizontal/vertical size illusion. Error bars represent the standard
deviation of the data across eight subjects.
3.2
Results
Figure 6(A) shows the results of experiment 1a. The rectangle disparity does not have a marked
influence on the performance. Ratios obtained in the presence of both a contour and a background
are consistently larger than ratios obtained in the other conditions. Subjects reported difficulties in
doing the task in the case of a rectangle disparity of 0.5 deg when there was a background but no
contour around the rectangle (leftmost black diamond in panel (A) of fig. 6). They reported that it
was difficult to fuse the rectangle at this small disparity; it was not always clear to the subject
which dots at the leftmost and rightmost parts of the rectangle's rim belonged to the background
and which to the foreground. Experienced subjects reported that complete correspondence was
not accomplished within the display duration. We attribute subjects' significantly different

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12
performance in this case, including the large error bar to a fusion problem. (In experiment 3 we
will encounter a similar situation. See figure 7(B) for a demonstration of a similarly difficult
stimulus.)
Figure 6(B) shows that performance is veridical for short display durations. The results of
experiment 1a and 1b are consistent with each other: the values in figure 6(A) at a disparity of 1
deg should be similar to the values of figure 6(B) at a display duration of 2 sec because in
experiment 1a we used a display duration of 2 sec and in experiment 1b we used a square
disparity of 1 deg. Within one standard deviation the two values are similar.
In all conditions (except for the shortest presentation duration) of experiments 1a and 1b
subjects perceived the rectangular stimuli as elongated in the vertical direction. We attribute this
to the well-known horizontal/vertical illusion (HVI, see Discussion section).
4
Experiment 2: Influence of the fixation depth on perceived aspect ratio
In normal situations when different depth planes are present observers probably change their
fixation between the different depth planes when judging aspect ratio. The depth of the fixation
point is an interesting variable because it can have a significant influence on perceived aspect ratio
(fig. 2). We investigated this possibility in this experiment.
4.1
Methods
Experiment 2 is essentially identical to experiment 1a. The difference is the fact that fixation in
various depth planes was required. Figure 5(B) shows a schematic drawing of the rectangular
stimulus. The rectangle was always presented with a disparity of 0.5 deg. There was a circular
window (diameter 1.8 deg) in the center of the rectangle. In the window a fixation dot was
presented with three possible disparities (see fig. 7 for a schematic example). The disparity of the
fixation dot was 0, 0.5 or 1 deg (which means that the distortions predicted based on the
alignment-enclosed aspect ratio were 0, 9.6 or 19.2%, respectively). The diameter of the fixation
dot was 0.3 deg. In order to minimize eye movements that are conducted to search for the depth of
the fixation dot during the onset of the rectangle, the fixation dot had already been shown for 1 sec
prior to the presentation of the stimulus. When strict fixation is required in vision research stimuli
are generally presented with a short duration, like 75 msec, so that completion of a vergence eye
movement is impossible. In pilot experiments, however, we found that conducting the task with
such a short display duration is not feasible. On the other hand, locked fixation is not really
necessary in our experiment because it is sufficient for the subject to keep his/her fixation in one
depth plane. Given that the fixation dot has a diameter of 0.3 deg it is relatively easy to keep
fixation on the dot. We did not measure eye posture during the experiment. On the basis of
subjective impressions during preliminary participation in the experiment we consider it unlikely
that occasional unintended vergence eye movements acted as significant contributors to our
results. In experiments 2 (like experiment 1a) the subject performed 1188 aspect ratio judgments.
4.2
Results
Figure 6(C) shows the results of experiment 2. The responses do not depend on the disparity of
the fixation dot. They are somewhat closer to veridical than in experiment 1a. This implies that the
horizontal dimension of the rectangle is perceived to be somewhat larger (relative to the vertical
dimension) than in experiment 1. There is no HVI in cases where both a background and a contour
are present. Apparently the presence of a circular window in conjunction with the presence of a
background and a contour is sufficient for a veridically perceived aspect ratio. While we cannot
absolutely rule out the possibility that occasional unintended eye movements contributed to our
results, our results provide no support for the hypothesis that the fixation depth is important for
the stereoscopically perceived aspect ratio of planar surfaces. Figure 7 provides a demonstration of
the findings.

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Figure is too large
See paper in perception & Psychophysics
Fig. 7: The Influence of fixation depth on the perceived aspect ratio of the foreground. Fixation is in
the plane of the background (A), in the foreground (B) or in front of the foreground (C). For most
observers there is no significant influence of the fixation depth on the perceived aspect ratio of the
foreground rectangle if fusion is established. (Make sure that the viewing distance is large enough for
fusion to be established).
5
Experiment 3: Perceived aspect ratio of an occluded pattern
Now that we know a number of the essential characteristics of stereoscopic aspect ratio
perception in different depth planes of non-occluded patterns, we will investigate the influence of
an occluding bar on the perceived aspect ratio of the rectangle.
5.1
Methods
Figure 5(C) shows a schematic drawing of the stimulus used in experiment three. Except for the
presence of the occluding bar the experiment is similar to experiment 1a. The disparity of the
rectangle was either 0.2, 0.6 or 1.0 deg. The occluder consisted of a vertical bar (width 1.6 deg,
height 30 deg). The occluder was always presented in the center of the screen with a disparity of
1.2 deg. It always contained a surrounding contour in order to facilitate correspondence and
fusion. In experiment 3 we also ran a control condition regarding the influence of eye posture
(fixation depth). In this control we required the subjects to fixate the foreground bar both with and
without contour around the rectangle. We did not calculate predicted distortions based on
alignment enclosed aspect-ratio because there is no tested model for multi-layered environments
on which such calculations should be based. However, distortions predicted by any model based
on alignment enclosed aspect ratio are similar to the predictions in experiments 1 and 2.
In experiment 3 the subject performed 924 aspect ratio judgments: The first experimental series
(no fixation, with contour) consisted of 396 trials: 3 disparities of the rectangle, 11 aspect ratios, 12
repetitions per trial. Two other experimental series (fixation on the bar) consisted of 264 trials: 2
disparities of the rectangle, 11 aspect ratios, 12 repetitions per trial.
5.2
Results
Figure 8 shows the results of experiment 3. Again, our results provide no support for the
hypothesis that the fixation depth is important for stereoscopic aspect ratio perception. When a
contour is present the rectangle disparity has no influence on the subjects' performance. As in
experiment 2 we found that the HVI is somewhat weaker in the case of fixation in the plane of the
rectangle.

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Perceived Aspect Ratio [%]
88
90
92
94
96
98
100
102
0.2 0.4 0.6 0.8
Rectangle Disparity [deg]
1
Result of Experiment 3
Contour
With
Bar
Fixation,
Bar
Fixation,
Contour
With
No
Contour
Without
Fixation,
Fig. 8: Result of experiment 3. The mean perceived aspect ratios (vertical/horizontal dimension) as a
function of the rectangle disparity. In the with-contour conditions the rectangle disparity had no
significant effect on subject's performance. The distinguished value of the open diamond at a
disparity of 0.2 deg is due to the above-mentioned fusion problem (see text). No data were collected
in the bar-fixation/with-contour condition and in the bar-fixation/without-contour (open symbols)
condition in the case of a rectangle disparity of 0.6 deg. Error bars represent the standard deviation of
the data across eight subjects.
We attribute the value of the open diamond in the left part of figure 8 to the above-mentioned
fusion problem. The open diamond represents the condition in which there was no contour
present around the rectangle. In the case of the small rectangle disparity (0.2 deg) subjects
reported that they found it difficult to fuse the rectangle. Regan & Hamstra (1994) reported a
similar finding; they found that discrimination thresholds in aspect ratio judgment of disparity-
defined rectangles rose significantly for small disparities. Our subjects reported that it was often
not clear which dots at the leftmost and rightmost parts of the rectangle's rim belonged to the
background and which to the rectangle. Fusion problems are not reflected merely by increasing
standard deviations in the data; they are also reflected by the fact that subjects perceive the
horizontal dimension to be significantly smaller than when there is fusion. We speculate that this
finding reflects a mechanism similar to that found by Erkelens et al. (1996): The location of the
rightmost (leftmost) part of the rim is determined by the right (left) eye7. Figure 9 provides a
demonstration of the findings.

7 The compression in the horizontal dimension caused by the absence of fusion can be readily observed
when one holds one's naked fore-arm vertically in front of the nose at a distance of 15 cm. The effect is
strongest when one fixates in the background.

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Figure is too large
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Fig. 9: Lack of fusion causes horizontal compression of perceived aspect ratio. (A) The rectangle has
such a disparity that it is relatively easy to fuse when observers fixate on the foreground bar. This
example illustrates the 1 deg disparity condition in experiment 3. (B) The disparity of the rectangle
relative to the background is so small that it is hardly fusible when fixation is in the plane of the
foreground bar. Lack of fusion causes the square to be perceived as a rectangle the vertical dimension
of which is larger than the horizontal dimension. This example illustrates the 0.2 deg disparity
condition in experiment 3 in the absence of the surrounding contour. (C) Fixation should be on the
black symbol in front of the rectangle. The disparity of this symbol is large. As in figure (B), lack of
fusion causes the square to be perceived as a rectangle the vertical dimension of which is larger than
the horizontal dimension. The effect is most pronounced when the viewing distance is short so
disparities are relatively large. This example represents most of the conditions in the experiment
performed by Liu and Kennedy (see Discussion).
6
Experiment 4: Aspect ratio of an occluded and an unoccluded pattern
This experiment is a variation on experiment 3. In this experiment we do not investigate the
aspect ratio of an occluded pattern (as in experiment 3) but we study the perceived aspect ratio of
an occluded pattern relative to a non-occluded one.
foreground
70
deg
8 deg
1.6
deg
4.5
deg
Screen
70 deg
60 deg
18.5 deg
1.6 deg
Stimulus of Experiment 4
Fig. 10: Stimulus of experiment 4. The vertical dimension of the rectangles was fixed at 4.5 deg. The
horizontal dimension of the left rectangle was 28.0, 29.8 or 31.6 deg. The horizontal dimension of the
right rectangle varied randomly between 86% and 114% of the dimension of the left rectangle.
Subjects were instructed to discriminate whether the rectangle behind the occluder was wider and
narrower than the non-occluded rectangle. The center of the left (right) rectangle was always
presented 18.5 deg to the left (right) hand side of the center of the screen (marked by the grey cross).
Whenever the vertical foreground bar (size 1.6 * 8 deg) was present it was presented 18.5 deg to the
left hand side of the screen (that is, on top of the center of the left rectangle). The bar was part of a
foreground pattern that was 60 deg wide. The boundaries of the screen (size 70 * 70 deg) and the cross
in the center were not visible. Although the figure shows different textures, the textures in the
experiment were identical (Julesz stereogram). There was no background present.