Perception, 2001, volume 30, pages 411 ^ 429
DOI:10.1068/p3091
The backward inclination of a surface defined by empirical
corresponding points
Philip M Groveô½, Hirohiko Kaneko#, Hiroshi Onoô
ATR Human Information Processing Research Laboratories, 2-2 Hikaridai, Seika-cho, Soraku-gun,
Kyoto 619-0288, Japan; ô also at Centre for Vision Research, York University, 4700 Keele Street,
Toronto, Ontario M3J 1P3, Canada; # also at Imaging Science and Engineering Laboratory,
Tokyo Institute of Technology, Tokyo, Japan
Received 30 May 2000, in revised form 20 November 2000
Abstract. Three experiments were conducted to investigate whether a locus of binocular correspondence extends eccentrically from the vertical horopter. In experiment 1, we investigated
whether the backward inclination of the vertical horopter was manifested in the angle at which
readers prefer to orient the page. All observers preferred a page inclined backwards to any other
orientation. This backward inclination was less than predicted from previous psychophysical
reports, however. In experiment 2, we investigated the extent of binocular correspondence, defined
by minimal apparent interocular horizontal motion, in the central 24 deg of the binocular field. Our
data define a planar surface inclined top-away from the observer as a locus from which psychophysical corresponding points are stimulated. In experiment 3, we measured vertical adjustments
required to eliminate apparent vertical motion. Together, the pattern of results from experiments
2 and 3 is most consistent with a planar surface, inclined top-away from the observer. This is
consistent with Helmholtz's account of the backward inclination of the vertical horopter and
expands the locus of zero horizontal disparity from a single line in the median plane to eccentric
loci extending at least 12 deg in the central binocular field.
1 Introduction
Imagining our two eyes as identical globes with longitudes and latitudes by analogy
to the globe of the Earth, we consider points that have the same longitude and latitude
in the two eyes to be geometrical corresponding points. In other words, corresponding
points have the same positions on two idealised spherical retinas that are superimposed.
When one is lying on one's back and looking at the stars, fixation is at optical infinity
and the principal rays entering the eyes are effectively parallel. Therefore, the images
of each visible star in the sky project to the same distance and direction from the
centres of the foveas in the two eyes. Consider symmetrical convergence nearer than
infinity in the transverse plane, passing through the nodal points of the two eyes (hereafter referred to as the horizontal plane of regard).(1) The locus projecting to geometrical
corresponding points is a circle containing the nodal points of the two eyes and the
fixation point, and a line, perpendicular to the horizontal plane of regard, intersecting
that circle at the fixation point. These are known as the Vieth ^ Mu« ller circle and the
vertical horopter, respectively. Empirically, however, numerous psychophysical investigations, primarily using the criterion of collinear Nonius lines or an analogous method,
have shown that the empirical horizontal horopter is characterised by a shallower
curve than the Vieth ^ Mu«ller circle, known as the Hering ^ Hillebrand deviation
(Hillebrand 1929; Ogle 1964), and the empirical vertical horopter is inclined top-away
½ All correspondence and requests for reprints should be addressed to Philip M Grove, Graduate
Programme, Department of Psychology, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3,
Canada; e-mail: [email protected]
(1) The terms horizontal and vertical are specified with respect to gravity. The median, transverse, and
frontoparallel planes are specified with respect to the body (Howard 1982). To maintain congruence
with the literature, however, we will use horizontal to denote locations in the transverse plane, and
vertical to denote lines perpendicular to the transverse plane contained in the frontoparallel plane.
412
P M Grove, H Kaneko, H Ono
(see, for example, Helmholtz 1910/1962; Ogle 1964; Nakayama 1977; Nakayama et al
1977; Ledgeway and Rogers 1999; Siderov et al 1999).
1.1 Definitions
Let us define five horopters relevant to our discussion: the geometrically defined point
horopter and four empirical horoptersöthe Nonius, minimum-motion, fusion, and
apparent frontoparallel horopters. The geometric point horopter is the locus of zero
disparity in both horizontal and vertical directions stimulating positions with the same
longitude and latitude in the two eyes, referring to our analogy with the globe of the
Earth, above. By definition, it can only be a circle intersecting the nodal points of the
eyes and the fixation point, plus a vertical line intersecting the fixation point. It is
never a surface, except when the visual axes are parallel, ie when fixation is at infinity.
The empirical horopter may take one of many forms, depending on the psychophysical
criterion used to measure it. The definition of an empirical horopter may be based on
zero binocular disparity, zero horizontal disparity, equal perceived distance from the
observer, binocular fusion, to name but a few criteria. In each case the analytic form
of the horopter is unique because different geometries and tasks are involved in its
definition. Measuring the locus of identical visual directions in the two eyes, with a
subjective criterion of collinear dichoptic Nonius lines (Ogle 1964; Siderov et al 1999),
yields a locus called the Nonius horopter or longitudinal horopter. The locus of minimum
apparent motion between successively presented dichoptic lights or computer-generated
dots (Nakayama 1977; Ledgeway and Rogers 1999) has been dubbed the minimummotion horopter by Howard and Rogers (1995). Under these two procedures, empirical
corresponding points are operationally defined as the point between collinear dichoptic
Nonius lines or the point at which zero apparent motion between images presented
alternately to each eye is perceived. The empirical horopter, found with the Nonius
or minimum-motion criteria, is a shallower curve than the Vieth ^ Mu«ller circle with
a backwardly inclined vertical horopter. The fusion horopter is a three-dimensional
volume around the empirical horopter denoting the range in space where objects in the
two eyes appear single (Helmholtz 1910/1962; Cogan 1979). The apparent frontoparallel
horopter is a mapping of locations in space that appear to lie in a plane parallel to
the observer's forehead, if his or her visual axes lie in the horizontal plane of regard.
This type of judgment involves activity at a higher level of the visual system, which
may add compensations, distortions, and plasticity relative to the assumed invariant
form of the binocular-correspondence horopter. The apparent frontoparallel horopter
is therefore not as fundamental a concept as the binocular correspondence horopter.
It is defined here because of its possible implications in interpreting the results of
experiment 1 and is not to be confused with empirical horopters discussed elsewhere in
this paper. For a complete review of all types of horopters, see Tyler (1991), and Howard
and Rogers (1995).
1.2 Rationale
While the magnitude of the discrepancies between geometric and empirical horopters
may differ across observers, (see, for example, Helmholtz 1910/1962; Nakayama 1977;
Ledgeway and Rogers 1999), the qualitative nature of these deviations from the geometric
ideal is appreciated among all investigators. Until very recently, horopter investigations
shared another common feature. That is, the search for locations in space from
which empirical corresponding points are stimulated has been restricted to the horizontal plane containing the visual axes, for horizontal-horopter measurements, and
the median plane of the head, for vertical-horopter measurements. This restriction is
due, at least in part, to the fact that, for nonzero convergence angles, vertical disparities exist between the two eyes everywhere in space except along the horizontal plane
containing the visual axes and the median plane. There are no vertical disparities along
Empirical corresponding points
413
the horizontal meridian because points have no vertical extent and thus project to the
same horizontal meridian in the two eyes. Points in the median plane of the head are
equidistant from the two eyes and also project to the same horizontal meridians in
the two eyes. Therefore, in previous work on the horizontal and vertical horopters,
researchers have obeyed these geometrical restrictions, ignoring the possibility that
there may exist points in space at oblique locations, which may come close to stimulating empirical corresponding points. Two exceptions are cited below.
Banks and Backus (1998) measured both empirical corresponding points, with a
modified Nonius procedure, and the fusional volume over the central visual field. They
report data denoting the shape and inclination of a minimum disparity or `most fusible
surface'. Ledgeway and Rogers (1999), using a minimum-apparent-motion technique,
report data showing that corresponding vertical meridians are tilted templeward at
eccentricities out to 16 deg. Theoretically, points in space defining a zero-disparity or
minimum-disparity surface could be derived from Ledgeway and Rogers's data, but
were not reported by the authors. Nevertheless, previous claims in the literature have
hinted at the existence of a minimum-disparity locus that defines a surface inclined
towards the vertical horopter. Three such claims are outlined below.
When Helmholtz (1910/1962) first suggested that the empirical vertical horopter is
inclined top-away from the observer, he offered an explanation for this phenomenon
based on adaptive principles of evolutionary development. That is, at viewing distances
beyond, say, 2 ^ 3 m the vertical horopter is effectively horizontal and coincident with
the ground plane. Objects of most interest to our prehistoric ground-dwelling ancestors,
such as food, prey, and predators, were most likely to be situated on the ground as
well. A horopter coincident with the ground plane would, therefore, be advantageous.
Nakayama (1977) acknowledged Helmholtz's explanation and added his own, contemporary, behavioural consequence of the backward inclination of the vertical horopter:
``Only a moment's reflection is required to recall that this is, indeed a very typical case
in which we use our eyes, either looking at a book slanting backwards from our line of
sight or when we are walking along the ground and looking ahead.'' (page 7, our italics)
One can hardly assume that evolutionary pressures favoured a visual system that could
read books with greater facility, yet it is an attractive behaviour to cite as a consequence of the backward inclination of the vertical horopter.
Two more references to reading and the vertical horopter can be found in Tyler's (1991)
chapter on the horopter and binocular fusion. The context of this quotation is a discussion on how extorsional eye movements affect the inclination of the vertical horopter:
``... extorsion accompanying down-gaze ... will further incline the vertical horopter in
relation to its position in down-gaze if no cyclovergence were to occur. This is a common
occurrence; it occurs, for example, when a person reads a book.'' (page 28, our italics)
Later in that chapter, in the context of the adaptive advantage to the increasing size of
Panum's fusional area with eccentricity, Tyler argues that this renders larger portions
of planar surfaces fusible. Of interest here, however, is his assertion about the optimal
angle at which to view a planar object, which is consistent with Helmholtz's claim
that the vertical horopter is a line intersecting the fixation point and the feet of the
standing observer:
``... the optimal angle to view any planar object is such that its plane would pass about
5 ft below the eyes (through the feet when standing) ... and is the angle to adopt for
maximum comfort when viewing computer and display screens.'' (page 32)
The explicit suggestion in the above statements of Helmholtz (1910/1962), Nakayama
(1977), and Tyler (1991) is that the vertical horopter is inclined top-away, resulting
from evolutionary pressures to bring the ground plane into binocular correspondence.
414
P M Grove, H Kaneko, H Ono
This can be observed in everyday behaviours like reading. Therefore, experiment 1
analysed reading behaviour to determine systematically if we hold our reading material
to bring the page close the orientation of the vertical horopter.
The implicit suggestion contained in the above statements is that central and oblique locations in the binocular field of view that stimulate empirical corresponding
points may be mapped out to define a `plane' that is inclined backwards, including the
vertical horopter. If retinal correspondence is sheared horizontally to bring the ground
plane to within fusional limits of binocular correspondence across the retinas, it is
possible that points within the central binocular field defining a minimum-disparity
surface may be derived from psychophysical data. This implicit suggestion is supported
by a recent study by Nakamizo et al (1999). These authors presented data consistent
with the idea that points can be identified which define a surface lying within the fusional
limits of the binocular system. In their experiments, observers viewed a uniform display
of dots, 21 cm wide and 31 cm long, at various inclinations and reported seeing a
`subjective staircase' consisting of dots on treads. The number of `steps' was a function
of the angle of inclination of the display of dots. The least number of steps was
reported when the inclination of the display was close to that predicted from Helmholtz's report that corresponding vertical meridians are tilted templeward by 1:258.
The fact that each perceived step spanned the entire stimulus and appeared
and disappeared as a function of inclination is further evidence that binocular correspondence is horizontally sheared across the retina. Moreover, it suggests the existence
of a planar locus of minimum disparity that is inclined backwards towards the empirical
vertical horopter. In experiments 2 and 3 we measured the horizontal and vertical
adjustments required to stimulate corresponding points, psychophysically defined, in
the two eyes using a motion-nulling technique for fifty locations over the central
24 deg of the binocular field. In this report, we refer to ``psychophysically defined
corresponding points'' as points in the two eyes which, when stimulated in succession,
give rise to zero apparent interocular motion.
2 Experiment 1
As we have outlined above, the horizontal shearing of retinal correspondence may be
the result of evolutionary pressures to make the horopter coincident with the ground
plane. Furthermore, the backward inclination of the vertical horopter may be reflected
in behaviours such as reading. This experiment tested this hypothesis directly. The
question this experiment was intended to answer was: ``What is the preferred page
orientation, relative to the visual axis, for reading?''.
2.1 Method
2.1.1 Observers. Five observers, na|« ve to the purpose of the study, with normal or
corrected-to-normal binocular vision, participated.
2.1.2 Apparatus and stimulus. The apparatus for this experiment, illustrated schematically
in figure 1, was modeled after that of Breitmeyer et al (1977). It consisted of the passage
of English text, in 12-point Times font, mounted on a surface which rotated about
a horizontal axis. Each observer stood with his or her head in a chin-and-forehead
rest. An aperture, subtending 21 deg621 deg restricted the field of view. This passage
of text contained a fixation point at the centre of the page. The page was illuminated
equally from above and below to reduce the change in luminance of the page with
orientation. Viewing distance was 65 cm.
2.1.3 Procedure. We used the method of adjustment to determine observers' preferred
orientation for reading a page of text. A given trial began with the page of text oriented
at 408 relative to the visual axis. All observers reported that this position was an
Empirical corresponding points
415
Text
Rotating screen
Aperture
12
cm
12 cm
65 cm
Figure 1. Schematic diagram of the apparatus used in experiment 1.
undesirable orientation for reading. The experimenter slowly rotated the passage towards
vertical (and past, if necessary) until the observer reported that the page was in .....an
`optimal' position for reading, and the angle relative to vertical was recorded. Observers
completed 24 trials, 12 ascending and 12 descending, in random order.
2.2 Results and discussion
We calculated the mean angles preferred for reading, relative to vertical, for each
observer. These are presented in figure 2. Although individual differences in magnitude
of backward inclination are apparent, all observers preferred a backwards-inclined
page relative to vertical. The mean backward inclination of all observers was 9.818
(SD ˆ 8:328). This overall value is much less than the inclination angle predicted from
the psychophysical data reported by Helmholtz (1910/1962) and, later, by Nakayama (1977).
20
Inclination=8
15
Backwards tilt
10
5
0
ÿ5
MT
DC
PZ
RK
Subject
AC Grand
mean
a
Figure 2. Mean settings of preferred
orientation for reading, 1 SEM, for five
observers. The right bar indicates the
mean of all observers, 1 SEM. Note
that all observers oriented the passage
with the top away.
416
P M Grove, H Kaneko, H Ono
If it is assumed that corresponding vertical meridians, intersecting the fovea in each
eye, are tilted templewards by 28 relative to vertical, the vertical horopter at 65 cm
should be inclined back from vertical by 358.
How, then, can we explain this difference between our data and those predicted
from previous studies? One possible account is that reading does not necessarily
require binocular correspondence over a large area. This may be the reason, combined
with other possible mitigating factors like perspective cues contained in the text or the
subjective bias that the page `should' be perpendicular to the line of sight.(2) As one
anonymous reviewer of this paper pointed out, Cogan (1979) verified that the
fusion horopter for a single bar was inclined backwards by approximately 308 at
100 cm. Five of the eight observers' judgments of apparent vertical for the same bar
were inclined backwards by between 28 and 78. It is possible that observers may have
been setting the passage of text as close to apparent frontoparallel as possible within
the range of fusion, similar to what observers might do when measuring the apparent
frontoparallel horopter. As mentioned above, observers were standing when viewing
the page of text. Our common experience when reading in this posture is that the
text is presented on frontoparallel surfaces, such as bulletin boards and billboard
advertisements. Therefore, interpreting our test of Nakayama's (1977) and Tyler's (1991)
observations about reading and the backward inclination of the vertical horopter is
not as straightforward as we expected. Orienting the page so as to reduce binocular
disparity and increase fusion over a larger portion of the page is one of several criteria
observers may have used, given the instructions they followed. Observers may have
been orienting the page to satisfy criteria associated with a zero-disparity horopter, a
fusion horopter, or an apparent-frontoparallel horopter, or in accordance with their
daily experience with billboards, computer screens, and other text presented on frontoparallel surfaces.
Nevertheless, the bias towards a backwards-inclined page for optimal reading is
consistent with the claims of Nakayama (1977) and Tyler (1991), among others, that the
vertical horopter is indeed inclined backwards. The explicit suggestion here is that a page
containing text is preferentially oriented backwards relative to the observer to bring it
close to the orientation of the vertical horopter so that text above and below fixation
falls within Panum's fusional area. This matching is not completely in accordance
with the psychophysical data collected under reduced conditions, however. In the
next two experiments we employed reduced psychophysical techniques to answer the
question: ``Where are the locations in the central binocular field that stimulate psychophysically defined corresponding points in the two eyes?''. We collected data from
50 point locations, regularly spaced over the central 24 deg of the binocular field.
3 Experiment 2
This experiment can be thought of as measuring the inclination of seven vertical
horopters: one in the median plane, three to the left, and three to the right. Our first
step was to ascertain how much we had to deviate horizontally each eye's image of
a single point, on a frontal surface, until it stimulated corresponding points in the two
eyes, according to the criterion of zero interocular motion. Our method, adapted from
Nakayama (1977) and Nakayama et al (1977), consisted of measuring the horizontal
adjustments required to eliminate subjective motion of a single dot, presented alternately
to each eye at each of 50 locations, regularly spaced over the central binocular field.
We used zero apparent interocular motion as the psychophysical criterion of identical
visual direction and inferred from this that the dichoptic images were stimulating corresponding points, psychophysically defined, in the two eyes.
(2) It
was the PG's impression, during pilot experiments, that he was orienting the page at a right
angle to his visual axes, or parallel to his forehead, even though the page was inclined top-away.
Empirical corresponding points
417
3.1 Method
3.1.1 Observers. Three male observers participated, consisting of two of the authors and
one observer, na|« ve to the purpose of the study. One observer, HK, wore corrective
spectacles during the experiment. The other two reported normal acuity and binocular
vision. All were experienced in psychophysical experiments.
3.1.2 Apparatus and stimuli. Stimuli were generated on a Macintosh computer and
presented on two 40.4 cm630.2 cm Sony Trinitron colour graphic displays (model
number GDM-F500). The monitors were carefully adjusted such that the two screens
faced one another and were aligned and parallel. Observers viewed the screens via
two front-silvered mirrors oriented at 458 to the median plane of the head. The alignment of the monitors and mirrors was checked by comparing a computer-generated
grid, presented on each of the screens with a hand-drawn grid presented 65 cm directly
in front of the observer, centred on the median plane. Minor adjustments were
made to bring the computer grid to as close alignment as possible with the real grid.
Deviations of the computer grid from the real one were less than 1%, 22 cm (half the
width of the stimulus display) from the centre. More importantly, differences between
the grids of the two displays were negligible. When viewed in a dark room, the fused
image of the two zero-disparity grids appeared as a flat surface. The fronts of the two
monitors were cropped with flat black apertures and a flat black curtain enclosed
the entire stereoscope. During the experiment, only the stimuli on the screens were
visible to the observer.
For observers PG and HK, measurements of the horizontal adjustments required
to null apparent motion were taken at azimuths of 0, 4:05, 8:07, 12:01 deg and
elevations of 0, 2:98, 5:96, 8:9, 11:8 deg. NU completed measurements from a
smaller sample of locations (horizontal azimuths of 0, 6:07, 12:01 deg; elevations
of 0, 4:5, 11:8 deg). Point locations at which measurements of horizontal and vertical adjustment required to eliminate apparent motion were made were determined by
the intersection of the azimuth and elevations.
Each observer sat with his head in a chin-and-forehead rest and viewed stimuli in
a stereoscope at a distance of 65 cm. The height of the chin rest was adjusted for
each observer such that the centre of the fixation stimulus was located at the intersection of the median plane and the horizontal plane of regard. The fixation stimulus
consisted of a binocular circle flanked by dichoptic Nonius lines, oriented vertically
and horizontally, to monitor observers' horizontal and vertical vergence. (3) The target
stimulus was a bright dot (diameter 3 deg) that was presented alternately to each eye.
For horizontal measurements, the dots had a different horizontal Cartesian coordinate
in each eye, but had the same vertical coordinate.
3.1.3 Procedure. An observer sat in a dark room with only the fixation and target
stimuli visible. His task was to fixate the centre of the display and null apparent
motion by adjusting the horizontal positions of the dichoptically presented targets
with leftward and rightward movements of a trackball. On a given trial, the starting
horizontal positions of the target dots were offset from the actual point of interest by
0:75 deg. For example, if the point of interest was located in the median plane and
2.98 deg above the fixation point, the starting position of the dot visible to the right
(3) During
this experiment as well as the pilot studies, several Nonius stimuli were used to monitor
possible cyclorotary eye movements. In one case, extended horizontal Nonius, with the image for
each eye subtending over 12 deg, were employed to ensure that torsional eye movements were
minimal during the experiments. In addition, four small binocular dots were positioned around the
fixation stimulus to act as a fusion lock. We are, therefore, confident that observer's fixation was
stable during the experiment and that the data are not contaminated with systematic cyclorotary
eye movements.
418
P M Grove, H Kaneko, H Ono
eye would be 0:75 deg away from the median plane. The image for the left eye was
positioned an equal distance to the other side of median plane. The image for the right
eye had an initial positive offset on half the trials, randomly determined, and a negative
offset on half the trials, while the opposite was true for the image for the left eye.
Schematic views of the dichoptic images and the perceived motion are illustrated in
figure 3a.
Image for the
left eye
Image for the
right eye
Perceived motion
t1
t2
(a)
Image for the
left eye
Image for the
right eye
Perceived motion
(b)
t2
t1
t1
t2
Figure 3. (a) Illustration of the fixation
stimuli and dichoptic images used to measure the horizontal adjustments required to
eliminate apparent interocular motion.
Upper and lower image pairs alternated
at 0.63 Hz to produce interocular apparent
motion, as illustrated in the single frame
on the right. At time t1 (upper two panels),
the image for the left eye contained a bright
dot and fixation stimulus while the image
for the right eye contained only the fixation
stimulus. At time t2 (lower two panels), the
image for the right eye contained the bright
dot, laterally displaced while the image
for the left eye contained only the fixation
stimulus. (b) Illustration of the fixation
stimuli and dichoptic images used to measure the vertical adjustments required to
eliminate apparent vertical motion. Upper
and lower image pairs alternated, as in (a),
to produce interocular apparent `seesaw'
motion, illustrated in the single frame on
the right.
The two dots could be adjusted in equal and opposite directions, centred on the
point of interest. In the case of the above example, the motion of the dichoptic dots
would be centred on the median plane. On a given trial, leftward movement of the
trackball moved the dots away from each other, and rightward movement brought
them closer together. The dots could be adjusted with enough freedom so that they
could pass over the `zero point'öthe point where they were positioned at the same
Cartesian coordinate on each monitor öand continue to separate in the other direction. Therefore, the adjustment apparatus did not limit motion-nulling adjustments.
When the observer was satisfied that he had nulled the apparent motion, he pressed
a button. This procedure was immediately repeated at another, randomly chosen,
location. Observers made six motion-nulling adjustments at each location.
3.2 Results and discussion
The actual linear deviations of the images for the left and right eyes from the objective
points of interest on the computer screens are illustrated in figure 4. This graph shows
the direction and relative magnitude of horizontal adjustments required to stimulate
psychophysical corresponding points from a frontal plane viewed at 65 cm. Each
symbol, diamonds for the right eye, circles for the left eye, represents the mean of
six observations. Objective locations defined by Cartesian coordinates are indicated by
`plus' symbols. The most striking feature of this illustration is the pronounced shear of
Empirical corresponding points
15
419
PG
10
5
0
ÿ5
ÿ15
15
15
HK
10
10
5
5
0
ÿ5
ÿ10
ÿ15
15
NU
10
5
0
ÿ5
Vertical distance from fixation point=cm
Vertical distance from fixation point=cm
ÿ10
Frontoparallel surface
0
ÿ5
ÿ10
ÿ15
15
Inclined surface
10
5
0
ÿ5
ÿ10
ÿ10
ÿ15
ÿ20 ÿ10
0
10
20
Distance from median plane=cm
ÿ15
ÿ20 ÿ10
0
10
20
Distance from median plane=cm
(a)
(b)
Figure 4. (a) Linear horizontal shifts required to eliminate apparent motion at each of the fifty
locations in the central binocular field. `Plus' symbols (‡) indicate the Cartesian coordinate of the
point on the display; diamonds (^) indicate the location of the image for the right eye; circles (*)
indicate the location of the image for the left eye. (b) Horizontal-adjustment vectors predicted
from geometry. The top graph shows the predicted adjustment vectors for a frontoparallel surface;
the bottom graph shows the predicted vectors for a surface inclined top-away by 308.
horizontal adjustments across the entire field, not predicted from geometry. For positions
further above the plane of regard, the image for the right eye was positioned further
and further to the right of the objective location, whereas the image for the left eye
was positioned further and further to the left of the objective location. These adjustments correspond to real locations in space further behind the plane of fixation (the
computer screens). This pattern was reversed for positions below the plane of regard.
That is, the image for the right eye was positioned further and further to the left of
the objective location whereas the image for the left eye was positioned further and
further to the right of the objective location for elevations below the fixation point.
These adjustments correspond to real locations in space in front of the plane of
fixation. This pattern is exhibited most clearly in the median plane but is still quite
pronounced for eccentric locations, particularly for observers HK and NU. This
complete pattern of horizontal adjustments defines a surface inclined top-away. This is
apparent if we inspect figure 4b. Here we have illustrated the adjustment vectors,
derived from geometry, required to eliminate the retinal disparity of a frontoparallel
surface and one inclined top-away by 308 at 65 cm. It is evident that our psychophysical
420
P M Grove, H Kaneko, H Ono
data more closely resemble the geometrically predicted data for an inclined surface
than for a frontoparallel surface.
Additionally, we determined the magnitude of horizontal adjustments, in angular
terms, by calculating the mean angular displacement of the image for the left eye from
the median plane and subtracting it from that of the image for the right eye. This analysis
is shown graphically in figure 5. We calculated the angular difference between the images
for the left and right eyes by subtracting the monocular subtense between the fixation
point and its image on the display for the left eye from that for the right eye. This
value gave us the difference in angular deviation from the vertical meridian containing
the fovea on the retina of the images for the left and right eyes that was required to
eliminate apparent motion at the point defined by Cartesian coordinates on the displays.
75
50
HK, eccentricity
0 deg
PG, eccentricity
0 deg
NU, eccentricity
0 deg
25
0
ÿ25
ÿ50
ÿ75
ÿ15ÿ10 ÿ5
75
50
0
5
10 15 ÿ15ÿ10 ÿ5
HK, eccentricity
4:05 deg
0
5
10 15 ÿ15ÿ10 ÿ5
0
5
10 15
PG, eccentricity
4:05 deg
Horizontal adjustment=min of arc
25
0
75
ÿ25
50
ÿ50
25
ÿ75
0
75
50
HK, eccentricity
8:07 deg
PG, eccentricity
8:07 deg
25
NU, eccentricity
6.07 deg
ÿ25
ÿ50
ÿ75
0
ÿ25
ÿ50
ÿ75
75
50
HK, eccentricity
12:01 deg
NU, eccentricity
12:01 deg
PG, eccentricity
12:01 deg
25
0
ÿ25
ÿ50
ÿ75
ÿ10
ÿ5
0
5
10 ÿ10
ÿ5
0
5
Elevation=8
10
ÿ10
ÿ5
0
5
10
Figure 5. Individual plots of mean angular horizontal adjustments required to null apparent
interocular motion as a function of elevation for three observers. Each column of graphs is
for one observer. Each row of graphs is for each absolute eccentricity. Open circles represent
data at positive eccentricities, closed triangles represent data at negative eccentricities, and
closed squares represent data at zero eccentricity. Error bars represent 1 SEM.
Empirical corresponding points
421
Conceptually, the locations of real points in space that would project these images
onto the two retinas can be ascertained by drawing a line through the nodal point of
each eye and its respective image on the computer screens and locating the point in
space where these two lines intersect. An analysis of the data allows us to do this
and find the angle of inclination of each of the vertical horopters defined thus far.
In our analysis, shown graphically in figure 6, we first calculated the location in depth
of each objective location. We did this by subtracting the angular separation of the
images for the right and left eyes from the angle of convergence at the point of
fixation. This gave us the angle of convergence required to fixate a real point yielding
the disparate points measured on the displays. We then took the tangent of half this
angle and divided it into half the interocular distance, yielding the distance at which
that point needed to be positioned in order to stimulate corresponding vertical meridians in the two eyes. We repeated this calculation at each elevation and eccentricity.
We next plotted the resultant data, with distance to the actual location with zero
horizontal disparity along the x axis and linear elevation along the y axis. Finally,
using the slope of the best-fitting line, joining all the points at a given eccentricity, we
calculated the angle of inclination away from vertical. It is apparent from the figure that
the empirical vertical horopter in the median plane is inclined, top-away, by approximately 308, for all three observers (33.98 for PG, 30.98 for HK, and 29.48 for NU).
50
Inclination=8
40
30
20
10
PG
HK
NU
0
ÿ15 ÿ10 ÿ5 0
5 10
Eccentricity=8
Figure 6. Inclination of the zero horizontal disparity locus
plotted as a function of eccentricity. See text for details.
15
This inclination is fairly constant at the other eccentricities measured for two of
the three observers. For one observer, PG, however, the inclination of the vertical
horopter at eccentric locations to the left and right of the median plane drops off and
approaches vertical, most notably at 12 deg to the left of the median plane where it is
almost vertical. One anonymous reviewer of this paper pointed out that this pattern
is reminiscent of Owens et al's (1987) report where inclination of the vertical horopter
was greater on one side than on the midline, and less on the other side. Such a twist
could be accounted for by opposing prismatic distortions in the two eyes. We do not
know of any optic distortion that would account for a flattening of the inclination on
either side of the median plane. This observer is not aware of any optical distortions
in his eyes and does not wear corrective spectacles. We speculate (along with our
reviewer) that his data represent a local inflection in the inclination of the more
eccentric vertical horopters. The pattern of these adjustments, for two of the three
observers, agrees with the report of Ledgeway and Rogers (1999) that corresponding
vertical meridians are tilted top-away, relative to each other in the two eyes. To
check further our compliance with Ledgeway and Rogers, we calculated the relative tilt
of the primary vertical meridians for each of the three observers in experiment 2.
The vertical meridians are tilted templeward by between 1.68 and 1.98 (1.98 for PG,
1.78 for HK, and 1.68 for NU). The magnitudes of the relative tilt of corresponding
vertical meridians reported here are consistent with the previous reports of Helmholtz
422
P M Grove, H Kaneko, H Ono
Horizontal adjustment=min of arc
(1910/1962), Nakayama (1977), and more recently Ledgeway and Rogers (1999) and
Nakamizo et al (1999). These four investigations report a relative tilt in the range of
1:258 to 28. The angle of inclination of the vertical horopter is directly related to
the relative tilt of the vertical meridians (Howard and Rogers 1995). Therefore, it is
clear that the relative tilt of corresponding vertical meridians is fairly constant to
eccentricities of 12 deg, for observers HK and NU in particular, further supporting
the report of Ledgeway and Rogers (1999).(4)
Finally, the data from experiment 2 show that points defining a planar surface
65 cm from the observer's eyes, inclined top-away by approximately 308 has zero horizontal disparity, psychophysically defined. This inclination angle is significantly larger
than that reported recently by Siderov et al (1999). These authors report that the
vertical horopter is inclined by between 38 and 48 in the median plane at a viewing
distance of 50 cm, determined with the Nonius criterion. How can we account for
this tenfold difference? One possible explanation is the differences in experimental
procedures. Therefore, to address this possibility, we modified our displays to comply
with Siderov et al's paradigm as closely as possible. In this replication, the stimuli
were two bright thin bars positioned one above the other. We employed Siderov et al's
alternating Nonius method, measuring at 2.5 deg intervals above and below the fixation point. Owing to the limitations of our displays, our measurements extended only
10 deg above and below the fixation point. We manipulated observers' vergence angle
to simulate two different viewing distances. In the near-vergence condition, measurements were taken at 65 cm with the eyes converged at that distance. In the far-vergence
condition, the fixation point for each eye was deviated away from the median plane
to simulate vergence at 200 cm. The Nonius lines alternated at 5 Hz, equivalent to the
200 ms duration reported by Siderov et al. Observers adjusted the relative offset of
the Nonius lines using a trackball. They made six measurements at each elevation above
and below the fixation point. A bracketing procedure was used where the observer
pressed a button when he was confident that the Nonius lines were positioned one on
top of the other. Data for the two observers are presented in figure 7. Clearly, the
magnitude of horizontal adjustments required to eliminate the apparent offset between
the Nonius lines is quite similar to the data we report for our minimum-motion paradigm
with two bright dots. The data are in close agreement at both fixation distances.
60
PG
HK
30
0
ÿ30
ÿ60
ÿ15 ÿ10 ÿ5
0
5
10
15 ÿ15 ÿ10 ÿ5
Elevation=8
0
5
10
15
Figure 7. Mean angular horizontal adjustments, 1 SEM, required to null apparent motion of
dichoptic Nonius lines obtained with the same experimental paradigm as that used by Siderov
et al (1999). Vergence angles equivalent to 65 cm (open squares) and 200 cm (solid circles) are
shown. The lines are best fit. The separation between the two lines represents a small fixation
disparity. See text for details.
(4) In fact, calculating the relative tilt of corresponding vertical meridians is merely a transform
of the calculation of the inclination of the vertical horopter. Therefore, the pattern of relative
tilt of corresponding meridians is identical to the pattern of inclination of the vertical horopter at
different eccentricities.
Empirical corresponding points
423
We, therefore, cannot account for the discrepancy between our data and Siderov et al's
on the basis of these methodological differences. At this point we conclude that our
horizontal-adjustment data are most consistent with the report of Ledgeway and Rogers (1999).
4 Experiment 3
Geometry dictates that points in oblique locations project to vertically noncorresponding
points in the two eyes. For example, a vertical line viewed in the median plane of
the head, at a given viewing distance, projects images of equal vertical extent in the
two eyes because the line is equidistant from the two eyes. If the line is moved
laterally, the relative vertical size of the images in the two eyes changes. If the line is
moved to the right, for example, its image in the right eye increases in vertical size
relative to the left eye because the object is now closer to the right eye than the left.
The end points of the line are now in oblique locations and have a vertical disparity.
By extension, points away from the median plane and the horizontal plane of regard
on a frontal surface, such as the computer screens used in this study, project to vertically disparate points in the two eyes. Rogers and Bradshaw (1995) reported that
frontal surfaces project a specific pattern of horizontal and vertical disparities onto the
two retinas. They termed this pattern `differential perspective', because these disparities
arise from viewing a frontoparallel surface from the vantage points of the two eyes.
In this experiment we determined the locations in space which project to vertically
nondisparate points, psychophysically defined, in the two eyes. To do this, we measured
the vertical adjustments required to null apparent vertical motion of a pair of dots,
presented at the same objective locations at which horizontal measurements were taken.(5)
4.1 Method
4.1.1 Observers. The observers were the same as in experiment 2.
4.1.2 Apparatus. The apparatus was the same as that used in experiment 2.
4.1.3 Stimuli. We employed different stimuli for the measurement of vertical adjustments
required to stimulate corresponding points. Instead of a single dot, presented alternately to each eye, we used a pair of dots, separated horizontally and equidistant from
the median plane, shown in figure 3b. The dots in the image for one eye were tilted
in the opposite direction to those in the image for the other eye so that when the
image pairs alternated at 0.63 Hz the observer perceived a `seesaw' motion, as depicted
in the figure.
On a given trial the dot pairs were presented, one on each side of the median plane
at one of three eccentricities (4:05, 8:07, 12 deg). For observers PG and HK,
measurements were taken for three dot separations and seven elevations (8:9, 5:9,
2:9, and 0 deg) for a total of 126 trials each. For observer NU, measurements were taken
for two dot separations (12.14 and 24 deg) and five elevations (8:9, 4:5, and 0 deg), for
a total of 60 trials.
4.1.4 Procedure. The procedure was nearly the same as in experiment 2. The only
difference was the number and position of the dots in the alternating images. As in
experiment 2, observers viewed dichoptic images, one a blank field and the other
(5) In
principle, experiments 2 and 3 could have been performed at the same time. That is, nulling
of vertical motion could have followed nulling of horizontal motion on each trial. Owing to fatigue
and the subjective difficulty of the task, however, we chose to conduct two separate experiments.
A subset of data was collected from PG where nulling of horizontal motion was followed by
nulling of vertical motion. Moreover, we employed similar stimuli to those used by Banks and Backus
(1998). Their data matched the pattern of results recorded from the same locations in experiments
2 and 3.
424
P M Grove, H Kaneko, H Ono
contained the dot images. This time, however, observers were required to adjust the
relative tilt of the alternating images until no vertical motion was seen. Images for the
left and right eyes began with an initial offset from horizontal. This was done in a
similar manner to experiment 2. One dot was shifted 0.75 deg up or down from its
objective location, while its mate on the other side of the median plane was shifted
0.75 deg down or up from its objective location, respectively. Shifts in the opposite
direction were added to the image for the other eye and the direction of the shifts
was alternated between the eyes randomly throughout the experiment. Observers saw
a pronounced `seesaw' motion at the beginning of each trial and were required to
eliminate any vertical motion with upward and downward adjustments of a trackball.
As in experiment 2, observers could adjust the two images past the `zero point' so as
not to limit the position of the dots when trying to eliminate apparent motion. When
the observer was satisfied that he had nulled the apparent motion, he pressed a button.
This procedure was immediately repeated at another, randomly determined, elevation
and dot separation. Observers made six vertical-motion-nulling adjustments for each
elevation and dot separation.
4.2 Results and discussion
The actual linear vertical deviations required to stimulate psychophysical corresponding
horizontal meridians from a frontal plane viewed at 65 cm are illustrated in figure 8
together with the horizontal adjustments from experiment 2. Deviating the images for
the left and right eyes from their objective location on the displays according to the
data from experiments 2 and 3 will result in the stimulation of corresponding points,
psychophysically defined. Each symbol represents the mean of six observations. For
positions to the right of the median plane, above and below the horizontal plane of
regard, observers set the distal image for the left eye further away from the horizontal
plane of regard than the Cartesian coordinate. For the same positions, observers set
the distal image for the right eye closer to the horizontal plane of regard than the
Cartesian coordinate. The opposite was true for positions further to the left of the
median plane.
We calculated the retinal projections predicted for a surface inclined by 308 at a
distance of 65 cm. Howard and Rogers (1995) also illustrate the predicted pattern of
horizontal and vertical retinal disparities of an inclined surface. Our data are consistent,
in pattern, with an inclined surface and a frontoparallel surface. At this time we do not
know of a method precise enough to distinguish between the vertical adjustment
patterns of a frontoparallel surface and one inclined by 308. In the context of this paper,
however, where the horizontal-adjustment data from experiment 2 define a
surface stimulating psychophysical corresponding points as inclined backwards by 308
at 65 cm, the data from the present experiment are consistent with an inclined surface.
Additionally, we determined the magnitude of vertical adjustments, at the retinas,
by calculating the mean angular displacement of the image for the left eye from the
horizontal plane of regard and subtracting it from that of the image for the right eye.
We calculated the angular difference between the images for the left and right eyes by
subtracting the monocular subtense between the horizontal plane of regard and its
image on the display for the left eye from that for the right eye. This value gave us the
difference in angular deviation from the horizontal plane of regard, containing the
fovea, of the images for the left and right eyes that was required to eliminate apparent
motion at the point defined by Cartesian coordinates. The mean angular vertical
adjustments which were required to eliminate apparent motion at each of the three
elevations (0, 8:9 deg) and two dot separations (16.14 and 24 deg) are illustrated in
figure 9. Data for an additional dot separation of 8.1 deg are illustrated for observer PG.
One observer's data, NU, show that vertical adjustments were required to eliminate
Empirical corresponding points
15
425
PG
10
5
0
ÿ5
ÿ15
15
10
5
0
ÿ5
ÿ10
ÿ15
15
10
15
HK
NU
5
0
ÿ5
Frontoparallel surface
10
Vertical distance from fixation point=cm
Vertical distance from fixation point=cm
ÿ10
5
0
ÿ5
ÿ10
ÿ15
15
Inclined surface
10
5
0
ÿ5
ÿ10
ÿ10
ÿ15
ÿ20 ÿ10
0
10
20
Distance from median plane=cm
ÿ15
ÿ20 ÿ10
0
10
20
Distance from median plane=cm
(a)
(b)
Figure 8. (a) Linear vertical and horizontal shifts required to eliminate apparent motion at each of
the 50 locations in the central binocular field. `Plus' symbols (‡) indicate the Cartesian coordinate
of the point on the display; diamonds (^) indicate the location of the image for the right eye;
circles (*) indicate the location of the image for the left eye. (b) Vertical and horizontal adjustment vectors predicted from geometry. The top graph shows the predicted adjustment vectors
for a frontoparallel surface, and the bottom graph shows the predicted vectors for a surface
inclined top-away by 308. See text for details.
apparent vertical motion at zero elevation. We interpret this as a possible cyclophoria.
We normalised this observer's data by subtracting the values for his zero-elevation
adjustments at each eccentricity from the elevated locations at the same eccentricities.
After this correction, NU's data are very similar to the other two observers' data. The
data from this experiment confirm that points away from the median plane and the
horizontal plane of regard project images to vertically disparate points in the two eyes.
Banks and Backus (1998) reported vertical-adjustment data suggesting a finite distance, closer than infinity but more distant than the viewing distance they employed,
for locations in space that project to minimally disparate points on the two retinas.
The pattern of vertical adjustments measured in this experiment is similar to the
pattern of geometrically predicted vertical adjustments for a frontoparallel surface at
65 cm, as shown in the top graph of figure 8b. The pattern of vertical adjustments
indicates that the surface stimulating psychophysically defined corresponding points is
farther than 65 cm. If the vertical displacements of the images for the right and left
Vertical adjustment=min of arc
Vertical adjustment=min of arc
426
P M Grove, H Kaneko, H Ono
60
45
Elevation=deg
PG
NU (raw)
HK
NU (normalised)
8.9
0
ÿ8:9
30
15
0
ÿ15
ÿ30
ÿ45
ÿ60
60
45
30
15
0
ÿ15
ÿ30
ÿ45
ÿ60
ÿ15 ÿ10 ÿ5
0
5
10 15 ÿ15 ÿ10 ÿ5
Eccentricity=8
0
5
10
15
Figure 9. Three observers' angular vertical adjustments, 1 SEM, required to eliminate apparent
vertical interocular motion for dot separations of 16.14 and 24 deg and elevations of 0 and
8:9 deg. See text for details.
eyes are zero everywhere on the screen, with the images for the left and right eyes both
coinciding with the Cartesian coordinates on the displays, this would indicate that
the surface stimulating psychophysically defined corresponding points is at 65 cm.
If the vertical displacement pattern coincides with the pattern of geometrically predicted
vertical adjustments for a frontoparallel surface at 65 cm, as shown in figure 8b, geometry dictates that a surface stimulating geometrically defined corresponding points is
at optical infinity. Our results, as shown in figure 8a, are in the direction of the latter
case. We are unable to make a quantitative inference on the specific distance of the
minimum-disparity surface, defined by our data, at this time. In this paper we wish
to conclude only that vertical disparities are also needed to stimulate psychophysically defined corresponding points. We are continuing to investigate more precise
methods for determining the vertical component of psychophysically defined corresponding points.
5 General discussion
Our investigation into binocular correspondence for retinal locations other than the
primary meridians of the two eyes began with a behavioural investigation about how
readers orient a page for most comfortable reading. The data from experiment 1 illustrate
that readers preferentially orient a page of text such that it is inclined top-away. This
supports previous claims by Nakayama (1977) and Tyler (1991) that behavioural consequences of the inclination of the empirical vertical horopter may be manifested in
reading. While agreeing with these previous claims in pattern, the magnitude of these
adjustments were less than predicted from previous psychophysical reports. The magnitude of these adjustments being less than predicted from previous psychophysical
reports may be due to our choice of stimulus and instructions, although our stimulus
conforms to everyday reading stimuli as was suggested by Nakayama and by Tyler.
The entire text in this experiment was 12-point font. Therefore, observers were unlikely
Empirical corresponding points
427
to be able to resolve the text a few degrees away from where they were fixating.
Moreover, when reading English text, it is not necessarily functional to resolve the text
vertically since we move our eyes primarily from left to right when reading. With our
stimulus, resolving the text on the right side of fixation is functional for moving our
eyes to the next word, but there is little functional advantage to resolving the passage
too far up or down. We hypothesised that a stimulus that can be resolved and is functional to resolve the upper and lower visual fields (like in reading large Japanese or
Chinese characters written top to bottom), may yield results that agree more with
the predicted inclination from psychophysical data. Pilot data with Japanese native
speakers, reading large Kanji print from top to bottom, are similar to those presented
in experiment 1, however. Still, our data show that observers' unanimously preferred a
backwards-inclined page for reading. We interpreted observers' unanimous preference for
a passage of text, inclined top-away, to suggest that eccentric loci, inclined backwards,
may also project to corresponding points, psychophysically defined, in the two eyes.
In experiment 2, we employed reduced psychophysical conditions to ascertain if
the inclination of eccentric vertical horopters corresponded to the inclination of the
vertical horopter in the median plane. The horizontal-adjustment data from fifty locations
in the central binocular field define an approximate planar surface, inclined top-away
by about 308 at 65 cm, about a horizontal axis intersecting the fixation point; this
is considerably more than the data from experiment 1 suggest. In experiment 3 we
measured vertical adjustments required to eliminate apparent vertical motion.
Taken together, the pattern of horizontal and vertical adjustments, reported above,
defines a planar surface inclined top-away. Our everyday visual experience presents us
with a persistent asymmetry of the disparity field. We see more outwardly sheared images
(top outwards) than inwardly sheared images since there are more ground plane surfaces
than ceiling surfaces. It is reasonable to suppose that evolution has selected for a visual
system in which binocular correspondence is shaped by early visual experience. Thus,
the distribution of corresponding points in the two eyes is organised such that the
locus of zero disparity is biased towards the ground plane. This hypothesis is supported
by several visual-deprivation and neurophysiological experiments performed on cats.
Hanny and von der Heydt (1982) reared one set of kittens in an environment in
which visible contours were confined to a floor plane below eye level, producing only
outwardly sheared disparity fields. A second set of kittens was reared with only a
ceiling plane visible, producing only inwardly sheared disparity fields. When tested, the
binocular cortical cells of the kittens preferred orientations that differed in the two
eyes in accordance with the type of disparity that the kittens had experienced. Shinkman
and Bruce (1977) showed that kittens fitted with prisms that disjunctively rotate the
images in the two eyes through a small angle (approximately 88 in each eye) developed
a full complement of binocular cells. The preferred stimulus orientations of these
cells, for stimuli presented to each eye in turn, were found to be relatively rotated by
the amount of the induced optical rotation. These two studies suggest that the visual
system adapts during development to best accommodate the predominant visual stimuli.
Cooper and Pettigrew (1979a, 1979b) determined the vertical horopter in the cat and
burrowing owl by mapping the receptive-field positions of binocular cortical neurons
at various elevations along the zero azimuth meridians. In the cat, they found that
the primary vertical meridians were tilted templeward by 58 on average. In the owl, the
primary meridians were found to be tilted templeward by approximately 48 on average.
Using these figures and an average interpupillary distance for each species, the authors
calculated that the vertical horopter would form a line intersecting the fixation point
and a point in the corneal plane near the animal's feet. This is strong comparative
evidence in support of Helmholtz's account of a backwards-inclined vertical horopter.
428
P M Grove, H Kaneko, H Ono
While Helmholtz hypothesised that this adaptation took place on an evolutionary
scale rather than during the developmental period of our visual system, the general
idea is the same. That is, retinal correspondence is established to process the prevalent
disparity field with maximum efficiency. Our horizontal-adjustment data define, and
our vertical-adjustment data are consistent with, a minimum-disparity surface, inclined
by a similar amount to that predicted from Helmholtz's original data. This is further
support for the idea that evolution has selected for a visual system which organises
itself to place the locus of binocular correspondence near the ground plane.
It is possible that observers in experiment 1 oriented the text such that fusion was
achieved in the midline alone. In this case the inclination of more eccentric vertical
horopters cannot be inferred. As we mentioned before, our data are also consistent
with Nakamizo et al's (1999) data indicating that corresponding vertical meridians are
tilted templeward by 1:38. This paper is relevant to our data for two reasons. First,
using a different methodology (recording the inclination of a uniform display of dots
when the number of steps in the subjective staircase was at a minimum), these investigators reported data very close to our own. They reported that the vertical horopter
is inclined top-away by 408 at a viewing distance of 120 cm. This corresponds to a
backwards inclination of 248 at a viewing distance of 65 cm; here, we report a backwards inclination of approximately 308 at 65 cm (equivalent to a mean declination of
binocular correspondence of 1.78). Second, the horizontal extent of their stimulus was
21 cm, implying that the relative tilt of corresponding vertical meridians extends across
the retina and is not confined to the meridian intersecting the fovea. This is further
data in agreement with our claim that points defining a surface inclined top-away by
308 at 65 cm project to psychophysical corresponding points in the two eyes.
The congruence between binocular correspondence, as measured in this study,
and the disparity projections of points comprising surfaces, like the ground plane for
example, lend support to Helmholtz's (1910/1962) original interpretation of his data
on the backwards inclination of the vertical horopter. Our data further support the
later findings of Nakayama (1977), and, more recently, Banks and Backus (1998),
Ledgeway and Rogers (1999), and Nakamizo et al (1999).
Acknowledgements. The experiments in this paper were conducted while the first author was
a visiting intern researcher at ATR ^ HIP laboratories. Their hospitality and collaboration are
warmly acknowledged. Thanks to the anonymous reviewer for his/her helpful comments on an
earlier version of this paper. This research was supported by the Natural Sciences and Engineering
Research Council of Canada.
References
Banks M S, Backus B T, 1998 ``The shape of the most fusible surface'' Investigative Ophthalmology
& Visual Science 39(4) S617
Breitmeyer B, Battaglia F, Bridge J, 1977 ``Existence and implications of a tilted binocular disparity
space'' Perception 6 161 ^ 164
Cogan A I, 1979 ``The relationship between the apparent vertical and the vertical horopter''
Vision Research 19 655 ^ 665
Cooper M L, Pettigrew J D, 1979a ``A neurophysiological determination of the vertical horopter
in the cat and owl'' Journal of Comparative Neurology 184 1 ^ 26
Cooper M L, Pettigrew J D, 1979b ``The decussation of the retinothalamic pathway in the cat,
with a note on the major meridians of the cat's eye'' Journal of Comparative Neurology 187
285 ^ 311
Hanny P, Heydt R von der, 1982 ``The effect of horizontal plane environment on the development
of binocular receptive fields of cells in cat visual cortex'' Journal of Physiology 329 75 ^ 92
Helmholtz H von, 1910/1962 Treatise on Physiological Optics volume 3 (New York: Dover, 1962);
English translation by J P C Southall for the Optical Society of America (1925) from the
3rd German edition of Handbuch der physiologischen Optik (Hamburg: Voss, 1910)
Hillebrand F, 1929 Lehre von den Gesichtsempfindungen (Vienna: Springer)
Howard I P, 1982 Human Visual Orientation (Toronto: John Wiley)
Empirical corresponding points
429
Howard I P, Rogers B J, 1995 Binocular Vision and Stereopsis (New York: Oxford University Press)
Ledgeway T, Rogers B J, 1999 ``The effects of eccentricity and vergence angle upon the relative
tilt of corresponding vertical and horizontal meridia revealed using the minimum motion
paradigm'' Perception 28 143 ^ 153
Nakamizo S, Ono H, Ujike H, 1999 ``Subjective staircase: A multiple wallpaper illusion'' Perception
& Psychophysics 61 13 ^ 22
Nakayama K, 1977 ``Human depth perception'' Society of Photo-Optical Engineers 120 2 ^ 9
Nakayama K, Tyler C W, Appleman J, 1977 ``A new angle on the vertical horopter'' Investigative
Ophthalmology & Visual Science 16 Supplement, 82
Ogle K, 1964 Researches in Binocular Vision (New York: Hafner)
Owens L R, Kooi F L, Tyler C, W, 1987 ``A new twist on the vertical horopter'' Investigative
Ophthalmology & Visual Science 28 Supplement, 295
Rogers B J, Bradshaw M F, 1995 ``Disparity scaling and the perception of frontoparallel surfaces''
Perception 24 155 ^ 179
Shinkman P G, Bruce C J, 1977 ``Binocular difference in cortical receptive fields of kittens after
rotationally disparate binocular experience'' Science 197 285 ^ 287
Siderov J, Harwerth R S, Bedell H E, 1999 ``Stereopsis, cyclovergence and the backwards tilt of
the vertical horopter'' Vision Research 39 1347 ^ 1357
Tyler C, 1991 ``The horopter and binocular fusion'', in Binocular Vision Ed. D Regan, volume 9
of Vision and Visual Dysfunction Ed. J R Cronly-Dillon (London: Macmillan) pp 19 ^ 37
ß 2001 a Pion publication printed in Great Britain