Subjective staircase: A multiple wallpaper illusion
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Perception & Psychophysics
1999, 61 (1), 13-22
Subjective staircase: A multiple wallpaper illusion
SACHIO NAKAMIZO
Fukuoka University ofEducation, Fukuoka, Japan
HIROSHIONO
ATR Human Information Processing Research Laboratories, Kyoto, Japan
and
HIROYASU UJIKE
National Institute ofBioscience and Human- Technology, Tsukuba, Japan
When observers binocularly fixate on an inclined sheet of paper with equally spaced dots, an appar-
ent "staircase" is seen. Wevaried the inclination ofthe sheet, the spacing among the dots, and the view-
ing distance. The results indicate that (1) as the space and the inclination decreased, the number of ap-
parent steps increased and the height of apparent steps decreased, and (2) as the distance and the
inclination increased, the number of apparent steps decreased, and eventually the illusion disappeared.
The nearest-neighbor rule and the extent of the vertical horopter inclination explain the characteris-
tics of the illusion.
When one fixates binocularly on a slanted sheet of that the subjective staircase does follow the geometry,
paper covered with equally spaced dots (see Figures la and, therefore, so do the binocular moire fringes. We elab-
and 2), lines, or gratings, an apparent three-dimensional orate on this point in the General Discussion section.
(3-D) staircase is perceived (see Figure 1b). The apparent An illusion conceptually similar to the one described
steps above and below the fixation point appear above above is the wallpaper illusion, which was first described
and below the actual stimulus plane, respectively (see by Smith (1738) and elaborated on by Brewster (1844). In
Figure lc). Moreover, the apparent treads (the horizontal this illusion, a horizontally repeating pattern presented
part of a step) appear parallel to the stimulus plane, and on a frontoparallel plane (e.g., wallpaper) appears in
the apparent risers (the vertical part ofa step) decrease in front of or behind the actual location when the intersec-
height the higher up they appear. (The reader can experi- tion of the visual axes are in front of or behind the plane,
ence this illusion by viewing Figure 2 actually inclined, as respectively.' The staircase illusion is a special case ofthe
shown in Figure la.) wallpaper illusion (i.e., a multiple wallpaper illusion)
When the lines or gratings are in parasagittal planes, and thus can be explained in the same way-namely, by
an apparent 3-D staircase consisting oflines on treads is the nearest-neighbor rule (for a discussion of this rule,
perceived. The proximal stimulus for this stimulus is al- see Howard & Rogers, 1995). In the following, we describe
most identical to the one created by lines with opposing how the nearest-neighbor rule explains the wallpaper il-
orientation in two fields presented in a stereoscope. lusion and how we believe it can be modified to explain
Therefore, the apparent staircase with the lines or grat- the staircase illusion.
ings is the same as what has been called the binocular The rule applies to the wallpaper illusion in the fol-
moire fringes in depth (piggins, 1978; Tyler, 1980, 1991), lowing ways:
in which the lines or gratings appear to be segregated 1. When the intersection of the visual axes is on the
horizontally into layers at different depths. Although the stimulus plane, the retinal images (one in each eye) ofa
binocular moire fringes are said not to follow directly the given element in the pattern fall on binocularly corre-
geometry of the situation (Tyler, 1980), this paper shows sponding points (and close by for images produced by a
peripheral element). The two images fuse, because they
are "nearest neighbors," and they appear as one element
This research was supported by a Grant-in-Aid for Scientific Re- on the stimulus plane (i.e., elements are seen in their
search (Ippan-kenkyu C: 0780 I016) provided by the Japanese Ministry
veridical locations).
of Education, Science, and Culture. The second author, Hiroshi Ono,
worked on this study while on a sabbatical from York University and 2. As the intersection of the visual axes deviates from
was an invited researcher at ATR Information Processing Research the stimulus plane, the retinal images ofa given element
Laboratories, Kyoto, Japan. The authors thank Ian Howard, Ali- also deviate binocularlyfrom correspondingpoints and be-
stair Mapp, Martin 1. Steinbach, and anonymous reviewers for their come closer to the retinal images ofa neighboring element.
helpful comments on an earlier version of the present paper. Corre-
spondence should be addressed to S. Nakamizo, Department of Psy-
When the retinal images of the neighboring elements be-
chology, Kyushu University, 812-8581,6-19-1, Hakozaki, Higashi-ku, come closer than the retinal images from the given ele-
Fukuoka, Japan (e-mail: nakamltm@mbox.nc.kyushu-u.ac.jp). ment, they fuse and appear closer or farther than the stim-
13 Copyright 1999 Psychonomic Society, Inc.14 NAKAMIZO, ONO, AND UJIKE
(a)
R-eye
L-eye
(b)
~ Franta-parallel Plane
..........
..........
(c) ----'-------""';ooi;;;=---------~+____1 EyeSUBJECTIVE STAIRCASE 15
···· ..... point has larger crossed disparity than the one in the back,
···· ... and an apparent dot in back of another located lower than
the fixation point has larger uncrossed disparity than the
···· .... one in front.
Moreover, due to the inclination ofthe vertical horopter
··· ... (see Cogan, 1979; Helmholtz, 1909/1962; Nakayama,
1977), what constitutes a nearest neighbor is different
····· .....
when we consider the elements away from the transverse
(horizontal) plane. When the stimulus inclination corre-
···· .....
sponds to the extent ofthe horopter inclination at a given
convergence distance, the retinal images of stimulus el-
·· ements located near the midsagittal plane fall on binoc-
·:·: : : : : : : : : : : : : : : : : : ::+: ::::::::::::::::::...
ularly corresponding points, and thus, as described in
Process 1, they appear in their veridical locations; con-
··· ... sequently, the illusion should disappear.
The main concern of this study was to empirically test
···· ...
..
our contention that the staircase illusion is a special case
of the wallpaper illusion, and, thus, it can be explained
··· . by the nearest-neighbor rule and the extent of the verti-
··· .... cal horopter inclination. To achieve this, we used a stim-
ulus consisting of black dots on a white sheet of paper,
···· .... as shown in Figure 2, and measured the quantitative char-
acteristics of the illusion at a fixed distance or at varying
··· . .. distances. (We chose a dotted stimulus rather than one
····· .....
consisting oflines or gratings, because it allowed for eas-
ier measurements of the perceived extent ofthe apparent
treads and risers.) In Experiment 1, viewing distance was
· . fixed, and we measured the number of apparent steps as
a function of the inclination of the stimulus and the sep-
Figure 2. An example of the dotted stimulus with a fixation aration ofthe dots. In Experiment 2, viewing distance was
point (+). The illusion can be attained with either ofthe following
procedures: (1) Place the sheet flat on a desk, place your chin on
varied, and we measured the number of apparent steps
a chinrest made by your fist located 15 em from the edge ofthe as a function of the inclination of the stimulus. In Ex-
sheet, and then fixate on the "+" in the middle of the sheet. periments 3 and 4, viewing distance was fixed, we mea-
(2) Place the stimulus flat on a wall, rest your forehead on your sured the number of rows of dots in each of the five ap-
fist placed 15 cm above the edge of the sheet, and fixate on the "+" parent steps and the height of each of the five apparent
in the middle. In either case, keeping a steady fixation is critical
to seeing the illusion.
risers above the stimulus plane, respectively,and compared
the obtained means with the predicted from the nearest-
neighbor rule.
1960, for discussions of the hypothesis. There is also a
size illusion in what we call the "subjective" staircase: EXPERIMENT 1
The higher the stimulus element is in the visual field, the
smaller the apparent size and the apparent separation, In Experiment 1, we examined the effects of dot sep-
which also can be explained by the hypothesis. Questions aration and stimulus inclination on the number of appar-
regarding the size illusion, however, are not addressed in ent steps. Both the physical size of the dots and the view-
this study.) ing distance were held constant. We choose not to vary the
For the subjective staircase or the multiple wallpaper physical size of the dots because, in our preliminary ob-
illusion, the processes described above occur simultane- servation (Nakamizo and Ono, 1993), we found that the
ously. Note in Figure 1c that the dots above (or below) the number of apparent steps was independent of dot size (at
fixation point are farther away (or closer), and more so least for the sizes we tested, 1-4 mm).
for the dots located higher up (or lower). The apparent The nearest-neighbor rule predicts how the two ex-
riser in the illusion occurs when the closest neighbor perimental variables would produce different numbers of
changes; when the new neighbor moves away from the apparent steps. The smaller the dot separation for a given
old neighbor by one column, an apparent riser is created. angle of inclination or the smaller the angle of inclina-
The apparent tread in the illusion is produced by the fu- tion for a given separation of dots, the smaller the hori-
sion of dots from a pair of columns with different retinal zontal retinal separation is, and what constitutes the
disparity values in the same neighborhood. One apparent nearest-neighbors changes more frequently. Because an
dot in front of another located higher than the fixation apparent step is seen every time the nearest neighbor16 NAKAMIZO, ONO, AND UJIKE
Figure 3. A schematic representation of the apparatus used in the experiments.
changes from the adjacent one to one apart and to two angles) was run. The presentation order of different inclinations in
apart and so on, the more frequent the changes the more each separation was randomized, and the order of presentation of
apparent steps should be seen. the separations was changed in each session for each observer.
Observers. Ten observers (including the first author), all with
normal binocular vision, participated in the experiment. All ob-
Method servers except for the first author were naive as to the purpose of
Stimulus and Apparatus. The stimulus and the apparatus are il- the experiment.
lustrated schematically in Figure 3. The stimulus consisted of an
equally spaced 2-D dot pattern printed with black ink on a sheet of
white paper, 21 em wide and 30 em long. Each dot was 2 mm in di-
Results and Discussion
ameter. The experimenter was able to turn the crank shown in the The results of Experiment 1 confirmed our expecta-
figure to incline the stimulus about its midhorizontal axis with a tion that the number of apparent steps would decrease as
resolution of 1°. (The angle of inclination refers to the angle be- the stimulus inclination and dot separation increased. The
tween the horizontal plane and the plane of the sheet of paper; see mean number of apparent steps averaged over the 10 ob-
Figure Ic.) A white cardboard screen with a rectangular aperture, servers plotted as a function of the stimulus inclination,
3 em high x 8 em wide, was placed in front of the observer's eyes.
separately for the different dot separations, is shown in
Another white cardboard screen was placed behind the stimulus as
a background. These screens and the stimulus were diffusely illu- Figure 4.
minated by a fluorescent lamp situated 30 em above the stimulus. Not only did the results confirm our expectation, the
A rod, 0.5 mm wide and 3 mm long, was used as a fixation stimu- strength of association between the independent and de-
lus, which was located at the center of the stimulus and in the trans- pendent variables was very high for the two main effects.
verse plane at eye level. The distance from the corneal plane to the The (j)2 scores, computed after performing a two-way (4
fixation rod was set at 30 ern. dot separations X 4 inclination angles) repeated mea-
Procedure. Prior to the first experimental session, the height of
the chin-and-forehead rest was adjusted so that the fixation rod was sures analysis ofvariance (ANOVA) on the mean number
at the observer's eye level. The observers viewed the stimulus of apparent steps averaged over the three trials for each
through the viewing aperture while maintaining their gaze on the subcondition and for each observer, were high except for
fixation stimulus. They were required to report the total number of that ofthe interaction. The computed (j)2 values for the two
steps they perceived. main effects, dot separation and inclination angle, were
A 4 x 4 factorial design was used: four dot separations (3.3, 4.8, .52 and .22, respectively [for dot separation, F(3,27) =
6.3, and 7.8 mm) and four inclination angles (20°, 30°, 40°, and
50°). (The range of these two independent variables was based on the
103.81,p < .001; for inclination angle, F(3,27) = 193.41,
results of the preliminary study.) Each observer completed three 16- P < .001], and the (j)2 value for the interaction was .008
trial sessions. Within each session, I trial for each of the 16 combina- [F(9,8l) = 4.43,p < .01]. The main effect of dot separa-
tions of the two independent variables (4 separations X 4 inclination tion is depicted by the different relative heights of theSUBJECTIVE STAIRCASE 17
Procedure. A 4 X 4 factorial design was used: four viewing dis-
t/)
--.
C. 8 Separation of dots (mm) tances (30, 45, 60, and 120 em) and four inclination angles (20°,
CI)
en • 3.3
0 4.8
30°,40°, and 50°). Each observer completed three 16-trial sessions.
Within each session, I trial for each of the 16 combinations of the
C
CI)
6 • 6.3
0 7.8
two independent variables was performed. In each trial, the ob-
servers were asked to report the number of apparent steps on the
stimulus plane .
as
c. Observers. Nine observers (including the first author), all with
c.
-..
normal binocular vision, participated in the experiment. All observers
ct 4 except for the first author were naive as to the purpose of the ex-
0 periment.
CI)
~ 2 Results and Discussion
E
:::s
The results of Experiment 2 confirmed our expecta-
tion that the number of apparent steps would decrease as
Z
0 the viewing distance increased. The mean number of ap-
.0 20 30 40 50 60 parent steps averaged over the 9 observers plotted as a
function ofthe viewing distance, separately for each angle
Inclination Angle (deg) of inclination, is shown in Figure 5.
Figure 4. Mean number of apparent steps plotted as a function Not only did the results confirm our expectation, but
of the inclination angle for the four different extents of separation the strength of association between the independent and
of dots in Experiment 1. dependent variables was also very high. The 0)2 scores,
computed after performing a two-way (4 distances X 4
inclination angles) repeated measures ANOVA on the
four curves shown in Figure 4, and the main effect of the mean number of apparent steps averaged over the three
inclination angle is depicted by the negative slope of all trials for each subcondition and for each observer, were
four of the lines. The small significant interaction of the high except for that of the interaction. The computed 0)2
two variables was a consequence of the extent of change values for the two main effects of viewing distance and
in the retinal separation of dots deviating slightly from inclination angle were .54 and .33, respectively [for view-
linearity as a function of the angle of inclination and the ing distance, F(3,24) = 101.34,p < .001; for inclination
separation of dots. angle, F(3,72) = 208.93,p < .001], and the 0)2 value for
the interaction was .02 [F(9,72) = 8.50, p < .01]. The
EXPERIMENT 2 main effect of inclination angle is depicted by the differ-
ent relative heights of the four curves shown in Figure 5,
In Experiment 2, we examined the effect of viewing and the main effect ofthe viewing distance is depicted by
distance on the number ofapparent steps. Our expectation the negative slope of all four of the lines.
was that the number of apparent steps would decrease as
the viewing distance increased and, furthermore, that the
staircase illusion would disappear when the angle of the
stimulus inclination matched that ofthe vertical horopter tn 10
--...
inclination (see Cogan, 1979; Helmholtz, 190911962; Co Angle of Inclination (deg)
Q)
Nakayama, 1977). Because the inclination of the vertical C/)
horopter varies with distance (Helmholtz, 190911962), the 8 • 20
o 30
nearest neighbor would also vary with distance. When cQ) • 40
the inclination of the stimulus corresponds to the extent o 50
of the vertical horopter inclination at a given viewing as 6
Co
-...
distance, dots located on the midsagittal plane stimulate Co
corresponding points in the retinal meridian. As a result,18 NAKAMIZO, ONO, AND UJIKE
As we expected, the staircase illusion disappeared at a and di is binocular disparity produced by a dot in the ith
viewing distance of 120 em and when the stimulus incli- row. We compared the number of rows of dots obtained
nation was 50° for 7 out of the 9 observers. In this sub- when observers perceived the nth step while introducing
condition, 7 observers perceived only one plane-namely, a single row of dots at a time with the predicted number
the plane of the stimulus. This disappearance of the illu- ofrows calculated by using Constraint 1.
sion indicates that the stimulus plane and the vertical
horopter inclination matched in this subcondition for these Method
observers. That is, for these observers, the angle ofthe ver- Stimulus and Apparatus. The apparatus was the same as that
tical horopter inclination at 120 em was approximately used in Experiment 1. The experimenter attached a scale to the
stimulus plane as an aid for the observers to change fixation on the
50°, or the angle of declination of binocular correspon-
frontoparallel plane (see Figure 3). The scale was always on the
dence was estimated to be 1.3°. (Note that Helmholtz's frontoparallel plane at a distance of 30 em from the eyes irrespec-
estimate of the angle of the declination was 1.25°, using tive ofthe angle of inclination. Therefore, the convergence distance
the nonius method.) Since the results conformed to our was held approximately constant as long as the intersection of the
expectation, the method used in Experiment 2 can be visual axes was on the scale. This scale, in millimeter units, was
modified to estimate the angle of declination ofbinocu- printed on a translucent board.
lar correspondence. By having the observer adjusting the Procedure. We used the following method to record when a riser
occurred (or to determine the number of rows of dots in a given
inclination of the stimulus or by the experimenter using tread): At the beginning of each trial, the experimenter covered the
a finer gradation of stimulus inclination, we can estimate area of the stimulus higher than the fixation rod with a blank sheet
more precisely and accurately the angle ofthe declination of paper. He then uncovered the dots one row at a time by moving
ofbinocular correspondence in the vertical retinal merid- the sheet away from the fixation rod; the observer was asked to re-
ian. In fact, at this time, we are performing such an ex- port when a new riser appeared. When the observer reported a riser,
periment in our laboratory. the experimenter recorded the number of uncovered rows of dots
from the fixation rod. The trial was continued until the observer re-
Although the results of Experiments I and 2 were ex- ported the fifth apparent step. (The results of Experiment 1 showed
plained by the nearest-neighbor rule and the vertical that the observers perceived approximately five apparent steps in
horopter inclination, we did not compare the predicted and the subconditions used in Experiment 3.) To make the observer's
obtained number of apparent steps. Our reason for not judgment of the appearance of a new step easier, they were allowed
doing so was that, in most of the conditions, the illusion to change their fixation on the attached scale. Three trials were run
did not extend to the whole stimulus. The top and the bot- for each angle of inclination. The presentation order for the angle
of inclination was random and was changed for each observer.
tom of the stimulus appeared blurred, and the limit of the
Observers. Three observers participated. Two of them (the first
spatial extent of the illusion was difficult to specify. To author and M.K.) were experienced in psychophysical experiments,
demonstrate the power ofthe nearest-neighbor rule to ac- and the other one (Y!.) was not. Observers M.K. and Y.l. were naive
count for the illusion, however, a high agreement between as to the purpose of the experiment.
predicted and obtained values would be helpful. Accord-
ingly, we measured the number of rows of dots in each of Results and Discussion
five apparent steps and the height of each of the five ap- The results of Experiment 3 confirmed the prediction
parent risers in Experiments 3 and 4, respectively, and the from the nearest-neighbor rule. The obtained mean num-
obtained values were compared with the predicted values. ber of the uncovered rows of dots when the observers
perceived each riser for each inclination angle was very
EXPERIMENT 3 close to the predicted number of rows, as calculated by
using Constraint 1. Table 1 shows the predicted and ob-
In Experiment 3, we tested the quantitative predictions tained mean location (nth row of dots from the center of
from the nearest-neighbor rule by measuring the number the stimulus), averaged across the 3 observers, at which
of rows of dots in each of five apparent steps as a func- a riser appeared as a function ofthe angle of inclination.
tion of the stimulus inclination. According to the nearest- Two different statistical analyses showed the closeness of
neighbor rule, a riser appears when binocular disparity the obtained to the predicted. First, the predicted number
produced by the dots in the ith row is just greater than ofrows for each inclination angle and for each riser was
halfthe horizontal separation ofthe images of the dots in within the 95% confidence interval (CI) calculated from
that row. When the retinal images of the neighboring el- the mean and standard deviation obtained for the same
ements become closer than the retinal images from the riser and for the same inclination angle. (Values of the
given element, they fuse and appear either closer or farther 95% CI are not shown in Table 1.) Second, correlation
than the stimulus plane depending on which (left or coefficients between the means and the predicted values
right) neighbor produces the closer retinal image. The nth for each observer and for the group were highly statisti-
step should appear when the following constraint is met: cally significant. The mean correlation coefficients aver-
aged over the three angles of inclination for Observers
I(n - I) X Si>- dil >Si/2, (1) S.N., M.K., and Y.!. were .998, .998, and .999, respectively,
where n is a positive integer representing the order of ap- and .998 for the group.
parent steps from the fixation point, Si is the horizontal Not only were the obtained values in each subcondition
separation ofdots, expressed in visual angle, ofthe ith row, very close to the predicted values, the strength of asso-SUBJECTIVE STAIRCASE 19
Table I
Predicted and Obtained Mean Locations (nth Row of Dots
From the Fixation Point), Across the 3 Observers, at Which
a Step Appeared as a Function of the Inclination Angle,
Along With Standard Deviations ofthe Obtained Means
Stimulus Inclination
20° 30° 40°
Obtained Obtained Obtained
Step Predicted M SD Predicted M SD Predicted M SD
1st 3 3.3 0.5 3 3.5 0.4 4 3.8 0.2
2nd 8 8.2 0.3 9 9.0 0.3 10 10.3 0.3
3rd 13 13.2 0.3 14 14.3 0.5 16 16.3 0.3
4th 18 18.7 0.3 19 20.0 0.6 22 22.8 0.3
5th 23 24.3 0.5 25 25.7 0.7 28 28.6 0.9
ciation between the independent and dependent variables height of each step. The same scale was attached to the stimulus.
was very high. The (f)2 score, computed after performing Since the scale was always on the frontoparallel plane at a distance
a two-way (3 inclination angle X 5 risers) repeated mea- of30 em from the eyes, irrespective of the angle of inclination, the
observer's convergence was held approximately constant, and, con-
sures ANOVA on the mean number of the uncovered
sequently, a fused neighbor did not change as long as the intersec-
rows of dots from the three trials for each observer, was tion of the visual axes was on the scale. The observers' task was to
high, but the (f)2 scores for the main effect of inclination read the height of each of the five apparent steps above the stimu-
angle and the interaction were not. The (f)2 value for the lus plane on the scale and to report it to the experimenter. The ob-
five risers was .97 [F(4, 8) = I,OB.20,p < .001], and the servers were allowed to change fixation on the scale. Three trials
(f)2 values for the main effect of the inclination angle and were run for each angle of inclination. The presentation order of the
for the interaction were .02 and .004, respectively [for in- angle of inclination was random and was changed for each ob-
server. The observers were the same 3 who had participated in Ex-
clination angle, F(2,4) = 106.56, P < .001; for the inter- periment 3.
action, F(8,16) = 22.16,p < .01].
The results of Experiment 3 showed a high agreement Results and Discussion
between the obtained quantitative values and their pre- The results of Experiment 4 confirmed the prediction
dicted values. This agreement shows the power of the from the nearest-neighbor rule. The obtained mean height
nearest-neighbor rule more than the results of Experi- ofeach apparent step was very close to the height predicted
ments 1 and 2, in which only ordinal predictions were from Equation 2. Figure 6 shows the obtained mean height
confirmed. In Experiment 4, we examined another quan- of each ofthe five apparent steps, averaged over the three
titative prediction. trials and plotted as a function of the stimulus inclination,
separately for each observer. The predicted values for each
EXPERIMENT 4 observer are shown as solid lines in each panel ofFigure 6.
As is clearly indicated in the figure, all the data points
In Experiment 4, we tested the quantitative predictions are very close to the solid lines. The two main effects of
on the height of apparent steps from the nearest-neighbor inclination angle and the five apparent steps are depicted
rule by measuring the height of each of five apparent in the figure by all five curves in each panel having a
steps as a function ofthe stimulus inclination. (The height positive slope and in the different relative heights of the
ofapparent step refers to a distance of each apparent step, five slopes, respectively. The height of each of the five
or a top ofeach apparent riser, from the center ofthe stim- apparent steps predicted from Equation 2 was contained
ulus.) The height of each apparent step is given by the in 95% CI computed from the mean and standard devia-
following equation: tions across the 3 observers shown in Table 2. The re-
H = n X S X D X tan(1 (2) sults ofthe statistical analysis and Figure 6 show the va-
n I+nXS ' lidity ofEquation 2 in describing the height ofan apparent
where n is a positive integer representing the order of the step and provide confirmation of our contention that the
perceived step, S is the physical separation of dots, D is staircase illusion is a multiple wallpaper illusion and that
the convergence distance, I is the interocular distance, and it can be explained by the nearest-neighbor rule.
(1 is the inclination angle. We compared the height of each Not only were the obtained values in each subcondi-
of the five apparent steps obtained for each inclination tion very close to the predicted values, the strength of as-
angle with that predicted from Equation 2. sociation between the independent and dependent vari-
ables was very high. The (f)2 scores, computed after a
Method two-way (3 inclination angles X 5 steps) repeated mea-
The stimulus and apparatus were the same as those in Experi- sures ANOVA on the mean height of each apparent step
ment 3. We used the following method to measure the apparent averaged over the three trials for each subcondition and20 NAKAMIZO, ono, AND UJIKE
E
-
E
• 1st Riser
Sub: V.1. Sub: S.N. Sub: M.K.
80 0 2nd
-
s:::
CI)
"-
as
60
A 3rd
o 4th
•
5th
Q. 40
c.
--
«
o
J:
en 20 30 40 50 20 30 40 50 20 30 40 50
CI)
J:
Inclination Angle (deg)
Figure 6. Mean heights of five apparent steps above the stimulus plane plotted as a function of the inclina-
tion angle for each observer in Experiment 4.
for each observer, were high, but the w2 score for the apparent step was located with respect to the location of
interaction was not. The computed w2 value for the inter- the rows in Experiment 3 and the height of each apparent
action was .06 [F(8,16) = 476.45,p < .001], and the co 2 step in Experiment 4.
values for the two main effects of inclination angle and The rule and the extent explain which elements of the
the five apparent steps were .39 and .54, respectively [for stimulus fused with which elements and the relative depths
inclination angle, F(2,4) = 3,557.06,p < .001; for the five among rows of the apparent dots. They do not, however,
steps, F(4,8) = 7,007.66,p < .001]. explain how far an apparent staircase or an apparent wall-
paper appears to be-that is, they do not predict the ap-
GENERAL DISCUSSION parent absolute distance. Traditionally, convergence ofthe
two eyes is thought to determine the apparent absolute dis-
The nearest-neighbor rule and the extent of the verti- tance of the wallpaper illusion (Helmholtz, 1909/1962;
cal horopter inclination explain the quantitative charac- Lie, 1965). If we were to generalize from the results of
teristics of the staircase illusion. When the viewing dis- Rogers and Bradshow (1993), however, the apparent ab-
tance was fixed, the variation of the number of apparent solute distance is a function ofboth vertical disparity and
steps was successfully described by the nearest-neighbor convergence. Evidence that the visual system uses verti-
rule in Experiment I. When it was varied, the variation cal disparity in processing absolute distance in the wall-
was successfully described by the extent of the vertical paper illusion was shown in a recent study (Susami & Ono,
horopter inclination, together with the rule, in Experi- 1995). It is likely, therefore, that the apparent distance of
ment 2. The rule also successfully predicted where each the staircase is also determined by both.
Table 2
Predicted Heights of Apparent Step From Equation 2
and the Obtained Means, Across the 3 Observers, as
a Function of the Inclination Angle, Along
With Standard Deviations ofthe Obtained Means
Stimulus Inclination
20° 30° 40°
Obtained Obtained Obtained
Step Predicted M SD Predicted M SD Predicted M SD
1st 7.6 7.6 0.29 12.2 13.0 0.48 17.7 17.9 0.37
2nd 14.3 14.3 0.29 22.7 23.7 0.54 33.0 34.4 0.49
3rd 20.2 20.2 0.22 32.0 32.7 1.10 46.5 48.5 1.00
4th 25.3 25.2 0.24 40.1 41.4 1.34 58.2 60.7 1.10
5th 29.9 29.3 0.50 47.4 49.0 1.41 69.0 70.9 0.68
Note-The predicted values are based on the mean interocular distance, 63.6 mm,
across the 3 observers.SUBJECTIVE STAIRCASE 21
Our contention in the introduction that what has been GOGEL, W. C. (1993). The analysis of perceived space. In S. C. Masin (Ed.),
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paper illusion. Vision, 5, 77-80. (Abstract in Japanese)
the grating stimulus in the staircase illusion. In our study,
NAKAYAMA, K. (1977). Geometric and physiological aspects of depth
the number of apparent steps increased as a function of perception. Proceedings ofthe Society ofPhoto-Optical Instrument
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frequency, orientation, field size, and viewing distance. Perception &
fringes in depth follow the geometry in the same way as Psychophysics, 42, 140-149.
the subjective staircase. PIGGINS, D. (1978). Moines maintained internally by binocular vision.
But why the name binocular moire fringes in depth? Perception, 7, 679-681.
Perhaps this name was given because of an assumption ROGERS, B. J., & BRADSHOW, M. F. (1993). Vertical disparities, differ-
ential perspective and binocular stereopsis. Nature, 361, 253-255.
that periodic patterns, such as gratings, presented di-
SMITH, R. (1738). A compleat system ofopticks (Vol. 2).
choptically generate perceptually the same interference SPILLMANN, L. (1993). The perception of movement and depth in moire
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1987, and Kaufman, 1974, for the likelihood that this as- TYLER, C. W. (1980). Binocular moire fringes and the vertical horopter.
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ent stimulus is moved above another stimulus in the 2-D
moire fringe (Piggins, 1978). This movement, however, NOTE
is not due to a change in the interference pattern, but,
rather, it is due to a change in which element becomes a I. The same illusion was reported by Dixon (1938) with a floor or
nearest neighbor. The major difference between the way pavement as a stimulus. When the two eyes converge above the floor
with a pattern such as "a checkerboard in light and dark squares," the
we examined staircase illusion and the way Piggins (1978) pattern appears to be "raised above the level of the floor" and seen in "a
and Tyler (1980) examined the binocular moire fringe in smaller scale than that of the actual pattern."
depth is in the methodology. We used a fixation point, and
our procedure led to a stable perception of a subjective APPENDIX
staircase. If we were to change our fixation deliberately The Geometric Relation Between the Staircase IUusion
to a different row of dots, however, the locations of the and the Traditional Wallpaper IUusion
steps would change. In contrast, they did not use a fixation
point, and "the layers are not completely spatially local- The geometric relation between the staircase illusion and the
traditional wallpaper phenomenon is shown in Figure A I. The
ized, appearing to move vertically at times" (Piggins,
schematic illustration of the staircase illusion viewed from the
1978, p. 679). Our informal replication of their studies side is shown in Figure A Ia, where AI= the stimulus plane, (J=
using a stereoscope with a fixation point produced a sta- the extent of stimulus inclination (the angle of inclination),
ble perception, just as in a subjective staircase with a fix- TE = the transverse plane at eye level, I = the fixation point,
ation point. FI= the frontoparallel plane at the fixation distance, PI= the
plane perpendicular to the stimulus plane, and each of the five
REFERENCES consecutive dots represents each apparent tread of the staircase
illusion. The schematic illustration of the wallpaper illusion
BADCOCK, D. R., & DERRINGTON, A. M. (1987). Detecting the dis- viewed from the top is shown in Figure A Ib, where W = the
placements of spatial beats: A monocular capability. Vision Research,
stimulus plane in the frontoparallel plane at a distance of D
27,793-797.
BREWSTER, D. (1844). On the knowledge of distance given by binocular
from the eyes, S = the separation of repeating elements, I and
vision. Transactions ofthe Royal Society ofEdinburgh, 15,663-674. r = the nodal points of the left eye and the right eye, respec-
COGAN, A. I. (1979). The relationship between the apparent vertical and tively, 1= interocular distance, and d = depth of the images lo-
the vertical horopter. Vision Research, 19,655-665. cated when an image ofelement for the right eye was fused with
DIXON, H. H. (1938). A binocular illusion. Nature, 141, 792. an image of the right neighboring element for the left eye.22 NAKAMIZD, DND, AND UJIKE
A
F
(a)
E
(b)
I'w- - - - - 0 -----1
Figure At. Geometric relation between the staircase illusion and the traditional wallpaper phenomenon. See text for details.
Brewster (1844) proposed the following formula to compute Formula Al is also equivalent to the inverse square law of
depth (d) of the wallpaper stimulus: binocular disparity if 8 = SID, where 8 is disparity, as follows:
d=SXD. (AI) if 8 =~,thenS = 8D
I+S D
(Note that Formula A I is formally equivalent to that reported by d= 8DxD
Odom & Chao, 1987.) A more general form of Formula Al can I+8D
be obtained as follows: d(l +8D) = 8D 2
d = n X SX D (A2) dI=8D 2-d8D
n I+nXS'
dI = 8(D 2 -dD)
where n is a positive integer. n - I represents the number ofthe
elements between a pair ofelements ofwhich images are fused. 8= Id =K. (A3)
That is, if n = I, then the two images of adjacent elements are D(D-d) D 2
fused; if n = 2, then the two images of a pair of elements apart Formula A3 is a formal expression of the inverse square law of
one element are fused, and so on. Rows of dots shown in Fig- disparity.
ure Alb indicate an assumed apparent location of the stimulus
for a given n. An extent of d increases with an increase of n, as (Manuscript received August 26, 1996;
shown in Figure A Ib. revision accepted for publication December 16, 1997.)You can also read
