Subjective staircase: A multiple wallpaper illusion

 
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                              Eye
SUBJECTIVE 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 neighbor
16        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 the
SUBJECTIVE 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 and
20      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.),
 called the binocular moire fringes in depth (Piggins, 1978;                 Foundations ofperceptual theory (pp. 113-182). Amsterdam: Elsevier.
                                                                          HELMHOLTZ, H. VON (1962). Helmholtz s treatise on physiological op-
Tyler, 1980, 1991) is another instance of the staircase il-
                                                                             tics (Vol. 3; J P. C. Southall, Ed. and Trans.). Rochester, NY: Optical
 lusion is based on the fact that it is also governed by the                 Society of America. (Original work published 1909)
nearest-neighbor rule. That is, the experimental variables                HOWARD, I. P., & ROGERS, B. J. (1995). Binocular vision and stereopsis.
that affect the characteristics of the illusion also affect                  New York: Oxford University Press.
the binocular moire fringes in depth, or vice versa. For ex-              ITTELSON, W. H. (1960). Visual space perception. New York: Springer-
                                                                             Verlag.
ample, Piggins (1978) observed that "the number of lay-                   KAUFMAN, L. (1974). Sight and mind. New York:Oxford University Press.
ers is a direct function of orientation difference" (p. 679).             LIE, I. (1965). Convergence as a cue to perceived size and distance.
A greater orientation difference in the two fields of a                     Scandinavian Journal ofPsychology, 6, 109-116.
stereoscope corresponds to a greater inclination angle of                 NAKAMIZO, S., & ONO,H. (1993). Subjective staircase: A multiple wall-
                                                                            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
the angle of inclination, as it did in a separate experiment                Engineers, 120,2-9.
using a grating stimulus having an equal horizontal sep-                  Oooa, J. v.,& CHAO, G.-M. (1987). A stereo illusion induced by binoc-
aration as the dots in this study. The binocular moire                      ularly presented gratings: Effects of number of eyes stimulated, spatial
                                                                            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
pattern as that from the two patterns superimposed to the                   patterns. Perception, 22, 287-308.
same eye (see, e.g., Spillmann, 1993, for perception of                   SUSAMI, K., & ONO, H. (1995). Wallpaper illusion and vertical dispar-
two patterns superimposed; see Badcock & Derrington,                        ity. Japanese Journal ofPsychonomic Science, 14, 33. (Abstract in
                                                                            Japanese)
 1987, and Kaufman, 1974, for the likelihood that this as-                TYLER, C. W. (1980). Binocular moire fringes and the vertical horopter.
sumption is incorrect). Probably, this assumption was                       Perception, 9, 475-478.
made because the binocular moire "fringe" appears to                      TYLER, C. W. (1991). The horopter and binocular fusion. In D. Regan
move with a vertical eye movement, as when a transpar-                      (Ed.), Binocular vision: Vision and visual dysfunction (Vol. 9, pp. 19-
                                                                            37). London: Macmillan.
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
Next slide ... Cancel