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by Latha K. Ta’eed, 0. Ta’eed, and J. E. Wright
T h e University of Leeds, Leeds, England
The study is concerned with evaluating interactions at the organic level within the
visua1 perception subsystem of living systems. The reported work focuses on the
identification of some of the determinants of multistable perception by experimentally
testing a nonlinear dynamical systems (catastrophe) model of the Necker Cube. This
technique serves as an advantage over linear threshold models which cannot effectively
study multivalued functional relationships. It was proposed that manipulation of two
independent control parameters (bias or changing shape by continuously varying
perspective lines and selective stimulus shading) was compatible with the subjective
dichotomy of bistable perception of the Necker cube. One hundred and twenty naive
subjects, categorized by age, sex, and optical aids, were presented with a computergenerated sequence of 63 stimuli (7shading levels X 9 perspective levels) to which they
had to respond as to whether they saw a “hollow” or “solid” image. The work revealed
that bias and shading exerted their effects in opposition and that each influenced the
other. Both were decisive factors involved in the perception of the cube. These findings
are supported by topological and psychological evidence.
KEY WORDS:organism, catastrophe theory, multistable perception, Necker cube.
types of visual illusions is centrally organized. Theorists also agree on a few funHE MAIN function of perception is to damental points:
decide the transient two-dimensional
(1) The illusions are not conceptual but
retinal image into a three-dimensional rep- perceptual (the knowledge that an illusion
resentation to achieve constancy, which is exists does not diminish the strength of
to perceive the external world in terms of that illusion).
(2) That illusions do not result from eye
its stable and intrinsic characteristics. A
problem occurs when the expected corre- movements, and do not originate in the
spondence between an object (stimulus) retina.
(3) That prior and past experience play
and percept is violated. The result is an
illusion or a perceptual distortion. Visual a part in resolving the equivocal sensory
illusions can be classified as geometric, am- image when the resolving information is
biguous, or reversible (Fisher, 1968), and in absent, present, or reduced.
Further understanding of the mechaeach class of illusionistic phenomena there
are great differences of opinion regarding nisms underlying the phenomena of illuthe nature and explanation of the illusions. sions, such as the effect of physical charResearch into visual illusions has concen- acteristics of the figures (illumination contrated on presentation procedures, prior ex- trast, color, spatial frequency) is, however,
posures, and clinical studies of the influ- limited.
It is clear from a survey of the literature
ence of sex, age, motivation, along with
cognitive, personality, and developmental that for most part there has been a piecevariables upon features of perceptual orga- meal approach to problems of investigating
nization, (Gillam, 1980; Gregory, 1971). the causes of visual illusions. The source of
While none of the studies are definitive, in the problem does not seem to be the lack
general terms, it is agreed that the principle of imaginative theorization and experimenmechanism underlying perception of all tal study, but simply that the methods and
Behavioral Science, Volume 33, 1988
techniques presently employed can only reveal explanations of linearly related psychological processes. The same investigative tools cannot be utilized in analyzing
perceptual behavior which, as a consequence of the interaction of variables, needs
to be studied as a single unified structure.
Underlying the problem is the more basic
and false assumption that most psychological processes (including perception) are
either smooth continuous changes in behavior, or piecemeal step functions or
thresholds. In experiments, therefore, data
analysis involves the use of linear mathematics, and the resulting graphs are described in imprecise qualitative language.
Some psychological processes which are invariably nonlinear (e.g., elations of mood
in manic depressives, fasting and gorging
by anorexia nervosa victims, learning and
performance, and the multistable perception of reversible figures) cannot be fully
understood using the limitations of linear
mathematics. In this respect, the handicap
motivates the use of nonlinear models in
dynamical systems theory which has contributed substantially to identifying and
representing both qualitative and quantitative interactions of psychological behavior, while at the same time allowing for the
characterization of the dynamic evolution
of the interacting variables. The emergence
of interactionism as a framework for understanding psychological phenomena has
been a relatively recent development,
mainly as a result of the impact of general
systems theory (Thorn, 1975; Zeeman,
1977; Stewart & Peregoy, 1983). Most applications to psychological phenomena
have been concerned with bifurcation theory or its subset, catastrophe theory (CT),
particularly with the second elementary catastrophe-the “cusp,” which involves the
properties of catastrophic change: hysteresis, bimodality, inaccessibility, and divergency in behavior (Wright et al., 1984;
Ta’eed, 1984b; Baker & Frey, 1980; Flay,
1978; Poston & Stewart, 1978b; Isnard &
Zeeman, 1977).
The present paper has three aims: first,
t o investigate one group of visual illusions-reversible
figures-by building a
cusp catastrophe model of the Necker cube
(Necker, 1832; see Figure 1)after postulating some of the determinants of its perception; second, to identify the conditions
which must be met for the model to hold;
third, to consider some of the implications
of the model for further theorization and
research, and to invoke greater insight into
the general determinants of multistable
perception and related perception phenomena.
1. Rationale for Study
In the field of perception psychology various authors have illustrated that certain
FIG. 1. Two views of the Necker Cube.
Behavioral Science, Volume 33, 1988
ambiguous figures possess CT properties
such as hysteresis and bias effects (Attneave 1971; Hill, 1915). Poston & Stewart
(197813) argue that multistable perception
and other such phenomena could be modeled by canonical cusp geometry. Using
stimuli suggested by Fisher (1967) they postulated that Fisher’s sequence of girl/man
figures could be biased (to give hysteresis)
by embedding the figure in a sequence, thus
viewing it in different directions would be
statistically different. This sequence could
include a second dimension where “details
of the figures are selectively shaded toward
the bottom row, . . . providing the shading
is done smoothly-not in the strict mathematical sense but in the heuristic sense of
varying greatly on the scale we are using.”
Thus, they predicted that the relative degree of girl/man perception is controlled by
the geometrical parameters 3 and b, according to the cusp catastrophe in canonical form. Stewart & Peregoy (1983) had
only limited success in confirming this prediction, mainly due to poor experimental
In the present investigation, the Necker
cube (Necker, 1832) was chosen experimentally to verify Poston & Stewart’s tentative
nonlinear model. Apart from the naturally
observed features of multistable perception,
the Necker cube (NC) has appealing properties appropriate for experimentation.
First, it is a traditional perceptual figure
which gives rise to spontaneous perceptual
changes independent of physical properties
(color, etc.), thus these can be added one or
two a t a time to study the effect on perception. Fisher’s girl/man figure is subjective
and relies upon the observer’s familiarity
with similar images in order to recognize
either the girl or the man. In contrast, a
report of changes in cube perspective is a
more direct measure of perceptual change
than are most of the behavioral indices
which have been so far used. Furthermore,
the NC has the distinct advantage in that
it relies upon very few parameters on which
to base one’s perception, thus allowing for
greater experimental control.
2. Formulation of the Cusp Model
In view of the excellent properties of the
Necker cube (NC) as the subject for CT
modeling, it was proposed that the manipBehavioral Science, Volume 33, 1988
ulation of two dimensions of the Necker
cube, (that is, changing shape by continuously varying perspective cues and selective
stimulus shading) was compatible with the
subjective dichotomy of bistable perception, thus indicating a reasonable setting
for CT application.
The Necker. cube model Is based on certain psychological assumptions (framed as
hypotheses) which, although simple in
statement, implicitly make use of deep
theorems. The hypotheses were as follows:
Hypothesis 1
P is smooth and generic (where P, is the
parametrized family of distribution of perception). This “technical but harmless”
(Zeeman, 1977) mathematical assumption
enabled the use of CT in order to weld the
local psychological hypothesis below.
Psychological Hypotheses
For convenience, “BIAS” ( I )defines the
variation in shape of the NC resulting in
the viewer’s “forced” perception of the cube
solid or cube hollow. “SHADING” ( J )represents the change in detail of the cube
from a two-dimensional flat figure to a tridimensional figure.
Hypothesis 2
If shading ( J ) is minimized (top row of
plate l),then perception of cube hollow to
cube solid is a smooth continuous function;
an increase in bias ( I ) toward solid/hollow
will result in a greater probability of perceiving solid/hollow respectively.
bias hollow
bias solid
FIG. 2. Bias Perception Graph for Minimum
Hypothesis 3
If shading ( J ) approaches a maximum,
and bias ( I ) is minimal, then perception is
split between cube solid and cube hollow.
Note: Depth is perceived when the most
distant surface appears the smallest.
at all, the Necker cube can be observed to
flip spontaneously from the solid aspect to
the hollow one, and vice versa (hysteresis).
The cube, however, remains stable in one
state for a reasonable period of time. We
may expect the initial percept to approximate perfect delay.
Hypothesis 4
If shading ( J ) approaches a maximum
and bias ( I )is a t a near maximum in either
direction, then bistable perception will not
occur, and only one aspect of the cube is
Theorem: The graph G’ of perception
is a cusp catastrophe with bias ( I ) as
the normal factor, and shading ( J ) as
the splitting (bifurcation) factor, as
shown in Figure 4.
Hypothesis 5
Depth reversal in the ambiguous Necker
cube is operated by the Delay Convention,
which appears to be the most suitable one
of the two governing system dynamicsDelay and Maxwell conventions-for psychological applications (Ta’eed, 1985). Several reasons exist for this preference of
convention, but the two primary ones are:
first, “inertia”-it takes time to cause a
change in behavior; second, under the Delay rule, recent behavior is critical in determining the current behavior.
It is important to note that the Delay
Convention is a quasistatic approximation
to dynamics behavior, not the dynamic behavior itself. It is never precisely true for
any reasonable dynamic (Poston & Stewart, 1978a), although the approximation is
very good for many systems. It is not precisely true here either, since with no change
This statement is a theorem because it is
an immediate corollary of the classification
theorem, and the graphs of Figures 2 and 3
arise from the hypotheses stated. It should,
however, be noted that formulation of the
model is heuristic, and a large amount of
stochastic statistical noise is to be expected.
1. Subjects
One hundred and twenty individuals
(equal number of females and males) participated in the experiment. They were selected randomly from student and nonstudent population. Ages ranged from 15 to 56
bifurcation set
bias hollow
FIG. 3. Bias Perception Graph for Maximum
Behavioral Science, Volume 33, 1988
Control Factors
a = asymmetry
[ normal 1
Factors - _d p_
b = bifurcation
( splitting i
4. Predicted Cusp surface for the Necker
years and all were “naive” with respect to
the purpose of the study.
2. Apparatus
The experimental equipment consisted
of a 10” monochrome VDU with a Polaroid
antiglare screen and a microcomputer. The
approximate distance between the observer
and the screen was 90 centimeters.
3. Method
The design and procedure of the investigation revolved around a computer program which was first tested out in a pilot
study. The program presented the subject
with a list of instructions, displayed the set
of stimuli, and interactively read the responses. The advantages of this is that
unlike photographic slides or tachistoscope
cards, the sequence of presentation is
smooth and animated due to the use of
high-resolution computer graphics displayed on the screen. Also, experimentersubject interactions are kept to a minimum,
thus minimizing unusual behavior from the
subject or unwarranted influence by the
experimenter (Silverman, 1977).
3.1 Experimental Design
(a) Stimulus material
Computer graphics were utilized to interpolate enough pictures to give a sequence
of 7 levels of shading and 9 levels of bias
(see Plate 1-specific pictures in the array
can be referred to using a combination of
the I and J values).
The Necker cube is usually drawn as
either an isometric cube or a conceptual
cube (See Figure 1).The former is accepted
as the more realistic representation, but the
conceptual cube is the current standard
drawing of a cube in perspective using the
vanishing point system (Helson, 1967). The
sides of a conceptual cube are transformed
into trapoids and quadrangles, so more
than half of the parallel relationships, and
all the right angles that exist in the real
case, cease to be. In the contruction of the
Necker cube stimuli orthogonal perspective
cues were, however, not employed. Instead,
the isometric cube was biased by keeping
all sides, except one, the same length. By
varying the length of one side, the perspective of a hollow cube or a solid cube was
w 3
j >, I “,r2,
PLATE1. Photograph of the Necker Cube stimulus array.
Behavioral Science, Volume 33, 1988
induced in the observer. (Figures 5a/5b illustrate the difference). Thus, it was possible to achieve smooth biasing in the stimulus arrayj according to the laws of cusp
Shading was represented by the Jdimension J1 d7.
An increase in shading (darkA
FIG. 5a. Orthogonal drawing of stimulus cube
FIG. 5b. Stimulus IlJ7.
Behavioral Science, Volume 33. 1988
ening) increases the three-dimensionality
of the stimulus, because of the combinational contrast effect of the sepamte shaded
surfaces. J7,therefore, forms the row with
the figures having most detail, and J1 the
row with the least detail, where the figures
consist of construction lines. With the options available two surfaces of the figures
were sequentially shaded.
(b) Response to stimuli
In order to record the set of tesponses, a
matrix was defined in software of dimension 9 (levels of bias) x 7 (levelslofshading).
One matrix 9 x 7 was used to record the
responses of 50 subjects.
The empirical testing of bistable perception was ashie+ed in a concrete way by
introducifg ‘a relationship between a deand the capital
fined left-seeking arrow c,
letter “H”. Subjects were instructed to indicate, by pressing keys 1 or 0, if H was in
front of the arrow (representing the solid
cube) or if H was behind/rear of the arrow
(representing the hollow cube). Not being
part of the ambiguous stimuli and independent from it, the letter k and the arrow
thus enabled a direct and reliable way of
testing the proposed model, by informing
subjects of the three-dimensionality of the
figures, the changing shape and detail of
form, but not iflforming them that the stimuli were cubes. hdeed, the pilot study verified the notion that stating that the figures
were cubes would immediately bias the subject’s perceptioh by implying solid cubes
(Postman & Bruner, 1949; Solley & Santos,
3.2 Procedure
The volunteer subject was initially
briefed and familiarized with the equipment, with his/her personal characteristics
interactively fed into the computer. In the
set of instructions automatically displayed
next on the screen the subject was first
informed of the nature of the experiment,
and instructed to identify the position of
the letter H in relation to the arrow (behind
or in front). Each response (1 or 0) made
was registered under the particular stimulus it flas given for. This gave a matrix 9 x
7 comprised of 0’s and 1’s in response to 63
stimuli on one level. The screen automatically displayed the next stimulus after the
subject had responded to the preceding one
(by depressing a key), until all 63 stimuli
had been responded to.
Exposure time of each stimulus was kept
at two seconds, enabling the subject to respond with the first percept (Chastain &
Burnham, 1975). Distraction and biasing
were avoided by asking the subjects to close
their eyes after responding to a stimulus.
After ten seconds an audible cue was given
for them to open their eyes in order to
respond to the next stimulus. Figural aftereffects did not occur after this period of
time. The session was terminated by debriefing the subject as to the nature and
aims of the experiment. Questions were
answered, and the subject was asked for his
or her comments.
Three experiments were run: Experiment
1consisted of subjects viewing the stimulus
array from one direction while Experiment
11 subjects viewed it from the opposite direction in order for the Delay rule to be
applied. The same set of subjects could not
participate in both experiments owing to
the possible influence of prior knowledge of
the induced components. Each experiment,
therefore consisted of 50 subjects each,
making two sets of data to fit the bifurcation set of the Cusp catastrophe (each set
comprising one-half of the cusp).
Expt. I sequence:
1 1J i , 1 2 J i
. - .I9 J I
1. Data Analysis
1.1 Preliminary Manipulation of Results
Individual subject matrices were added
together to form a single global matrix for
each experiment (I, I1 and 111).These global
matrices were reduced to six minor ones
differentiated by sex (male/female), age
(over 25 years/under 25 years), and optical
aids (with/without), so as to give comparable and possible independent variables to
the cusp for all three experiments. Incorrect individual responses (due to misinterpretion of instructions) were deleted
from the raw data, and the matrix reduced
to a common denominator so that each
global matrix represented an equal number
of subjects.
Smoothing Raw Data
In order to reveal the underlying trend
movement from irregular data, a one-sided
exponential filter was used. If I denotes raw
variable (bias) at position t, then the
smoothing function is given by:
I, = (1 -
x (It + XI,,, + X21t+z + . . . )
Although irrelevant short-term effects on
the biasing factor are eliminated, a damping effect also occurs which possibly hides
the effects of faster dynamics. As a comparison, X was set to 0.8 (strongly smoothing) and X = 0.4 (milder smoothing). The
two arrays were evaluated for each experiment.
1.2 Contour Maps
Expt. I1 sequence:
For a quick visual comparison of raw
unprocessed (by Cobb’s 1980 algorithm)
data, three different elementary techniques
were employed. First, a 2D representation
of the 3D behavioral response surface was
Second, the flat contour of the
& J I , & J i , I ~ J I ,... I i J i
above surface was drawn for additional inTwenty additional subjects were run in a formation on the steepness of the catastrothird experiment as a control.
phe, and heights at individual points. FiExpt. 111 sequence: I,J, where n and m nally, the cross-section at J 1 and Jl were
are computer generated random numbers. plotted to show smooth and fold catastro-
Behavioral Science, Volume 33, 1988
phes respectively (for reasons of brevity,
these are not shown here).
1.3 Statistical Analysis
Almost all parameter estimation procedures are based on the minimization of the
departures of empirical observations from
the predictions of a model. Therefore,
models which make several different predictions for the same configuration of independent variables cannot be analyzed
without fundamental modifications to the
usual concept of estimation. As a descriptive statistic, the “mean” loses most of its
usefulness since it is expected to fall between the modes of a bimodal frequency
distribution. Similarly, the standard deviation fails to describe the peakedness of
multimodal data. Thus, multimodal members typically require at least four descriptive statistics (i.e. bifurcation, asymmetry,
location, and scale) as compared with two
for the unimodal density (mean and standard deviation-which have little to do with
location or scale).
In an attempt to apply classical statistics
to CT, a series of unpaired t-tests were
carried out to indicate if sudden jumps had
occurred at different points in an array,
along the a control variable (to indicate the
occurrence of hysteresis). The total or
global matrices for experiments. I and I1
were chosen to demonstrate validity. In
order to find the largest catastrophe point
(as a likely indication of the actual catastrophe point), unpaired t-tests were carried
out at each level (J1- J 7 ) ,and between 8
sets of sequenced pairs of sets of data on
that level: e.g. for J1:
- 19
1112 v. 1 3 - 1 9
11 v.
mining whether observed data are described by the theory, (Sussman & Zahler,
1978). This is mainly created by the topological principles on which CT is founded.
Cobb (1978) tackled this problem by developing a method which under certain circumstances, fits observed data to a probability distribution defining a cusp surface.
Briefly, the numerical algorithm uses the
method of moments to achieve an initial
estimate (IE) in order to start an NewtonRaphson iterative search for the maximum
likelihood estimates (MLE), which, if
found, yield the Chi-square approximation
to the MLE. The MLE is then used to
compare a linear regression model to the
equivalent cusp model to obtain the most
likely cusp from which the data may have
been drawn (Cobb, 1980). The test statistics are asymptotically Chi-square distributed, with degrees of freedom equal to the
number of independent variables plus 2.
The approximate nature of the test status
limits its use (tested with a very high significance level 0.001). Unfortunately, due
to the numerical stability of the algorithm,
MLE’s are not always found, in which case
the program works on the IE.
The cusp surface equation is
y is the dependent variable
a is the asymmetry variable
b is the bifurcation variable
and each control factor is represented by a
linear combination of the independent variables; that is,
a = (ao+ alxl + a 4
1.4 Cobb’s Cusp Surface Analysis
A criticism of applied CT has been the
absence of statistical procedures for deterBehavioral Science, Volume 33, 1988
+ b2x2)
(bo + bixi + bzxz)Y
= (bo +
+ -y3
I1Z& v. I4 - 19.. . etc.
This provided 8 t-tests at each level, for 7
levels per experiment; a total of 112 unpaired t-tests. A high significance level was
used p < 0.001 and p < 0.01, because of the
approximate nature of t-tests.
+ by + a = 0
+ (a0 + alxl + a2x2)= 0
and for a standardized variable z =
(Y - N
X = location parameter for data points
u =
scale parameter.
+ -Z3
+ (bo + blxl + bzx2)z
+ (a0 + alxl + a2x2)= 0
Equation 2 is the final form of the standardized cusp surface. The canonical cusp
measures the state variable on each side of
the zero point.
Cobb’s 1980 algorithm also, and more
usefully, gives the raw (unstandardized)
factor coefficients as (for our data):
+ b2J) yh-
+ (a,, + a l l + azJ)= 0
Y = observed value of the dependent
variable (0-50).
u =
A =
bo =
bl =
I =
J =
calculated scale parameter.
calculated location parameter.
constant related to splitting factor.
weight that variable I loads on
splitting factor.
weight that variable J loads on
splitting factor.
constant related to normal factor.
weight that variable I loads on normal factor.
weight that variable J loads on
normal factor.
hollow/solid independent variable,
I = 1-9.
2D/3D solid independent variable,
J = 1-7.
The first and last solutions of Equation 3
correspond to the upper and lower surfaces
of the cusp surface (called modes), and the
middle solution is called the antimode, corresponding to the middle sheet (i.e., the
inaccessible region) of the cusp surface.
The program reports a “pseudo” - R2value
(pseudo variance because it uses an improper prediction), by calculating all of the
modes and antimodes. Cobb (1980) interpreted this as “the proportion of variance
accounted for by the model if the nearest
mode is used as the predicted value of y
(i.e. nearest because it is on the same side
of the antimode as the observed value of
y)”.Thus, in CT terminology, the predicted
Behavioral Science, Voiume 33, 1988
value of y is a t the bottom of the domain
of attraction in which the observed value
of y was found. The linear R2 is also calculated for comparison.
An additional error term analysis not in
the original program was supplemented in
order to calculate the corrected raw score
standard error of estimate.
S,,,(l - R 2 ) l ”
where S is the standard deviation of y
(dependent variable), and R2 is the linear
or pseudo variance.
1.5 Bifurcation Set
The bifurcation set was obtained by solving the equation of the bifurcated cusp to
obtain the raw (unstandardized) factor
coefficients (Ta’eed et al, 1984a).
27 a’ = 4b3
+ a: + a2J)2= 4 (bo + bzJ)3
-+ 27 (a,,
The bifurcation set was evaluated for all
matrices concerned, and plotted graphically.
2. Experimental Findings
2.1 Experiments I and
The visual data consisting of 3-D surface
graphs (examples are shown in Figures 6
and 8), established the fact that bistability
associated with the Necker Cube does indeed take place. The corresponding contour
graphs (e.g., Figures 7 and 9) show lines
which represent the response surface a t
FIG. 6. 3-Drepresentation of Expt. I-hollow to
solid sequence (raw data displaying two modes of
Z s l 1 . 0 TO 42.0
FIG. 7. Contour map of Expt. I (the closer the
lines, the steeper the catastrophe).
210.0 TO 19.0 CONTOURS AT H U T I R E S OF
FIG. 9. Contour map of Expt. 111.
ing through the catastrophe region (proequivalent heights or values (in multiples). ducing a peak around the I9 J9 corner).
Both techniques visually illustrate the raw
The attempt statistically to locate catasdata required to form each arm of the bi- trophe points consisted of drawing a guesfurcation set, and hence the cusp model.
timate line joining all array points (for ExOn a basic level, the figures indicated periments I and 11) above the 0.001 signifthat the behavioral surface of Experiment icance level resulting from the histograms
I rises above that of Experiment I1 (shown for 112 unpaired ‘ttests carried out on the
by the close proximity of lines), which also array, (Figure 10). This was predictably
represented a wider range of values on the unclear as to what result signifies the lodependent axis.
cation of the bifurcation set cusp, and conThe catastrophe point (at I = 4) on Ex- straints the technique for use only when
periment I1 occurred (as predicted by the none other exists.
cusp model) a t an earlier bias level than
The results of the initial application of
Experiment I, thus creating a bimodal re- Cobb’s program to find the most likely cusp
gion where two planes exist on the com- surface given the data of the above study is
bined or “complete” matrix. With increas- displayed in Table 1.
ing shading both experiments had increasFor the complete matrix, the calculated
ing dependent variables, but in the latter
there appeared to be a definite skew, pass-
FIG. 8. 3-D representation of Expt. 111-random
Behavioral Science, Volume 33, 1988
FIG. 10. ‘%test” results on Expt. I and I1 ( p =
0.001 - 5.405 forming the bifurcation set).
Behavioral Science, Volume 33, 1988
zero position of the canonical cusp; and
that each I unit changes the normal factor
by a multiple of 0.527 and each J unit
changes normal factor (a) by a multiple of
0.487. The calculated weights on the control normal variable a are given by (-4.73
+ 0.5271 + 0.487J). Thus, both independent variables influence a, although, as expected bias variable had the strongest influence on the normal factor. Similarly, the
weights on the splitting factor (b) given by
(-2.31 + 0.3641 +0.337J), indicate that the
predicted cusp began at -2.31 z units from
the zero position of the canonical cusp and
FIG.11. Bifurcation set from Expt. I and 11; using that the bias variable changes b by multithe Maximum Likelihood Estimates (MLE). Both
ples of 0.364 opposed to 0.337 for shading.
arms of the cusp lie in the control space.
This indicated that once again, both bias
and shading influenced the splitting factor,
cusp is described by the equation:
almost to the same degree.
The resultant bifurcation set, shown in
+ (-2.31 0.364 1
Figure 11 for Experiments I and I1 represents the independent variables 1and J as
two almost parallel lines/pleats-an indiy - 23.6
+ 0.337 J) 7.17
cation of a very large cusp. Figures 12 and
13 present the results of the smoothed data.
+ 0.527 I + 0.487 J) = 0
Figure 12 (A = 0.8 asymmetry) clearly
This equation was solved to give the cal- shows that the underlying trend is the inculated values of the dependent variable x fluence of the bias control variable on the
in relation to the independent variables I normal factor, and that of the shading facand J . For example, an observed dependent tor over the splitting factor, although both
variable of 16.00 at 17J1has the calculated have much smaller influences on the upper
asymmetry factor -0.03, a bifurcation fac- variable. Strong smoothing at X = 0.8
tor 0.94, a mode 16.52, an antimode 23.77 tended to obliterate any trends in the data,
and a higher mode 30.43. The equation but with X = 0.4 more realistic and useful
showed that the predicted cusp began results were obtained. Figure 13 shows the
at-4.73 z (standardized) units from the
[y 112735.13 +
[ ]+
FIG.12. Bifurcation set for smoothed data array
(A = 0.4) from Expt. I and 11.
Behavioral Science, Volume 33, 1988
- I
FIG.13. Bifurcation set for smoothed data array
( X = 0.8) from Expt. I and 11. Note that the control
set is translated by 8.
cusp, complete with smooth-way round the
back ( b < 0), cusp point ( b = 0, a = 01,
convergence and bimodality ( b > 0).
2.2 Random Control Experiment
The random experiment (111) gave results which were similar to, and almost a
replication of, Stewart & Peregoy’s (1983)
study. The standardized data suggested
that a is strongly influenced by bias and
negligibly by shading. Similarly forb, shading is the stronger influence although the
influence of bias is significant. The expected “ideal” cusp is evaluated by Figure
14 which shows the cusp to be symmetrical
around 15, confirming the visual data analysis above. Once again, no smoothness or
continuity at levels was found a t J1,as
expected for a “perfect” cusp. Also, no
“skew” was evident. Bimodality and catastrophe may have been due to the fault of
Cobb’s algorithm fitting the largest cusp
possible, but this is only conjectural.
2.3 Auxiliary Findings: Sex, Age, Optical
A ids
A secondary study of subject characteristics, in particular sex and age, conducted
in the experiments resulted in some interesting facts. The use or freedom from use
of optical correction aids (spectacles, contact lenses, etc.) displayed such random
data that cusp fitting became meaningless.
The difference between male and female
cusps was quite drastic in their implications, although less so when viewed stochastically. Generally, males and females
FIG.15. Bifurcation set for female subject,s.
differed only in the relative size of the
catastrophe (the switch in perception) at a
particular bias and shading, (Figure 15
shows the cusp obtained for female subjects). Both cusps displayed the skew,
which tends to obscure any real differences
between females and males in their perception of the Necker cube. More sensitive
experimentation is needed to eradicate any
differences created by errors in the cusp
fitting, and discover any underlying subtle
differences between the sexes.
Age appeared to make an impact on perception. For example, those under 25 years
of age were influenced to a greater degree
by the biased sequence, (Figure 16), than
were those over this age. Some difference
in age was observed simply in the degree of
active response given to the stimuli. Older
people tended to try and give responses
FIG.16. Bifurcation set for subjects aged under 25
FIG.14. Bifurcation set from random data.
Behavioral Science, Volume 33, 1988
expected for a certain stimuli. For example,
some verbalized that they saw a solid cube
but “knew” that the particular cube was
biased toward cube hollow and therefore
gave a response which conflicted with their
perception. Data for the over-25s fitted a
linear model better than a cusp model, and
thus comparison is difficult.
The visual data implied that the bias
from solid to hollow cube has an influence
less significant to the bias from hollow to
solid. There seemed to be a general tendency by subjects immediately to register a
reduction in shape and/or size of the figures, rather than an increase in them. It is
contended that the size of the cube, owing
to varying perspective lines and angles, may
be a factor involved in the perception of a
multistable figure. The unexpected presence of a skew is not readily interpretable
in terms of a simple conception of the proposed hypotheses. The cusp was expected
to be symmetrical around a = 0 asymmetry
line, but the data indicated that it lay diagonally (toward 19J7).On smoothing
which to a degree forces the above result, it
was seen that for the normal factor, the
bias variable had a stronger influence, while
for the splitting factor, shading exerts the
stronger influence. In other words, the perception of the solid cube maintained itself
longer, even when a cue of perspective indicates that the cube is hollow.
As can be seen, the raw unstandardized
coefficients obtained by an initial application of Cobb’s (1980) program imply that
for the complete (unsmoothed) data there
is little choice differentiating which variable influences the control factors on the
predicted cusp. In terms of the standardized
independent variables (which should be
comparable), on Table 1, the bias variable
I had by far the stronger influence on both
normal and splitting factors (i.e., the cusp
lay diagonally, and not symmetrically as
indicated by the visual data analysis). The
psuedo-R2 (for the cusp model) at 0.715
was greater than that for the linear (multiple regression) model of 0.617, indicating
that the cusp model yielded a better fit, a
small error mean between the two. Thus,
Behavioral Science, Volume 33, 1988
the pseudo-standard error of estimate for
the cusp model on the dependent axis is 4.9
on the scale of 0-50. Checking with the
Chi-squared approximatiod to the MLE,
between the linear and cusp models, gives
~ ‘ ( 4=
) 306, p < 0.001, for the cusp (critical
&4)p = 0.001 is 18.46).
It appears from the results of the bifurcation set analysis, as displayed by Figure
11, that the differences between hollow and
solid surfaces remained constant and were
negligibly affected by independent variables. The stress or diagonal set may have
been due to the equi-dependence of the
dependent variable y, on the independent
variables I and J . This means, that rather
than perspective cues and shading compounding to produce either the cube hollow
or cube solid, these independent variableswere acting in opposition to produce perceptual behaviour. Figure 11 also indicated
the absence of non-catastrophe areas, i.e.,
the transition from solid to hollow and vice
versa is smooth (for b < 0).
The presence of the skew on the cusp
(from corner to corner) apparent through
out all the bifurcation sets, introduces a
new element in the discussion-the consideration that the general trends exhibited
were inconsistent with the original cusp
predicted by the model, although there is
convincing evidence for the existence of a
cusp. The empirical study did, however,
provide some valuable clues such as the
possible influence of the size of the cube
and the independent, mutually exclusive
influence of the independent variables,
each of which was thought to relate to each
of the two control factors a and b. A reasonable solution was, therefore, to seek to
attribute the observed performance in the
experiments to alternative hypotheses
which may elucidate the “real” reason for
the skew.
It is evident from the results that the
skew cannot be solely attributed to unidirectional learning in that subjects learned
to predict the sequence presentation; that
is, the effect of showing a subject a sequence
of alternatives may enable her/him to predict and alter her/his perception of the
figure just presented. Strong expectations
of the possible sequence would cause a shift
of responses to the right (indicated by the
skew to the right, diagonally). If learning
was cause of the skew, then it is clear that
the resulting graph should show two lines
crossing because subjects would have equal
chances of learning in both sequential directions.
Resorting back to a review of CT, it
seemed legitimate to consider that the perceptual behavior observed in the study indicated the operation of two conflicting factors, as opposed to a normal factor and a
splitting factor (Zeeman, 1977). In the final
analysis, the evidence for this speculation
has to be derived from postulating a model
which explains the trends discovered under
the conditions of experiment, and for which
topological and psychological evidence
must exist. It is then possible to make
empirically grounded inferences regarding
the components of the perceptual behavior
In some applications of CT, the control
factors are seen as two normal factors
either side of the cube, such that, a = b +
a, and, @ = b - a. In this case, aB are
conflicting factors influencing x. a tries to
“push” behavior onto the upper surface,
while B tries to ‘push’ it onto the lower.
Inside the cusp the two controls conflict.
Figure 4 shows how conflicting factors
scaled on an a/b bifurcation set exhibit the
characteristic skew inherent in the above
data. Thus, tentatively, the standardized
conflicting factors may be redefined as J I - ( k - t ) and J + I - ( k t ) ,where t =
5 for random data and k will depend on
subject characteristics. The new hypothesis
can be stated as follows: the graph G of
perception is a cusp catastrophe with bias
and shading as conflicting factors, a and
8. The topology of conflicting factors reaffirms observation that the dependent variable x, solid/hollow perception, increased
with an increase in the dependent variable,
J. This is acceptable in terms of the new
hypothesis, since with increasing shading,
the response surface changes from the
lower sheet to the upper.
The following discussion of theoretical
validation for the relationships suggested
by the alternative model requires some understanding and familiarity with percep-
Behavioral Science, Volume 33, 1988
tion concepts. The first point is understanding the human capacity to segment a
picture in many different ways, and hence
build up many different structural descriptions in order to interpret a picture in a
meaningful way (Weale, 1982). Thus, for
example, the Necker cube stimulus may
have only had two variables being manipulated (shading and varying perspective),
but the human visual system also integrates
information about objects from different
sources including relative and contextual
cues. For example, the brightness of the
surfaces, the adjacent effects of shaded surfaces, the overall size of the cube, its apparent orientation, etc., are all factors that
affect the magnitude of the ambiguity of
depth perception.
In the light of this, the interpretation of
the experimental findings in terms of conflicting factors therefore sets two further
questions-first, the existence of a bimodal
region. The answer to this seems to lie in
current research of how the visual system
discerns contrast: the differences in brightness of adjacent areas. Without entering
into the complex phenomena of spatial frequencies, edge effects, and reflectance the
critical fact is that the visual system is less
sensitive to contrast when the contrasting
areas of a figure decrease, rather than a
change in the relative brightness of adjacent areas. When an object recedes from
the viewer, it becomes smaller and details
with low contrast become difficult to perceive, (Campbell & Maffei, 1974). Thus, if
contrast is discerned from viewing the three
different shades of the surfaces of the cube,
it follows that when progressing from hollow to solid the net areas of contrast of the
whole cube decreases, the retinal image becomes less detailed and less sensitive to the
influence of shading. The reverse is also
true: when progressing from solid to hollow,
the relative influence of bias decreases with
the increase in contrasting areas. Interpreted back into the CT model, the interplay of these two factors results in the
phenomenon of bimodality; therefore, reversibility occurs both in the sequence hollow to solid, and in the sequence solid to
The above reasoning begs the second
question as well as being allied to the effect
of the skew in the original cusp model. The
cusp point begins at the solid bias, rather
than at the neutral point or the hollow bias.
As discovered, subjects tended to see a solid
cube a t high shading even at, and despite,
extremes of hollow bias. The reverse was
also found where the hollow cube was observed a t low shading and extremes of solid
bias. Shading (or gradients of illumination)
appears to be a strong cue for solidity according to several authors, (Haber & Hershenson, 1980; Attneave, 1972). In general,
adding color (shading) imposes three dimensionality on an ambiguous figure. Thus
at high shading, despite perspective bias,
depth is not observed because it is countermanded by competing depth information.
Hence the effect of solidity predominates,
and ambiguity is greatly reduced.
It is interesting to note that although the
top row of Plate 1 (IIJ1shows the
absence of one feature-that of shadingcurrent theory would indicate that the arrangement of angles and lengths of the
sides of the cube should not affect its tridimensionality. This is true even if the
magnitude of the latter are being varied,
because the actual number of angles and
the number of continuous lines remains
constant. If the model, however, assumes
conflicting factors are a t play, then the
three-dimensional appearance of the cube
is affected. Instead of seeing 3-D forms,
there is a tendency to see the figures as
simple flat patterns made up of geometric
lines. This might be explained as a special
case of the Minimum Principle in operation
(Hochberg, 1974) which states that the perceptual system organizes the perceived objects of the world in such a way as to keep
changes and differences to a minimum.
Thus, the lack of any other variant renders
a two-dimensional shape to the figures of
rows 1 and 2 in Plate 1. This effectively
pushes the skew to the left since the figures
are denied depth and the only alternative
response to subjects can make is that the
preceding figures are hollow.
The “law of simplicity” is a powerful one
if one can prove its operation in perception,
but the objective and quantitative organization of simplicity is a difficult problem.
Behavioral Science, Volume 33, 1988
How can one measure the complexity of
alternative forms of line drawings of the
Necker cube in varying perspectives on one
hand, and the strength of the tendency to
see that form on the other? An experiment
conducted by Attneave and Frost (1969)
might help further understanding of this
problem. They studied the effect of angles
and side lengths of an outline cube on its
apparent tri-dimensionality, and concluded
that subjects perceived a three-dimensional
arrangement only when the angles and side
lengths of the stimulus cube were most
uniform as a three-dimensional object. Interpreting this finding within the model
supports the argument for the source of the
skew at solid bias. Figure 11shows that the
catastrophic point begins at IJ1, which
resembles the isometric line drawing of the
Necker cube. Subsequent figures appear
then to be affected by the bias variable. The
significant feature of the remainder of the
array is shading which begins to assert its
strong influence.
It is also plausible that some unconcious
perceptual learning had occurred to reinforce the observed skew. The initial stage
of recognition of the Necker stimulus is
followed by the second state involving the
focus of attention upon distinctive features
of the stimulus. As the subject responds
systematically to the stimuli she/he is
aware of both alternatives of depth reversal, and able to judge which was the better
interpretation in any given picture. TOdisgress briefly in order to clarify, Weale
(1982) argues that an urban (as opposed to
a rural) society is more exposed to illusions
involving perspective cues (such as the
Muller-Lyer and Ponzo illusions; Gregory,
1986), and viewing certain classes of objects
in tri-dimensional configurations (e.g.,
boxes). Subjects in the present study were
part of an urban population and therefore
should be more biased toward the solid cube
in the experiment. An appreciation of this
point makes more comprehensible the role
of “past experience,” or “recognition” or
“meaning” in perceptual learning and subsequent perceptual organization. This determinant, together with short term memory, is bound to play a critical role in subject
responses, and is difficult to control. But to
say that is not enough. The basic problem
is to understand precisely how mental preconceptions interact with immediate sensory input to create visual experience.
The conclusions reached from the auxiliary findings are compounded by the fact
that the original proposed model’s hypothesis seemed no longer to hold. This, however, does not deter any further research in
this area. Indeed, some research, mostly
undervalued, has found that the rate of
spontaneous reversals of ambiguous figures
is markedly reduced in old people (Welford,
1958). Attneave (1971) reported that damage to one frontal lobe decreased reversal
rate. Any theoretical link between this and
Welford’s finding can only come through
possible neurophysiological explanations.
The task of this paper has been to identify some of the determinants involved in
multistable perception, by studying the
Necker cube, a reversible figure, the perception of which results in two alternative and
mutually exclusive aspects. A nonlinear catastrophe theory model enabled the experimental manipulation of two variables, selective stimulus shading and varying perspective cues, in a systematic manner in
the current study. The investigation resulted in an interesting but unanticipated
situation where the proposed model’s hypotheses were found to be redundant, but
the cusp catastrophe was maintained.
Our discussions to this point have reinterpreted the model tentatively in terms of
the operation of conflicting factors. This
may appear paradoxical-How
can the
original model generate hypotheses which
would organize components of the perception of the Necker cube so that the parameters of selective stimulus shading and
varying perspective cues do not in fact interact, but act in opposition to each other?
The cusp catastrophe model is an analytical
one which picks out observed patterns, and
keeps open ways of conceptualizing variables and relationships between variables.
It is intrinsic to the mathematical construction of Cobb’s algorithm that it linearly
combines the independent variables. This
it does in order to test for the most likely
Behavioral Science, Volume 33, 1988
direction and size of the cusp. It is this very
nature of catastrophe theory which implies
that differing hypotheses can ultimately
lead to the same experimentation. As far as
the current research goes, shading and bias
are decisive factors in the perception of the
Necker cube, exerting their effects in opposition, and neither reinforcing the other.
The topological and psychological evidence
for the new model does not, however, have
the immediate and dubious honour of producing the second theorem of conflicting
The present state of affairs leads to the
reasonable conclusion that ambiguity in reversible figures stems from several different
factors which may be of both peripheral
and central origin. The perceptual mechanisms involved in the spontaneous structuring and restructuring of both reversible
and ambiguous figures (figures which have
alternating appearances but remain in the
same depth plane, e.g., Rubin’s “chalice/
faces” figure; Fisher, 1968), are closely related. Estimation of the relevant importance of the contribution made through
different factors to the Necker cube remains largely unchallenged, perhaps because perception of the Necker cube is affected by influences intrinsic to the perceiving individual. This can lead to the
negation of its importance since it can be
argued that there are structural rules intrinsic to the visual system that determine
relevant aspects of the environment.
The phenomena examined here show
that shading and perspective bias are both
units and part of the larger totality of the
perception of the Necker cube. Further
studies could be made of which features are
significant for multistable perception. For
example, Chastain and Burham (1975) discuss the perception of a multistable man/
rat figure and suggest that which is perceived depends on which part of the figure
falls on the fovea of the eye. Non-foveal
parts sometimes also play a role that is still
unclear. Experiment may provide a useful
pointer for the study of perception of sequentially presented multistable figures
which can be affected by controlling the
segment that is first viewed (Olson & Orbach, 1966). One can then find out how
instrumental different presentation procedures, prior exposure to similar patterns,
varying degrees of sophistication of the figure, spatial and illumination differences,
set, etc., are on perceptual organization.
One might also note that lines in ambiguous geometric/reversible figures represent
edges which also influence perception, because of abrupt changes in image intensity
and boundaries of objects. For instance, it
is known that long lines are brighter than
short lines; end-end connected lines are
brighter than end-middle. This may be a
direction of research into line drawings of
the Necker cube which may take into consideration sizes of the cube. Largely unchallenged is research into neurophysiological mechanisms that determine multistable perception.
So far this article has emphasized the
legitimacy of using catastrophe theory in
handling a psychological problem where a
bimodal distribution exists, a technique
serving as an advantage over linear threshold models. This does not eliminate the
need for measurement and statistical elaboration that actually exist in the field of
perception-merely the need to utilize appropriate mathematical techniques in the
study of various aspects of perceptual behavior. It is of particular relevance that
nonlinear cusp geometry is capable of generating new hypotheses of scientific relationships which are also capable of being
experimentally tested. For example, one
could complicate the Necker cube model by
demonstrating the influence of three or
four independent variables on a given dependent variable. Their combined or individual effect could be determined in relation to two control variables. Further still,
catastrophe modeling may be applicable to
other reversible figures such as the Stanford figure (Stanford, 1897), Ames trapezoidal window, triangular prisms, and reasonably to other rectilinear objects. Higher
dimensional catastrophes (e.g., the butterfly) may be utilized in modeling to understand the complexity of an event (providing
that the use of cusp catastrophe proves
successful). This, at the very least, must be
encouraging, and motivate further exploration of t,he process underlying stimulus
extraction and percept formation.
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