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код для вставкиDETERMINANTS INVOLVED IN THE PERCEPTION OF THE NECKER CUBE: AN APPLICATION OF CATASTROPHE THEORY 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. r+.Y INTRODUCTION 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 T 97 Behavioral Science, Volume 33, 1988 98 TA’EED, TA’EED, AND WRIGHT 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. CATASTROPHE THEORY MODELING OF THE NECKER CUBE 1. Rationale for Study In the field of perception psychology various authors have illustrated that certain ISOMETRIC CUBE ORTHOGONAL CUBE FIG. 1. Two views of the Necker Cube. Behavioral Science, Volume 33, 1988 PERCEPTION OF THE 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 procedure. 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 NECKERCUBE 99 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. X t HYPOTHESIS 2 bias hollow 0 bias solid FIG. 2. Bias Perception Graph for Minimum Shading. TA’EED, TA’EED, AND WRIGHT 100 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 observed. 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 X 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. EXPERIMENTAL METHOD 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 XI L i 3 + il t 3 2 bifurcation set BIFURCATION SET 4h3 27a2 a L ,a bias hollow 0 - FIG. 3. Bias Perception Graph for Maximum Shading. Behavioral Science, Volume 33, 1988 Control Factors Conflittinq a = asymmetry [ normal 1 Factors - _d p_ b = bifurcation ( splitting i FIG. Cube. d:d+b L? =b-a b 4. Predicted Cusp surface for the Necker PERCEPTION OF THE NECKERCUBE 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). 101 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 n a I w 3 2 1 2 3 & 5 j >, I “,r2, A, 6 . T PLATE1. Photograph of the Necker Cube stimulus array. Behavioral Science, Volume 33, 1988 c TA’EED, TA’EED,AND WRIGHT 102 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 catastrophe. Shading was represented by the Jdimension J1 d7. An increase in shading (darkA a 5a a FIG. 5a. Orthogonal drawing of stimulus cube (1157). 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, 1958). 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 PERCEPTION OF THE NECKER CUBE 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 1 103 RESULTS 1. Data Analysis 1.1 Preliminary Manipulation of Results Grouping 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 drawn. 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- I Behavioral Science, Volume 33, 1988 TA’EED, TA’EED, AND 104 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. WRIGHT 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 -y3 where 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 I2 b (1) 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 bixi + (a0 + alxl + a2x2)= 0 and for a standardized variable z = (Y - N U where X = location parameter for data points u = scale parameter. PERCEPTION OF THE + -Z3 (2) + (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): LI + b2J) yh- + (a,, + a l l + azJ)= 0 where Y = observed value of the dependent variable (0-50). u = A = bo = bl = b2 = a0 a1 = = a2 = 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 NECKER CUBE 105 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,,, = 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 II 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 behaviour). TA’EED,TA’EED,AND WRIGHT 106 7 6 7 2 5 4 L 2 , ‘ 1 2 3 4 BIAS Z s l 1 . 0 TO 42.0 5 -I 6 7 8 a M T a R s AT IILLTIPLES Cf 9 ‘ 1 2 3 4 5 6 7 8 9 BIAS - I 5 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 2 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- BIAS FIG. 8. 3-D representation of Expt. 111-random data. Behavioral Science, Volume 33, 1988 - I FIG. 10. ‘%test” results on Expt. I and I1 ( p = 0.001 - 5.405 forming the bifurcation set). PERCEPTION OF THE NECKER CUBE Behavioral Science, Volume 33, 1988 107 TA'EED,TA'EED, A N D WRIGHT 108 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 9 ' 1 2 3 4 5 6 7 0 9 predicted cusp began at -2.31 z units from BIAS - I 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 (-4.73 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 + [ ]+ 'I '1' ' 1 2 3 4 BIAS 5 -I 6 7 8 9 FIG.12. Bifurcation set for smoothed data array (A = 0.4) from Expt. I and 11. Behavioral Science, Volume 33, 1988 I' 3' 4l 5' BIAS 6' - I 7' 8' pt 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. PERCEPTION OF THE NECKERCUBE 109 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 1 ‘I1 \ *I 3l 4’ 5l 81A3 -I 7l 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 F 5 6 7 8 9 BIAS 1 FIG.16. Bifurcation set for subjects aged under 25 years. ’1 BIAS - I FIG.14. Bifurcation set from random data. Behavioral Science, Volume 33, 1988 2 2 3 4 - TA’EED, TA’EED, AND WRIGHT 110 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. DISCUSSION 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 PERCEPTION OF THE 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 involved. 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 NECKER CUBE 111 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 hollow. The above reasoning begs the second 112 TA’EED,TA’EED,A N D WRIGHT 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 PERCEPTION OF THE NECKERCUBE 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. CONCLUSIONS 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 113 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 factors. 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 114 TA’EED, TA’EED, AND 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. Behavioral Science, Volume 33, 19B8 WRIGHT REFERENCES Attneave, F. & Frost, R. The discrimination of perceived tri-dimensional orientation by minimum criteria. Percept. and Psychophys., 1969,6,391396. Attneave, F. Multistability in perception. Scientific American, 1971, 2256, 62-71. Attneave, F. Representation of physical space. In A. W. Melton and E. J. Marlins (Eds.), Coding processes in human memory. Washington, D.C.: N. H. Winston, 1972. 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