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Geometrical illusions: Study and modelling

Article in Biological Cybernetics · December 1997

DOI: 10.1007/s004220050399 · Source: DBLP

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Biol. Cybern. 77, 395–406 (1997) BiologicalCyberneticsc© Springer-Verlag 1997

Geometrical illusions: study and modellingA. Bulatov, A. Bertulis, L. Mickien e

Department of Biology, Kaunas Medical Academy, Mickeviciaus 9, 3000 Kaunas, Lithuania

Received: 18 June 1996 / Accepted in revised form: 24 June 1997

Abstract. The phenomena of geometrical illusions of extentsuggest that the metric of a perceived field is different fromthe metric of a physical stimulus. The present study inves-tigated the Muller-Lyer and Oppel-Kundt illusions as func-tions of spatial parameters of the figures, and constructed aneurophysiological model. The main idea of the modellingis based on the uncertainty principle, according to which dis-tortions of size relations of certain parts of the stimulus, so-called geometrical illusions, are determined by processes ofspatial filtering in the visual system. Qualitative and quanti-tative agreement was obtained between psychophysical mea-surement of the strength value of the illusions and the pre-dictions of our model.

1 Introduction

We report here our psychophysical measurements of twogeometrical illusions of extent and discuss our reasoningconcerning the illusions’ nature. We take Muller-Lyer andOppel-Kundt figures as objects of our study (Fig. 1). Thetwo different figures produce a similar effect on perceivedsize: the two halves of the figures seem to be differentin length when in fact they are equal. In the case of theMuller-Lyer figure, the half that has inward-facing wingsseems to be shorter than the half with outward-facing wings.In the Oppel-Kundt pattern, which has no crossing linesand no flanking contextual figures, the shaded half seemsto be longer than the unshaded one. The Muller-Lyer il-lusion is one of the most famous illusions and has beenstudied intensively since the end of the nineteenth century.The Oppel-Kundt illusion has been less well studied, thoughwell-documented and fairly well known.

The question arises whether the same neurophysiologicalmechanism is responsible for the two types of perceivedsize distortions, and whether the same theoretical approachmight be applied to the similar effects produced by dissimilargeometrical patterns.

Correspondence to: A. Bertulis (Fax: +370-7-2207-33)

Fig. 1. Facsimiles of the Muller-Lyer (A–D), and Oppel-Kundt (E, F) fig-ures.a, vertical line length;b, d, lengths of the reference and test halvesof the figure respectively;w, wing length;α, wing tilt angle

The experimental data in the literature show a rather con-sistent picture in the magnitude of distortions in the Muller-Lyer figure. The illusion has been measured as a function of:the shaft length (Fellows 1967; Brigell et al. 1977; Restle andDecker 1977; Gilam and Chambers 1985; Mack et al. 1985),

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the wing length (Heymans 1896; Lewis 1909; Nakagawa1958; Dewar 1967; Erlebacher and Seculer 1969; Fisher1970; Restle and Decker 1977), the wing tilt angle (Biervliet1896; Heymans 1896; Dewar 1967; Erlebacher and Seculer1969; Davies and Spencer 1977; Pressey et al. 1977; Restleand Decker 1977), the gap between the shaft and the apicesof the wings (Yanagisawa 1939; Fellows 1967; Worrall andFirth 1974; Pressey et al. 1977; Predebon 1992) and theframing ratio, i.e. the ratio of the total figure length to theshaft length (Brigell et al. 1977; Brigell and Uhlarik 1979;Schiano 1986). The contrasting effect of the inward-facingand outward-facing wings was examined simultaneously inthe double figure, and each effect was also measured sep-arately by presenting a half-figure with a comparison linethat had no wings (Lewis 1909; Pieron 1911; Erlebacherand Seculer 1969; Fisher 1970; Restle and Decker 1977).

The data accumulated demonstrate the regularities suchas: the illusion strength is proportional to the shaft length;the illusion strength varies as an inverted U-shaped functionof the wing length, reaching a maximum when the wingsare approximately 30–40% of the length of the shaft; largeinward-facing wings have a size reduction effect on the shaft;the illusion decreases with the tilt angle of wings up to 90;the angle exerts its effect within a small region near the tip ofthe shaft, the region extending about one-seventh of the shaftlength; the reverse direction of distortions can be achievedby moving the wings away from the ends of the shaft, andthe gap at which the maximum reversed effect occurs isinversely related to the tilt angle; the peak overestimation ofthe shaft is obtained when the framing ratio is 3:2.

The experimental data have been analyzed qualitativelyand quantitatively because it was expected that the observeddistortions in the visual field would elucidate the fundamen-tal properties of visual processing. As a result, numeroushypotheses have been proposed and various explanations ofthe illusion have been offered.

Originally, the Muller-Lyer illusion was explained as acombination of two opposing factors: confluxion and con-trast (Muller-Lyer 1896a,b; Heymans 1896; Lewis 1909;Pieron 1911). Confluxion means that two points are seencloser than the objective display would justify, and contrastmeans that they are seen too far apart. It is suggested thatthe two factors are implicated at various levels of processinga stimulus through lateral inhibition and contour repulsion,which have a fixed range of action (Nakagawa 1958; Wagner1968; Coren 1970).

Another type of explanation is given by Gregory (1968,1970), according to which, primary cues of depth and per-spective elicit perceived length distortions. The explanationis supported by an idea that depth perception operates on asize constancy mechanism (Day 1972; Gauld 1975; Lester1977; Ward et al. 1977; Smith 1978). According to Gillam(1980), size constancy is based on the linear-perspectivescale and not on a depth response to the scale. All thesestatements are reflected in one way or another in cognitivetheory (Gregory 1972; Rock and Anson 1979), which as-sumes that perception results from the system trying to findthe most likely explanation for given sensory data.

Nevertheless the presence of illusions in figures display-ing no apparent depth cues, such as Baldwin’s and dividedline figures (Baldwin 1895; Obonai 1954; Day 1965; Zan-

forlin 1967; Coren and Girgus 1972; Brigell and Uhlarik1975; Brigell et al. 1977), is used as evidence against theperspective theory. Brigell and co-authors (1977) came tothe conclusion that the ratio of the total figure length to thefocal shaft length determines the relative magnitude of theillusion.

The illusions on lines flanked by contextual figures suchas rectangles, arrows and lines are also interpreted in theterms of the adaptational level theory (Helson 1964; Greenand Hoyle 1964; Restle and Merryman 1968; Restle 1971,1977), which states that judgments on length are directlyrelated to the size of a focal stimulus and inversely relatedto the adaptation level of the observer.

In parallel, Pressey’s (1967) assimilation theory is ap-plied: the focal shaft assimilates to (or is averaged with)the magnitude of the contextual figures that are flanking theshaft (Pressey 1971; Pressey and Bross 1973). A qualita-tively similar weighted averaging model is proposed by An-derson (1974).

Other theories, such as a ‘confusion’ theory (Chiang1968; Erlebacher and Seculer 1969) or receptive field models(Walker 1973), say that the size of proximal figures shoulddetermine the magnitude of distortions.

The concept of the center of gravity, though, is in someconflict with these theories. It is based on feedback fromefferent commands for eye movements (Judd 1905; Festingeret al. 1968; Kaufman and Richards 1969; McLaughlin et al.1969; Virsu 1971; Coren and Hoenig 1972). The conceptexplains that perceptual distortions might occur because offeedback resulting from an inappropriate tendency to fixatethe center of gravity of contextual figures when attemptingto fixate the end-points of the focal shaft.

These opposing concepts taken together yield furtherproposals. For instance, among the receptive field modelsdeveloped, the idea of oriented line detectors in the visualcortex is recognized. The detectors presumably measure anyorientation as the ratio of vertical to horizontal extent, whichcauses overestimation or underestimation of the outward-facing or inward-facing wings (Caelli 1977). Then the ideaof oriented receptive fields acting as spatial frequency filtersis also taken as an interpretation of the length distortions.Kawabata (1976) and Ginsburg (1984, 1986) present evi-dence that filtering processes caused by lateral inhibitionand producing a certain amount of blurring of the retinalpicture are the reason for the Muller-Lyer illusion.

The similarities and differences among the numerous the-oretical approaches do not assign priority to any of them.Comparison of the effects of various analyzed models andexperimental data fail to establish a quantitative match:Eijkman et al. (1981) found that a model consisting of asize-constancy operator triggered by depth cues predictedeffects larger than those actually observed; a filter modelseemed partly responsible for the observed illusion, but asufficient magnitude of illusion could not be obtained; ori-ented bar detectors were even less effective in explaining theobserved length illusions. Therefore some investigators (Day1972; Restle and Decker 1977; Pressey and Di Lollo 1978;Eijkman et al. 1981) suggest that simple straightforward ex-planations are questionable and that there may be more thanone mechanism contributing to the observed effect. Others,though (Predebon 1992), still emphasize the lack of neces-

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sity of invoking different causal mechanisms. One may cometo the conclusion, then, that a satisfactory reliable model ofthe Muller-Lyer illusion has not been found yet.

The same is true of the Oppel-Kundt illusion. Accord-ing to Craven and Watt (1989), the essential explanation isthat the averaged contour density makes the filled half ofthe Oppel-Kundt figure appear wider than the empty one.The question of what in an ordinary image would corre-spond to contour density may be answered by stating thatthe zero-crossing numbers within a range of spatial scales areadequate and contribute quantitatively to the illusion (Watt1990). Craven (1995) measured the Oppel-Kundt illusionafter adaptation of the subject to spaced parallel lines, andobtained no aftereffect, which led him to conclude that theillusion was not a product of a continuous spatial calibrationmechanism.

So, what is the origin of the perceptual distortions gen-erated by the two different geometrical patterns? Overallthe experimental investigations on illusions and modellingapproaches provide evidence that understanding of percep-tual distortions may be helped by signal theory. The uncer-tainty principle, defined by terms of the Fourier transform,indicates a certain interrelation between the extent of a sig-nal and its spectral characteristics. Therefore spatial filteringprocesses unavoidably produce distortions in the size andshape relations of various parts of the image. The magni-tude of these distortions depends directly on the parametersof the filtering system. This leads to the idea that length il-lusions, such as the Muller-Lyer and Oppel-Kundt illusions,are determined by filtering processes in the neural networksof the visual system. This gives grounds for the filter theoryto regain its importance, and for filter models to be examinedagain.

Therefore we have constructed a filter model that em-ploys the neurophysiological data on the properties of retino-cortical pathways, the principles of spatial organization ofsimple and complex cortical receptive fields, and the schemeof their distribution in the retina – which makes our modelessentially different from that of Eijkman et al. (1981) orGinsburg (1975). The variability of the parameters of thefilters of our model depends on the eccentricity of the vi-sual field. To test our model, we performed a series of ex-periments with the two illusions under constant observationconditions with the same subjects. The results were similarto those in the literature, but also add quantitative detailsand reveal individual differences between the subjects. Ourmodel is able to predict the absolute values of the magnitudeof the illusions obtained in all our experiments, thereby pro-viding an estimate of the importance of the filter mechanismin the origin of the illusions.

2 Methods

The experiments were controlled by a computer program of our designthat arranged the order of the stimuli, implemented alterations accordingto the subject’s command, recorded the subject’s responses, and computedthe results. Solid, bright, sharp-edged Muller-Lyer and Oppel-Kundt figureswere generated on a computer monitor. The figures were oriented horizon-tally with the two halves lying side by side or one below the other. Forcomparison vertical figures were also tested. The display frame could notbe seen in the experiments. The width of the lines was about 0.8 min arc.

Fig. 2. Bisection as a function of the length of the reference half for twosubjects: ER (filled circles) and UL (open circles). The height of the verticallines is fixed at 28 min arc.Abscissa, reference half length, in min arc (theleft half of the figure);ordinate, absolute error, in min arc.Continuouscurves, predictions of the model with various gaze fixation points: on thecenter of the left half of the stimulus (upper curve), on the center of theright half (lower curve), on the center of the whole stimulus (middle curve)

Subjects viewed the stimulus monocularly. Viewing distance was 180 cm.An artificial pupil 2 mm in diameter used. We considered the absolute errorof a subject as the illusion strength value. We measured the Muller-Lyerillusion as a function of: (i) shaft length of the reference half of the figure,b, which varied from 0.1 to 1.5; (ii) the length of the wings of the arrows,w, (0.1 to 1.5); (iii) the tilt angle of the wings from the shaft,α, (10 to170). The Oppel-Kundt illusion was measured as a function of: (iv) lengthof the reference half of the figure,b, (0.1 to 1.5); (v) number of stripesin the shaded half,n (2 to 50).

2.1 Procedure

Subjects estimated the perceived length of the test half of the figure byadjusting it to be equal to the length of the reference half. No instructionsconcerning gaze fixation point were given. Subjects were provided withthree buttons, and instructed to press button 1 if they wanted to make thetest half of the figure longer, and button 2 if they wanted it shorter. Asingle press varied the size by one pixel, which corresponded to 0.8 min arcin our experiments. Auditory feedback was provided. Observation time wasunlimited and the subject manipulated the figure with buttons 1 and 2 untilthe desired length equality was achieved. Then the subject pressed button 3to transfer the response into the computer. In the next presentation a certainparameter of the reference half of the figure was changed and the procedurerepeated. In presentations the subjects were asked to disregard the randomchanges in the reference value. The length of the test half of the figurewas also randomized and subjects did not know in advance whether thecomputer would make it longer or shorter and how much different it mightbe from the length of the reference half. Two hundred and fifty presentationswere included in a single experiment, i.e. 50 values of each parameter wererepeated five times. The experiment was repeated two or three times duringa session. In various sessions different parameters of the figure were tested.

2.2 Subjects

Data were collected from 16 subjects (female and male). None of the sub-

jects had a history of visual disorder. Their visual acuity was 1.0. The

results obtained were qualitatively the same for all of them.

3 Results

In pilot experiments we introduced a bisection procedure totest the subject’s ability for spatial interval discrimination

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Fig. 3. The Muller-Lyer illusion (ordinate) as a function of the length ofthe reference half of the figure (abscissa). Three pairs of wings and theshaft line of the figure were present as in Fig. 1C;w is fixed at 28 min arc,andα at 45. One can notice that the experimental data for subject ER(filled circles) fit well the upper curve of the model’s predictions, and thatthose for subject UL (open circles) fit the lower curve. This leads to theassumption that subject ER keeps looking at the left half of the stimuluswhile estimating size equality, while subject UL concentrates on the righthalf

Fig. 4. The Muller-Lyer illusion as a function of the length of the referencehalf. The shaft line is absent, as in Fig. 1D. Further details are as in Fig. 3

and to estimate the left-right asymmetry value. Three ver-tical lines, each 0.8 min arc wide and 28 min arc high, weregenerated on the monitor. Subjects were asked to equalizethe intervals between the left-hand line and the center lineand the right-hand line and the center line by pushing oneof the flanking lines to the left or right. The difference be-tween the test and the reference intervals was plotted against

Fig. 5. The Muller-Lyer illusion as a function of the length of the referencehalf. Two pairs of wings are present as in Fig. 1B;a is fixed at 7 min arc.Further details are as in Fig. 3

Fig. 6. The Muller-Lyer illusion as a function of the length of the referencehalf. Only one pair of wings is present, as in Fig. 1A. Further details are asin Fig. 3 and 5

Fig. 7. The Muller-Lyer illusion as the function of wing tilt angle. The shaftline is present;b is fixed at 70 min arc,w at 28 min arc

Fig. 8. The Muller-Lyer illusion as a function of wing tilt angle. The shaftline is absent; Further details are as in Fig. 7

Fig. 9. The Muller-Lyer illusion as a function of wing length. The shaftline is present;b is fixed at 70 min arc,α at 45

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the reference interval length (Fig. 2). Absolute error value ofperceived equality increased with the reference interval andcould reach 6–8 min arc at a size of 100 min arc. Some ofthe subjects showed a preference for one side, while othersdid not.

The data collected on Muller-Lyer figures showed errorsgreatly exceeding those of bisection. Illusion strength valuereached 16–20 or even 35 min arc at a size of 100 min arc(Fig. 3). Individual differences were observed, illusionstrength value differing by a factor of 2 or even 3 betweensubjects. Removal of the shaft line showed some alterationsin the b function (Fig. 4): the difference in illusion strengthvalue between observers decreased (compare Figs. 3 and 4).Removal of the wings weakened the illusion, e.g. figureswith two pairs of wings showed smaller errors (Fig. 5), andthose with one pair still smaller errors (Fig. 6). On the whole,each pair of wings had its own influence on the strength ofillusion, and some additivity was found.

Illusion as a function of the tilt angleα showed curveswith positive and negative values (Fig. 7). The curve was notquite symmetrical around the zero point. In the first place, thezero point for some subjects was shifted from 90. Secondly,the absolute values of the negative and positive data pointswere slightly different. Removal of the shaft line changedthe character of theα function (Fig. 8). The difference in thedata between subjects decreased and some extreme valuesappeared on the curves.

Variations in wing length,w, demonstrated relatively flatcurves while the shaft line was present (Fig. 9). Some suspi-cion of a maximum point at smallerw values (15–30 min arc)arises. The maxima became clearly visible on thew curvesif the shaft line was absent (Fig. 10).

Vertical orientation of the Muller-Lyer figure producedan illusion of almost the same strength as horizontal orien-tation (Fig. 11).

In summary, the Muller-Lyer illusion extreme pointschanged their position accordingb, w, andα.

The Oppel-Kundt illusion also increases gradually withthe length of the figure’s reference half,b (Fig. 12). TheOppel-Kundt illusion as a function of the number of stripes,n, had local maximum, somewhere between 6 and 15 stripes,(Fig. 13).

4 Discussion

The data reported above suggest that the perceived size ofan object is a result of compound interaction of spatial pa-rameters of the object. Each part of the stimulus plays itsrole in creating the so-called psychophysical phenomenonof an illusion. Consequently, any shape is enriched by moreor less noticeable misrepresentations. Recent data on shapeperception thresholds have indicated a dominant role of fil-tering processes of spatial frequencies in perceptual perfor-mance (Ginsburg 1984, 1986; Fogel and Sagi 1989; Sagi1991; Chen et al. 1993). Considerable psychophysical andneurophysiological evidence has been accumulated that thevisual system up through the cortex may be carrying outspatial frequency filtering of visual information (Kulikowskiand Bishop 1981; De Valois et al. 1982; Glezer et al. 1990).Neurophysiological data suggest that simple cortical cells

Fig. 10. The Muller-Lyer illusion as a function of wing length. The shaftline is absent;b is fixed at 70 min arc,α at 45

Fig. 11. The Muller-Lyer illusion as a function of the length of the refer-ence half (A) and wing tilt angle (B), with the Muller-Lyer figure orientedvertically. A w fixed at 28 min arc,α at 45. B b fixed at 70 min arc,w at28 min arc

may be described by the product of Gaussian envelope andperiodic functions (Marcelja 1980; Kulikowski and Bishop1981; Kulikowski et al. 1982). Therefore they might be con-sidered as two-dimensional spatial frequency filters (Gaborelements). The weighting function of the Gabor element iscalculated by the equation

w(x, y) = A · cos(ω0x + θ) · exp

(− x2

2σ2x

− y2

2σ2y

)(1)

whereω0 is the optimal frequency,θ is the phase shift, andσx, σy are parameters that determine the characteristics ofthe Gaussian envelope in the preferred orientation and or-thogonal to it. In addition,σy = 2σx. A is the coefficientof contrast sensitivity, which does not depend on peak fre-quency (De Valois et al. 1982).

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Fig. 12. The Oppel-Kundt illusion as a function of the length of the refer-ence half. The length of the vertical stripes forming the stimulus was fixedat 28 min arc. The number of the stripes in the shaded half of the figureequals 10

Fig. 13. The Oppel-Kundt illusion as the function of stripe number in theshaded half of the figure.b is fixed at 70 min arc,a at 28 min arc

Fig. 14. Relative modulation (f1/f0) in responses of two simple cells to adrifting sinusoidal grating plotted against the spatial frequency of the grating(c/deg). Squares, responses from a close-to-linear cell; circles, responsesfrom a nonlinear cell. (From Movshon et al. 1978)

Fig. 15. Distribution of simple cells in the foveal projection as a functionof peak frequency (c/deg). Squares, from De Valois et al. (1982); circles,prediction of our model, calculated by (9)

According to the literature data (Pollen and Feldon 1979;Glezer et al. 1990), the peak frequencies of cortical simpleunits are rather different and produce geometrical progres-sion with a step of 0.5 of an octave. The highest frequencyis found to be about 20 cycles per degree (c/deg). The band-width of the Gabor element equals 1.2–1.4 octaves (De Val-ois et al. 1982). The bandwidth (in octaves) does not dependon peak frequency. Therefore the weighting functions of var-ious cortical neurones have about the same shape. A singleGabor element is tuned to both frequency and orientation.The complete set of them shows a continuum of spatial fre-quency peaks and orientations over a relatively wide range.

The nonlinearity of neuron output characteristics mustalso not be overlooked in the modelling. In the studies(Movshon et al. 1978; De Valois et al. 1982; Henry 1985)of simple and complex cortical units, the specific estimationmethod of neuron response to drifting sine gratings was ap-plied. Authors measured amplitude and mean value ratios.If simple cell output was considered as a half-wave rectifierwith linear characteristics, the ratio should be

f1/f0 = π/

∫ π

0sin(x)dx = π/2≈ 1.57 (2)

We assume that simple neurons have quadratic outputcharacteristics rather than the linear ones:

f1/f0 = π/

∫ π

0sin2(x)dx = 2 (3)

This assumption provides a proper description of neuronoutput nonlinearity and gives good correspondence betweenthe calculated values and the experimental data (Movshonet al. 1978) (Fig. 14).

The next assumption of our modelling is concerned withsignal phase. The phase shift of the periodic function of aGabor element may produce significant phase alterations inthe output signal. To avoid this, the nervous system possessescertain mechanisms of conjugated units. The simple cells,which are located close together within the same corticallayer and have overlapping receptive fields of the same size,orientation, and peak frequency, are joined in pairs. Theweighting functionsw1(x) andw2(x) of the two members ina couple are conjugated by the Hilbert transform:

w1(x) = A(x) · cos(ω0x + θ)

w2(x) = A(x) · sin(ω0x + θ) (4)

whereA(x) describes the Gaussian envelope,ω0 is optimalfrequency, andθ is phase shift.

The outputs of the two members converge on the nextunit. Characteristics of the output unit resemble those ofthe complex cells: it is tuned to spatial frequency and ori-entation, and it is not selective to stimulus position in thereceptive field (Glezer et al. 1973, 1980). Therefore the re-sponse of the complex unit does not depend on the phaseshift of the periodic component of Gabor element:

r21(x) =

[∫ ∞

−∞f (x− ξ) ·A(ξ) · cos(ω0ξ + θ)dξ

]2

= [cos(θ) · hc(x)− sin(θ) · hs(x)]2

r22(x) = [sin(θ) · hc(x)− cos(θ) · hs(x)]2

r(x) =√r2

1(x) + r22(x) =

√h2c(x) + h2

s(x) (5)

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Fig. 16.The main scheme of the model.F (ωx, ωy)is the input signal spectrum;Hi(ωx, ωy) are thespectral characteristics of simple receptive fieldsof i-th size;Φ−1: complex inverse Fourier trans-form;Pi(x, y) are the fields ofi-th size distributionfunction; and∇2 is the Laplacian operator

where

hc(x) =∫ ∞

−∞f (x− ξ) ·A(ξ) · cos(ω0ξ)dξ

hs(x) =∫ ∞

−∞f (x− ξ) ·A(ξ) · sin(ω0ξ)dξ (6)

The complex units are optimized for extraction of theenvelope of signal and weighting function convolution. In-direct evidence of this might be found in the literature (Ku-likowski and Bishop 1981). Filters conjugated by the Hilberttransform might be organized by excitatory and inhibitorypathways of a neural network.

An alternative mechanism of elimination of phase alter-ations might be also suggested. A large number of simplecortical units with overlapping receptive fields of the samesize and orientation may converge on the output unit. Eachof them is considered to have an individual phase shift ofthe periodic component. If one suggested an even phase dis-tribution within the range 0 to 2π and quadratic output char-acteristic, then the output of the assembly would not dependon the phase shift of the periodic component of Gabor ele-ments:

r2(x) =∫ 2π

0dθ

[∫ ∞

−∞f (x− ξ)

·A(ξ) · cos(ω0ξ + θ)dξ

]2

= π · [h2c(x) + h2

s(x)]

(7)

The next assumption of our model deals with contrastelevation in the cortical excitation patterns. To describe theseprocesses in the excitation pattern of complex units we usethe two-dimensional Laplacian operator:

∇2f (x, y) =∂2f (x, y)∂x2

+∂2f (x, y)∂y2

(8)

The Laplacian operator has been applied to the mod-elling of retinal excitation patterns by Kovasznay and Joseph(1953), and Kelly (1974).

The last assumption is concerned with receptive fielddistribution in the retina. Without taking into account the re-spect of magnification factor (Daniel and Witteridge 1961;Guld and Bertulis 1976; Schwartz 1980; Dow et al. 1981),it is well known that the number of cells per square unitis constant over all the striate cortex (Hubel and Wiesel1974). The foveal projection contains the maximum num-ber of units, tuned to higher frequencies. With eccentricitythe high-frequency representation diminishes, but lower fre-quencies increase, as does receptive field size. We make anassumption that the distribution of fields of the same sizewithin the retina may be determined by a law ofχ2 type:

pi(ξ) = ki · (ξ + a)n · exp

[− (ξ + a)2

2σ2i

](9)

402

A B

Fig. 17a–f. The output patterns of the model at stages (iv) (top) and (v) (middle), and the normalized transverse sections of the patterns on axes X-X′(bottom) of the Muller-Lyer figures (A, B, C) and the Oppel-Kundt figures (D, E, F) with various gaze fixation points: in the center of the left half of thefigure (A, D), in the center of the figure (B, E), and in the center of the right half of the figure (C, F). The tilt angle of the wings of the Muller-Lyer figureis 60, and the length of the wings is 24 min arc. The number of the stripes in the filled half of the Oppel-Kundt figure is 8

whereξ is eccentricity,pi(ξ) is the distribution density ofthe fields with given size,ki is the normalization coefficient,σi = a ·20.5i, a, the parameter, is determined by the smallestfield size, andn is the power exponent. We assume thatnequals 2 because it gives the best correlation of calculationswith experimental findings (Dow et al. 1981; De Valois etal. 1982) (Fig. 15). The parameters of the distribution lawvary with the size of the receptive field. Consequently, theseries of receptive fields different in size produce the systemof overlapping concentric rings organized around the fovea.We come to the conclusion that in spatial frequency termsthe visual field is not a homogeneous system, and the param-eters of spatial filtering both in the visual system and in ourmodel depend on: (a) spatial properties of the image, and(b) its position in the visual field. Therefore various partsof the retinal image are perceived with different accuracy.Due to attention and eye movements, the gaze fixation pointjumps and the spatial filtering parameters in various parts ofthe visual scene change, which in turn determines the indi-vidual deviations of illusion strength value, especially whenlarge objects are observed. One can imagine two alternativesituations in the experiments: (i) the left half of the figure

is localized in the fovea and the right half in the parafovealarea, and, (ii), on the contrary, the left half of the figureis situated in the parafovea and the right half in the fovea.Indeed the experiments with geometrical illusions providesome indications that the observer usually keeps looking in-voluntarily either at the left or at the right half of the figure(e.g., Figs. 3, 4). Therefore nonhomogeneity of spatial filter-ing parameters in the visual field may influence both illusionstrength and its individual variations, as well as giving rise toleft-right asymmetry. Errors of asymmetry are much smallerthan those of illusion. The firsts are caused by visual fieldproperties only and the second also by differences in thespatial organization of the objects.

To test the idea that spatial filtering causes geometricalillusions we passed the patterns of Muller-Lyer and Oppel-Kundt figures through the model (Fig. 16). There are fivemain stages of image processing in the model:(i) calculation of the two-dimensional spectrum of an image,F (ωx, ωy);(ii) production of the two-dimensional spectrumF (ωx, ωy)and the spectral characteristicsHi(ωx, ωy) of the filters ofvarious size conjugated by Hilbert transform;

403

C D

Fig. 17. (continued)

(iii) inverse Fourier transformΦ−1 of the productsF (ωx, ωy)·Hi(ωx, ωy);(iv) summation of the modules of the results ofΦ−1, usingweighting functionsPi(x, y);v) application of the two-dimensional Laplace operator∇2.

In stage (i) the two-dimensional spectrum of the Muller-Lyer figure was calculated by the following equation:

ML(Ω,Ψ, α,w) = K(Ω,Ψ, α,w) · exp[iΩ · cos(Ψ ) · b]+K(Ω,Ψ, 180− α,w) (10)

+K(Ω,Ψ, α,w) exp[−iΩ · cos(Ψ ) · d]

whereK(Ω,Ψ, α,w) is the spectrum of a single arrow of thefigure with wing tilt angleα, and wing lengthw; Ω, Ψ arepolar co-ordinates; andb, d are the lengths of the referenceand the test halves of the figure respectively.

K(Ω,Ψ, α,w) =2 sin

[Ω cos(Ψ − α)w/2

]Ω cos(Ψ − α)

× exp[−iΩ cos(Ψ − α)w/2

]+

2 sin[Ω cos(Ψ + α)w/2

]Ω · cos(Ψ + α)

× exp[−iΩ · cos(+α) · w/2

](11)

The two-dimensional spectrum of the Oppel-Kundt fig-ure is

OK(Ω,Ψ, n) = R(Ω,Ψ, n) +2 · sin

[Ω · sin(Ψ ) · a/2

]Ω · sin(Ψ )

· exp[−iΩ · cos(Ψ ) · (d + b/2)

](12)

whereR(Ω,Ψ, n) is the spectrum of the grating located atthe coordinate center, and consisting ofn stripes;a is thestripe’s height;b is the length of the shaded half, andd isthe length of the test half.

R(Ω,Ψ, n) = (13)

2 · sin[Ω · sin(Ψ ) · a/2

] · sin[Ω · cos(Ψ ) · b · n · (n− 1)−1

]Ω · sin(Ψ ) · sin

[Ω · cos(Ψ ) · b · (n− 1)−1

]At stage (ii), the spectral characteristics of the conjugated

filters are the two-dimensional Gaussians of certain orienta-tions, and shifted according to the optimal frequencies. Theoptimal frequency and the parameters any of the Gaussiansare determined by the size of the simple cortical cells.

The output patterns of stages (iv) and (v) of the modelwere shown on the monitor. Some of the patterns are shownin Fig. 17. In all cases, the output patterns contained distor-tions equivalent to the illusions observed in the experiments

404

EF

Fig. 17. (continued)

(Fig. 2 to Figs. 10, 12, 13). In other words, calculated sizedistortions were found to be in a good agreement with all ex-perimental data. Individual variations between subjects werefollowed by the model as well.

Our experimental data and modelling furnish groundsto consider the physiological processes of two-dimensionalspatial filtering as the primary determining factor of the phe-nomenon of so-called visual illusions of extent. In the taskof judgment on length, the retinal pattern of an object isprocessed as a single unit, and the perceived size is a resultof the interaction of the spatial parameters of all parts of theobject. From this point of view, no conflict occurs betweenthe filtering hypothesis and the other theories: confluxionand contrast, assimilation, adaptational level, framing ratio,center of gravity or averaging of weights, size constancy,and orientation line detectors. The theories do not fail; theirstatements cope with and flow together in the filtering model,which yields an explanation of the illusions in neurophysio-logical and mathematical categories.

The present findings do not suggest that it might be un-necessary to invoke some other causal mechanisms of theillusions. On the contrary, our experiments and the mod-elling show that eye movements determine the individualdeviations of illusion strength value, especially when large

objects are observed. Apparently adaptational properties ofthe spatial filters, visual field anisotropy, attention, or mem-ory might act as the additional factors or corrections of per-ceived distortions. Even blurring of the retinal image dueto optical aberrations of the eye shows a quantitative influ-ence on certain visual illusions (Chiang 1968; Coren 1969).But reduction of the aberrations in the experiments does notdecrease the Poggendorff illusion by the anticipated propor-tion.

In general, two types of factors may be having an effect:physiological properties of low-level visual pathways andpsychological events at higher centers of the nervous sys-tem. The number and weight of the factors may be variablefrom illusion to illusion. The existence and analysis of somany types of illusions (Gregory 1970, 1990) indicates thatany level of the visual system may contribute to perceptualdistortions.

We have applied the filtering model only to the two illu-sory figures, and discussion of misperceptions of other types,such as the illusion of orientation, curvature or area, is be-yond the scope of the present communication. On the otherhand there is evidence that filtering processes are involvedin these cases as well. For instance, the magnitude of variousillusions of size and area, such as the Delboeuf (Ikeda and

405

Obonai 1955) and Titchner (Zigler 1960) illusions, dependson the ratio of contextual size to focal stimulus size. InvertedU-shaped functions have been found when contextual sizewas varied for the Ebbinghaus (Zigler 1960) and Delboeuf(Ikeda and Obonai 1955) figures as in the Muller-Lyer (e.g.Lewis 1908), Ponzo (Fisher 1969; Pressey et al. 1971) anddivided line (Obonai 1954) illusions of extent.

In addition, the Kanizsa triangle was tested by two-dimensional spatial filtering (Ginsburg 1975). The resultssuggest that the low spatial frequency attenuation charac-teristics of the visual system contribute to the formation ofthe illusory triangle. However, Ginsburg applied an ideallowpass filter using 16 low spatial frequencies. The vari-ability of the filtering characteristics of the visual pathwayswas not considered, which might have caused the failure toexplain other subjective contour patterns by filtering. A pos-sible neural mechanism for illusory contour perception hasbeen suggested (Peterhans and von der Heydt 1991) em-ploying end-stopped cells. It seems to be reasonable then,to organize a model of spatial filtering which possesses theproperties of end-stopped cells. Furthermore, the short la-tency of the illusory contour responses (von der Heydt andPeterhans 1989), orientation-specific adaptation and the dis-appearance of anomalous contours at equiluminance (Gre-gory 1987; Brigner and Gallagher 1974) seem to confirmthe low-level mechanism.

In conclusion:

1. The Muller-Lyer illusion strength is measured as a func-tion of: (i) the length of the figure, (ii) the length of thewings, (iii) the tilt angle of the wings. The Oppel-Kundtillusion strength is measured as a function of: (iv) lengthof the reference half, (v) number of stripes in the shadedhalf of the figure.

2. The neurophysiological model of spatial information fil-tering is applied to the Muller-Lyer and Oppel-Kundtfigures. The spatial properties of simple and complexunits of striate cortex are used in modelling.

3. The output patterns obtained by the model contain distor-tions equivalent to the illusions, observed in psychophys-ical experiments.

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