Ingoing versus outgoing wings. The Müller-Lyer and the mirrored triangle illusion

Transcription

1 Ingoing versus outgoing wings. The Müller-Lyer and the mirrored triangle illusion W.A. Kreiner Faculty of Natural Sciences University of Ulm

2 . The Müller-Lyer illusion Context elements, their shape, their size and their position, exert an influence on the perceived length of a line (Brigell, Uhlarik, and Goldhorn, 9). Since the Müller-Lyer illusion (Fig. ) was first published in 889 it has been studied by several authors (eg, Yanagisawa, 99; Fellows, 9; Pressey, Di Lollo, and Tait, 9; Predebon, 99). Weidner and Fink (00) investigated the neural mechanisms underlying the illusion. Weidner, Boers, Mathiak, Dammers, and Fink (00) studied the dynamics of the brain processes using magnetoencephalography. Using fmri, Murray, Boyaci, and Kersten (00) showed that the retinotopic representation in V of an object changes in accordance with its perceived angular size. In the Müller-Lyer illusion, the apparent length of the shaft depends primarily on the geometric properties of the wings (their length and their angle) and on the framing ratio FR (= overall length of the stimulus divided by the length of the shaft; Fig. a, b). The effect that the perceived length of the shaft is modified depending on the range spanned by the context elements is called assimilation. It is explained by a theory of Pressey (9) which states that the line is averaged with the contextual magnitudes. Carlson, Moeller, and Anderson (98) discuss the possibility whether assimilation can be explained from low pass filtering. Fig.. Drawing by Müller-Lyer, published in 889 in the Archiv für Anatomie und Physiologie.

3 The perceived length of the shaft strongly depends on the gap size and the orientation of the wings (ingoing or outgoing). On the version with ingoing wings (tips pointing outwards), Pressey et al. (9) found that the perceived length of the shaft as a function of the framing ratio FR first increases (reversed Müller-Lyer effect), but then decreases again. A smooth maximum was observed when the wings were positioned near the ends of the shaft (Fig. a, top). However, for outgoing wings the apparent length of the shaft was found to decrease continuously when the gap was increased, starting from zero on (Predebon, 99; Fig. b, top). The question arises what the apparent length would be when the wings were positioned at even shorter distance, i.e., within the length of the shaft (Fig. a, b, bottom) and whether the shapes of the curves for ingoing and outgoing wings would be similar or significantly different. Further, it was the goal of this work to derive algebraic expressions suitable to describe both versions of the illusion as a function of the framing ratio. Fig. a Ingoing wings Fig. b Outgoing wings. Mirrored triangles Pairs of triangles were arranged as shown in Fig.. Although in both examples the objective separation of the apices is the same, in case of the long triangles (bottom) the separation of the tips appears to be shorter than for the tall ones. Fig.. The separation of the tips appears to be different although their distance is objectively the same.

4 . Experiments. Experiment.. Subjects. healthy volunteers, age above 0, took part, among them the author. Vision was corrected to normal. 0 participants were naive... Stimuli. transparencies showed the variant of the illusion with outgoing wings. On a DIN A print, the red shaft was mm long and.mm wide. The inside angle of the blue wings was 0 0 and the length of their fins mm. transparencies gave the version with ingoing wings. On one of them the line measured mm instead of mm. Two more of the transparencies showed a red line without wings, and mm long, respectively. In the lower half of the transparencies, to the right, seven blue lines were plotted for comparison. Fig.. Examples of transparencies with outgoing (left) and ingoing wings. L0 G G L0 W Fig.. Examples of stimuli with different framing ratios. With outgoing wings the illusion peaks at zero gap (middle row, right). In case of ingoing wings, the framing ratio is FR= (L0 + G)/L0 (G=gap), for outgoing wings FR= (L0 + G + W)/L0.

5 L(perceived)/L0 L(perceived) / L0 Their length decreased from top to bottom. On different transparencies, different sets of lines were presented. Examples are given in Fig.... Procedure. The transparencies were projected with a beamer and magnified linearly by a factor of six. Distance of observation was m ( subjects) and.m ( subjects). The length of the shaft (mm on a DIN A plot) subtended an angle of 0 - rad and rad, respectively. The stimuli were presented for seconds, then, for another seconds, the standard lines were added, followed by a blank of three seconds. Fig.. Perceived length of the shaft as a function of the framing ratio / ingoing wings Function: y=d+a (x-) exp(-b abs(x-)) Parameters fitted: D= 0.9() A= 0.00() B=.8(),0,,0,,0,0,00 0,9 0,90 0,,0,,0,,0 Framing ratio x Fig.. Perceived length / outgoing wings. Function: y=d+a(x-f) + C exp(-b abs(x-f)) Parameters fitted: D=.0(8) A= -0.0(8) C= 0.90() B=.() F=. (fixed),0,,0,,0,0,00 0,9 0,90 0,,0,,0,,0, Framing ratio x.. Evaluation. All data pertaining to the same transparency were averaged and divided by the average perceived length of the shaft without wings [.() mm]. The framing ratio is

6 defined as indicated in Fig.. For ingoing wings, FR= (L0 + G)/L0 (G=gap). FR- equals G/L0. For outgoing wings FR= (L0 + G + W)/L0. F means the framing ratio when the tips of the outgoing wings (tips pointing inwards) are attached to the shaft without leaving a gap. FR-F is equal to G/L0 in this case... Results. Figs. and give the experimental values plotted as a function of the framing ratio. There is an obvious difference between the two variants of the illusion: With ingoing wings there is a flat maximum while for outgoing wings a sharp peak is observed, leading to a pronounced peak intensity. In addition, there is a stronger average illusion, too. Both curves peak at a framing ratio around.. This is the maximum FR value where overlap still occurs. The Figs. 8 a, b show the stimuli where the illusion is close to its maximum. Fig. 8a. Curve exhibits a sharp peak. Fig. 8b. Curve exhibits a flat peak. Outgoing wings overlap with the shaft -> elongation. Ingoing wings overlap with the shaft -> shrinkage. No overlap -> illusion decreases rapidly with increasing framing ratio Partial overlap -> illusion decreases gradually with increasing framing ratio In the function fitted to the values obtained for outgoing wings, the parameter F was set to. which is the value of the framing ratio when the wings are attached to the ends of the shaft without leaving a gap. A fit of this parameter produced a slightly higher value. However, the difference is not significant.. Experiment Fig. 9 gives examples of mirror like triangles. Irrespective of their shape, their area is kept constant. Although the separation of the tips is always the same, their apparent distance first increases with decreasing height, but then decreases again when the triangles expand more and more in horizontal direction at the expense of their height. Other than in the first experiment, where the perceived length of a physically present line was to be estimated, the target is just a gap in this case.

7 Illusions of apparent size are attributed to a change in the size of the conspicuity range which, in turn, is determined by the dimensions of the object. A small object will trigger a reduction of the conspicuity range. As a consequence, small objects will appear to be enlarged and vice versa. The question is whether the vertical extension of the object would have the same influence on the size of the conspicuity range as the horizontal dimension. As already Cornish (9) pointed out, a large vertical extension of the range taken in by the eye will cause an object to appear small. So, not only for a horizontally, but also for a vertically long shaped triangle the conspicuity range will expand, causing the target gap to appear short. Reducing the height, the perceived size of the gap will first increase, but then decrease again as soon as the triangles become long shaped horizontally. In between a maximum is expected. It was the purpose of this experiment to investigate the relative influence of the vertical and the horizontal extension of the stimulus, respectively, on the apparent separation of the tips... Subjects most elderly people took part, among them the author. Sight was normal or corrected to normal. 0 of them were naive. x00/.= x8/.= 0 x/.= 80x9/.= Fig. 9. Examples of transparencies showing mirrored triangles.

8 Y Perceived separation of the dips/ mm.. Stimuli transparencies were presented. On a DIN A print the separation of the tips was 0mm. The horizontal dimension of the triangles varied from to 9mm, their height from 00 to mm. The area of the triangles was kept constant. Length times height was 00 millimeter squared. Seven red standard lines were shown in the lower half to the right, there length varying between and 8mm. On different transparencies different sets of standards were employed... Procedure. Linear magnification and distance of observation was the same as in... First, the stimulus was presented for four seconds, then, for another six seconds, the lines of comparison were added, followed by a blank of two seconds. The participants indicated the perceived separation of the tips by selecting one of the standard lines... Evaluation. All results were averaged and plotted on a diagram, together with the standard errors (Fig. 0). The function derived from a conceptual model (chapter.) was fitted to the experimental values. 80 Fig. 0 Mirrored triangles X Horizontal dimension of the triangle/ mm Function: y= D+A/(*x+0 +B*00/x) D=.9() A= 89(9) B= 0.8(8) Chi^/DoF=.0 R^ = Results. The function could be fitted within the standard errors. The value of the constant B=0.8 shows that, at least due to this conceptual model, the vertical extension of the stimulus has less influence on the illusion than its horizontal dimension. Its only 8% of the influence a horizontal expansion or contraction of the stimulus exerts.. Algebraic functions.. Müller-Lyer with gap. The functions employed for fitting are: y(x)=d + A (x-) exp[-b abs(x-)] (ingoing wings) and (Eq. ) 8

9 y(x)=d + A (x-f) + C exp[-b abs(x-f)] (outgoing wings) (Eq. ).. Constant term D. This term gives a basic value of the perceived length and takes also into account influences on the apparent length of the shaft independent of the framing ratio. For example, in order to measure the apparent length of the shaft within its specific context, a set of lines is presented for comparison. The way how these lines are arranged and grouped, provides another context, which may change the perceived length of the standards. This effect is included in the constant term D... Term in proportion to the gap size. It is assumed that the apparent size of the shaft increases or decreases in proportion to the size or the distance of the contextual magnitudes (averaging effect). In the Müller-Lyer illusion averaging is observed in the sense that the apparent length of the shaft varies with the (positive or negative) gap size. In case of ingoing wings (dips pointing outwards), the expression (x-), with x being the framing ratio, is equivalent to twice the length of the gap in relation to the length of the shaft, L0. For x <, this expression becomes negative, which means shrinking of the apparent length. To first order of approximation, the averaging effect can be described by the term A (x-). In case of outgoing wings, (x-f) is equivalent to twice the gap size, divided by L0, too (with F being the framing ratio, where the tips of the wings coincide with the ends of the shaft). A negative sign of the term A (x-f) indicates shrinking as well.... Nonlinear decay terms. Depending on the position of the context elements with respect to the target, averaging may be more or less effective. With ingoing wings, the overlap reduces gradually as soon as the tips of the wings move away from the ends of the shaft. As a result, averaging becomes less effective. This is described by an exponential decay term, exp[-b abs(x-)]. A different situation arises with outgoing wings. Starting from a small framing ratio, the shaft appears to be elongated up to the point where the tips of the wings touch the ends of the shaft (FR=F). However, with increasing gap, the context breaks down suddenly. The illusion shows a sharp maximum (discontinuity) at the framing ratio FR=F and reduces rapidly when the elements are detached from this position. This can be described by a term of the form C exp[-b abs(x-f)]. A sudden decrease of the context occurs with two variants of the Baldwin (89) illusion, too, each of them exhibiting a similar kind of discontinuity. In one version, squares serve as context elements which are moved apart from each other gradually, starting from positions within the length of the shaft. In the other version the squares are replaced by full circles (dumbbell variant; Delboeuf, 89; 89). Both illusions show a sharp maximum (Kreiner, 009; 0). Expressions similar to equation can be applied to both of these variants.. Mirrored triangles The function fitted to the results of experiment is y(x)= D + A/(*x+0 +B*00/x). ( Eq. ) 9

10 y(x) means the perceived distance of the tips of the triangles, x is the length of one triangle (horizontal expansion), the expression 00/x means its height (on a DIN A sheet the number 00 means length times height in square millimeters). It is assumed that the width of the conspicuity range is a) proportional to the horizontal extension (x +0) of the object plus b) some unknown fraction B of the object s vertical extent (B*00/x). This fraction B can be determined from fitting. The perceived separation then is given by the value of D plus an expression inversely proportional to the width of the conspicuity range, which, in turn, is assumed to be proportional to the term in brackets in Eq... Summary The Müller-Lyer illusion with ingoing and outgoing wings has been investigated. The framing ratio x was varied from values below x= up to about x=. Plotting the results as a function of the framing ratio shows that, for ingoing wings, the intensity of the illusion first increases and then decreases gradually, while, for outgoing wings, a sharp maximum is observed. In both cases the illusion peaks near the framing ratio where overlap still occurs. There is a significant difference in the average as well as in the peak intensities of the illusions. From the mirrored triangles experiment it is concluded that a change in the vertical dimension of the stimulus contributes less to the size of the conspicuity range than a horizontal expansion or contraction. Citations Baldwin, J.M. (89). The effect of size-contrast upon judgments of position in the retinal field. Psychol. Rev.,, -9. Brigell, M., Uhlarik, J., and Goldhorn, P. (9). Contextual influence on judgements of linear extent. Journal of Experimental Psychology: Human Perception and Performance,, 9, 0-8. Carlson, C.R., Moeller, J.R., and Anderson, C.H. (98). Visual illusions without low spatial frequencies. Vision Research,, 0-. 0

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