The Mirrored Triangles Illusion. On the perceived distance between triangles in mirror image arrangement. W.A. Kreiner

Transcription

1 The Mirrored Triangles Illusion On the perceived distance between triangles in mirror image arrangement W.A. Kreiner Faculty of Natural Sciences University of Ulm

2 1. Illusions on perceived length There are several illusions where the apparent length of a line depends on the size, the shape or the type of context elements. Examples are the Oppel-Kundt illusion (Oppel, 1854; Kundt, 1863; Spiegel, 1937; Surkys, 2007; Surkys, Bertulis, Bulatov, and Mickiene, 2008), the Müller-Lyer (1889) or the Baldwin (1895) illusion (Brigell, Uhlarik, and Goldhorn, 1977; Wilson and Pressey, 1988; Kreiner, 2011). In addition, there are variants of these illusions, where the context elements are not attached to the target line, but leaving a gap in between (Pressey, Di Lollo, and Tate, 1977; Kreiner, 2012). In the illusion discussed here, not the length of a line serves as a target, but an empty space, ie, the distance between the tips of two isosceles triangles. In case of the Baldwin illusion, the apparent length of the target line decreases with increasing size of the adjacent boxes. In case of the mirrored triangles, one finds that the apparent distance of the dips is a function of the radius of the outer circle. It decreases with increasing radius, and vice versa. This can be interpreted as due to a size constancy effect. Originally, the term size constancy referred to the observation that the perceived size of an object as a function of its distance does not follow the laws of geometrical optics (Schur, 1925). With increasing distance, it appears rather larger than one would expect it from the size of the retinal image. While the retinal image decreases with distance d according to a power function d -1, the apparent size does not decrease as rapidly. It rather follows a function d n-1. Schur s observation has been confirmed by Gilinsky (1955). In addition, she has found a corresponding effect on objects of different size presented at the same distance of observation. In comparison with larger objects, smaller targets appear rather larger than one would expect it from their true dimension. This can be described by a similar function as in the case of observation at varying distances. Both variants of size constancy are explained such that in case of a smaller retinal image the visual angle is reduced, which, in turn, causes subjective magnification (Kreiner, 2004). It should be mentioned that, in case the field of sight is somewhat restricted by artificial frames, a significant influence of the restriction on the apparent length of vertical or horizontal lines has been observed (Gavilán et al, 2017). 2. The experiment 2.1 Stimuli: 14 pairs of isosceles black triangles (Fig 1) above bright background were presented in the top half of transparencies. The triangles were arranged such that their dips faced each other. The distance of the dips was held constant. It served as the target. In the original drawing on a DIN A4 sheet of paper the distance was 60 mm. The area of the 2

3 triangles was kept nearly constant to 756(10) units square (mm 2 ), while the ratio between the trangles height (horizontal dimension) and their base was varied between 0.15 and A comparison scale of seven horizontal red lines was presented in the lower right half of the transparency. The lengths of the comparison lines decreased from top to bottom by a factor of 1.3. On different transparencies, different response scales were presented, where the absolute length of the standards varied up 29% (Fig 1) x100/1.=3 7 22x68/1.= x27/1.=5 7 80x19/1.=7 7 Fig 1 Examples of pairs of triangles, where the separation of the dips is kept constant. Their apparent distance was determined from comparison with the reference lines (Kreiner, 2012). 2.2 Subjects: 11 healthy volunteers took part (among them the author), all of them age above 54. Vision was corrected to normal. 2.3 Experimental procedure: The transparencies were projected with a beamer. First, the triangles were shown for 4 seconds, then, for another 6 seconds, the standard lines were added. An empty transparency followed for 2 seconds. 6 participants were seated at a distance of 3 meters. The target subtended an angle of 0.081rad. 5 participants observed from a distance of 4.5 meters, the target subtending 0.054rad. 3

4 R(outer circle = red dots) [mm] y Apparent distance / mm Fig. 2a Apparent distance of the equilateral triangles apices as a function of their height (=horizontal extension). The horizontal line at 60 mm marks the true separation of the tips x Horizontal dimension of a triangle [mm] [1/R]* Horizontal extension [mm] R Fig 2b The red curve gives the radius R of the outer circle, the blue one its inverse, multiplied by some arbitrary number. The intensity of the illusion seems to be correlated with the inverse of the radius. The vertical line indicates the minimum radius which coincides with the maximum of the illusion, as obtained from the fit (Fig 6). 3. Results Fig 2a gives the result of the experiment. Plotting the apparent separation of the triangles as a function of their horizontal extension (their height), one finds that it first increases and then, after a maximum, steeply decreases towards the triangles becoming more and more elongated horizontally. Depending on the triangles shape, the apparent distance was found to be larger or smaller than their true distance. The perceived length oft he target appears to be negatively correlated with the radius R of the outer circle (Fig 2a,b). R is determined after the theorem of Pythagoras, as shown in Fig 3. The solid line at x= 22 indicates the maximum of the function given by Eq (1). See Fig 6. 4

5 Fig 3 The radius R of the outer circle is found after the theorem of Pythagoras. T means the target length. x The smallest apparent distance corresponds to the largest circle, and vice versa. In the following, from a conceptual model, an algebraic function is derived. The mathematical function The conceptual model is based on the idea of size constancy. Originally, size constancy meant that, with increasing distance, the decrease in apparent size is less pronounced than one would expect it from the size of the retinal image [Schur (1925)]. While the retinal image follows the function 1/d (d= distance) which can be written as d -1, the apparent size follows a function d n-1, where n is a so called size constancy parameter. A similar law applies to the case where not the distance, but the target s size r (ie, any linear dimension of the target) is varied. Small objects appear enlarged, and vice versa. This has been shown by Gilinsky (1955) on triangles. With respect to the retinal image, the d and the r are inversely proportional to each other: Both, a large distance d as well as a small dimension r of an object result in a small retinal image. Therefore, at constant distance, the apparent size is proportional to (1/r) k-1. k stands for the size constancy parameter in case the object s linear dimension is varied instead of its distance. This is shown in Fig 4. There, R means the radius of the outer circle of the triangles, its radius being varied between R =1 and R =3: 5

6 d/d0 Fig 4 At the left, a small circle is seen (R=1). Another circle, three times as large, appears to be reduced in size (red dashed circular line). So does the apparent distance of the triangles, surrounded by the circle. Drawing not exactly to scale. The perceived length of the radius is Rperc= (1/R) (k-1). If there were no size constancy effect (k=0), one had Rperc= (1/R) (0-1) =R. Division yields Rperc/R = (1/R) k or R -k. The expression R -k gives the factor by which a larger circle appears smaller than one would expect it from the size of the retinal image. Here, R=1 means the reference circle (Table 1). This reference is an aribitrary choice. One could take the larger circle as the reference as well. In that case, the smaller circle would appear enlarged. Function: R(perceived) = (1/R) (k-1) R R(perc) [k= 0.4] R/R(perceived) Table 1 Perceived size of circles due to the size constancy effect. The column on the the righthand side gives the amount of shrinking. For example 1.275/1.5 equals See Fig 5. 1,1 1,0 0,9 0,8 0,7 0,6 Fig. 5 Reduction of the apparent size of circles with radii between R=1 and R=3 (arbitrary units) due to the size constancy effect. A target of linear extent within the circle (eg, the separation of the traingles dips) appears to shrink by the same amount. 1,0 1,5 2,0 2,5 3,0 Radius r 6

7 Apparent distance [mm] Assuming the apparent distance of the dips to be in proportion to the apparent radius of the outer circle R (Fig 3), the function to be fitted to the experimental results is y (apparent distance) = 60*A*[Rperceived/R0] = 60*A*[((30 + x) 2 + (756/x) 2 )/ ] -k/2 Eq (1) 60 means the true distance in millimeters. k and A are the parameters to be fitted. The expression in square brackets is the square of the radius R of the outer circle, divided by the square of the smallest radius which can occur (62.3mm). It corresponds to a horizontal extent of one triangle of 22.2 mm. (30 + x) means the sum of half the target s length plus the horizontal dimension x of one triangle. A can be called the illusion factor Horizontal extension of triangel [mm] Fig 6 Fitting Eq 1 to the results of the experiment. The values obtained for A and k are given in Table 2. The maximum oft he curve is around x=22mm. Underneath, there are examples of stimuli: A pair of the tallest triangles (left), of the longest ones (right), and a pair of the ones which are nearly rectangular. 7

8 756/x means half of the triangle s base (its vertical dimension), as 756 units (mm 2 ) is the triangle s area. The exponent contains a square root (1/2), which means that the apparent distance is assumed to be a function of the linear extent of the outer circle. In Fig 6, and in Table 2, the results of the fitting procedure are shown. The value obtained for the size constancy parameter k= 0.477(33) can be compared with the values derived from the results reported by Gilinsky (1955), which are between 0.359(23) and 0,500(42) (Kreiner, 2004), depending on the distance of observation, which was larger in her experiment by more than two orders of magnitude. A / Illusion factor k /size constancy parameter 2 red Table 2 Result of fitting Eq (12) 0.477(33) Discussion The mirrored triangles illusion is interpreted as due to a size constancy effect. The result shows that the intensity of the illusion correlates inversly with the diameter of the outer circle. From this, it is concluded that the size of the visual angle is in proportion to the outer circle of the stimulus. This can be compared with the Baldwin illusion, where the intensity of the illusion appears to be correlated with the size of the outer circle as well [Kreiner (2011)]. In contrast to the latter, in case of the triangles illusion it is not the area of the context elements which is altered, but their shape. 5. Size constancy as a consequence of limited data processing capacity The apparent size is not in proportion to the size of the retinal image (Lühr, 1898; Cornish, 1937; Schur, 1925; Gilinsky, 1955; Kreiner, 2004). Comparing two retinal images of different size, the smaller one appears somewhat enlarged, or the larger one reduced in size. Concerning the illusion, it does not matter whether it is the image of a small object nearby or of a large object at far distance [Schur, 1935; Gilinsky, 1955). Size constancy can be interpreted as due to the limited information processing capacity of the visual system (Kreiner, 2004). Figuratively speaking, there is only a certain number of pixels which can be processed by the brain per unit time, finally producing the perceived image. Choosing a wide visual angle and collecting the pixels from a large area will result in an overview of the scenery, but at low resolution, while a narrow visual angle will improve resolution on the expense of overview. A conceptional model of the size constancy effect is based on the following assumptions: 8

9 - The information processed by the visual system is retrieved from only part of the retinal area. - The size of this section increases or decreases with the size of the object one gives a close inspection. - The information processing capacity of the visual system is limited. Usually it makes use of its full capacity, independent of the object s size and the visual angle chosen. - The information collected, processed and transformed to give finally the perceived image, is projected onto kind of an internal visual memory screen which always exhibits constant size (Kreiner, 2004). This leads to enlargement, in case a small visual angle had been chosen. Small or large have to be understood as relative to a standard size. The internal visual memory screen is just an expressive comparison. It means that the visual system always uses its full data handling and storage capacity, regardless of the size of the visual angle. In the following, the consequence of a limited channel capacity on the perceived image is illustrated. It is a trade off between the size of the image and the resolution achieved. Fig 7 shows a photo taken in the French town of Pontrieux (Pontrev)/ Côtes-d Armor. The image consists of pixels. Lets assume that, within a certain time given, the visual system can manage only pixels (= /25). Now, if each 25 pixels (squares of 5 times 5 =25 pixels) are replaced by one large pixel, one gets the situation to be seen in Fig 8: One cannot recognize details. However, it is possible to achieve high resolution again from sacrifizing overview: Concentrating the pixels on a smaller section where each side has been reduced by a factor of 5, one can achieve the resolution of the original photo. For this purpose, the visual angle has to be reduced by a factor of 5 too, horizontally as well as vertically (Fig 9). 9

10 Fig 7 Scenery in the town of Pontrieux (France). Picture taken with a total of pixels. The frame indicates the area one will soon give a close inspection. Legends to the figures on next page: Fig 8 (At the top oft he following page) In case the capacity of the system amounts to only pixels (per time given), the image appears blurred. High resolution can be achieved only by sacrifizing overview, ie, retrieving all the pixels from a smaller area (white rectangle). Fig 9 All the pixels have been concentrated on an area smaller by a factor of 25 compared to the overview (a factor of 5 on each side). Resolution has been improved considerably. it matches the resolution presented in Fig 7. However, the improvement is achieved on the expense of gaining knowledge on what s going on around. 10

11 Fig 8 Fig

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