tion model to account not only for the metrical load of angles, but ... So our model departs from a continu- .... fits less well with line a (Boselie & Wouterlood, 1992;.
Psychol Res (1993) 55:191 - 199
Research chologischeForschung P ychological
© Springer-Verlag1993
Metrical information load of lines and angles in line patterns A. Hanssen, E. Leeuwenberg, and P. v. d. Helm The NijmegenInstitute for Cognitionand Information,Universityof Nijmegen, P. O. Box 9104, 6500 HE Nijmegen, The Netherlands Received September24, 1992/AcceptedFebruary4, 1993
Summary. Acute angles and long lines are more complex than obtuse angels and short lines. This source of complexity is called metrical information, for which a measure was proposed by Leeuwenberg (1982). According to this measure, the metrical information of a static pattern is specified as the sum of the mechanical impulses necessary to move an object according to this pattern. The perceptual relevance of this metrical load was tested in an experiment in which judged complexities were gathered for a number of simple line patterns.
Introduction Mackay (1969) makes a main distinction between two types of pattern information based on the situation in which a pattern is perceived. The first type of information refers to a pattern as an element of a set: a pattern is specified in terms of the alternative patterns that could have occurred and is thus quantified by its selective information content (Attneave, 1957). The other type of information is related to the construction of a pattern or to its representation by an observer in the light of constructive principles. In this paper we shall focus on the latter type of information. We are mainly interested in how a single pattern will be organized perceptually, and not in how a pattern is distinguished from various alternatives. Mackay (1969) makes a subordinate distinction between two sorts of information that are involved in pattern construction. One is structural information, the other is metrical information. The former specifies distinguishable dimensions or degrees of freedom that characterize the "structure" of a pattern. Metrical information, on the other hand, refers to quantitative measures of pattern elements
Correspondence to: E. Leeuwenberg
within dimensions and contributes to the "weight of evidence" for the structural components. Mackay's distinction between structural and metrical information, is in our opinion, rather close to the distinction we shall make. We shall therefore use the same terms. We describe the structural information by the amount of irregularity in a pattern. This amount roughly agrees with the number of different angles and lines in a pattern. Structural information has been the subject of perception research by Simon (1972), Restle (1970), Vitz (1968), Leeuwenberg (1969), Buffart, Leeuwenberg, and Restle (1981), Collard and Buffart (1983), Van der Helm and Leeuwenberg (1986, 1991). Metrical information deals with quantitative aspects of a pattern such as the size of angles and the length of lines. These sizes do not affect structural information, except when they are identical. In the latter case the identity may lead to a reduction in the structural information. For instance, a rectangle's length is different from its width, whereas all sides of a square are equal. Therefore a rectangle has one more unit of structural information than a square. Yet the fact remains that whether a rectangle is short or long in length, its structural information will not be different for that reason. The length difference is purely a metrical one. There are reasons to believe that structural information is more decisive for an interpretation of a pattern than metrical information. A theoretical argument might be that angles and lines themselves are the elements of the structural-information specification. That is, the identity or nonidentity of these elements determine the structural information. Hence structural information may be superordinate to metrical information. However, on the presumption of a bottom-up process, metrical dominance is more plausible. Thus the above theoretical argument for the structural dominance does not seem satisfactory. We therefore present a visual illustration to support the idea of structural dominance. Figures 1 A and 1 B are structurally identical: both are zigzag patterns. Both patterns can be described by: srsl srsl srsl srsl = 4 x (srsl)
192
A
assume that an acute angle has a greater metrical load than does an obtuse angle. Research into structural-information theory has supported the conviction that a pattern organization having the least amount of information is perceptually preferred. This agrees with the minimum principle. According to this principle, preferences for short lines and for obtuse angles are merely due to the relatively high metrical load of long lines and acute angles. For these preferences innumerable demonstrations are given by Gestalt psychologists. These have been summarized by the laws of proximity and good continuation. Especially for more or less continuous angles, Attneave (1957) has shown that acute angels are judged to be more complex than obtuse angles. These findings, however, do not provide us with a measure of metrical-information load, which applies to angles or to line lengths. Our aim is to develop one measure that applies both to angles and to lengths, as angles and line lengths perceptually interact. A demonstration of this interaction will be given. Figures 2A, 2B, and 2C comprise x and y parts. Within each figure the x and y parts are structurally identical. Nevertheless the x parts in Figures 2A and 2B are perceived as foreground, whereas in Figure 2C there is some ambiguity about the part that is taken as foreground. We shall explain these interpretations as follows: each figure can be organized in terms of an x or a y foreground and a global rectangular background. The simplest parts, x or y, stand for the foreground, given the assumption that there is a preference for the simplest organization. In Figure 2 A the average lengths of the x parts are equal to the average length of the y parts, but the angles of their contours differ. It can be shown that the sum of the angular deviations or turns (t) of the contour of each x part of Figure 2A equals: ~t = 360 °. All their angles are convex. However, each y part has some concave turns (c). It can also be shown that of each of these partly concave y patterns the sum of the turns equals: Zt = 360°+2 (c). Hence the x parts contain a smaller sum of angular turns than the y parts, i.e., the x parts have less metrical-information load than the y parts (Chadwick, 1983; Hoffman, 1983; Kanizsa, 1979; Koenderink & Van Doom, 1982). This agrees with the preference, previously cited, to interpret the x part as foreground in Figure 2A. In Figure 2B the lengths of the y parts are larger than the lengths of the x parts, i.e., the metrical information load of the y parts is greater than that of the x parts. And this agrees with the preference, pre-
B
Fig. 1. The patterns A and B have a common structure, but differ with respect to metrical information. The reverse applies for patterns A and C: common metrical, different structural information. Patterns A and B are perceptually more alike
where s stands for a line segment, r for a right turn, 1 for a left turn (Leeuwenberg, 1969). However, these patterns differ with respect to their metrical aspects: in Figure 1B the angles are more acute and the line segments are longer. Figures 1 A and 1 C differ structurally, but have equal angles and line segments. Yet these patterns are judged to be more different than the patterns 1 A and 1 B.
Metrical aspects Mackay (1969) considers metrical information to be a positive component of redundancy or "weight of evidence" for some structure. For instance, a long line is easily seen as a line. This is because a long line contains "more" points than a short line, and thus it more redundantly represents a line. This redundancy is specified by the number of visual elements (points) on the one hand, and the structure (line) on the other hand. The visual elements can be considered to be arguments for the presence of a structure. The ratio of the number of visual elements and the components of the structure determines the redundancy or evidence for the structure. Thus, in a given structure, metrical information enhances its structure. However, in assessing this structure, metrical information contributes to the complexity of this structure. For the moment we assume that a long line has a greater metrical load than does a short line. Similarly we
A
()()(I x
Y
B
x
I
Y
x
C
I(
v
)( y
Fig. 2. In figure A the convex x parts have contours with less angular change than the y parts. This may explain why the x parts are preferably interpreted as foreground. In figure B the x parts are shorter than the y parts and will be perceived with perference as foreground. In figure C the angle acuteness and line length both play a role and counteract each other
193 viously cited, to perceive the x parts as foreground. In Figure 2C the angular information favours the y parts, but the length information favours the x parts. Hence both types of metrical information are in rivalry. Our question is whether these kinds of rivalries can be predicted from a metrical measure of information. We are therefore seeking a measure that applies both to angles and to lengths from one point of view.
Towards a model
Dember (1965) has argued that several types of information have the aspect of change in common. We shall make use of this idea and look for the conditions by which this concept of change applies both to angles and to lengths. If change is taken literally, it is a property of a motion and will become manifest whenever a series consisting of lines connected by angles is traced from the beginning to the end. It may be evident that, in tracing such a path, the change involved at an obtuse angle is less than that at an acute angle. However, an assumption is made if this evidence is accepted. At an angle the simplest turn is taken as the relevant one. For instance, if one line is horizontal and the subsequent line has a tilt of 30 ° (upward or downward), the simplest turn is taken to be 30 ° and not 360 ° -30 °. One can view the absurdity of the latter angle as support for the use of the minimum principle in our concept of a turn. This principle is also applicable for distances between points. One considers the shortest distance undoubtedly as the distance. We include this minimum principle in the model for metrical information because otherwise turns and distances have no unique meaning. We now return to the concept of change. Changes stem from forces. These forces can be related to a pattern in two ways. As has been said, a pattern can be traced. In that case forces are elicited, for instance, by the muscles of the hand. A similar pattern can also be produced by external forces namely, by pushing a body at various moments. Notice that the internal forces caused by the hand are of equal size as the external forces exerted on the body. It is obvious that fewer forces of both kinds are involved in producing obtuse angles than in producing acute angles. The question is, which forces are involved in producing a single line? Our aim is to find the force that applies both to angles and to lines. In tracing one line there is no turn; hence no change is involved between its beginning and end. For the moment we shall focus on the end of the line motion, since our conclusion has an implication for the beginning of a line. At the end the motion seems to stop. Since there is no sideways motion left at the end, the amount of change involved in the stop is half the amount of the change involved in turning backwards with an angle of 180 °. Any collision can be decomposed into two equal changes: one for stopping and one for returning with the same speed as before the end. We have therefore expressed the amount of change in a motion at the end of a line in terms of the amount of change involved in angles, namely, half the maximal turn. However, there is a complication, if the motion is linear, i.e., if it is of constant speed. Then the change involved at the end of a long line trajectory is equal
to that of a short trajectory. We seek the simplest information model to account not only for the metrical load of angles, but also for line lengths, according to which long lines have more load than short ones. This is achieved with one additional assumption, namely that the motion has a constant acceleration instead of a constant speed. Then a motion at the termination of a long line will have greater speed and so that change at this point is greater than at the end of a short line. So our model departs from a continuously accelerating body, for instance a rocket, tracing a series of angles and lines in a space with no resistance. Now we assume that the sum of changes generated by, or exerted on, a constantly accelerating rocket for a special trajectory is the metrical load of that trajectory as a static pattern. Underlying this idea is an important assumption, namely, that the information load of static patterns is based on that of motion patterns. We suppose that the information load of a static pattern is not incomparably different from that of the same pattern as a motion. Swanston (in preparation) has provided experimental support for the idea that static and motion patterns have a common structure. He showed that the Hering and Poggendorf illusions also occur if they are shaped by motions. In the text above we have used the terms force, push, and change interchangeably. We started with the term change and shall from now on use it as the basic term. However, we define the term change for motions, as in physics: in mechanics the amount of change is determined by impulse and not by force. However, before this idea is outlined, a remark is in order: our hypothesis is that the metrical-information toad of a static pattern equals the sum of impulses exerted on a constantly accelerating rocket in tracing the pattern. This might suggest that such a scanning occurs during the perception process of a static shape. This is not intended. Rather, we hypothesize that the metrical information, as it is stored in the representation of a pattern, is expressed in terms of impulses necessary to generate a pattern. In fact we assume that a pattern representation is a procedural description of how one constructs a pattern. It does not mean that the process of perception, from pattern to representation, occurs in a constructive way. Constructive information, as opposed to selective information, merely implies, according to Mackay (1969), that the terms of the final pattern representation are appropriate for pattern reconstruction. A similar point is made by Runeson and Frykholm (1981). They have shown that subjects are quite good at estimating the mass of objects portrayed only visually. According to Runeson and Frykholm, people make such estimates on the basis of physical laws. This does not imply that real physical events take place in the brain during perception.
Impulse load In this section we examine a technical specification of the impulse load of a specific pattern, namely that in Figure 3. This pattern we conceive as a trace of a constantly accelerating rocket, which starts at the left side. Acceleration is a basic property of the model and is considered to be a given. Therefore only impulses needed to keep the rocket in a
194
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u? "'~
L3
P3
/ Fig. 3. The arrows indicate the impulses exerted on a moving object. Because of these impulses the object traces the specific pattern
special path contribute to the pattern's metrical load. The arrows in Figure 3 represent the impulses (P1, P2, P3) that are exerted on the rocket at these points. Assume that the rocket begins from the leftmost point. No impulse is involved at the start. Once the acceleration is given, the speed of the projectile grows over time along the given path. When it has covered the distance L1, an elastic collison takes place, against a hypothetical wall, causing the projectile to deflect. The impulse exerted by the wall has two vectorial components. One refers to a partial return, i.e., to stopping a body, the other to a motion in the direction of L2. Both vector components depend on the angle at the end of L1. After the first collision the rocket continues. At the end of L2 its speed is greater than at the end of L1. In fact its speed will be some function of L1 + L2. The second collision also has two components, since it is also elastic. At the end of L3 the projectile does not return, but merely stop. Hence there is no elastic collision, and only one vector component of the impulse is involved. This component is a function of L1 + L2 + L3. So for Figure 3 the total metrical load (M) is specified by the sum of impulses needed for the complete motion trajectory:
Some further specifications are required with respect to the magnitude of the lengths L1, L2, and L3. Whether a pattern is perceived from a perspective that is near or far from the display is irrelevant; that is, the absolute size of the pattern is unimportant. Only relative lengths within a pattern are decisive. It is this information that must be specified. We have several options. For example, we may take as our frame of reference either the first line of the pattern or the shortest line or the longest line. Given a particular reference frame, then all other lines are conceived as proportions of this frame. The answer is straightforward. The longest has to be taken as a frame of reference, i. e. as a unit length. This is because the metrical load is minimal when the longest line is used as a reference frame. The minimum principle is thus an important working assumption of the model. In Figure 4 some simple line patterns are presented with their concrete impulse load computations. In each of the six patterns the length values are quantified as parts of the longest line segment in a each given pattern. Notice that the patterns B and E; whose segmented lengths are equal, appear to have the relatively highest impulse load. To an observer this may seem strange, since these two patterns appear to be simple. This is true, but not in a metrical sense. Both patterns have less structural information than the others because of their equal lengths. In summary, the main assumptions we have proposed are these: (a) a static pattern is represented in terms of principles of pattern reconstruction; i.e., in terms of change as defined in mechanics; (b) preferred organizations are described by a minimum principle that here involves minimizing change; (c) pattern construction involves a motion that is assumed to have constant acceleration.
Pilot experiments
M=2P1 +2P2+P3 P stands for one impulse component specified by: P=mv m = mass and v = velocity. The mass is not considered a distinctive characteristic of motions in the context of this approach. Therefore it is taken as a constant: m = 1. The velocity equals: v = at, where a = acceleration, t = time. The time can be expressed by: t = ~ / ~ ) . L stands for a distance. Thus v = ~/(2aL). In this expression the term 2 a is considered an irrelevant value, since it refers to the starting condition of the model: the acceleration of the rocket. Hence v becomes a simple function of L: v = {L. Thus one complete impulse component can be simplified by: P = {L. At a collision which is not frontal, only some portion of this impulse is involved. This portion will be given by cos (A/2), where A stands for the inside angle caused by the collision. Thus the impulse at an angle A at the end of a length L equals: P = 2 "~(L) cos (A/2) The total metrical information involved in Figure 3 is: M = 2 {(T1) cos (A/2) + 2 x / ( L ~
cos (B/2) + ~](LI+L2+L3)
Before we describe the main experiment, we shall discuss a few pilot experiments, testing the impulse-load hypothesis. The reason that we start by discussing these explorations is that they clarify - at least in part - the choice of the main experiment. The explorations reveal an unexpected abundance of potentially intervening variables that impose a high constraint not only on the ideal experimental task, but also, and even more, on the choice of the stimuli. For instance, the patterns in Figure 2 are not useful in combination with the task of indicating the foreground parts. These patterns comprise surface parts. Within each of these parts, lengths and angles cannot be varied independently. Nor can the lengths of the parts be specified uniquely. It is not clear whether the longest diameter or the average length of each of these surface parts has to be considered as their lengths. So surface stimuli are avoided in the following pilot experiments. In one pilot experiment, stimuli are used as depicted in Figure 5 A. The task was to indicate which line, b or c, fits less well with line a (Boselie & Wouterlood, 1992; Wouterlood & Boselie, 1992). The outcomes showed consistency within subjects, but not between subjects. Several subjects had a preference for the least turn. These subjects
195
c/
/ /--
A
M=3.3 2 cos(60) (',/0.5 + "41.5) + "42
M=4.1 2 cos(60) (~11 + "/2) + "43
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[1
M=3.8 2 cos(60) ('41 + ",/1.5) + "]2.5
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,X M=4.6 2 cos(30) ('40.5 + ~/1.5) + "42
M=5.8 2 cos(30) (~/1 + "/2) + ~/3
M=5.3 2 cos(30) ('~1 + ",/1.5) + ~/2.5
Fig. 4. Patterns with their impulse load computations
A
B
C/~
a'
D
b'
E
L3 Fig. 5. Stimuli used in the piiot experiments
may have interpreted the pair of the most continuous lines as a sequence of lines. Other subjects preferred the inside angles. These subjects may have interpreted the lines as rotating around a centre. In order to avoid this ambiguity, patterns as depicted in Figure 5 B are used in a second pilot experiment. The task was to pick the line that fits less well with line b. The outcomes showed categories of reactions that are not yet mentioned, but also appeared in the previous experiment. Some subjects provided outcomes that agree with our predictions. They were able to disregard one line segment and preferred to connect the middle line with the shortest line (c). However, several other subjects systematically preferred the connection of the middle line with the longest line segment (a). We can explain this as follows: these subjects were not entirely able to imagine the absence of one of the line segments, so they preferred the line segment with the highest load of evidence. This is the longest line. In a third pilot experiment the imaginative capacity of the subjects was less challenged. The task was to indicate which of two pairs of connected line segments is preferred.
For a pattern such as Figure 5 C subjects had to make a choice between a and b or a' and b'. Whatever subjective criterion is used for the best choice is not clear, but we assumed that it would agree with simplicity. At least we hoped that the best choice might evoke consistent responses within subjects. This was the case within subjects, but again great individual differences were observed. Some subjects preferred the pattern with the shortest trajectory. For patterns whose total trajectory length was constant, several subjects preferred pattern a and b" in Figure 5 C over the others. Spontaneous remarks by these people were revealing: many reported that they interpreted the patterns as "mountains" and "valleys." Given this image, their preferences reflected an inclination to "climb the mountains" from the steepest side (on the assumption that the patterns are traced from left to right). Any pattern may give rise to associations. This can hardly be avoided. If they are based on relative angles or turns, these associations are perhaps not completely detrimental from the perspective of our model, which emphasizes relative angles. However, the patterns in Figure 5 C are ambiguous in a structural sense. They can be fixed in two ways: one is fixed by: L1, d, L2 (see Figure 5D); the d stands for a turn or a relative angle. The other is fixed by: e, L 3 , f ( s e e Figure 5D); L3 is the length of the base line and e andfstand for the absolute angles with respect to this base line. The last code is to be preferred, since it includes the horizontal axis, which is a natural frame of reference. The associations of "mountains" and "valleys" are probably based on the latter code with absolute angles. In order to avoid this structural ambiguity in the stimuli, the line patterns (depicted in Figure 5 E) were used in still another pilot experiment. Again subjects had to indicate which pattern (a or b) they preferred. A problem here involves the determination of the metrical load of such patterns because the computation of this load varies depending upon the way a pattern is scanned (i. e., from left to
196
/
/ Fig. 6. One of the experimental stimuli in the main experiment
LENGTH PROPORTIONS
165
,
~
,
_
_
pattern is depicted in Figure 6. The objections mentioned earlier do not apply to these patterns. Those objections are related to : (a) rotation angles, (b) absolute angles, (c) induction of scanning behaviour caused by metrical information load. We also decided not to let subjects select a part of a pattern, but chose the simplest pattern out of a series of patterns. We regard this experiment as an extension of the experiments done by Restle and Decker (1977) and its analysis by Leeuwenberg (1982). In their experiments the same stimuli were used as in our final experiment. The aim was to explore Mfiller-Lyer effects as a function of wing lengths and angles. In the analysis by Leeuwenberg (1982) the same metrical load model was used and predicted the results rather well.
Experiment Material. Each stimulus consisted of a horizontal middle line and two
Fig. 7. One set of stimuli; this is a D W B set (see text)
right or from right to left). The problem is that according to the minimum principle our model favours a certain scanning direction. However, we do not want to test scanning behaviour as such, being a motion, but the metrical load of a static pattern. The outcomes did not show a useful consistency either. Our tentative conclusion is that the metrical load equals the average of the loads in both ways, i. e., from left to right and from right to left. On the basis of these pilot experiments, we decided to use patterns that show a scanning symmetry. One such
wings of equal lengths (see Figure 6). The lines, 0.5 m m wide, were drawn on a white plastic sheet of 10 × 10 cm. The minimal distance of the stimulus to the edge of the sheet was 1 cm. A dot in the lower left corner was placed for orientation. There were 36 different stimuli, representing combinations of six angles and six length proportions. Altogether eight different sets of 36 stimuli were used. One set is illustrated in Figure 7. Sets are introduced to control three binary variables: (1) one variable is related to the absolute pattern size. In the set indicated by D, the diagonal (being the distance between the most distant stimulus ends) is kept constant. So these patterns cover an equal visual angle. In the set indicated by T, the total length of the three line segments or the trajectory is kept constant. No prediction is made about the relative influence of these conditions; (2) a second variable deals with wing orientation. The writing direction - being from left below to right above - is indicated by W and the reading direction - being from left above to right below - is indicated by R; (3) the third variable involves wing position. The left wing m a y be either above or below the central line (indicated by A or B respectively). Thus the 8 sets are: DWB DRB
DWA DRA
TWB TRB
TWA TRA
Procedure. There were 176 subjects, all of them students in social
sciences, twenty-two subjects for each of the eight sets. Each subject was given a set of 36 patterns in a different random order. W e decided not to present the 8 x 36 stimuli to each subject. Such a task would place too heavy a burden on each subject. Only one pattern was visible at the start of each experiment. The task was to take the stimuli one by one and to place them in a long row: at the left side the simplest pattern, at the right side the most complex, so that all patterns were rank ordered according to
Table 1. Predictions and experimental data summated over eight sets. The rows refer to the angles, the columns to the length proportions. The left number in each cell represents the rank of the metrical information load. The right number represents the experimental rank order. The sums of the rank orders are indicated at the end of each row and column Length proportions 1/8 - 1 - 1/8
1/4 - 1 - 1/4
1/2 - 1 - 1/2
1 - 1/2 -
1
1 - 1/4 -
1
1 - 1/8 -
1
165 140 115 65 40 15
1 6 9 16 181/2 20
1 5 13 18 17 24
2 8 13 21 23 24
2 8 15 20 30 22
3 10 15 25 27 29
4 10 19 29 23 31
7 14 22 31 35 36
7 9 21 26 34 36
5 12 181/2 28 32 34
3 11 16 27 32 35
4 11 17 26 30 33
6 12 14 25 28 33
Total
701/2
78
91
97
109
116
145
143
1291/2
124
121
118
Theory
Experiment
22 61 941/2 147 1651/2 176
23 55 98 145 164 181
666
666
197 complexity. Subjects had to take care that the dot on each stimulus sheet was placed in the left corner below. There was no time limit and corrections were allowed. No positive definition was given for the meaning of complexity. Subjects were told that complexity is not related to the effort required to remember or to copy a pattern. Furthermore the subjects were warned not to place stimuli next to each other on the basis of similarity per se and not to judge patterns because of their associations with meaningful objects.
Table 2. In the upper row the cosines (all divided by cos 15/2) for some angles are presented. In the lower row the summated rankorders (S) produced by the subjects (all divided by S(15) are given. This Table shows the resemblance of the S values with the cosines Angles (cos A/2)/(cos 15/2) (S)/(S(15))
15 1.00 1.00
65 0,86 0,80
115 .55 .54
165 .14 .13
Results Table 1 presents data for the 36 basic stimuli in the eight conditions. The rows refer to the angles and the columns to the length proportions. Cell entries contain the ranks of the metrical information values (left values) and the experimental rank orders (right values). The latter were obtained as follows: for each stimulus the ranks as given by all subjects are summed, after which the stimuli are rank ordered according to these summed ranks. The Kendall rank correlation yields .95, the Spearman rank correlation r yields .97. About 20% of the subjects individually ordered the stimuli very well according to the metrical values. Each of these subjects produced a correlation r = .90. Furthermore, some of the subjects (3%) each produced r = .97. After each experimental session the subjects were interrogated. At least 40 of the 176 subjects said that they had judged the complexity on the basis of associations or similarities. With data deleted from these individuals, the Spearman rank correlation between the judgements and the impulse loads becomes r = .99.
Angles. If we focus simply on the effect of the angles, the following observations are evident. According to the theory, the metrical load should be monotonically increasing with the angular deviation or turns, thus decreasing with the inside angle° This inside angle is indicated in Table 1. The summated experimental rank orders, represented at the right side of the last column, show this decrease. According to variance analysis, the experimental rank orders reflect this effect (p
