A Pattern Measure for Arrays
School of Engineering and Applied Science
University of California at Los Angeles
Los Angeles, CA 90095 USA
Nikos A. Salingaros
Division of Mathematics
University of Texas at San Antonio
San Antonio, TX 78249 USA
A simple model quantifies two-dimensional designs and arrays according to their visual impact on viewers. It distinguishes simple from high-variation cases by measuring their degree of complexity and pattern. The method used to define these measures combines symbol variety with symmetry calculations, and employs hierarchical scaling to assess the relative impact of substructure on different levels of scale. Modeling intensity values by symbols, and using substructures within global metrics of image complexity distinguishes between disorganized complexity and organized patterns in 2x2 and 6x6 arrays. Finally, the psychological relevance of the measures described here to human perception is discussed.
- Background: Information Measures
- Binary Example
- Hierarchical Generalization
- Four-Level Example
- Psychological Responses
Any system -- in this case a visual symbol array -- is easily recognizable as being either simple or complex, yet measuring its complexity has remained an open problem. Part of the trouble is the lack of a satisfactory complexity measure (Maddox 1990). Some would like to measure complexity by the total number of different subunits. While useful, this does not account for connectivity and internal organization. Others consider the code that generates a pattern, and measure complexity by the length of such a code; thus self-similar fractal structures such as a fern leaf are simple, whereas random structures requiring a code as large as themselves are complex. This latter approach identifies complexity with randomness, which neglects complex systems that are highly ordered.
On the other hand, man's visual system is especially receptive to patterns. The complexity of images determines their effect on the human perceptual system. For millennia, mankind has produced structures with different subsymmetries on many different levels of scale. We apparently enjoy the input from patterns, and this enjoyment often increases with the complexity of a pattern; however, this is true only for complex patterns that have some sort of ordering. The precise nature of this effect remains imprecise and largely intuitive.
We propose to combine these and other ideas into a pattern measure that distinguishes between disorganized complexity and organized patterns. The model will be applied to two very simple examples: first to 2x2 binary arrays; then to 6x6 four-element arrays. The results obtained clearly distinguish simple from complex arrays; they further differentiate the more complex cases according to their degree of internal organization. By extracting information that is not otherwise obtainable, we believe that this approach offers a more useful measure of the visual impact of arrays.
The notion of simplicity has always been problematic. Here we identify simplicity with uniformity, and not with brevity of code. (Problems with the latter as a definition of complexity are pointed out in the next section). Deep, or profound geometrical simplicity such as is found in biological and physical structures (or, for that matter, the greatest of human creations such as religious buildings throughout the ages) depends on substructure, and thus corresponds to organized patterns. For this reason, the often-used phrase "reduce complexity through organization" is ultimately misleading.
By focusing on visual organization, we obtain a quantitative description of variation in two-dimensional arrays. There is no commonly-accepted quantitative means for describing images. The logarithmic measures of communication theory and physics simply do not apply to the varied practical situations or designs in diverse cultural areas (textiles, buildings, computer data and displays). Models of variation in binary strings, time signals, and more complex situations usually employ symbol probabilities (see texts on information theory, e.g. Cover & Thomas 1991). We show here that three other practical issues predominate in a quantitative description of visual complexity: 1) symbol diversity; 2) presence of symmetry; and 3) regional organization.
The measures described below involve both content (information) and relationships (organization). Typical content operations involve identifying coherent units; finding the number of repeated occurrences; and measuring the extent of a field property. Relationships concerning symmetry and other mutual qualities then come from comparing pictorial entities or elements. One goal of this study is a method to compare visual images viewed on a computer screen, and more generally in human-computer graphic-user interfaces. Nevertheless, the measures and calculations we define here apply to many diverse domains. They provide a numerical indication of whether visual images are immediately understandable, based on the ease by which their information can be processed.
2 Background: Information Measures
A vast literature and a historic description (Spence 1984) show how events can be organized by hierarchy in order to assist human memory. The psychological limit on the number of distinct items people can easily recall (Miller 1956) underlines the need for chunking or grouping of data to aid both memory and perception. Computer technology -- e.g. the large number of icons and pull-down menu alternatives in a graphic-user interface -- stimulates what should be measured to describe visual data. While compression schemes take advantage of identical data in portions of an array (condensing uniform data in a row by run-length encoding as in JPEG and GIF), they do not provide usable indices for comparisons.
More to the point, distinct encoding schemes will likely treat the same visual image differently, which is an intrinsic limitation of complexity measures relying on length-of-program. For instance, a fern leaf is simple according to the Fractal Image Compression (FIC) algorithm, see (Barnsley & Hurd 1993; Fisher 1995), but is complex according to the Graphics Interchange Format (GIF). The reason is that GIF works in a rectangular geometry. This approach is clearly program-specific and cannot be made universal.
Prior work on this topic has focused on one-dimensional data describing information in terms of symbol sequence variation in communication (Cover & Thomas 1991; Hartley 1928; Shannon & Weaver 1949). In physics a parallel term, entropy, describes the lack of organization, but without any geometrical reference. The communication and physics order-oriented measures contrast with visual concepts such as texture (Haralick et al 1973; Julesz 1981), a term from image analysis. However, both information/entropy and texture fail to describe key aspects of visual images.
The information theory model of symbol transmission has us learning the most from receiving a specific sequence when element values are random and equally likely. The Figure 1 sequence shown below (exactly fifty ones followed by that number of zeroes) labeled by (1) conveys little additional information at the thirtieth element after observing the first twenty-nine. But if (1) is replaced by the random sequence (2), the thirtieth symbol conveys a great deal of information.
11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
00100 01111 01010 11000 01011 10010 11111 10101 11111 11010 01110 00110 10001 00101 10111 01100 10111 11001 11101 01100
Mathematica generation rule
Figure 1. Two sequences of fifty binary digits.
If information is being received one digit at a time, then the probability that a 1 appears as the thirtieth digit is 1/2 for both (1) and (2), but that does not concern us here: we want to know how information is organized in the ensemble.
Shannon (Shannon & Weaver 1949) extended a logarithmic measure introduced by Hartley (1928) that proposed a numerical way to describe information based on the probability of each individual symbol in a transmission. Weaver (1948) termed the symbol randomness to be disorganized complexity. While it is impossible to recognize disorganized complexity (such as sequence (2)) since it does not match something we remember, and cannot fit into a perceptual framework, there are other situations where previously-seen structures enable recognition. We describe two measures below that do this in quantitative terms.
In (Salingaros 1997) one of us introduced two pragmatic measures, termed temperature (T) and harmony (H). Temperature describes symbol variation. Originally an index ranging from zero to ten, it sums five partial measures, each ranging from zero to two. The temperature components are: 1) intensity and size of details; 2) differentiation density; 3) line curvature; 4) color-intensity; 5) color-contrast. Harmony is a similar five-part sum composed of symmetry values: 1) vertical and horizontal reflections; 2) translations and rotations; 3) shape-similarity; 4) form-connectedness; and 5) color-matching.
Another approach looks at hierarchy. Figure 3 given later shows the four-way and nine-way decomposition of 6x6 arrays into non-overlapping blocks. Measurements could compute complexity within perceptual regions such as lines/stripes, blocks, and rings. Uniformity is easy to see in such areas. A set of six thirty-six-element four-level arrays in Figure 4 below shows these perceptual regions. A method for aggregating such measures was introduced by the other co-author (Klinger 1980). Suppose a perceptual region with n elements has uniformity index 0.7, and another with m points scores 0.9. An overall value could be the size-weighted average:
(0.7n + 0.9m)/(n + m)
An early application of information theory to aesthetics was undertaken by Moles (1966). While this book is ground-breaking, no pattern measure for visual arrays was ever developed. One of the few authors to distinguish between organized and disorganized complexity (Papentin 1980) introduced a quantitative model that is entirely distinct from ours. Later work by Landsberg (1984) uses entropy in an essential manner to measure the complexity of systems; nevertheless, that model was not applied to visual arrays. We discuss the details of Landsberg's model below.
2.1 Complexity and Patterns
In order to describe relationships that cover multiple factors we propose two linearly independent descriptors: T, a simplification of traditional measures of information; and H, a representation of symmetry. T measures the number of different subunits. Symmetry is a factor analyzed in (Attneave 1954; Berlyne 1971), though we give here a practical means of computing it. The main comparisons will involve two measures derived from T and H:
L = T H
C = T (Hmax - H)
The combination of T with H follows an analogy with thermodynamics (Salingaros 1997) where potentials are products such as TS (where T is the physical temperature and S is the entropy). H measures the negative entropy, since the presence of symmetries corresponds to the absence of visual disorder, and S = Hmax - H. In practice, it is much easier to measure H than S directly. With the constant b = Hmax , the two measures L and C differ by a constant times T , i.e., C + L = bT from Eq. (6).
Landsberg (1984) clarified the role of entropy as distinct from order. Labeling a system's maximum entropy as Smax = b implies that Hmax = b. Landsberg's "complexity" S(b - S)/b2 (which is closest to our L and C measures) uses only S, whereas our model combines the two complementary measures of disorder T and S = Hmax - H.
Other authors have proposed similar measures. Birkhoff (1933) was clearly searching for something like this when he tried to quantify beauty mathematically. He did not, however, address the question of organization. Eysenk (1941) followed that work with a better model and derived statistical correlations. (This early work is reviewed in the book by Berlyne (1971)). The model presented here extends their approach to measuring the appearance and characteristics of general physical entities.
Another group of researchers pursue a different tack altogether. They classify emotional reactions to physical environments in terms of opposite psychological relationships (e.g., exciting versus gloomy) corresponding to opposite geometric points on a circle (Nasar 1989; Russell 1988; Ward & Russell 1981). These characteristics parallel human responses to different degrees of complexity and organization. Earlier work in this vein describes emotional states associated with points on a circle with reference to attention versus rejection and pleasantness versus unpleasantness (Schlosberg 1952). We are going to propose an essential link between that work and ours.
3 Binary Example
We begin with four-element, two-by-two binary arrays shown in Figure 2, below. (Rotation, reflection, or taking complements can transform each array into others not shown here). We will later present extensions to more general arrays.
Figure 2. Four four-element binary arrays.
Here and below we simply use T and H for the temperature/harmony measures. In the general case of arrays with different entries, T reduces to just the number of different element-symbols (smallest units) minus one. The basic criterion is that a uniform surface represented by a single symbol has T = 0. Since there are only two symbols on the Figure 2 arrays, T takes binary values. In terms of the Figure 2 labels the uniform-symbol case I has T = 0. For the others, we have:
T(I) = 0, T(II) = T(III) = T(IV) = 1
H measures the presence/absence of symmetry by assigning a binary value (present corresponds to one). Six possibilities are described:
- h1 = reflectional symmetry about the x-axis
- h2 = reflectional symmetry about the y-axis
- h3 = reflectional symmetry about the diagonal y = x
- h4 = reflectional symmetry about the diagonal y = -x
- h5 = 90° rotational symmetry (either +90° or -90°)
- h6 = 180° rotational symmetry
Summing the six hi causes H to range from 0 to 6. The Figure 2 arrays have H values:
H(I) = 6, H(II) = H(III) = 1, H (IV) = 3
With Hmax = 6, Equations (5), (6), (7) and (8) give the Table 1 summary:
Table 1. Indices of four-element two-value arrays.
We are proposing that the Table 1 entries correspond to human perception. This has been tested and validated informally. Overall, L measures the level of interest of each array, whereas C measures the internal complexity. The two indices are entirely distinct. Array IV is subjectively "interesting". It has L three times those of II and III. The other tabulated numbers also correspond to intuitive feelings about interest in the arrays. E.g., uniformity in I leads to zero values for L and C, but all the other cases have a comparable number for measure C.
Without getting into the theory behind psychological perception, this example serves to show how the proposed clearly measures differentiate between uniform and varied arrays. As discussed in Section 6 below, the psychological response to array complexity needs to be further verified by rigorous controlled experiments.
4 Hierarchical Generalization
Using thirty-six-element arrays allows us to incorporate hierarchy and combine small region measures. (While this could be via (4), in the simple example below averaging calculations are weighted equally since all are based on the full thirty-six elements). If four 3 x 3 arrays or nine 2 x 2 arrays are adjoined via the spatial relationships shown in Figure 3 below, the result is a 6 x 6 array. We call a small region within an array a subblock if it is square and if the overall entity can be composed from adjoining many such units (tiling, or tessellation, are related terms). We now seek measures that describe symbol diversity and the presence of symmetry within subblocks.
a b c d
Figure 3. Subdividing a 6X6 array into 3X3 and 2X2 subblocks.
We begin by explaining how subblock calculations can introduce recursion into large arrays. A square thirty-six-element array is four nine-element or nine four-element subblocks. T is the number of different symbols minus one, and for thirty-six elements it can be evaluated on three scales. The first is all thirty-six elements at once. The other two are based on either nine- or four-element subblocks. To simplify matters, we ignore all other regions that overlap such as 4x4's, and other 3x3's and 2x2's different from the ones shown in Figure 3. We use the subblock labeling following letters and numbers in Figure 3.
The T (or H) values are computed as averages of the different size subblocks.
T(3 x 3) = [Ta(3x3) + ... + Td(3x3)]/4
T(2 x 2) = [T1(2x2) + ... + T9(2x2)]/9
Each of the measures T(6x6), T(3x3), and T(2x2) expresses something about the overall array. Since there is no built-in preference for one over the other, the three combine as:
T = [T(6 x 6) + T(3 x 3) + T(2 x 2)]/3
H was previously defined for 2x2 arrays in terms of six different symmetry operations. Now, similarity-at-a-distance measures are required to account for interactions between subblocks. This adds three measures of translational symmetry:
- h7 = similarity to another element (yes or no gives a 1 or 0)
- h8 = relation to another element by translation plus a reflection about either the x-axis or the y-axis (glide reflection).
- h9 = relation to another element by translation plus a rotation by either +90°, -90°, or 180°.
In cases of high symmetry h8 and h9 sometimes double-count h7. Two different subblocks may be similar as oriented and also so after a reflection or a rotation. If a subblock is glide-reflection-related-to-another and similarly associated to one more by glide-rotation, it counts as 2. (We do not consider glide-rotation by multiples of 90°, as that would lead to a more complicated model. Since empirical experiments show that glide-reflections about the two diagonal axes do not provide a strong visual connection, they are not counted here).
H sums the nine hi, i = 1, ... , 9, each a binary value, so it ranges from 0 to 9. As in the case of T, these computations have to be done on three different levels, 6x6, 3x3, and 2x2 (the last two by eqs. (9, 10)). The results are then combined via a form of eq. (11).
The role of the symmetry measures hi is to determine the degree of visual pattern in the arrays. H is therefore a direct pattern measure. The hierarchical subdivisions (Equations 9, 10 and 11 for H) combine the pattern measures on each individual level of scale into an overall symmetry measure. Even so, our model does not use H alone, but requires the product of the two factors T and H. The reasons for this will become clearer below, where we distinguish between disorganized complexity and coherent patterns.
When elements on one scale are related through symmetry, they create an element on a higher scale. This process is taken into account here by the hierarchical decomposition of H. While we do not address the Gestalt process of grouping individual elements so that a single percept emerges, that is consistent with our model.
5 Four-Level Example
A game proposed by Sackson (1969) led one of us (Klinger 1980) to design six thirty-six-element square arrays shown in Figure 4. Each array element is one of four symbols, visually indicated here by 0, #, +, and *, but easily shown as colors or gray-values on a computer screen. [A numeric equivalent for these symbols, e.g., 0 = 0, * = 1/3, + = 2/3, # = 1, enables one to identify such game-arrays with zones of pixels and their actual intensities in a digitized image.]
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + 0 0 * *
+ + 0 0 * *
# # # # # #
# # # # # #
* * 0 # + +
* * + * + +
+ # * # * #
# 0 # * # *
* # + + * #
# * 0 0 # *
* # * # + #
# * # * # 0
+ 0 # * 0 #
* # 0 + * *
# 0 * # 0 +
0 # + 0 + #
+ * + # * *
# 0 + * 0 +
+ + + + + +
+ 0 0 0 0 +
+ 0 * # 0 +
+ 0 # * 0 +
+ 0 0 0 0 +
+ + + + + +
+ # * * # +
# 0 # # 0 #
* # + + # *
* # + + # *
# 0 # # 0 #
+ # * * # +
Figure 4. Six thirty-six-element four-value arrays.
As in Figure 2, array I is uniform: it presents no perceptual information beyond the 6 x 6 frame and the single symbol + (equivalently, any other single symbol). Array II has information strongly organized by 2 x 2 subblocks. Array IV was constructed by an algorithm based on outer and middle ring structures. Array V, which vividly displays those rings, is visually simpler than IV. Array VI varies III by taking its 3 x 3 subblock a and reflecting it four ways; it has high internal symmetry.
Computations for T and H are straightforward and they readily lead to the L and C values. Results appear below as Table 2. Unlike the exhaustive 2x2 four-element arrays, the tabulated cases are solely for comparison among differently-structured designs (the structure leads to some high entries in Table 2). These sample computations use the definition L = TH, and require writing eq. (6) as:
C = T (9 - H)
Table 2. Indices for six thirty-six-element four-symbol arrays.
Some comments on how each array gets its values are instructive. Even though T is high for VI the number of internal symmetries lowers C and raises L. In contrast, IV is high-T, low-H; it possesses little internal organization which raises C and lowers L. The table values can be the basis of rank orderings such as the following alternatives:
Decreasing L: VI -> V -> II -> III -> IV -> I
Decreasing C: IV -> III -> V -> II -> VI -> I
Inspecting the H values in Table 2 demonstrates why we did not choose H alone as the pattern measure: it is highest for the pattern-less uniform array I. Multiplying by T gives a measure that corresponds more closely to our intuitive assessment of degree of pattern. To clear up the question of using T alone as the measure of disorganized complexity, note that the ranking by decreasing T is:
Decreasing T: IV -> VI -> III -> V -> II -> I
The two rankings (14) and (15) would coincide if VI were deleted, which shows how C differs from T. It is essential to measure symmetries (and not just visual temperature or symbol variation) as can be seen from the role of VI in the above ranking: intuitively, VI has low disorganized complexity, validating (14) rather than (15). The combination of measures in C is therefore useful as a way to detect visual disorder or disorganized complexity. Further, the calculations display differences between similar arrays such as III, IV and VI, which are superficially alike due to their substructures and symmetries.
The measures could be refined further using hierarchy, but even now they distinguish the visual designs on the basis of internal structure. We see that L and C differ in essential ways. L corresponds to organized non-trivial structure of visual patterns, whereas C measures disorganized internal structure. We have succeeded in clarifying the role of symmetry in these two complementary measures of complexity. Preliminary work (a survey of people in a talk audience) indicates that the numeric measures accord with human perception of the six patterns. Similarly, the following imperfection example supports using the L and C measures to study image degradation.
5.1 The Effect of Imperfections
Suppose the design II in Fig. 4 were made more symmetric by removing the imperfection in subblock 8 (see Fig. 3). How does that change become reflected in the measures we have proposed? Figure 5 shows the two alternatives, II and the remedied version VII.
+ + 0 0 * *
+ + 0 0 * *
# # # # # #
# # # # # #
* * 0 # + +
* * + * + +
+ + 0 0 * *
+ + 0 0 * *
# # # # # #
# # # # # #
* * 0 0 + +
* * 0 0 + +
Figure 5. Locally imperfect and symmetric versions of a design.
Although VII changes just three elements from II (an amount that is only about 8% of the data), it raises L about 28% and lowers C approximately 26%. (The indices for VII are: T = 1.67, H = 3.67, L = 6.11, C = 8.89). This shows a significant impact on the L and C measures from added symmetry introduced by changing a small amount of data. Other questions, not addressed here, concern the degradation of images and how far objects with very high L are still recognizable even with imperfections.
6 Psychological Responses
This section is conjectural, based on empirical observations by the authors. A circular diagram documenting opposite psychological responses to physical environments (Nasar 1989; Russell 1988) prompts us to adopt the same descriptors for the response to visual images: unpleasant versus pleasant; gloomy versus exciting; sleepy versus arousing; relaxing versus distressing. As outlined in (Russell 1988; Schlosberg 1952; Ward & Russell 1981), this work establishes a two-dimensional field of responses which is a mixture of these variables, depending on their relative proportion.
We believe that easily-absorbed visual information uses a structural organization of complexity and that the metrics we have devised can quantify that organization. Structures generate a psychological response on a viewer because human beings judge the content and organization of environmental information.
It appears that our indices L and C are consistent with the above psychological descriptors. Going from a low to high C visual image or environment tends to alter one's response from sleepy/relaxing to arousing/distressing. On the other hand, going from low to high L changes one's response from unpleasant/gloomy to pleasant/exciting. An examination of the Figure 4 and Figure 5 arrays illustrates this to some extent. More dramatic support comes from using this model to estimate L and C for twenty-five famous buildings (Salingaros 1997). The results agree with most people's responses to those buildings.
The psychological model helps to establish that observed states are two-dimensional combinations of theoretical states. Using two separate measures such as L and C for the perception of information represents a departure from previous one-variable approaches to complexity and organization.
This paper developed useful pattern measures for visual arrays. Several examples demonstrated the capability of such an approach for differentiating array complexity and organization. We showed the need for measurement to be based on internal substructures. By combining symbol and symmetry measures, we enabled a quantitative characterization of the complexity of two-dimensional arrays, and distinguished between uniformity, organized patterns, and disorganized complexity. Our result is based on only two examples. It doesn't attempt to answer the numerous questions about human psychological perception viewing such two-dimensional arrays. It is clear, however, that our model is directly relevant to these points and that it identifies the basis for measuring the degree of pattern and complexity. We contend that the new measures described here open a useful path to the systematic investigation of array data.
Computations were performed by Trask Stalnaker, Department of Mathematics, UCLA. N.A.S. is supported in part by a grant from the Alfred P. Sloan Foundation.
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