LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN cop The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN LSEP 2 1 19 DEC 9 i Red L161 — O-1096 ~-?ruSCi mucDcs-R-71-^90 COO-2118-0027 A PROBLEM IN FORM PERCEPTION: ODD SHAPE DETECTION By Kiyoshi Maruyama December , 1971 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS COO-2118-0027 UIUCDCS-R-T1-J+90 A PROBLEM IN FORM PERCEPTION: ODD SHAPE DETECTION By Kiyoshi Maruyama December, 1971 Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 6l801 This work was supported by Contract AT(ll-l)-21l8 with the U.S. Atomic Energy Commission. Digitized by the Internet Archive in 2013 http://archive.org/details/probleminformper490maru -111- ACKNOWLEDGMENT The author wishes to thank Dr. Bruce H. McCorniick, Professor of Computer Science at University of Illinois, for suggesting the problem and for his helpful discussion and comments. Thanks are also extended to Mrs. Judy Arter who both typed and helped edit this report, -IV- A PROBLEM IN FORM PERCEPTION: ODD SHAPE DETECTION Abstract A procedure to determine an odd shape or form in a given set of shapes, each set consisting of more than two shapes, is described. The "oddness" of a shape in a set of shapes on the basis of a specified feature is defined and then the "most odd" shape on the basis of many- detected features is defined. To make the system response like an average human response, modification of an assumed weight vector by means of an "attention mechanism" is considered. Some of the results attained for various criteria by the system simulation are as follows: (l) A comparison of Linear Separation and Euclidean Distance Separation methods indicates that the Linear Separa- tion model adequately models psychological form perception. (2) The attention mechanism described in this paper gives better system responses than those given without it. (3) The gestalt measure (emphasized in the psychology literature) which is defined as the function of the peri- meter and area of a given shape becomes less important. (k) The shape representation used in this paper, called the pattern sequence, S (x ), has many advantages in the analysis of geometric shapes. Index terms shape, form, oddness, feature, pattern sequence, linear separation, Eucledian distance separation, attention mechanism, system response, human responce. -V- TABLE OF CONTENTS page 1. INTRODUCTION 1 2. DEFINITION 6 3. CONTOUR ABSTRACTION AND FEATURE DETECTION. ... 12 3.1 Contour Abstraction 12 3.2 Feature Extraction 2h 3.2.1 Description of Selected Features . . 2k 3.2.2 Feature Dependence on Circularity. . 30 k. A SYSTEM TO DETECT AN ODD FORM 39 k.l Analysis 39 k.2 Attention and Weight Modification k2 k.3 Decision and Selection hk 5. HUMAN RESPONSE AND SYSTEM RESPONSE 51 5.1 Human Response 51 5.2 Attention Function 51 5.3 Weight Vectors 53 5,k System Responses 59 5.5 Experimental Results 6l 5.5.1 Definition of Odd Form in Terms of D-interval or Dispersion 6l 5.5.2 LS & EDS 6l 5.5.3 Given Weight Vector "w 6h 5.5.^ D-table and T-table 65 5.5.5 Attention Mechanism 65 6. CONCLUSION 67 -VI- LIST OF REFERENCES 69 APPENDICES A. Twenty Sets of Four Shapes Used in the Experiment 72 B. Computer Extracted Feature Values for the Twenty Sets of Shapes Illustrated in Appendix A 78 C. Examples of System Response for the Decision of Oddness 89 D. Some Other System Responses Under Various Criteria 10 U -1- 1. INTRODUCTION Stimulated by the publication of Wertheimer 's "Principles of Perceptual Organization" in 1923 many psychologists have studied inten- sively human perception of shape and form. A good survey paper of these studies is found in Hake (1957) and a recent book, Visual Perception of Form by Zusne (1970), is an excellent summary and contains an extensive bibliography. A 1966 collection of papers edited by Leonard Uhr deals ■with various problems of pattern recognition by computers. However, many of the more crucial problems such as how humans encode shape infor- mation, detect near similarity or dissimularity of shapes, focus atten- tion upon particular aspects of shapes, and achieve perceptual invariance are not yet adequately understood. A basic weakness of shape/form recognition by present computer systems may stem from the observation that shape/form recognition by humans has not yet been solved on the psychological level. Zusne (1970) says: In the history of modeling pattern perception on the computer, the task is really one of modeling discrimina- tion rather than recognition. Recognition in a computer is recognition only in a trivial sense. Recognition by organisms implies previous experience, and with experience there accrue meanings to patterns that are uncorrelated in any important way with the physical dimensions of patterns. Whether a machine will ever be able to cope with this pro- blem is uncertain. (page 8U) The objectives of this report are to construct a computer sys- tem in which the system response is demonst rat ably similar to an average human response on a specified task and to provide clues for exploring 1. task: detection, discrimination, recognition, identification, judgment (Hake 1957). -2- some unsolved problems of shape perception in psychology. Let us first investigate the problems involved with defining shape and form. Webster's Third New International Dictionary (l96l) defines shape as "the visual make-up characteristic of a particular item or kind of item: characteristic appearance or visual form, or spatial form or contour that is used". The term "form" is somewhat more nebelous— it is used by different people to mean different things and by the same person to mean different things on different occasions. Shape, figure, structure, pattern, order, arrangement, configuration, plan, outline, and contour are similar terms without distinct meanings. This indefinite terminology is a source of confusion and a more rigorous terminology may b e very much needed. The Webster's Dictionary defines form as "the shape and structure of something as distinguished from the material of which it is composed", and it defines gestalt as "a structure or configuration of physical, biological, or psychological phenomena". A number of explicit definitions of visual forms have been given by Gibson (1951); they were proposed out of a conviction that "psychologists should come down to earth to say exactly what they mean when they talk 2 about form". According to his proposed definitions solid forms and sur- face forms are realities. However, there is no such thing as form-in- general with the universal characteristics ascribed to it by gestalt theorists The only kinds of visual form we shall undertake to deal with hereafter are those associated with or derived from physical objects. 2* Solid form: the closed physical surface enveloping a substance of some kind. The surface may be curved or it may be composed of adjoining flat surfaces with edges. 3 Surface form: a flat physical surface with its edges. -3- Let us describe briefly what kind of experimental studies have been made on human shape perception by psychologists. They have first tried to define what is the quality of "goodness" in a shape or form. If a line forms a closed or almost closed figure we may no longer see merely a line on a homogeneous background, but a surface figure bounded by the line. It seems that a straight line is a more stable structure than a broken one and that therefore organization will occur in such a way that a straight line will be perceptually continued as a straight line (this idea was implemented by Guzuman ( 1968) in his "Decomposition of Scenes Into Bodies"). • Koffka (1935)' generalizes this as: "Any curve will proceed in its own natural way; a circle as a circle, an ellipse as an ellipse, and so forth". Attneave (195*0 says that many of the gestalt principles of perceptual organization pertain essentially to 4 information distribution and the good gestalt is a figure with some high degree of internal redundancy. It may be that an important criteria for "goodness" of form, however, is the relative importance of simplicity and complexity — a "good" configuration being simple. The theory is that our perceptual and memorial processes tend toward simplification of forms. To get a "good" figure from a given figure, Mowatt (1940) obtained the statistical result that the large majority, about 9^%, of all changes to a given figure consist in the addition of lines or dots to the figure o He also shows that open figures are changed much more frequently than closed figures , and he concludes that closure and symmetry are con- sidered "good" properties of a figure. Another way to determine goodness k. It was demonstrated by Attneave ( 195*0- that information is concentrated at points where there is a change in an otherwise continuous gradient; at the contours of a form which mark the change from ground to figure; and at any inflection along the contour where the direction of the contour changes most rapidly. -u- of shapes is to determine some kind of threshold function (Bohbitt 19U2). For example, the threshold may he stated in terms of the percentage of the total perimeter of the figure present at the point in the series represent- ing the transition between the forms having the quality of twoness and those having the quality of oneness. In 1955 Deutsch observed that "the perceived form or shape of patterns must be invariant with changes in location and orientation of the patterns, and even minor changes in shape itself". However, recognition of forms and comparative judgment of forms deteriorates when forms are rotated with respect to the observer. In fact, there is considerable evi- dence to indicate that orientation plays a part in the definition of what ve mean by the form of an object (Dodwell 1961). Thus it may be so that the form of an object may not be invariant with rotation of the object and is partly defined in terms of orientation characteristics (see Zusne). Some of the stimuli, or target shapes, which have been used by psychologists to study the problem of human form perception are: Dot Patterns (French 195*0, Metric Figures ( Pitts and Leonard 1957), Random Polygons (Attneave and Arnoult 1956) and Computer-Generated Luminous Poly- gons (Polidora and Thompson 196U). Because of the simplicity of generation by computers, polygons are used quite frequently in psychological experiments. While it is true that a curved shape may be approximated with straight line segments without appreciable loss of information, some information is irretrievably lost, depending on how crude or fine the approximation is. How such approximation affects perceived complexity is not known. The methods of polygonal shape representation which have been j studied are mostly independent of shape orientation (see Deutsch, 1962, Attneave and Arnoult, 1956). However, it may be that the perceptual -5- independence of shape orientation may not play as significant a role for form perception in psychology. For the analysis of shapes and forms, a large number of physical characteristics and form invariances have been suggested and studied. Some of these are: the ratio between area and peri- meter, maximum dimensions of geometric forms, number of angles and line seg- ments, number of elements in dot patterns, texture, regular-irregular, sim- ple-complex, and first, second, third and fourth moments of area. There have also been many other studies: theories of visual form perception, perceptual tasks, memory for form, form perception by animals, application, etc., have been done and being continued by thousands of psychologists (see Zusne 1970). However, as we stated earlier in this chapter, knowledge of how humans perceive shape and form on a psychological level is still exceedingly primitive. The plan of this report is as follows: In Chapter 2, we define shape and form. These concepts are then refined to provide a mechanism to identify the oddness of a shape in a given small collection of shapes on the basis of one or more features. In Chapter 3, we introduce a shape representation called a "pattern sequence", and a process to detect features appropriate for shape representation. In Chapter k 9 a system to detect an odd shape or form from a given set of shapes is described, and the comparison between an average humans response and the system response is discussed in Chapter 5. In Appendix A, we show stimuli which are used for the experiment and their features are listed in Appendix B. In Appendix C, an example of the system response on the decision of oddness is given; and other aspects of system response under various criterion are tabulated in Appendix D. -6- 2 . DEFINITION Let us first define shape Shape: a closed physical contour (or a closed physical surface where the texture is assumed not to influence our system perseption of shape.) Since texture can be considered as one of many features, we can simply modify the feature vector, which will be discussed in the following chapter, to compensate for the texture influences in shape perception. However, in this report, we assume that the texture is identical for shapes presented simultaneously and, thus, we simply omit it. We assume a system for shape perception which is somewhat similar to the one given by Attneave & Arnoult (1956)* That is, we assume that the first stage of shape or form perception is "contour abstraction", the second is "feature detection" and the third stage is "analysis and decision". Our working hypothesis is that neither shape nor form is affected by rotation, translation, or the relative position to other objects. In this chapter we describe briefly our model system to detect an odd shape in a given set of shapes. The details of the system will be elaborated in the following two chapters. We define an odd shape/form in a given set of shapes as the one different from the others on the basis of some features. Let us first define discrimination within a given set of shapes on the basis of one single specified feature. (We assume that the given set of shapes to be presented to the subject (man/machine) consists of more than two shapes.) We assume that the given set of shapes consists of n shapes, $2 , if . . . , 9. . . , il . -7- Assume that a specified feature, n>3, takes test results x n , x^, .... x ^= 1' 2 ' n for the n shapes respectively. Definition 1 Odd Shape/Form on C We define the D- interval, d . , for each ft, as J J n n d 4 = max { 6c } - min { xj- . J k=l * k=l ^ Then we define ft. as an odd shape/form in the given set of n J 2 n shapes on the basis of C if d.= min {cL} k=l k To see how the above definition of oddness reflects human response on the basis of a specified feature, let us consider the examples illus- trated in Figure 1. We assume, for simplicity, that the specified features are the number of vertices and convexity. Table I shows the extracted fea- ture values for the specified two features , and Table II shows the D-inter- vals, where we assume that the convex shape is denoted with value 1 and the concave one with 0. For the set A, ft, is said to be an odd shape on the basis of the number of vertices, while ft would be chosen as an odd shape on the basis of convexity since d = is the minimum. For set B, the number of vertices is not significant for the decision of oddness, since d., j = 1, 2, 3, k t assume identical values. Similarly, for set.B, J the feature of convexity does not play any significant role for the decision of oddness since ft and ft_ are concave and . 3_ and ft, are convex. The following is an alternative definition of oddness based on a single specified feature: 2. It is easy to see that d. is minimum only if x. is an extremal. J J -8- a a. a- a 4 (a) Set A of Four Shapes a. &. a-. a. (b) Set B of Four Shapes Figure 1 Trial Sets of Shapes -9- Table I Feature Values of Shapes in Figure 1 ftj ft 2 "3 jty Set A Number of Vertices 5 6 6 3 Convexity 1 l 1 Set B Number of Vertices 5 k 5 U Convexity 1 1 Table II D-Interval by Definition 1 n i a 2 3 ft u Set A Number of Vertices 3 3 3 1* Convexity 1 1 0* 1 Set B Number of Vertices 1 1 1 1 Convexity 1 1 1 1 Table III Dispersion by Definition 2 ft 1 ftft 2 ft 3 °U Set A Number of Vertices k 10/3 10/3 U/3* Convexity U/3 U/3 0* U/3 Set B Number of Vertices h/3 U/3 U/3 U/3 Convexity U/3 U/3 U/3 Ufl Definition 2 Odd Shape/Form on C We define dispersion , d , for each ft as n cL =2 |x.-x I where x = U x )/(n-l) Again we define ft. as an odd shape/form in the given set of shapes J n . on the basis of feature C if d = min l±). . J k=l ^ Table III shows the dispersion for shapes which are illustrated in Figure 1. Comparing Table II and Table III, we see. there is no significant difference between the odd shapes detected by Definition 1 and those detected by Definition 2 C The only difference that appears in these two tables is in the very first rows for the set A. If we consider detection of two shapes from each set of shapes; one for the most odd shape and the other for the second most odd shape, then the Dispersion approach gives the second most odd shape as either ft or ft for the set A on the basis of the number of vertices. However, the D-interval approach gives the second most odd shape as either ft , ft or ft„. At this stage, we have no criteria for which definition of oddness is preferable, especially when the number, n, of shapes ±° small (about k to 6). However, as n becomes large then Definition 2 is preferable to Definition l. These two definitions become identical when n is 3. Thus far we have discussed oddness of shape on the basis of a single feature, C. We will now expand the above definitions to enable choice of oddness on the basis of m features, C, . C_, ..., C, ..,, C . 9 1' 2* ' i' ' m Definition 3 Odd Shape/Form on C n , ..., C. " ... , C 1 l m Let us assume that d. . denotes the D-interval or dispersion of the j-th shape, ft., on the basis of feature C. , and let -11- H j = l.l (W i d ij )K for J =1 » 2 ---» a Then the shape J2 is said to be the most odd shape in the given set of n shapes if n H. = min {H, } J k=l k Here W = (V^, W 2 , , VL , , W ffl ) is called the modified weight vector, which can be adapt ively varied, and will be discussed in Chapter k. In the above definition, if K=l then the method is called the Linear Separation (LS); if K = 2 then this is called the Euclidean Distance 3 Separation (EDS); and if K>3 it is called the K-th degree Separation. 3. Attneave (1950) asserts that psychological distances could not be fitted in Euclidean space since the psychological difference between any two stimili is approximately equal to the sum of their differences with respect to the physical variables, not to the square root of the sum of the squared differences as demanded by an Euclidean space. A comparison between a Multiple Regression and Multiple Discriminant models can be found in Jones (1971). -12- 3. CONTOUR ABSTRACTION AND FEATURE DETECTION In this chapter we describe the representation of shape by a pattern sequence, the features which have been chosen to characterize a shape, and the process by which we extract features for each given shape. A simple model for contour abstraction and feature detection is illustrated in Fig. 2 and it will be discussed later. I 3,1 Contour Abstraction Let us first describe a class of target stimuli and then we will introduce the representation of this class. As many experimental psychologist: have used polygons for the study of form perception, we will restrict our class of stimuli to a class of polygons. A polygon fl with basic points p., i = 1, 2,..., v, is said to be an gularly simple if there exist a point x q a^ the line^ does not intersect any edge of a for all i. In other words, Q is angularly simple if and only if there exists x Q in fl such that at x Q the contour of Q is totally visible. Definition h A Pattern Sequence , S r (x Q ), is defined as follows: MV = S r' S r+l> # -- 9 B r+i*'"" S r+N-1 If r+i < then r+i is interpreted or r+i+N and then for all i, r+i (mod N). In the above definition, r denotes the rotation index such that r>0 mean.s clockwise rotation of S^) by angle r (2H/N), and r<0 means counter clockvis rotation by angle r (21/*). ^ is a point called the encoding point and N j is the number of elementary radii called the circularity of the pattern sequence. We will consider this pattern sequence representation of a class of polygons, especially angularly simple (AS) polygons. As one -13- Stage A Stage B Stage C Stage D Stage E Input dbjects X sequences of point coordinates) I Find an encoding point, x , for each object I Generate pattern sequence, S (x Q ), for each object I Normalize S (x ) and give an orientation to each S (x ) I Do n number of tests (experiments or character detections) for^each S (x ) , and form n vectors, X, of dimension m (reorientation may be necessary) . % Figure 2 Contour Abstraction and Feature Extraction -lU- can see that this representation for a stape simplifies rotation and trans- lation operations. The detailed discussion of the correspondence between polygons and pattern sequences can be found in Maruyama (1971 a, 1971 b). Stage A Input N number of polygon s , fl^, . . . , Sly « • « » ^ n We assume that each polygonal shape, Sly is given as a sequence of point coordinates, p^ p 2 , .... P v> where each adjacent pair of coordinates corresponds to an edge of the polygon. Moreover, this sequence is considered closed, i.e. the pair P v and V± are adjacent. An example of a polygon with a sequence of point coordinates, v^ .... P 15 > is illustrated in Figure 3. Stage B Find the encoding point, x Q , for each 0. To generate a pattern sequence, S (x Q ), from a given polygon, we must find the encoding point, x Q , for each polygon Sly To have a unique pattern sequence for SI. we must uniquely determine x Q . (x Q should not be affected either by the rotation or translation of Sly) The center of gravity of the polygon can be considered as the encoding point because of the uniqueness of the center of gravity. If we restrict our class of polygons to a class of AS-polygons, we can use a quite different approach of find- ing the encoding point. Since each angularly simple polygon defines a convex set, which consists of all angularly simple points inside the poly- gon, we select a point inside the convex set which maximizes, or minimizes, a given objective function (Maruyama 1971 b). n-1 n-1 _ 1. An example is to maximize Z s. , or to minimize I (s^s), n-1 i=0 1 i=0 s = ( I s.)/N, where the s.'s are obtained from the contour of the i=0 1 convex set instead of the given polygon. -15- A polygon illustrated in Figure 3 is an AS-polygon since at any point in the convex polygonal region which is denoted by dotted contour, the entire boundary of the given polygon is visible. An example of a non- AS polygon is illustrated in Figure k where x denotes the center of gravity. Stage C Generation of Pattern Sequences i) Class of AS-polygons as Stimuli The encoded configuration of the AS-polygon of Figure 3 is illus- trated in Figure 5. The vector which corresponds to the X-coordinate denotes the very first elementary pattern, s , of the pattern sequence, 5 (x ); the rotation index, r, is assumed to be at this stage, and each elementary pattern, s. , follows in the counter clockwise direction. In this example, the circularity, N, of the pattern sequence is kQ, Figure 6 shows error dependence on the circularity N for the polygon of Figure 3, where the values obtained at N = 180 are assumed to be true values. For small circularity, errors are heavily dependent upon the orientation of the given shape. However, for sufficiently large circularity, orienta- tion errors become essentially negligible. To generate each elementary pattern, s., detect the innermost inter- section between the line segment at angle i2ir/N and an edge of the poly- gon. The pattern sequence generated in this manner is called the interior (or primal) pattern generation. i±) General Polygons as Stimuli Let us consider the polygon which is illustrated in Figure U, where x_, the center of gravity, is used as the encoding point to generate a pattern sequence. -16- Figure 3 Input Representation of a Shape -17- Figure k An Example of A Non-Angular ly Simple Polygon (x n Denotes the Center of Gravity) -18- Figure 5 Example of An Encoded Representation By Pattern Sequence, S (x Q ) -19- V(180)-V(N) V(180) ♦ Sls.-sr/N i l K s=(Ist)/N 40 60 CIRCULARITY N 100 Figure 6 Error e vs. Circularity N for the Polggon in Figure 3 -20- Alternatively, to generate each elementary pattern, s., instead of measuring distance from x to the innermost radial intersection, the following method (which is not implemented in the model discussed), called exterior (or dual) pattern generation, is used. We first choose a circle of radius R which is large enough to con- tain the polygon, where the center of the circle is the center of gravity of the figure. Then we measure the distance, s . , to the outermost radial intersection and the "boundary of the encompassing circle of radius R. Finally the desired pattern sequence, S (x ) is derived by s. = (R - s.) for i = 0, ..., N-l. The above process of generating the pattern sequence, S (x ), of the polygon in Figure h is illustrated in Figure 7, where the dotted arrows show the s.'s and the solid arrows emanating at x show the s.'s. The contour obtained by dual pattern generation is also illustrated in Figure 7. As we can see from the contour of the pattern sequence, we lost some information about the polygon of Figure h. This loss of information can be compensated by the introduction of many pattern sequences (Maruyama 1971b7. However, we feel that the polygon denoted by a single pattern sequence is accurate enough for the study of geometrical shapes. From a study of the above two methods of generating pattern sequences, we see that sharp corners of polygons do not show up in the abstracted con- tour. One way to avoid such information loss is to increase the circularity N. However, the problem here is the estimated. error of encoding a given shape by a pattern sequence of circularity N. In Figure 8, we show abstracted contours obtained by the primal encoding method applied to the polygon of 2. An exact representation of the contour of a shape (i.e. a pattern sequence which is generated by following the contour of a shape) is described in Maruyama ( 1971b). -21- Figure 3 with different values of circularity N. If the circularity is larger than 12, then the abstracted contour holds the outline of the origi- nal polygon in Figure 3, even though some minor abrupt changes of the orig- inal polygon have disappeared from the abstracted contours. However, when the circularity becomes small, like 6, then the outline of the polygon given by a pattern sequence is completely different from the contour of the orig- inal polygon. The detailed discussion of the optimal choice of circularity for our experiments appears in Section 3. (See also Maruyama 1971b). Stage D Normalization & Orientation Each elementary pattern, s* > in the pattern sequence S (x ) of an object SI , is normalized by the maximum-valued elementary pattern h, namely N-l s = max {s . } • r i=0 x Thus, each value of the elementary pattern of S (x., ) ranges from to 1. This normalization process eliminates differences in size of shapes for form perception* (s will, however, be stored for consideration of the actual size of the shape.) To assign an orientation for each pattern sequence, S_(x ), r is considered to be the orientation of the pattern sequence and the sequence becomes S (x ). This orientation r of S ( x_ ) may be changed r o r if the shape ft to be nearly symmetric. Another way to assign an optimal J orientation for a given shape, using an idea of vari -valued logic, has been 3 given by Tareski & Michalski. Stage E (Feature Extraction) This stage will be discussed in the following section. 3. Tareski, V.G. and Michalski, R.S., "Interval Covers Synthesis: Compu- ter Implementation of and Experiments with Algorithm A^" , Department of Computer Science, University of Illinois, Urbana, Illinois, 1971. (in progress) -22- -^►x Figure 7 Dual Encoded Pattern Sequence of the Polygon in Figure k H" fD Oo o | -d o o o H o *«j c+ H- OS O 4 o O 6 & O Cfi P Hj > 4 o" H- *J en c+ H- e+ < e+ 3 4 P P H c+ M> W OS o o Q p (B p> »ti m OS p o CD H) O c+ 0) 0' fD It 53 II H ro 53 II ON -2l+- 3.2 Feature Extraction The performance of a pattern recognition system depends crucially on the design of the feature extractor. Methods by which features are selected, however, are often intuitive and empirical, and depend upon the designer's experience with the problem. Both feature selection (Jones, 1971, Brown and Owen, 1967, etc.) and dimension reduction (Shepard, 1962, etc.) are important aspects of the design of a shape recognition system. The main guides to feature selection are: (l) Features should be invariant to irrelevant variations, such as limited translation, rotation, changes of scale, etc.; and (2) Differences that are important for distinguishing between patterns of different types need to be emphasized. Five major features and physical form measures most highly correlated with these requirements are tabulated by Zusne (1970, p. 223). They are "Compactness", "Jaggedness" , "Skewness" of contours relative to the Y-axis, "Skewness" of contours relative to the X-axis and "Dominant Axis". However, these features are not completely independent of the orientation of the shape. 3.2.1 Description of Selected Features In Table IV, we list 22 features which are independent of the orien- tation of a shape. Furthermore, these features are valid not only for a class of polygonal shapes but also for the closely allied class of polynomial k. Jones (1971) classified measures into 13 major features. 5. The parameters of dispersion, symmetry and elongation are considered to be analogs of the second, third and fourth moments of distribution of the area of a shape (page 215). -25- Table IV 22 Selected Features NO. NAME DESCRIPTION Fl MAXAMP ( s -s . ) , where s =max is . \ and s . =min js . \ max mm max 1 l 1 ; min 1 U' F2 MEANVDN N-1 s = z s . /N i=n x F3 DIVE# w "* 1 I "I / w i=0 Tk DSQUARE N-1 i=ol ( V 8) l /N F5 AVDIVE* N-1 N-1 I l — i / Z . -A s.-s / . _ s. i=Oi i I i=0 i F6 ALT# The number of alternations in S (x„) wrt s. r FT MAXIMAL* The number of maximal points in S (xj. r F8 VERTEX The number of vertices. F9 CONVEXV# The number of convex vertices. F10 CONCAVEV# ] The number of concave vertices. Fll D-EDGES Divergence of edge length. F12 PERIMETER Perimeter P derived from S ,(x ft ) F13 AREA* Actual area A derived from S (x ) FlU P OVER A P/A F15 P**2/A P 2 /A Fl6 P-SRA P//A FIT P-A-S (1-2/^A / P) F18 CONVEXITY Near convexity or concavity. F19 SYMMETRY Near symmetry or asymmetry. F20 IDS Interval difference sum. F21 EXTONE Existence of only one shape which is neither convex nor symmetric. F22 N-SIMILARITY Near similarity of disimilarity. -26- curved shapes. Features Fl through F19 can be extracted for each shape; the rest, however, are dependent upon the given set of shapes - because of the nature of the feature to be extracted. Some of these features are self evident and for those which are not, brief explanations are given below: Feature F6: the number of times the sequence S (x ) crosses the circle of radius s. For the polygon which is illus- trated in Figure 9(a), the number of alternations is 8. Feature F7: the number of local maxima in the sequence S (x n ). (This is identical to the number of local minima in the sequence.) In the example of Figure 9, elementary patterns s , s, , So, and s , correspond to local maxima, and s , s^-, s , and s i , correspond to local minima. Features F8, F9, and F10: the number of vertices (or the number of convex or concave vertices) is clear for polygonal shapes. For example, the polygon of Figure 9(a) has 12 vertices; 8 convex vertices and k concave vertices. However, it is not self evident for the polynomial shape of Figure 9(b), which is represented by the same pattern sequence of the polygon. For polynomial curved shapes, the number of convex vertices is defined to be the number of maximal points in S (x_); similarly the number of concave vertices is defined as the number of minimal points in S n (x_). Thus the number of vertices is the sum of the number of local maxima and minima. In the example of Figure 9(c) there are h convex vertices and h concave vertices. -27- a) A Polygon and a Circle of Radius s (b) A Polynomial Curved Shape Sq Sj Sg S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 Su S 12 S 13 S 14 S15 (c) The Sequence, S (x ), of the Above Polygon V Figure 9 Examples of Shapesaarid Their Pattern Sequences, S (x.) X 0' -28- Feature Fll: for a polygonal shape, Feature Fll is defined as | e — e | / (actual perimeter) where e is the j-th edge length and e is the mean J edge length. This feature value for the polygon of Figure 9(a) is zero since all edge lengths are identical. For a polynomial curved shape, distances between adjacent pairs of extrema (maximal and minimal points) are considered to "be the e.'s. For example, in Figure 9(c), s_s ? is an edge and s s + s s ? is a curve length for the shape of Figure 9(h). In other words, the distance between two adjacent extrema or the arc length between two adjacent extrema is interpreted as e . . Feature 18: a shape Q is said to be nearly convex if for each J pair of adjacent maximal elementary patterns the minimal elementary pattern between the pair is greater than the minimal pattern obtained by assuming that the pair form a straight edge of the given shape. In Figure 9(c), for example, s and s> are a pair of adjacent maxima and s p is the minimal elementary pattern. Since the minimal pattern length which is derived by assuming that the pair s and s, is connected with a straight edge is greater than the length of s„, we conclude that the shape which is denoted by the pattern sequence of Figure 9(c) is near concave. -29- Featute P19: a shape fl, is said to be nearly symmetric for orienta- tion r if the following property holds for S (x rt ): r u 3 ... - s . <0.lU • max(s ._ , s . ) r+i r-i ' r+1 r-1 for i = 1, 2, . .., N/2 - 1, where if r-i < then r-i is interpreted to be N+r-i and r+i (mod N) for all i. Feature F20: let s. denote the i-th elementary pattern of the k-th shape. Then the 20th feature value of the j-th shape fi. , x pf) ., is defined as: L 20j N-l n n k E (max{s, } - min{s. }) / N i=0 k=l k=l X ¥1 *l Feature 22: the near similarity of shapes is defined as follows. Let us define a mask bit M. for each feature C. , i i i= 1, 2, ..., 21, where M. assumes either or 1 depending upon which features, C through C , should be satisfied by "near" similar shapes, e.g. M. = 1 for all i. Then we define shape ft to be the odd J shape in the set of n shapes only if the shape Q is detected as an odd shape for those features whose mask bit is one. 6. The parameter of symmetry is considered to be the analog of the third moment of distribution of area [Brown and Owen (1967)]. -30- 3.2.2 Feature Dependence on Circularity In the first section of this chapter, we briefly discussed feature value dependence upon the circularity N. Here, we will continue this discussion as well as illustrate how circularity affects the detection of the odd shape using the four shapes: ft , ft_, ft and ft, (illustrated in Figure 10 ). Tables V and VI show MEANVDN(F2) values and DSQUARE(f1+) values of these four shapes with respect to different choices of circularity N, respectively. In these two tables , those feature values extracted at circularity N=96, whose values are identical to those extracted at N=l80, are considered to be true feature values of the given shapes. To illustrate feature dependence on circularity, errors of MEANVDN and DSQUARE for shapes ft, , ft p , ft~, and ft, , with respect to different circularities, N, are illustrated in Figures 11, 12, 13, and 1^, respectively. Roughly speaking, errors decrease as the circularity N increases for N>i+8. To understand how these errors affect the detection of odd shapes with respect to differences in circularity, we show, in Table VII, odd shapes detected at different circularities on l6 features, again using the four shapes of Figure 10. As we can see from the table, for N>U8, the odd shape detected for each feature is stable. However, for smaller N, detected shapes vary. It is interesting to note that the variation of circularity does not influence significantly the detection of an odd shape on the basis of a single specified feature, especially for N>2U. Those features which are influenced by the reduction of circularity and which also affect seriously the detection of the odd shape are DIVE#(F3) and MAXIMAL#(F7) . From the above argument, we conclude that circularity N=U8 is sufficient for the design of our model. As examples of computer extracted features, feature values for the shapes of Figure 10 are tabulated in Table VIII. -31- a, a. a- a. Figure 10 An Example of a. Set of Four Shapes for the Detection of the Odd Shape -32- Table V MEANVDN (F2) With Respect To Different Circularity N. >-^L "} "2 "3 "1+ 6 o.too 0.1+89 0,992 0.368 12 0.597 0.1+67 0.970 0.1+15 18 0.61H 0.1+33 0.950 0.328 2k 0.601 0.1+39 0.916 0.350 36 0.615 0.1+33 0.91+9 0.335 1+8 0.607 0.1+305 0.911 0.329 72 0.608 0.1+29 0.910 0.321+ 96 0.608 0.1+38 0.910 0.323 Table VI DSQUARE (jk) With Respect To Different Circularity N "X. fi l "2 "3 % 6 0.0900 0.0538 0.0001 0.0800 12 0.079^ 0.0576 0.0001 0.1126 18 0.0799 0.0327 0.0009 0.01+06 2k O.0661 0.0359 p. 0017 0.0651 36 O.0806 0.0303 0.0009 0.0501 1+8 0.0731 0.0282 0.0012 0.01+36 72 0.0751 O.0265 0.0011 0.0381+ 96 O.0761 0.0259 0.0010 0.0365 1.0 0.8 Vu cr o o: cr UJ 0.6 0.4 -33- ■* DSQUARE * MEANVDN 0.2 40 60 CIRCULARITY N 80 100 Figure 11 Errors of Shape of Figure 11 With Respect to Different Circularities, N -3U- DSQUARE * MEANVDN 40 60 CIRCULARITY N 80 100 Figure 12 Errors of Shape fi ©tf Figure 11 With Respect to Different Circularities, N -35- • DSQUARE * MEANVDN 40 60 CIRCULARITY N Figure 13 Errors of Shape fi of Figure 11 With Respect to Different Circularities, N -36- 20 ♦ DSQUARE * MEANVDN 40 60 CIRCULARITY N 80 100 Figure Ik Errors of Shape R, of Figure 11 With Respect to Different Circularities, N -37- Table VII Odd Shapes Detected At Different Circularities, N, On Different Features Using the Shapes of Figure 9 (Those rows marked .with * indicate features which are influenced by the variation of circularity and which are also affected for the detection of the odd shape. ) C ^ ^^ 6 12 18 2U 36 1(8 72 96 MAXAMP Q 3 « 3 S H 3 "3 "3 §3 »3 MEANVDN "3 "3 s "3 "3 s s Q„ DIVE# * s fl 3 S S n 3 n l "1 "1 DSQUARE * "3. "3 h °3 °1 Oj "1 "a AVDIVE# "3 "3 "3 "3 s "3 « 3 °3 ALT# * - "2 "3 "3 fl 3 S "3 S MAXIMAL* * - % "i "2 "3 "3 "it "it VERTEX/? °1 °1 "i "l B l °1 "l "l D-EDGES * n li n i % "l n i "l "l "l PERIMETER * s °? "1 °1 "l "l °1 "l AREA# "3 °3 "3 "3 °3 °3 s "3 P OVER A % 4 n l, °lt a k a h n l» «1, P**2/A * °3 n 3 "l4 % n it n l, "it P-SRA * "3 n ii s "l, a k "i* "it a k P-A-S « 3 " 3 "3 "3 S "3 "3 s IDS "3 "3 "3 °3 n 3 "3 °3 "3 -38- Table VIII Computer Extracted Features for Shapes Q , ft p , n_ , and ft« of Figure XI **4 <$******• .******* ************************* OBJECT NC = 1 2 3 4 MAXAMF MEANVDN 0.7196 0,6072 C.6865 0.4305 0.1191 0.9111 0.8380 0.3289 DIVE* CSGUARE = 0.2482 0.0731 0.1253 0.0282 0.0273 0.0012 0.1504 C.0436 AVDIVE* ALT* 0.4087 8 0.2911 4 C.C299 16 0.4573 6 hAXI^AL* = VERTEX* 4 12 4 4 8 8 3 6 CCNVEXVfl - CCNCAVEV4= 8 4 4 8 3 3 D_EDGES = PERIMETER^ 0.3C45 7.4614 0.0047 4. 1925 C.0024 5.9203 0.0075 5.5440 AREA* P OVER A = 548.5852 5.5349 384.5632 6.4110 2081.8154 2.2750 226.7480 12.9339 P*r2 / A = P_SRA 41*2919 6.4263 26. 8779 5.1844 13.4688 3.67C0 71.7050 8.4679 P_ A_ S 0.4484 SYMMETRIC C.3162 SYMMETRIC C.0341 SYMMETRIC 0.5814 SYMMETRIC IDS CCNCAVE 3C.56C7 CONVEX 30.4554 CONVEX 18.1797 CONCAVF 24.1289 -39- k. A SYSTEM TO DETECT AN ODD FORM We described the outline of our model for the detection of an odd shape/form among a given set of shapes in Chapter 2. This chapter develops the model rigorously. The "block diagram of our model is illus- trated in Figure 15 and we shall discuss here the role of each block. Let us assume that for shapes ft., j = 1, . . . , n, that X., J J j = 1,..., n, are feature vectors of dimension m, respectively, where: A- — \ X Tn» X p]» * * * » X Qt"'! X ml ■A- • — \ X- . , X~ • s • • • J X ..*••• , X ./ J lj 2j ij mj * t n In' 2n' in' mn X = ( X^ , X- 5 • • • 9 X,..., X ) = [x. . ] , x. . e R ij mxn' ij m: the number of features considered, n: the number of shapes presented, x. .: the feature value of the j-th shape 1J ft . on the basis of the i-th feature. J k.l Analysis A D-Table is defined as: D = [d ij ] mxn where, Case 1 (Using Definition 1 of D- interval) a ij = \i - h uhere h = £? {x ik } d. . = min {x #1 } 1J k^j lk OJ T L CU' -Uo- (2) ATTENTION a WEIGHT MODIFICATION W x A x 2 (1) ANALYSIS (3) DECISION a SELECTION OUTPUT j €{l,2,3,-'-,n} Figure 15 A System to Detect An Odd Shape/Form Among A Given Set of N Shapes -la- Case 2 (Using Definition 2 of Dispersion) n d. = 2 x., - X. . "J k=l lk lJ 4\ n rd whe*e x = ( E x., )/(n-l) 1J k=l lK Thus, d. denotes the original D-interval or dispersion for the shape 0. on the basis of the i-th feature, C. Since ve are dealing with m different features, we will normalize the D Table, i.e., d. . = d.Vd. where d. = rnax-^d. .]. This is quite important because each individual feature value J is evaluated by quite different scales. The T Table, which is derived from the D Table, may be used to save storage for our model: T = [t,,3 ij mxn i .1 if d. . = min {d. n } where t. . = < ij . lk ij k otherwise Thus t. . is 1 if and only if shape ft. is defined as an odd shape, by -*■ J J either Definition 1 or 2, on the basis of the i-th feature C. l -U2- U„2 Attention and Weight Modification Since humans pay attention to some features more than others (these attention centering features varying -with the object) we will introduce a mechanism, called the attention mechanism, into our model. We call * the attention function and we assume is a mapping from the D table into a vector A of dimension m, called the attention vector. Thus = . for all i. Then the modified weight vector W w = (w x , w 2 ,..., w.,..., w m ) is defined as W. = w A. for all i where i ii to = (co,,Wp,...,oj.,...,to) is the originally given (or assumed) weight vector (e.g., w = (l, 1,..., l) implies that all features are considered to be equally significant). Let us give a simple example of such an attention function ; $ : (a., g.) ■*• A. for all i where a . = min {d.. } , ill l , ik k 3. = min {d.. ly*.}. i . ik ' l More explicitly A, . « it - a.) -U3- l.U < 0.8 K O o < li- 0.6 Z o h- z UJ o.4 < 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 16 An Example of Attention Function : (6. - a. ) •* A. Where A.e[0,l] -kk- and <» has a property such that whenever B ± - a. > B^ - a.' then (6. - a.) > 4 (3/ - a.») holds. A simple example of such a function is illustrated in Fig.i6 , where A ■ d> (6 - a.) = 3. - a. for all i. Many other attention functions based i i i i * on the value (ft. - a.) are discussed in the following chapter. U a 3 Decision and 'Selection The role of this block is to form a vector H of n dimensions and then select an odd shape according to Definition 3. H = (H x , Hg, ..., H , ..., H n ) where: Case 1 (B Table is used) K/m Case 2 (t Table is used! H. = / I (W.t. .) , K> 1 . J / i=l X 1J = The selection of the most odd shape or form is accomplished in the following manner: Case 1 n Q. is the first choice odd shape if H. = min {H } J J k=l * n tip is the second choice odd shape if H* = min {HiJ^,} ■A5- Case 2 ft is the first choice odd shape if H = max {H } J J k=l k n ttj is the second choice odd shape if Ep = min {Hj#I ; } k= * - ~r -k-7 To get some idea of how our model behaves, let us consider an example of detecting an odd shape or form from a set of four shapes; £2. , ftp, ft_ and ft, , which are illustrated in Fig. 10.. The computer extracted feature vectors X , X , X and X, for shapes ft , ft , ft and ft, respectively, are tabulated in Table IX. In this example, for the sake of simplicity, we consider 8 features labelled as C , C , ..., Co. In Fig. 17, we plot these feature values, which are normalized, for each shape to understand the variation of feature values for the considered 8 features. At this stage, it is not clear which shape our model will select as an odd shape. The D-table, given by Definition 1 (D-interval) in Chapter 2, which is derived from feature vectors of Table IX is tabulated in Table X. The value (6. - a.) for each feature C. is also shown in the very last column. In Fig. 18 , we plot D-intervals for each feature of Table IX and the value (B. - a.) for feature C. is indicated with a solid line segment for all i. From this, it is now clear to see which shape will be selected as an odd shape or form on the basis of a single feature C for i = 1,..., 8. This is also confirmed from T-table which is tabulated in Table XI. The selected most odd shape or form for each feature is shown in the very last column of T-table. -46- 1.0 0.8 - LlI 0.6 ID _) < > q: £ c < Ld u. 0.2 - 0.0J r T T t + t ? T + II <» 3C J! (I i: o 9E a O C3 C4 C5 FEATURES Cg C7 Cg Figure IT Plotted Feature Values of Table IV (Normalized} (The Symbol, •, Denbtes Q , x Denotes R , 6 iBanotes ft_ and o Denotes ft, . ) 3 U •Tatoie IX -1*7- The Feature Vectors 1^ X g , X y and X^ of the Shapes «!» ft 2 * n 3 » and "1+ Respectively, where the Sigmas are those illustrated in Figure 10. -^^^^^Shape Feature ' — — 1 "2 "3 n k C 1 DSQUARE 0.0731 0.0282 0.0012 O.0U36 C 2 DIVE# 0.21+82 0.1253 0.0273 0.1501* C MEANVDN 0.60T2 0.1+305 0.9111 O.3289 C. ALT# 8 It 16 6 C MAXIMAL^ 1+ 1+ 8 3 Cg P-SRA 6.1+263 5.181+1+ 3.6700 8.1+679 C ? VERTEX^ 12 I* 8 6 Cg AREA# 5 1*9 385 2082 227 Table X D-table Derived From the Feature Vectors in Table IX "i^ua-l^ ftn tt 2 "3 n u 3 . -a 1 i c i 0.59 1.00 0.62 1.00 0.02 C 2 O.56 1.00 0.56 1.00 0.00 C 3 1.00 1.00 0.1+8 0.83 0.35 C h 1.00 0.83 0.33 1.00 0.50 C 5 1.00 1.00 0.20 0.80 0.60 C 6 1.00 1.00 0.68 0.57 0.11 C 7 0.50 0.75 1.00 1.00 0.25 C 8 1.00 1.00 0.17 0.91 0.71+ -48- 1.0 - C > < i c l i : t ) l « > * < 0.8 i i > INTERVAL o < » i i < ► ' 0.4 Q < k 0.2 i k i 0.0 C2 C3 C4 C5 FEATURES C« C7 Cg Figure 18 Plotted D-Intervals of Table V (Solid Intervals Show Values (3. - a.); • Denoted ft 5 x Denbfces ft ; A Denotes ft_; and x Denotes Ri , ) -1*9- Table XI T-table Derived From the D-table of Table X "* s "*«^J3hape Feature^*^ «1 Q 2 s % The Most Odd Shape c i 1 n. C 2 1 1 ft. or ft C 3 1 ft C h 1 ft C 5 1 ft 3 C 6 1 % C 7 1 «1 C 8 1 S Table XII The H Vector is Evaluated The Most Odd Shape is Marked With * ^""-"^hape "l ft 2 S % £ w.d. » i i ij 6.65 7.58 u.ou* 7.11 I w.t. . l l ij 3 5* l 2 W.d. . i i ij 2.51 2.U3 0.91* 2.27 I W.t. . 0.27 0.00 2.19* 0.11 -50- For the decision of the most odd shape or form on the "basis of 8 features, in the given example, let us assume that all features are considered to be equally significant, i.e., the originally given weight vector u is: w = (1,1,1,1,1,1,1,1). We also assume that our attention function is simply A. =0(3. -a.) =3. - a . for all i , which is illustrated in Fig. 10. Thus our modified weight vector W becomes: W = (0.02, 0, 0.35, 0.5, 0.67, 0.11, 0.25, O.lh), which is shown in the very last column of Table x. Finally, vectors H are evaluated for various parameters; with or without our attention mechanism and D or T tables, are tabulated in Table XII. For example, the row of £ M.t. . indicates the values which are evaluated by using T-table without i i ij attention mechanism (i.e., the given weight vector is not modified), and the row of Z W.d. . indicates the values which are evaluated by using D-table i i ij with attention mechanism (i.e., the given weight vector is modified). For each evaluation, the most odd shape or form selected by the model is marked with *, and as the result, the third shape 0, is selected. -51- 5. HUMAN RESPONSE AND SYSTEM RESPONSE In this chapter we compare human response and our system response to the task of detecting an odd shape from a given set of four shapes. The data stimuli, 20 sets of four shapes each, are illustrated in Appendix A, and their computer extracted feature values are listed in Appendix B. 5.1 Human Response To get an average human response to the 20 sets of shapes, we obtained exactly 100 student samples on campus at the University of Illinois. We asked each student to pick an odd shape from each set of shapes and we counted the frequency of response for each shape. Table XIII shows the human response for each set. For the first set of shapes, for example, 57 students chose the second shape, ft , as an odd shape, 22 students chose ft as an odd shape, and 17 students chose ft as an odd shape. Thus the human response for the first set of shapes is ft p as the most odd shape and ft or ft_ as the second most odd shape. This response is denoted as ftp/ft, ft^ (for the sake of simplicity we write 2/1,3). Considering the 7-th set of shapes in Appendix A, ft , ft_, and ft, are identical except for their rotation. Therefore, shape ft should be chosen most frequently as the odd shape. According to Table VIII, 98 students chose ft and two students chose ft, as the odd shape. The average human response is ft p and is denoted by 2/ 5.2 Attention Function As an example of the system response, a complete computer printout for the 9-th set of shapes is found in Appendix C. Before we compare our system response to the human response we shall introduce the attention function, . is of the form A. = (y) = Y k , k > (2) ( Y ) = [C ± sin (ir + C^) + ]J where [X] means if X > 1 then (y) = 1 and if X < then (y) = 0, otherwise (y) = X. v The first type of attention functions, y , 1/2 ^c k « 4, are illus- trated in Figure 19. The second type of attention functions, with C = Q,k and 5.23 >_ C :> 3.93, are illustrated in Figure 20. Figure 21 -shows atten- tion functions of the second type with C. = 0.2 and 5.23 ^ C ? > 3.93. We will see later, after running our model for various attention functions, that the second type of attention functions, which consist of trigonometric functions and linear functions with C. = 0.2, are at least as good as those functions discussed above. For a comparison with other attention factors see [Maruyama 1971 b]. The following is notation and abbreviation used for the system response, (i) D-interval: oddness given by Definition 1 Dispersion: oddness given by Definition 2 LS: Linear Separation EDS: Euclidean Distance Separation D: D-table (See Chapter k) T: T-table (See Chapter k) (ii) HR: Average Human Response (See Table XIII ) CR: The percentage of Correct System Responses with respect to the average Human Response. 5.3 Weight Vectors Various weight vectors for two classes of features are tabulated in Table XIV. The first class of features consists of 16 features and the -5U- 1.0 •-0.8 cr o §0.6 o h- z UJ f- < 0.4 0.2 0.0 • lllf n 11 If >(y) = Y k , 1/2 S k < k -55- 1.0 .-0.8 O ^0.6 z o UJ < 0.4 0.2 0.0 J| \V M 0.2 0.4 0.6 0.8 1.0 Figure 20 Attention Function of the Form 4>(y) = [0.1+ sin(7t + C 2 y) + yljt 3.93 = (y) = [0.2 sin(iT * C 2 y) + y]J, 3.93 ^ C 2 < 5.23 -57- second consists of 8 features. The features in the second class are strictly limited to those features directly extracted from pattern sequences of objects. Weight vectors Number 1 and Number h assume that all features considered for feature analysis are equally significant. Weight vector Number 2 is estimated from the experience obtained while collecting human samples (relatively heavy weight is given for the feature of invariance - the perimeter divided by the square root of area - which has been considered an important gestalt measure by many psychologists). Weight vectors Number 3 and Number 6 are obtained by solving Linear Programming Problems (T-table, Weight Vector Number 1 of Table XV and T-table, Weight Vector Number h of Table XVII, respectively) to satisfy the average human response as closely as possible. The following is the process used to obtain Weight Vector Number 3. To satisfy the average human response for all 20 sets of shapes, a) = luip* w^> wj^s Wc» cogs uw » cjq» ojq » w ]_q» W 13' w l6' u l8' w 19 ' w 20' w 21* w 22 should satisfy at least the following set of inequalities: Cw > 0, where C = 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -58- Figure XIV Weight Vectors No.\ Weight Vector u> N0.1 No. 2 No. 3 No.ij- No. 5 No. 6 Fl F2 1 3 1 1 1 k F3 1 6 6 F^ 1 6 2 1 8 8 F5 1 5 2 f6 1 k 1 1 1 7 F7 1 8 3 1 30 23 f8 1 8 10 F9 1 2 13 F.10 1 2 li. Fll F12 F13 1 h 1 1 l 11 Fl^ F15 Fl6 1 8 2 FIT 1 2 2 Fl8 1 5 3 F19 1 5 3 F20 1 10 12 1 12 2 F21 1 1+ 1 F22 1 20 20 1 20 20 -59- As we can see that tie above system is ±l feasible, we eliminate the 20th row of C, and we try to find a feasible solution, to, which satisfies the remaining 19 inequalities. We can also see from C that we cannot evaluate the weights, w, q , Up, and u,pp> "these weights are assumed and assigned as to, o = w~q, where to-, is small and w is quite large (actually the weight of w does not effect the system response). Thus we obtain a feasible solution for Veight Vector Number 3. Weight Vector Number 5 is derived from Weight Vector Number 3 by taking into consideration the dependency between features. 5.U System Responses Some examples of our system response are tabulated in Table XV, where the LS method and the D-interval are assumed. Those attention functions are illustrated in Figures 19, 20, and 21. The first column indicates the set number of shapes which are illustrated in Appendix A. Columns indicated by D or T show the system response with various assumptions, The next column, denoted by H.R. , shows the Average Human Response for those 20 sets of shapes. To compare the behavior of the system with the average human response, the last row, denoted by C.R., shows how many times the sys- tem response matches the average human response for the first choice of the odd shape. For example, if the given weight vector is Number 1 and a D-table with no attention mechanism is assumed, then the system response matches the average human response exactly ik sets out of 20 (a 70% correct response), If the weight vector is Number 3, which is given by the solution of the Linear Programming Problem, and a T-table with an attention mechanism of C = 0.2 and C p = U.^9 are assumed, then the system response matches the human response up to 18 sets out of 20 (a 90% correct response). -6ou Table XV Some System Responses: LS, D-Interval SET No. WEIGHT VECTOF 1 No.l WEIGHT VECTOR No. 2 WEIGHT VECTOR No. 3 H.R. NO AM 1 Y /2 Y 3 Y C x =0.k C 2 =5-23 C l= C 2= 0.2 k.k 9 D T D T D T D T D T D T 1 3/1 3/1 3/2 3/2 2/3 3/2 2/3 2/3 2/3 2/3 2/3 2/3 2/1,3 2 3/1 3/2 3/1 3/1 3/1 3/1 1/3 1/3 3/1 3/1 3/1 3/1 1,5/ 3 3A 3A sh> 3A k/3 3A V3 h/3 3A 3A 3/k 3/k 1,5/ k 1/3 1/3 3A 3/1 3/1 3/1 3A 3A 3A 3A 3A 3/1 3/ 5 3A V3 V3 V3 k/3 V3 k/2 h/3 k/3 k/3 3/k 3A 3/ 6 k/2 k/2 14-/2 k/2 k/2 k/2 2/k 2/k 2/k 2/k 2/k 2/k 2/k 7 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 8 1/3 1/2 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3/1 1,2, 3 A 9 3/1 3/1 3A 3/1 3/1 3/1 3A 3/1 3A 3/1 3/1 3/1 3/1 10 V3 k/3 k/3 k/3 M5 V3 h/3 k/3 k/3 k/3 k/3 V3 V3 11 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 12 iA lA iA iA Vl l/k k/l k/i k/l k/l k/l k/i 3 A/ 13 Vi k/l Vi Vi Vi k/l k/l k/l k/2 k/l k/2 k/l l,2/k Hi 3/1 3/1 1/3 1/3 3/1 3/1 1/2 1/2 1/3 1/3 1/3 1/3 1,2,3A 15 1/2 k/2 k/2 k/2 k/2 k/2 2/k 2/k k/2 k/l k/2 k/2 3/1, 2, U 16 2/1 2/1 2/1 2/1 2A 2/1 2/k 2/k 1/2 1/2 1/2 1/2 1,2,3 17 k/2 k/2 Vi k/2 k/1 k/l Vl k/l k/l k/3 Vi V3 l,k/2 18 3A 3A 3/ 3/ 3/ 3/ 3/ 3/ 3/2 3/2 3/2 3A 3/2, k 19 iA iA iA iA l/k l/k l/k iA l/k l/k iA lA 1/2,3 20 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 l/k l/k l/k l/k iA c.r.i 70 65 65 70 75 70 80 80 85 85 90 90 -61- Table XVI lists the features which command the most attention in each of the 20 sets of shapes and also lists the three most heavily- weighted features (based on the example using weight vector number 3 of Table XV). We can see from the table that some of the features do not significantly contribute to the detection of the odd shape. Most attention is paid to features F8 (VERTEX^), F9 (C0NVEXV#) and F10 (C0NCAVEV#) Other examples of our system response with LS rnd Divergence are tabulated in Table XVII. Only half the features are considered. If weight vector number k with a D-table and no attention mechanism are assumed, then we get about 55 to 60% correct response. However, if weight vector number 6 with a D-table anu an attention mechanism of C, = 0.2 and C p = k.k9 are assumed, then we get about 70 to 75% correct response. Some other examples of our system response can be found in Appendix D. 5. 5 Experimental Results From our experiments we conclude that we can achieve a wide variety of system responses using different assumptions: different separation methods, definitions of oddness, weight vectors and attention mechanisms. We summarize our overall experimental results as follows: 5.5.1 Definition of Odd Form in Terms of D-interval or Dispersion The number of shapes, n, in the set decides which definition of oddness is to be used. From our experimental results, if n _< 6 either odd form definition gives almost the same system response. However, when n gets large the oddness definition in terms of Dispersion is preferable. 5.5.2 LS & EDS For the analysis of the system, the LS model is preferable to the -62- Figure XVI Features Which Command the Most Attention and the Three Most Heavily Weighted Features;. Based on. Example, Weight Vector No. 3 of Table XV Set No. Features With Large Attention Factors 3 most Havily Weighted Features 1 F8,Fl8,F10 F8, F10 , Fl8 2 Fl8,F7,F13 Fl8,F 8 , F 9 3 F18,F10,F 9 F10 ,Fl8 , F20 h F 6,F 7,F 8,F 9 F 9,F 8 , F 7 5 Fl6,F13,F 3 F 3,Fl6 , F13 6 F 8,F 9,F 2 F 9,F 8 , F 7 7 F3,F2,Fl6,F5,F13,F20,F22, F22,F20 , F 3 8 Fl8,F10,F20 F20,F10 , F18 9 F13,F 6,F 5 F20,F13 , F 8 10 F 7,F 8,Fl8,F 9,F10 F 9,F 8 , F10 11 F 6,F 7,F 8,Fl8,F 9.F10 F 9,F 8 , F10 12 F 7,F 8,Fl8,F10 F 8,F10. , F18 13 F 7,F 8, F 9,F 7. , F 8 lU F 6,Fl8,F10 F10,F 8. , F 3 15 F 6,F 7 F20,F 9, F10 16 F 6,Fl8,F10 F 9,F10. Fl8 17 F 7,Fl6, F 8,F 9, F20 18 Fl8,F10,F 8 F 8,F 9. F10 19 Fl6,F10,F 6 F20,F 8, F10 20 F 6.F18.F 9.F10 r 9,i r 8, F10 -63- Table XVII Some System Responses: LS, Divergence SET No. WEIGHT VECTOR No A WEIGHT VECTOR No. 5 WEIGHT VECTOR No. 6 H.R. NO AM 1/2 Y Y 3 Y C ? =5.23 Ci-0.2 C 2 =U.U9 D T D T D T D T D T D T 1 1/3 3/1 1/3 3/1 1/3 1/3 1/3 1/3 1/2 1/2 1/3 1/3 2/1,3 2 3/2 1/3 3/U 3/1 3A k/3 3A 3/k 3A 3/1 3/U 3/1 1,3/ 3 3/k k/3 3/1 3A 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3/1 1.3/ k 1/k 1/2 k/1 iA lA lA k/1 lA k/1 k/1 k/l k/1 3/ 5 U/3 k/1 k/3 k/3 3A 3A k/3 k/3 3/k k/3 3/k 3/k 3/ 6 k/1 k/ k/1 k/ k/1 k/ k/1 k/ k/ k/ k/ k/ 2A 7 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 8 1/3 1/2 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1.2,3,fc 9 3A 3/1 3/k 3/1 3/k 3/1 3A 3/1 3A 3/1 3/k 3/1 3/1 10 k/3 k/1 k/3 k/2 k/3 k/2 k/3 k/2 k/3 k/2 k/3 k/2 k/3 11 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 12 1/2 1/2 3/1 1/3 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3/1 3,V 13 1/2 1/k 2/1 lA 2/1 lA 2/k 2/k 2 A 2/k 2/k 2/k 1.2A Ik 2/k 2/3 2/k 2/k 2/3 2/3 2/k 2/k 2/k 2/k 2/k 2/k 1,2, 3,U 15 k/2 2/k 2/k k/2 2/k 2/k 2/k 2/k 2/k 2/k 2/k 2/k 3/1,2, U 16 k/2 2/k k/2 k/2 1/k 1/2 1/k 1/k 1/k 1/k 1/k 1/k 1,2,3 IT 1/k 1/2 1/k 1/k 1/3 1/3 1/k 1/3 1/k 1/2 1/k 1/2 l.U/2 18 3/k k/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 3/1 3/1 3/2, k 19 1/k 1/k lA 1/k lA lA 1/k 1/k U/l k/i k/1 k/1 1/2,3 20 2/1 2/1 2/1 2/1 2/1 2/1 2/1 2/1 1/2 2/1 1/2 2/1 1/1* c.r.: 1 60 55 6o 55 70 65 65 65 70 65 75 70 -6k- EDS model because of its simplicity. According to our computational results the two models have almost no significant difference in system response. Thus we may conclude that the Linear System is an adequate model for human behavioral psychology. 5.5.3 Given Weight Vector co It is clear that each feature should have a different weight to achieve a better system response. However, the system response obtained with weight w. =1, for all i, matches the average human response in about 60 to 75% of the cases. Weight vector number k, which is derived by solving the Linear Programming Problem to satisfy the average human response as closely as possible, gives a better system response than with any other weight vector. The system matches the average human response in 70 to 90% of the cases no matter what other parts of the system are changed. From weight vector number 3, we may conclude that form invariance - perimeter of the shape divided by the square root of the area - which has been thought important for a long time by many psychologists, is not very significant for shape perception. The most important features (judging by their assigned weights) are: (a) the number of vertices and the number of concave and convex vertices, (b) the degree of matching on (n-l) shapes, N-l __ and (c) the scattering of the shape, £ |s. - s| (if the shape is a circle, i=0 x then the value is 0). From weight vector number 6, we conclude that the following fea- tures which are extracted from a pattern sequence, S (x ) , are significant: (a) the number of maximal (or minimal) points in the sequence, (b) the number of alternations in the sequence with respect to s, and (c) the N-l dispersion of a shape, E (sj_ - s) . i=0 By comparing weight vectors numbers 3, 5 and 6 we see that: (a) the -65- number of alternations and the number of maximal points in the pattern N-l _ N-l sequences are independent features; (b) E Is. - s | and l (s. - s) i-0 X i-0 1 are quite dependent features; and (c) the number of maximal points in the pattern sequence strongly corresponds to the number of convex vertices. 5.5.^ D-table and T-table The system response given in terms Of the D-table is better than the system response given in terms of the T-table regardless of the other parts of the system (e.g. weight vector, attention mechanism, separation method, etc.). The system analysis for the T-table, however, is simpler than for the D-table. 5.5.5 Attention Mechanism We see that the system response using some kind of attention mechan- ism (in which the attention factor, A., is non-decreasing on the value g. -a.) is better than the system response using no such mechanism. Thus we may conclude that the attention function, , : (6 i - a ± ) -* A ± such that g. - a. > g! - a! implies (g. - a.) > (g! - a!), which modifies 1 11 X 1 1 — 1 1 the given weight vector, is not a bad choice for our model. It is sufficient to set k > 2 for the attention function of form, (g. - a. ) , but a slightly better function has the form [C 1 • sin (it + C 2 (g i - a.)) + (& ± - a ± )]J . The best values obtained from our experiment are C = 0.2 and C = k.k9 (illustrated in Figure 21), where the system response matches the average human response in 90% of the cases. -66- None of the students responded exactly the same as the average human response; their correct response ranged from 65% to 90% (mostly 80 to 85%). Therefore, we see that the system response given by our model reflects the average human response on the odd shape/form detection. The procedure described in this report was written in PL/1 and implemented on an IBM/360 model 75 computer at the Department of Computer Science, University of Illinois at Urbana-Champaign. -67- 6. CONCLUSION The procedure described in this report for the detection of an odd shape from a given set of shapes, is based upon the following: (1) A shape encoding method called pattern sequence, S (x ). This method allows us to extract some features which have not previously been considered by psychologists. All of the features listed in Table IV have been extracted from the pattern sequence rather than from the shape representation used as input data, i.e., a sequence of point coordinates. (2) Definition 1 (D-interval) and Definition 2 (Dispersion) of an odd shape in a given set of shapes on the basis of one single specified feature. (3) Weighting different features. (h) The attention mechanism in which attention factors are a func- tion of (g. - a- ) and which modifies the original weight vector. It has been known for some time that any system response for shape perception can be improved by assigning an appropriate weight for each fea- ture as well as by the selection of appropriate features. In this work we see that the above statement is again confirmed. However, even if assigned weights are not optimal, we can get some improvement on the system response by an attention mechanism which modifies the assigned weight vector accord- ing to the given set of shapes. We hope that the model of our attention mechanism will provide some clues for the solution of problems in human form perception: how humans pay attention and how humans reach decisions. The shape representation, a pattern sequence, which is an approximate contour abstraction of a shape unless it is in the class of angularly simple polygons, can be used extensively for the solutions of many other geometric problems -68- (a good example of an application of this representation can be found in Maruyama 1971 a). Some related problems are the construction of a system to decide whether a given pair of shapes are near similar or disimilar, shape cluster- ing, naming of corresponding shapes, and naming of geometrical regions of complicated figures using a given textbook or atlas illustrations. -69- LIST OF REFERENCES Arnoult , M.D. , "Shape Discrimination As a Function of the Angular Orientation of the Stimuli" , Journal of Experimental Psychology , vol.. U7, no. 5, 195^, PP. 323-328. Attneave, F. , "Dimensions of Similarity", American Journal of Psychology , vol. 63, 1950, pp. 516-556. , *Se*e IiiA»mat icaml Aspects of Visual Perception", Psychology Eeftev, vol. 61, no. 3, 195**, pp. 183-193. Attneave, H. , Arnoult, M.D. , "The Quantitative Study of Shape and Pattern Perception", Psychology Bulletin , vol. 53, no. 6, 1956, pp. U52-U71. Bartley, S.H., Principles of Perception , Harper and Row-, New York, 1969. Bobbitt , J.M. , "An Experimental Study of the Phenomenon of Closure as a Threshold Function", Journal of Experimental Psychology , .vol. 30, 19l*2, pp. 273-291+. Brown, D.R. and Owen, D.H. , "The Metrics of Visual Form: Methodolgical Dyspepsia", Psychology Bulletin , vol. 68, no. U, 1967 , pp. 2^3-259. Calvert, T.W. , "Nonorthogonal Projections for Feature Extractions in Pattern Recognition", IEEE Transactions on Computers , May 1970, pp. UVT-U52. Casperson, R.C., "The Visual Discrimination of Geometric Forms", Journal of Experimental Psychology , vol. Uo, 1950, pp. 668-681. Cohen, G. , "Pattern Recognition: Differences Between Matching Patterns to Patterns and Matching Descriptions to Patterns", Journal of Experi- mental Psychology , vol. 82, no. 3, 1969, pp. 1 +27- 1 +3 1 +. Deutsch, J. A., "A Theory of Shape Recognition", British Journal of Psych- ology , vol. U6, 1955, pp. 30-37. _, "A System for Shape Recognition", Psychological Review , vol. 69, no. 6, 1962, pp. U92-506. Dodwell, P.C., "Coding and Learning in Shape Discrimination", Psychological Review , vol. 68, 1961, pp. 373-382. Fitts, P.M. and Leonard, J. A. , Stimulus Correlates of Visual Pattern Recognition: A Probability Approach , Ohio State University, Columbus, Ohio, 1957. Freeman, H. , "On the Encoding of Arbitrary Geometric Configurations", IRE Transactions on Electronic Computers , vol. EC-10 , June 196l, pp. 260-268, -70- , "Apictorial Jigsaw Puzzles: The Computer Solution of a Problem in Pattern Recognition" , IEEE Transactions on Electronic Computers , vol. EC-13, April 1964, pp. 118-127. '"~"' Jones, L.E., "Pattern Recognition In Ciological and Technical Systems", Department of Computer Science Seminar, University of Illinois, Urbana, Nov. 17, 1971. Gibson, J.J., "What is a Form?", Psychology Review , vol. 58, 1951, pp. U03-Ul2. Guzman, A., "Decomposition of Scenes into Bodies", AFIPS Conference Proceed- ings 33, Part One , December 1968, pp. 291-30*+. Hake, H.W. , "Form Discrimination and the Invariance of Form", WADC Technical Report 57-621, October 1957, pp. 60-79. Koffka, K. , "Points and Lines as Stimuli", in Principles of Gestalt Psychology by Koffka, K. , Harcourt , Brace and Co,, Inc., 1935. Maruyama, K. , "A Method For Solving SOFA Problems by Computers", Department of Computer Science, University of Illinois, Urbana, Illinois, Report UIUCDCS-R-71-489 , 1971(a). Maruyama, K. , "A Study of Computational Aspects in Geometry", Ph.D. Proposal, Department of Computer Science, University of Illinois, Urbana, Illinois, may 1971 (b). Mowatt , M.H. , "Configurational Properties Considered 'Good' by Naive S Objects", American Journal of Psychology , vol. 53, 19^0, pp. U6— 69. Shepard, R.N., "The Analysis of Proximities: Multidimensional Scaling with an Unknown Distance Function I", Psychometrika, vol. 27, no. 2, June 1962, pp. 125-lUO. Shepard, R.N. and Jacqueline, M. , "Mental Rotation of Three-Dimensional Objects", Science , vol. 171, no. 3972, February 1971, pp. 701-703. Sleight, R.R., and Mowbray, G.H. , "Discriminability Between Geometric Figures Under Complex Conditions", Journal of Psychology , vol. 31, 1951, pp. 121-127. Somnapan, R. , "Development of Sets of Mutually Equally Discriminable Random Shapes", Journal of Experimental Psychology , vol. 76, no. 2, 1968, pp. 297-302. Uhr, L. (Ed.), Pattern Recognition , New York, Wiley, 1966. Wertheimer, M. , "Principles of Perceptual Organization", Psychol. Forsch. , vol. k, 1923, pp. 301-350. Zahn, C.T., "Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters", IEEE Transactions on Computers , vol. C-20, no. 1, January 1971, pp. 68^86^ -71- Zigler, M. J. , and Northup, K.M. 9 "The Tactual Perception of Forms", American Journal of Psychology , vol. 37, 1926, pp. 391-397. Zusne, L. , "Moments of Area and of the Perimeter of Visual Form as Predictors of Discrimination Performance" , Journal of Experimental Psychology , vol. 69, 1965, pp. 213-220. , Visual Perception of Form , Academic Press, New York, 1970. -72- APPENDIX A Twenty Sets of Four Shapes Used in the Experiment -73- SET-1 SET- 2 SET- 3 SET - 4 -7U- SET-5 SET-6 SET-7 SET-8 -75- SET- 9 SET - 10 SET-li SET - 12 -76- SET -13 SET -14 SET -15 SET -16 -77- SET-17 SET -18 SET -19 SET -20 -78- APPENDIX B Computer Extracted Feature Values for the Twenty Sets of Shapes Illustrated in Appendix A -79- Set 1 .*$» -: »..«*»**-♦* ■* tity ***<■* S * * * •<•' *** * - ■* « •• *!* Oej ECT N O "max AMP MFANVDN DIVE" USQUARf AVDIVEfr ALT* MAX INAL* : VERTEX* ccnvexv* ccncavev^ D_cCGES P F o I M E TER AREA"" : P CVFR A P' ' 2 / A~i P_SRA PAS ICS v. 1662 0. 8 748 C.0395 ■<•>! 2 5 C.C4 52 __J 8" 8 8 r 5,7833 ^.2929 0.7.958 6.C728 (1.0076 r.~r'9i4 8 0.363* L.75 2 7 G.089r p.niiA ^•1182 8 0.2257 0.8362 (1.056 3 0. 00.47 ~.f 674 8 o.ccc 5,6621 4 8 4 :."doc5~ 5.6879 1922. <"*45 2.4-71 13.92 r 9 3.7311 0,0499 S Y 4 M F T K_I_C "CCNVEX' 4.CK n If 1.6157 2 . d 2JU 16.0131 4.0016 r . 1 1 4i _SY>_^_ETRJ_C_ CONVEX 5.8601 1441.7715 3.1560 17.9 512 4.2369 O.T'6 3 3 SYMMETRIC CONCAVE 3.7922 O.0^C9 5.690.5 176 1.2 19T 2.5848 14. 708 5 3.8352 0.0757 SYMMETR IC CONVEX 5.86*1 Set 2 CHJFCT Nl 1 2 3 4 MAXAMP •".2929 '.56 9 1 :.2 •t6 7 0.5159 MEANVCN = ">.8i 25 ' .6695 C • 8 7 3 7 0.7028 DIVE M = ».°680 0.H87 '".C6r2 r.iioe DSCUARE = 0.'>i fc5 ".02 2 3 f .0049 0.0183 AVCI VE* - r . )847 0. 1773 (■.-.6 89 r .1573 ALT* - 8 6 8 6 V A x i y A L * 4 4 u A Vt*TEX* = 5 7 12 7 CO V XV»f - 5 6 8 6 CCNCAVEV* t = r 1 A 1 D_tbGTS s '.3' -2 ^.62 f, 9 ' .9519 ".6358 PERIMETER | ;= 5.5629 4. 8950 6.156 3 5.04 36 AREA t 15 78.0' P9 1577. 7517 1525.8011 1 588. 8596 P EVER A - 2.7353 3.3322 2.5643 3.1521 P< 2 / A - 15.2162 16. 3112 15.7864 15.8979 P_S- « - 3.9' -;8 n. r 9 1 2 4.<'367 0.122 3 3.9732 3.9872 P_A_S • ; .10 7 8 C. 1109 SYMMETRIC SYMMETRIC SYMMETRIC SYMMETP IC CCNVEX CCNCAVh CCNCAVE CCNCAVE I OS = 5.8010 11 .4500 9.5254 12.79^4 -80- Set 3 OBJECT NO = L 2 3 4 MAXAMP MEANVDN = 0.4845 0.6792 0.5006 0.6344 0.3572 0.7377 C.5781 0.5758 DIVE* DSQUARE = C.1209 0.0213 0. 1128 C.0196 C.0816 C.0102 0.1269 0.0253 AVDIVE* = ALT* 0.1780 6 0.1779 6 0.1106 6 0.2204 6 MAXIMAL* = VERTEX* = 3 4 3 3 4 6 3 6 CCNVCXV* * CtNCAVEV¥= 4 n 3 6 3 3 C_tDGES = PERIMETER= C.3930 5.2476 C.0045 5.1985 0.0046 5.3310 0.0051 5.2192 AREA* P OVER A = 956.2048 3.4894 693.8628 3.9633 916.6936 3.0800 583.5623 4.7312 P**2 / A = P_SRA 18.3108 4.2791 20.6034 4.5391 16.4197 4.0521 24.6932 4.9692 P_A_S 0.1716 SYMMETRIC 0.219O SYMMETRIC 0.1252 SYMMETRIC 0.2866 SYMMETRIC IDS CCNVEX 7.7731 CONVEX 14.4031 CONVEX 12.5530 CCNCAVE 12.3601 Set k * *#* J ***>*»«** *# *4 4 ***** + ****** *4***»"*****#** OBJECT NO = 1 2 3 4 MAXAMP MEANVCN 0. 7196 0.6H7? T.2482 G.G731 C.4588 0.8498 0.5050 G.6568 0.2652 0.8937 DIVE* DSQUARE = C.1533 0.0275 0.1199 0.0211 0.0510 0.0047 AVDIVE* = ALT* 0.40 8 7 8 C.1804 8 r.1825 8 0.0570 16 MAXI VAL* = VERTEX* 4 12 8 12 4 8 8 12 CCNVEXV* = CCNCAVEV*= 8 4 8 4 4 4 8 4 U_EDGES = PER IMETER = C.3045 7.4614 0.1272 7.4242 COO 10 5.900^ 0.5377 6.7795 AREA* P OVER A = 548.5852 5.5349 1C17.0613 3.1906 1121.2937 4.2093 1336.4131 2.7057 P*"^2 / A = P_SRA 41.2979 6.4263 0.44 84 SYMMETRIC 23.6879 4.8670 C.2717 SYMMETRIC 24.8349 4.9835 18.3432 4.2829 P_A_S 0.2887 SYMMETRIC 0.1723 SYMMETRIC IDS CCNCAVE 17.4725 CCNCAVE 17.9097 CCNCAVE 22.3339 CONCAVE 21.8335 -8l- Set 5 0*4 , : , • ft*.********-! ** * »i» »»»afr*- »»»»*n»»»i|(»»»»4i»»»» OBJECT NG = 1 2 3 4 MAXAMP MEANVDN = 0.4596 0.6271 fi • C 8 24 0.0121 C.1315 8 0.3771 0.7053 0.4804 0.7389 C.5566 0.5446 DIVE* DSQUARE = C."7C3 0.0081 0.0996 8. C.154 8 •'••0283 C . i 5 8 0.0194 AVDIVE* = ALT* C.2095 4 0.1942 4 NAXINAL* = VERTEX* 4 4 4 4 4 4 4 4 CCNVEXV* = CGNCAVEV*= 4 "e.cielT 4.7210 4 n 4 4 d_edges = PERI w ibTER = r.nf>87 5. 1063 ".25 34 5.4684 C.0042 4.4446 APEA» P OVER A = 765.803C: 3.7377 17.6458 4. 20-0 7 0.1561 SYMMETR IC 961.8596 3.2356 16.5217 4.^647 0.12 79 SYMMETRIC 955.7C85 3.0517 576.62C1 4.5327 P**2 / A = P_S»A 16.68 80 4.0851 C.1322 SYMMETRIC 20.1460 4.4884 P_A_S C . 2 1 C 2 SYMMETRIC IDS CCNVEX 12.3572 CONVEX 10.5421 CONVEX 7.74 9") CONVEX 9.^530 Set 6 *« :.-3t ■■■ >• J: i # * -•■ * * ♦ j * * =« a********* * <* OBJ m"ax MEA DIV OSC AVD ALT "VAX VER cc;n CCN D_E PER APF P U P- P_S P A ECT_NO_ AMP ~ i NVCN F* "■ UAkE I VE* * I 2 [ML* : TEX* V E X V * i CAVEV*; DGES I M E T FRj VE« 2 / RA _S n . 3 5 2 1 0.791 2_ 0.C785 C. 0094 O.f.992 6 r . 4 1 4 8 Jj.^46 3 0.1183 Q.0189 0.1585 6 C.3521 .791 2 G.0785 0. 0094 0.0992 6 0.5355 C. 60 32 ~0.~1230 0.0 24 3 0.2040 6 4 5^ 5 p "CT'2720" 5.4 5 24 4 ^4 4 072908"" 5.4628 4 5 5 3 (.2 720 5.4524 ^.6376 4.7789 143 1.5H61 2. 7447 14.5652 3.8685 C.0 83 7 SYMMETRIC 1113.3801 3.0396 16.6046 4.^749 0. 13f 1 SYMMETRIC 1431.9861 2.7447 14.9652 _ 3._868_5_ "' i : . 8 3 7 SYMMETRIC 1016.0078 3.9560 18.9053 4_.34 8C_ " 0.1847 SYMMETRIC IDS CCNVEX 15.4281 CONVEX 1 1. 1467 CONVEX 15.4281 CONVEX 9.5501 ************* -82- Set 7 ***** i*******J ****************** OBJECT NO = 1 2 3 4 MAXAMP MEANVDN = 0.8380 0.3289 0.5781 0.5758 0.8380 0.3289 C.838C 0.3289 DIVE* DSOUARE = C.1504 0.0436 0.1269 0.0253 0.1504 G.0436 P. 1504 0.0436 AVDIVE* = ALT* 0.4573 6 0.22C4 6 0.4573 6 0.4573 6 MAXIMAL* = VERTEX* = 3 6 3 6 3 6 3 6 CCNVEXV* = CONCAVEV#= 3 3 3 3 3 3 3 3 C_EDGES = PERIMETER= 0.0075 5. 5440 r.oo5i 5.2192 0.0075 5.5440 0.0075 5.5440 AREA* P OVER A = 226. 748C 12.9339 583.5623 4.7312 226.7480 12.9339 226.7480 12.9339 P *»2 / A = P SPA 71.7C50 8.4679 24.6932 4.9692 71.7050 8.4679 71.7050 8.4679 P_A_S 0.5814 SYMMETRIC 0.2866 SYMMETRIC C.5814 SYMMETRIC 0.5814 SYMMETRIC IDS CCNCAVE 11.8503 CCNCAVE 0.0000 CONCAVE 11.8503 CONCAVE 11.8503 Set 8 **** $ *.**■#* ***#•*#*#******£****•****•***.*;******* OBJECT NO = 1 2 3 4 MAXAMP MEANVDN = 0. 1662 0.8748 0.2929 0.7958 0.2067 0.8737 C.2929 0.8025 DIVE* DSOUARF C.0395 0.0025 0.0452 ~ 8 0.^728 0.0076 0.0914 8 C.0602 ^.0049 0.0680 0.0065 AVDIVE* = ALT* 0.06 8 9 8 0.0847 8 MAXIMAL* = VERTEX* = 8 8 4 4 4 12 4 5 CCNVEXV* = CCNCAVEV*= 8 ft 4 8 4 5 D_EDGES = PERIMETER= C.0025 5.7833 0.0009 5.6621 C.9519 6.1563 0.3002 5.5629 ARFA# P OVER A = 1922.0845 2.4071 13.92C9 3.7311 16C 1.6157 2.8281 1525.8611 2.564 3 1578.6089 2.7353 P^2 / A = P_SRA 16.0131 4.0016 15.7864 3.9732 15.2162 3.9008 P_A_S 0.C499 SYMMETRIC 0.1141 SYMMETRIC O.1078 SYMMETRIC 0.0912 SYMMETRIC IDS CCNVFX 6.97b7 CONVEX 7.2632 CCNCAVE 3.9262 CONVEX 7.5856 -83- Set 9 **** ♦ A******;****** 4*+** ********** *********** OBJECT NO = 1 2 3 4 MAXAMP MEANVCN = C.7196 0.6072 0.6865 C.4305 C1191 0.9111 0.838C 0.3289 DIVE* DSCUARE = 0.2482 0.C731 0.1253 0.0282 C.0273 C0012 0.1504 0.0436 AVDIVE* = ALT* 0.40 8 7 8 C.2911 4 C.C299 16 0.4573 6 MAXIMAL* = VERTEX* 4 12 4 4 8 8 3 6 CONVEXV* = CCNCAVEV*= 8 4 4 - 8 3 3 D_ E D G E S = PERIMETER= C.304 5 7.4614 0. 10 4 7 4. 1925 T.00 24 5.9203 0.0075 5.5440 AREA* P OVER A = 548.5852 5.5349 384.5632 6.411^ 2081.8154 2.275) 226. 7480 12.9339 P' 2 / A = P_SRA 41.2979 6.4263 26.8779 5.1844 13.4688 3.6700 71.705C 8.4679 P_A_S 0.4484 SYMMETRIC 0.3162 SYMMETRIC 0.C341 SYMMETRIC 0.5814 SYMMETRIC IDS CCNCAVE 3C.56C7 CCNVEX 30.4554 CONVEX 18.1797 CONCAVE 24.1289 Set 10 * .« j» . -- l w > V af , * * * < * 1- -* # :* * ... i) » « «j < ■> :; -' |* b ^ * '** 4 i * * * * OBJECT NO = l 2 3 4 MAXAMP MEANVON = DIVE* DSCUARE AVDIVE « ALT* 0.2 9 29 0.7958 Q..0728 .C.0076 0.0914" 8 0.344 8 0.8011 0. 1074 0.0144 0. 1340 6 0.3771 0.7053 r.C-70 3 0.0C81 0.0996 8 0.5691 0.6695 0.118 7 0.O223 0.1773 6 MAXIMAL" = VERTEX* 4 4 4 r 5.6621 4 4 4 4 4 4 7 CONVEXV* = CCNCAVEV*= 4 6 1 D_FQGES = PFo IMETER= ^.lgsi 5.6389 f .0087 5.1063 0.6209 4.6950 AREA* P OVER A = 1601.6157 2.8281 16.0131 4.0c 16 o.ii4i SYMMETP IC 127C.8953 2.7'+98 15.5060 3.9378 " 0. '.'9 9 8 SYMMETRIC 961.8596 3.2356 1577.7517 3.3322 P* ? / A = P_SR£ 16.5217 4.0647 16.3112 4.0387 P_A_S C.1279 SYMMETRIC ^.1223 SYMMETRIC IDS CCNVEX 11.1352 CONVEX 10.5374 CCNVEX 11.7781 CCNCAVE 9.3491 -84- Set 11 * >,, r. i |rH * i >4t«^>«^i>MM »t I * ♦**** **-******** _r p jf MAXA MEAN PIVE DSCU AVHI ALT'/ NAXl" VFRT CLMV CCNC C_ED PEP I AHFA p rv prf - ? P_SR P A I OS MP = CT_ MP vcn ft ARF VE* VAL* : EX* F xv» AVEV4 G E S" ' : MET_ER; 1 ^.2929 _C . 8 1 5 0.06 in O.Cf 61 0.075^ 8 0.6316 0.6482 0.1413 0.0329 n.2181 8 0.5199 *._7C2 8 ^.1106 0.01 8 3 . 1 57 3 6 0.6687 C.6223 0.1477 0.0371 r .2 373~ 0.2 38 7 5.5507 4 7 6 1 n.5945 4.9446 4 7_ 6 1 0.63 5 8 5.0436 ER / A S 1544.46«-7 2.6775 " 14.6619 3.8551 C.0805 SYMMETRIC 1524. 7361 3.5094 1588.8596 3.1521 17.3484 4.1&C3 15.8979 3.9872 0.1479 SYMMETRIC C.1109 SYMMETRIC CCNVfcX 14.2190 CCNCAVE 12.3305 CCNCAVE 17.9738 4 7_ 6 J ('."584 5 4.8495 1482.3557 3.6685 17.79C4 4.2179 0.1596 SYMMETRIC CONCAVE 14.9869 Set 12 OBJECT NO ii n ii ii n ii ii * 1 • • ' . \ ■• 4 i "* 6 • * # if ' 2 3 4 MAXAVP MFANVUN PIVE* DSOUAKf- 0,b3ei C. 7065 " 0.1913 Q.042" H.27P8 4 (3.2929 C.7958 0.0728 o. v;76 (V )914~ 8 T.4845 •■■.6792 r.1209 C.0213 C.3636 0.7527 C.C890 0.0114 AVPIVE* ALT* C.1780 6 C.1182 8 N A X I f* A L * Vt PTEX« _ 4 4 4 P".34 74 5.3191 4 4 4 3 4 4 8 CCNVFXV4 CCNCAVFV* 4 1 4 4 D_ E D G F S PERIMETER 0.0009 5.6621 (.39 3? 5.2476 0.0005 5.6879 AREA* P OVER A = 79C.9614 3.1515 16.7632 4.T943 0.1342 SYMMETRIC 1601*6157 2.8281 16.0131 4. :°16 0.1141 SYM^ETklC 956.2CA6 3.4894 1441.7715 3.156C ?* •' ? / A P_ S R A 18.3108 4.2791 17.9512 4.2369 P_A_S 0.1716 SYMMETRIC 0. 1633 SYMMETRIC IPS _ CONVEX 9.0346 CONVEX 12.5681 CONVEX l r .4528 CONCAVE 13.0D77 -85- Set 13 ***#*♦*♦*#*»-*******♦** ****>#*♦*»***** ******* OBJECT NO = 1 2 3 4 MAXAMP MEANVDN = 0.5381 0.7065 0.7489 O.4260 0.6865 C.4305 0.7196 0.6072 DIVE* QSQUARE = C.1913 0.O4 2O 0.1491 0.0381 T.1253 0.0282 0.2482 0.0731 AVDIVE* = ALT* C.27C8 4 0.3 50C 6 C.2911 4 0.4C87 8 MAXIMAL* = VERTEX* 4 4 4 "0.3474 5.3191 3 6 4 4 4 12 CONVEXV* = CCNCAVEV*= 3 3 " o.oi2~5 5.3844 4 n 8 4 D_EDGrS = PERIMETER= 0.004 7 4.1925 0.3045 7.461 4 AREA* P OVER A = 79<*.9614 3.1515 16.76 32 4.0943 0.1342 SYMMETRIC 347.9614 8.1859 384.5632 6.4110 548.5852 5.5349 p**2 / A = P SKf- 44.0767 6.6390 26.8779 5.1844 41.2979 6.4263 P_A_S 0.4661 SYMMETRIC 0.3162 SYMMETRIC 0.4484 SYMMETRIC IDS CCNVEX 15.6791 CCNCAVE 1 8.8120 CONVEX 22.5479 CONCAVE 18.8217 • Set lk OBJECT NO = 1 2 MAXAMP = r :.2fb7 0.2929 r .3294 0.2257 MEANVDN = <*.8737 _ 0.8095 C.79?3 0.8362 DIVE* = ' C.C6C2 6.0640 0.0777 3.0563 DSUUAPF = 0.CO49 _ 0.0068 0.00 8 8 0_._OO_47 AVDIVE* = ~ 0.0689 " * 0.0790 0.C980 0.0674 ALT *_ = 8 6 8 8_ MAXIMAL* = 4 5 4 4 VERTEX* = 12 _ 5_ 4 8 CCNVEXV* - 8 5 4 8 CCNCAVEV*= 4 _0_ D.EOGES = "0.9519" 0.2552 '".0884 0.0009 PFRIMtTEk= 6.1563 5.5303 5. 5914 5.6905 AREA* = 1525.8611 1495.0222 1436.1384 1761.2190 P OVER A = 2.5643 2.6709 _ __ 2.8112 2.5848 P*>2 / A = 15.7864 ~ 14.7707 15.7182 14.7185 P_SRA = 3.9732 _3.8433 L-JL 64 ^ 3. 8352 P_A_S = C.1C.78 0.0776" 0.1059 0.0757 __SX y METRIC SYMMETRIC SYMMETRIC SYMMETRIC CCNCAVE CONVEX CONVEX CONVEX IDS = 7.3885 8.0639 7.0086 8.3878 -86- Set 15 *** * k-%» *** 4 •**•.« ♦ ♦ if*-** A ) + ;***+;***•*•***'*****.**#♦* OBJECT NO = 1 2 3 4 MAXAMP MEANVDN - 0.3636 0. 7527 0.2929 0.8095 0.2929 0.7958 0.6687 0.6223 0IVE# D SOU ARE o.0890 . 1 1 4 0.1182 8 C.Q640 0.0068 0.0728 0.0076 0.1477 0.0371 AVniVE* ALT* CO 7 90 6 C0914 8 0.2373 8 M A X I m A L n ■= V E R T E X ft 4 8 A A "C.'or 05 5. 6879 5 5 5 (\ C2552 5.53C3 4 4 4 7 CONVEXVft = ff'NCAvEVft = 4 6 1 C_FDGES = PEP IMETEP= ( .0009 5.6621 C5845 4.8495 AREA* P CVER A = 1441.7715 3.1560 1495.0222 2.6 7 09 1601.6157 2.8281 1482.3557 3.6685 P*«2 / A = P_ S R A 17.9512 4.2369 14. 77C7 3.8433 16.0131 4.0^16 17.7904 4.2179 P_A_S 0.163 3 SYMMETRIC 0.P776 SYMMETRIC C.1141 SYMMETRIC 0.1596 SYMMETRIC IDS CCNCAVE 13. 1888 CONVEX 12.5351 CONVEX 13.7367 CONCAVE 3.9435 Set 16 5, -' * *■ > '■' *■• : *• % A .-.'■ *>, ■V * * ■ > » "Hi 1 OBJECT NO = 1 2 3 4 MAXAMP M E A N V D N ■= IVEft D SQUARE 0.4588 f. 84 9 8 0.153 3 j. 0275 C 1804 8 0.3294 0.7923 O.f 777 0.0088 0*0980 8 0.4343 0.7064 0.1C49 0.^159 0.5781 0.5758 0.1269" 0.0253 __-.. AVCIVFft = ALT* 0.1486 8 0.2 204 6 NAXI^ALft = V E P- T E X ft 8 12 4 4 4 8 3 6 CCNVEXVft = CCNCAVEV*= 8 4 0. 1272 ~ 7.4242 4 n " C*0884~ 5.5914 4 4 3 3 D_EOGES = PERI VFTER= 0.CCC5 5.7656 0.00 51 5.2192 APFA* P OVER A = 1 17.06] 3 3.190b 23.6879 4.867^ 0.2717 SYMMETRIC 1426. 1384 2.8112 1281.0527 3.6006 583.5623 4.7312 P ' - 2 / A = P_SRA 15.7183 3. 9646 20.7602 4.5563 24.6932 4.9692 P_A_S 0. 1059 SYMMETR IC 0.2 2 20 SYMMETRIC 0.2866 SYMMETRIC - IPS CCNCAVE 15.4365 CONVEX 19.0 403 CONCAVE 16.6637 CCNCAVE 14.3453 -87- Set IT ♦ ****** **•*.**:***♦** ******** ***** ************* OBJECT NO = 1 2 3 4 MAXAMF MEANVDN - 0.5781 0.5758 0.5381 0.7065 C.3771 0.7053 0.2067 0.8737 DIVE* CSQUARE = 0.1269 O.C253 0.1913 0.0420 C.0 7C3 0.0081 C.0602 C.C049 AVOIVE* = ALT* 0.2204 6 0.27C8 4 C.C996 8 0.06 89 8 MXI^AL* = VERTEX* = 3 6 4 4 4 4 4 12 CCNVEXV* = CCNCAVEV*= 3 3 4 4 n 8 4 C_EDGFS = PERIMETER= O.CH 51 5.2192 C.34 74 5.3191 0.0087 5.106 3 o.9519 6.1563 AREA* P OVER A = 583. 5623 4.7312 790. 961* 3.1515 961.8596 3.2356 1525.8611 2.5643 p» .■ 2 / A = P_SRA 24.6932 4.9692 0.2866 SYMMETRIC 16.7632 4.0943 16.5217 4.0647 15.7864 3.9732 P_A_S 0.1342 SYMMETRIC 0.12 79 SYVVETRIC 0.1078 SYMMETRIC IDS CCNCAVE 14.5037 CONVEX 16.6524 CONVEX 19.3308 CCNCAVE 14.2746 Set 18 *.*< i * v» x t *; ..-.-** *J .*», I « **|M ill * * #i, • ^ ■». -* * -f * ft -** OBJECT NC = 1 2 3 4 MXAMP MFANVUN = '".5855 0.6C32 0.12 3C 0.0243 0.2040 6 0.6865 C . 4 3 5 0.7196 "".6072 0.3521 0.7912 DIVE* SQUARE = 0.1253 0.0282 €.2911 4 ".2482 C .0731 0.O785 0.OO94 AVOIVE* = A I T « C.4087 8 0.099 2 6 MAXIMAL* = VERTEX* = CCNVFXV* = CONCAVFV«= 3 5 5 0.6 3 76 4.7789 4 4 4 r.7*o~47 4.1925 4 12 8 4 4 5 5 O.EDCfcb = PERI *>FTER= 0.3045 7.4614 0.2720 5.4524 A R E A « P OVER A = K16.0C78 3.9560 18.9053 4. 3A80 0.1847 SYMMETRIC CCNVFX 20.1313 384.5632 6.4110 26.8779 5. 1844 0.3162 SYMMETRIC 548.5852 5.5349 1431.9861 2.7447 P*--2 / A = P_ S R A 41.2979 6.4263 14.9652 3.8685 P_A_S 0.4484 SYVPETRIC C . 8 3 7 SYMMETRIC IOS CONVEX 17.6391 CONCAVE 2■ * »• * * * i ■» » « « i * f »+*#■♦* i *<■***» OBJECT NO = 1 2 3 4 MAXAMP MEANVON W.505C n .6568 ~ 0.1199 0.0211 C. 1825 8 0.3448 0.8011 C. 1074 0.0144 (.1340 6 C .4845 0.6792 G.5^06 0.6344 DIVE* DSOUARE = C. 1209 0.0213 0.1128 0.0196 AVDIVE* = ALT* C.1780 6 C.1779 6 MAXIMAL* = VERTEX* = 4 8 4 _ 4 0.00 10 5.9C00 4 4 " o o. 1951 5.63b9 3 4 1 3 CCNVEXV* = CC\CAVEV*= 4 3 C_EDGES = PfcRI,MCTER = C.3930 5.2476 0.0045 5.1985 AREA* P OVER A = 1121.2937 4.209 3 1271.8953 2. 7h9P 956.2048 3.4894 693.8628 3.9633 P^~2 / A = P.S^A 24.8349 4.9835 . 288 7 SYMMETRIC 15.50 60 3.9378 18.3108 4.2791 20.6034 4.5391 P_A_S C.0998 SYMMETRIC C.1716 SYMMFTRIC 0.2190 SYMMETRIC IDS CCNCAVE 14.8014 CONVEX 12.7787 CONVEX 13.2050 CONVEX 14.6142 -89- APPENDIX C Examples of System Response for the Decision of Oddness -00-v IU CJ Lj |l/l 000 00 o o o O OO 00 O o, Itl II II II II II 1 11: II N II II II ' II II II II -~ — — —[-. — —, — •*>»-<»■>»•»»■< >»•«*■ • * » ►, ►•■ *l •■ • . . ► * . » • O-«r\jr o >frirj,0l^- ; <\J f«V * ITV «0 f** 'CO O s '-''-<'-'-'>-r:^ jo r- '1 11 II H II II II II II «*■ * •J- -J- -J -r II 11 II 11 11 11 II II 11 m r^ m m m rr pri m m 11 11 11 H 1 11 11 11 11 ciifiKnoiCKfin r jo r-- »< n fi-t— t— t— t— ►- t— r— . l/i' I/O V/1.00 00 OO'oO VI lfll/1 // OO'l/l OO CI cj o o c o o o e o o c o o o o o o <_) o o ...... o o o o c o o ■*• OOOOOC 00.000 ooo*+*++++* + 4- + LU ILI IU LU UJ LU IU LU LU LU LJ C O O O O O C O c o o o o o o o c o o o o o o o c o o o o o o o o o o o c o o o o o o o o o cj o o o 000 «•«••••• -* ;P H <-«ir-< o o o •» o o ;-< 0,0 o -* — 1; 1. 11 ,1 11 11 II II II II rn mini m r> 1 fl m PI m II II II II II • II II II (t> en f^'fi t^'fn fO'fi H (M,C1 >r lA >0 1^ tU'O* 1-1 rt «J rt ^ rfH ^ 1 II II II II II II II II II ' II II II II II II II II ->;——.-»——. -~rvj(\if\j(sj'<\jf\jfM r\J rsjc\jcMf\j<\ir\jr\j(\j.'\i •> " ■> - » •> » •> ► . ». «. » . . . ► a h m m nt ui >o f- ^(Njri^inoMiiO^H-irf _i,t-i — — 1 p«il u. LL LU LL u. LL. LL LL U. LL LL LL LL LL LL LL J. * u. U_ U. LL u_ LL U.O.X LL U. X LL U. LL LL LL * *• O C O O O D LJ O Q O 3 O O O O c c; O c; O O c c O C O O O O O O O ^ C/ c c- + + + -. t + ♦ ♦ ^ + + + + + + + •«■ + LU UJ UJ XI LU LU LU JJ UJ LU LU LU UJ LU LU LL' 'JJ c C C3 O O O C 1 C c? C O CI O O Cl C" O CJ CJ 00 c- O c c c CJ c_- O CJ ■— c- CJ c. f.' (_, G'LU c c, c «.? c C ' <_' c c O O C c c CJ f_; c c- O (-' c. 1- CJ (._ p-l —J •-< »-< z II II II II II II U » LU II II II II II II 11 11 11 C\J (\J rsi cu t\l fVJ "Nj (\l UJ f\J (\J CM (\i r\J (M f\i rvj rsi — « f\i f^l * m jo r- a. -^ (M ei -r ■x\ ^; MCO- ^^ ^-1 . — 1 1— 1 r-l -M — ' — » l_" C »_■ O O c,- c.- ?"• CU C Zi O u c.- O CJ OOO CJ O O CJ TC O v— • CJ c c O O i_ CJ c^ Cj O L» UJ O O ^. 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LL :i :.j o o o o o o o o o o a o o o o u o u o o r -s ra o a cj o l 1 ( j u n 1 J O LJ Q CI '.J O O ;-' rj i-J (J iJ 'J J U u u o u □ cj a o u cj u u a .:• r J w -91- !l a: ° Z LU X 2. a: U OC P uu oo CL H, < o X uj 00 -> a: O a O: t- U H LU OO t- ai U) (X a 1 uj! C x: Z U '< H UJ i a. o uj- Uj CC X < H Q. !oo a < o; OO UJ iUJ O _) uj > oc uj _j u 1 u- «■ — X. O 00 u ■j 10 UJ OO U. _) [J UJ U aj a 3 u- -i a UJ UJ ►- ac -3 CO LL CL *t LU _J a _j >- — • o CC < 00 : X' o LU -} CD O. ooi 3 < oo! u, X! lJ; Z ai i jj i oo 1 . I i < a. I oo Z i- <. U * UJ UJ •» i. -> ' tr Oi •t 111 CD » I t- X Q. ' * UJ >- Q 00 I-" Hi U) LO qc lUJZ If- U UJ O X I fl o- a CD O 3 < 00 o X u CO O z D X H 00 on a < X ° ►- O0 ct 1— o o OO UJ UJ -> X CO t- a QC X LU a > a i— OO Q_ LU JJ a t- LU UJ i. X — i H 0C UJ O o_ H z zl ■; Q- Q UJ o o UJ o o z o LU ae □ Z oo O UJ — . a. X o UJ X 0l O z < z: o El iol UJ: Hi 00 XI 2 ; o Hi si O LU a o UJ o z »— < V Q LU OC o z oo o LU « 1— 1 Q. * <3. y 00 X * >— OO « t— X ~c z LU O at X Q- ^-t J H DC O oo LU cc LU a o UJ z v LU •» oo > * *— •— « * X o oc a 5 Ul Z' o oo □ oo X X z < < a. a z Q UJ Q O LU LJ LU a ! - i °° i o 1 LU ! ° I Z 00 LU TT ►- Q- ■♦I < < 00 X *, •- VI «. X CL -. 3 ►— OC u oo> LU OC LU a. o f— 1 X 1 LU Z *, LU ♦ , OO > » ■— fcM »' X o *' f- CL X o u uu _i < > II 001 LU X 1 < X o UJ CD O LU X OO 3 < ujj Q3i UJ, OOl □ I X' Ui ujI LU z °! UJI x! »-| 00 •-i i CNJ' I III H CD a ►-• > JJ a O0 V- jj o 13 LU •c ~> oc CC LU U > •a. LL a LU X H i— OO JJ CL CL LU o LU k«4 X 1/5 1— Z LJ o o H „ Q UJ LL 0L — < 00. uu' oc D . o oo X z o LJ z Q LU LJ LJ UJ LJ O z LU QC O Z 00 O uj — a •c 00 X — 00 H- U. 2 U LU X O. — 3 0C LJ i' j or. a o LU Z » OO > «l X O -92- CJ IX' a. i OO Q O U a. a i z u a CJ CJ 2 00 < xj to ul tt-l 7T. Q O o! i/j a x Q UJ a: •a 'o. s: u u < 5» o It oo < x 03 a oo o x 'u LU -3 CO u ^ LU X ►- » - z > a Q LU a OO o UJ ot o i Z i/ji O ou —• O. •a oo X -• z u 3* z Q- — . O, ►— a: CJ oo UJ CL UJ a. U h- X UJ i: •» LUl ■*■ 00 > * •— • — 1 * X (J » >— 1 UJ X IZ i< o to 00 o 00 I z LU ^- < I V z •a > < a. CJ z Q LU a M o LU o - LU a z 00 Q LU at O Z oo * CJ UJ * •-« Q. * < *- 00 X » — oo >r »— u. —< z o LU * z a ■— z> »— cc a oo lu a. LU a O ^ X LU Z * LU # O0 > * *— *«• * X o UJ X 'o z o Z o oo oo X 2 a < z a z Q LU a o LU CJ \Ti 1 i o I JJ X a 1 7* 1/1 U> LUl — « 0l < y> X — oo -4 Z CJ' UJ * s. a — xv r- i U t/1 LU oc lu a. o, I- X LU Z! 3l O0 O0 «I X -3 < cj z LU oo u 'X 1° LU il3 JJ z o I II K- cj 03 ► I (J z Of 00! LU f-' u. <-> — -5 CJ OJ. < LUl > CJ O0 > X o LL, •— QC LU LU, X xi > iu )-: at Cj lu — • O t- oo iu! z z pel — a I ;lu IO0 , u >- c- '—* -'. c- o o •— o o c o c o r o o c ♦ + i i + + + i + LJ li- a, in l.J L' .11 11 11. 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H- ^ x: <_l Ui LJ _J CJ CU ,TJ LU, X < CJ' CO I— X <1 CO t— UJ ►— I— UJ cj a cu > U "5 CD X CO a »* — O co ^ o — «zw -£ CD -I _J LU ; LL Mj — U. cj *l cr; i O v X 00 3L o a o ! u p« n O- LU u. c» i. a - o cj| Z X cu' LU < Xl I uu| UJjCj X 1 ! *l t- jt « i ♦ I co > • ■4K * -10U- APPENDIX D Some Other System Responses Under Various Criterion -105- Table I Some System Responses: EDS, D-Interval SET No. WEIGHT VECTOR No. J WEIGHT VECTOR No. | WEIGHT VECTOR No. I H.R. r& T r z ^ r a D T D T D T D T D T D T 1 V, V, \Vz */z % 3 Vz */* Az *A Va 3 /i 3 A 2/1,3 2 ?/l 3 A V\ Vl S/i V\ y* y> Y Vi y> Y> 1,3/ 3 .*/* 3/u Vs V* % As n y> Aj y* y *A 1,3/ 4 '/3 1/3 Y« V\ sfy ¥* i/u 3/z 3/s ■A-/ 3fy V* 3/ 5 ¥* «/3 % fa Y* A % y> y* y^ y* V* 3/ 6 «/* «Vz % ¥j y* %. y« y* Z Y/ V* Yi V* 2/h 7 V V 2/ Z/ V V V V */ V V ■*y 2/ 8 V, »a V\ Y\ y V, V\ 1,2,3,4 9 V, V\ ¥ y* *A y* V/3 h/1 11 y* { A V* /3 V* y> a a y> Y y> y* 1/3 12 y* Y* y* // *A y y Yi y y y Y\ 3,V 13 V\ «A V\ y\ % % Yz M y* Yi Yz Y 1,2 A 14 V\ y> /3 y$ Vz Vz Vz Vz a. Vz i / Vz. 1,2,3,4 15 n y*. Vz Vz 2 A y* Z A Y/ y* V* y* Vs 3/1,2,1+ 16 V\ V\ *A S V\ y V\ y* -// *Y* V* y* y* 1,2,3 17 % y* "A *i % "A V\ y A Yi y Y\ 1,4/2 18 y* Y/ V V V y V >/ V V V Y 3/2,4 19 // y* /y /* A 'A/ // // y* Yi y* // 1/2,3 20 Vjl A ^ Vs Vz '/*, >A // w 'A y+ w 1/4 CR. 7P~ 6& t& 7* 7* 7o 7* 7S w 7s 7* V-r -106- Table II Some System Responses: EDS, D-Interval SET No. WEIGHT VECTOR No. Z WEIGHT VECTOR Ito.Z WEIGHT VECTOR No.Z H.R. r* r r z r* r? D T D T D T D T D T D T 1 y, 44 5* 44 y* *A y yk % y *A Vs 2/1,3 2 A ^ V\ 44 y y y* A* Vi y* y* y* 1,3/ 3 4-W ^ y y* % y> % y y> y-i y$ Ys 1,3/ 4 !/3 ^ d // V\ y* y 44 V/ i/f vu W ¥/ 3/ 5 3 /^ // # y % y % y "A % %.'«/* 3/ 6 % y* % «/z y-s y< y-s A* */¥ y* V* V/ 2/k 7 V V y V V y y y y y V V 2/ 8 V\ M y y Y\ y y y y y v/ y, 1,2,3,4 9 Yi Vl y y y* ■Vl Vf y 44 A y* V\ 3/1 10 % % y 3 y>. % y y> y y* As y* y* V3 n 'A '/S a As y^ y> y Vi A y* y> ys 1/3 12 A /* "A // y y y y y y y */\ 3,V 13 y ^ y y y y y y y y Y\ V) 1,2 A Hi 44 M y y 'A S3 A 14 Va y* 1,2, 3 A 15 y *s % y* '**/ /* S2 % y y* y* *// 3/1,2,4 16 y M J A y j // y y* y? y ¥ 'A y* y+ 1,2,3 17 y «i "A y % y y y y A «A */\ 1,4/2 18 yu 4^r V y V V y y V ¥ y y 3/2,4 19 y-4 /v y* y* /y y At V* y« x v+ 1/2,3 20 V* Y\ A> y Vs y A /a. A V* V* 1/4 C.R. 7o 75 H.R. ^ I r z r s ^^ D T D T D T D T D T D T 1 y* *i /3 % ^ % V* /3 y> y^ */a 2/1,3 2 Vx A y •/* »/i »// y* Va 'A 1/3 J/3 y.* 1,3/ 3 V\ M V, yi y* A y* y< y/ y/ y* 'A Vs ■y* 2/lj- 7 y V V / */ y V V V y y y 2/ 8 ^ 3 A & J A V/ Vz V* As »/y s /i yy y 1,2,3,4 9 M y+ y, */fc y A A- a M y ( ^ V\ 3/1 10 ^3 y* y» ^ y A> /3 a> y$ ^ */* y V3 11 J4 A y 1/ i/ A3 y> y> /J Vj> i/ Xl y 1/3 12 *i y« y % y y % # % M M y 3,4/ 13 ^ % A •y* y* n a* r< y* v /i ^ y< 1,2 A 14 M A y> 'A y y* >s i/ A 7 y^ y y^. 1,2, 3 ,4 15 ^ As A> y % % V* % %. y* 3/1,2,4 16 X 1/ A2. i/ i/ X X /a y* yl y y y\ 1,2,3 17 # A A a % % *j A y y> y *a 1,4/2 18 ^ A* y V »/ V */ V V y V V 5/2,4 19 /* t/ y+ y-f A Yi A i/ Y* 1/2,3 20 */ ^ */ y y A\ */ A y\ *f j*/i Vi 1/4 C.R. W 6 7 Ai 1,3/ h A 'A ¥/ ¥/ A/ y* Ay y* w y/ y* y? 3/ 5 ¥/ A> «A % As As As y& y y y* y> 3/ 6 % fa y. % y* y< y- // y*. A/ y 2/k 7 V V V y y y y y y V y v 2/ 8 V, /s V, A' y, A a y, y y y y 1,2,3,4 9 A y, ¥/ X As A a* y w A y< a 3/1 10 % #3 A* y* y >A> ?A V3 n /i 'A A A A3 A y y> y y y y 1/3 12 y* X A *4 // A y % y+ A y y y* 3,V _ /1 13 y V\ «/ 4 A %. y y y\ y^ y y y 1,2 A 14 y, y, y* As y* A X As i/ /-2 i/ /2 1,2,3,4 15 y y % "A % A* y y*. y y, ¥/ */** 3/1,2,4 lb y y y-> /z i/ 'A y, a 3./ y\ y. *A M 1,2,3 17 T O % a *A «A «A y y y y y v / ^4 1,4/2 lo As 3/yC Va Vt A* A- *A A-/ 'A y~ fy d // 3/2,4 19 // A* A* y* y-t y-t ft % ft /y Yi Y* 1/2,3 20 J6 /a A> A2 A> A? A Az A. y V* ^ iA C.R. ZL. 6£ 7° 7* 7* 70 7& ££ 7* 70 7t dt J -109- Table V Some System Responses: LS, D-Interval <3VT WEIGHT VECTOR No.£ WEIGHT VECTOR No.Z WEIGHT VECTOR No.Z T T Ti c,= O.Z c, = £>. Xa «A 1,3/ k Vs A "to x x« y* yy y\ X< Vy y X> As As y> As, Xs X* «A y* vs 3/ 6 #> X> ■to X*. to Xz. to yi. X x X* X "to X x/ X\ 3/1 10 % "A yi X X y* y* X X3 Xs Xi x* 4/5 11 X to i/ S3 'A X y* Xi, to X* to x 1/3 12 X # X to Y\ X X X X X, X\ X 5,4/ 13 X X X X X X X! X X X X X\ 1,2 A li+ X X "A A A A \y A \y Xa X yz Xi 1,2,3,U 15 %. %. to* x*- %, «A Xz «A %. X*. to 2 - /*- 3./l,2,>+ 16 X x *% y A /*. A *A X X X> X 1,2,3 1 /-. 17 n & *f «A % x X #/\ X X X\ X l,h/2 lo y* ¥/ y* w fa to y/ ¥JL c, ■ -p. # ys y w ¥s ¥/ y 3/ 5 *f 'A y y* y y> ys ys y y ys y 3/ 6 # tf. y ys y y Yy y y ys ys ys 2/4 7 y V y y y * y y y aV V y 2/ 8 y> y. y y y y y y y y y M 1,2,3,4 9 y< y Y, V\ Y/ y y y Yt Y\ y Y) 3/1 10 y y y y y y y y V* 9* *A y$ 4/3 n A /3 ys y y y y Yi /3 y* y$ 1/3 12 YA y* % y y y y y y y y y 3,4/ 13 y* y % y Y* y % y % y y y 1,2/4 14 M M y Ys y y y y y 1,2,3,4 15 YA *t Yz Yz & y y y %. % % %- 3/1,2,4 16 y i/ yz. V V i/ i/ \y Y2~ y. 1,2,3 17 fc y y y> y *A y y y % % 1,4/2 lo y* Yt y y* y y* Yz y y y. y. y^ 3/2,4 19 y? y* Yi y* /f Yt ys ys ft ft y Yf 1/2,3 20 y* y y* y* /y Y* Yf yi Y Y\ to w 1/4 C.R. at ^ *t 9o ft #■ ?e fo 7t & 7t fo -Ill- Table VII .Some System Responses: LS, D-Interval WEIGHT VECTOR No. | WEIGHT VECTOR No. Z WEIGHT VECTOR No. 3 c, =*/ c, «*tf C, *a % y X x V3 n }A y y y* X X 1/3 12 */\ */i y y y y 3,V 13 % f\ y y x y 1,2 A 1U y* /* y* y* A*. A2. l,2,3A 15 y* y+ y %- %. ¥ /L 3/1, 2 A 16 y< y % Y/ *y X X 1,2,3 17 y ¥\ y y X X 1A/2 18 y- y^ % y X X 3/2 A 19 y+ // y r y* X X 1/2,3 20 y± \l Aa V* *A V\ iA C.R. 7* 7? ft? %p 7* 7-t -112- Table VIII Some System Responses:' LS, Dispersion WEIGHT VECTOR No.| WEIGHT VECTOR No . | WEIGHT VECTOR No. c,= 0.Z = & y. 2/k 7 V y A A y A "■■'/ y v y y 2/ 8 A A 7\ A A A A A A A A A, 1,2,3,^ 9 A a A V\ A A */ r y A A y\ y\ 3/1 10 % A A-* A*. As A A> Ai As Vi Vs V 3 V3 11 A. A A As A> A As A As As a> y 1/3 12 A { A A A A % A A A A y y 3 ,kl 13 A A Aa A y A A) A V V v y 1,2 A Ik A A A A y a* 1,2, 3 ,k 15 % % V* A* y A* % % Y* 3/1, 2 A 16 A A*. A y y fy V\ y a y* y* 1,2,3 17 /J A 7\ A A) A % % y a % *A i,Va 18 ¥* Y* & Ax ft y X V* y y y/ // 3/2, k 19 y* A* A* 1/ V V y y 1/2,3 20 & y A3 y^ 2 1/ _/0 S A y* & a y* 'A y* iA C..R. 7* a- frv 7t t* 7t f* 7^r & c fv 7J- -113- Table IX Some System Responses: LS, Dispersion WEIGHT VECTOR No.# WEIGHT VECTOR No.y WEIGHT VECTOR Y\o .A * X* X X* X 1,3/ 3 *As ^ # ^ *f X X *f X X X x 1,3/ k y* 1/ A 2 'A % y* X ?i X X, X X 3/ 5 % rt Xs ft % X % *3 X X Xs x> 3/ 6 «a y X <*/ *A V X */ X X *4 y 2/k 7 y y y V V V V y y X y v A Ks 2/ 8 X A /^ ^ Xs X 1/ A3 1/ At ^ % X 1,2,3,^ 9 & M ^ X A X w H X* Vi X* X 3/1 10 yk yi ^ X- X> V* "/j X- x> x> % x> V3 11 X A a £ 1/ Ai A* X 1/ A* A4 X X X* X 1/3 12 1/ /a i/ ^ /9 y, X M X Y\ X X 3 /i 3,V 13 a* // y // X X # -A X* A X* A 1,2 A 1U A * // X X* X* X? ft ft A ¥/ X* 1,2,3,4 15 A X /-/ # X* X* yb X* X* A X* x>. 3/1,2,1+ 16 ft # #, # X X* X X X X X X\ 1,2,3 17 /v t/ A X* X* X* 1/ A* A A X* X* 1,4/2 18 y« A^ /* i/ X X* A ft A A Xs A 3/2 A 19 ft /* y« /* ft A ft ft* ft A %% 1/2,3 20 iL ^ * y X A* AJ. X X A3. A iA C.R. /« rftf /o &■ £S A & a /*> >rx ££* So -114- Ta"ble X Some System Responses: LS, Dispersion WEIGHT VECTOR No .4" WEIGHT VECTOR No.<£ WEIGHT VECTOR No S 1 / C,=a& C z -y.y? H.R. D T D T D T D T D T D T 1 % t/ 73 1/ A3 1/ 73 % JB As A v.* A )i 2/1,3 2 Ay % Ay % Ay As Ay Ay Ay A Ay Ay 1,3/ 3 a, y, A A A A A A V) A M A 1,3/ k yy A Ay yy Ay Ay A\ yy yy yy yy // 3/ 5 A, y* A A Ay Ay A3 A3 A* A As As 3/ 6 4/ A A A A A A **/ A A "A "A aA 7 y V A ,2/ V V y */ y v y y 2/ 8 y* A y> i / 73 A3 A3 1/ /3 [ A A A3 As y> l,2,3,!f 9 A A A* A Ay A\ Ay X Ay A Ay A 3/1 10 As A A A*- As A A A A* As A* V3 11 A y* a X, A A3 A Vs ys As As As 1/3 12 A ¥\ A A A A A % A A A A\ 3,V 13 A y> A* A/ Ay Ay Ay ft A> Ay Ay Ay 1,2 A 14 ^ y* y* ys A3 A3 Ay ft yy Ay Ay Ay 1,2, 3 A 15 Ay yy A/ Ay Ay yy fy -ft yy yy Ay Ay 3/1, 2, It 16 A* y* fly 1/ Ay 1/ /A A* ft A> A* yy A* 1,2,3 17 A A A> /j A A* Ay A A* Ay /3 As iA/a 18 A Ay /a y* A* y* A> /* /S /3 y* A 3 3/2 A 19 A* A 1/ /y A* Ay Ay ft y* a* At Ay 1/2,3 20 a A A f\ y, A A f\ A A A\ y lA C.R. 7c Jc yo /o 7° se /£■ & 7" 7" S3- 7t -115- Table XI Some System Responses: LS, Dispersion WEIGHT VECTOR No.<£ WEIGHT VECTOR No.£ WEIGHT VECTOR No.^ >/ C, ~a<{ C, =0. Z SET No. 7^ r ■r 3 CW-^ C^U.tf H.R. D T D T D T D T D T D T 1 X % y y % \ / y y y x» X X> 2/1,5 2 H x. y y> y. fr y* *A ¥•/ Xi ft X 1,3/ 3 ¥■/ y y-/ y ft x X X y X X Y\ 1,3/ k X r, X X/ X X x y y y % 3/ 5 yy Va yx % ft y> v y % x/ y yy x/ 3/ 6 y V y y V y •y «y */ y y y s/k 7 .?/ *>/ •>/ V V V V y -x y •y -/ 2/ 8 y /a /s /i ^ y i/ y y y y y 1,2,3,4 9 y* V\ y* y y y X/ Xi y x yv y 3/1 10 y y y> y % y % % y y x & U/3 11 /* /s y y y y y A1 y y >A x> 1/3 12 X x x y y y y y x y y M 5,V 13 y y y/ y y* y y* y y /y ys yy 1,2 A 1U y y* y> y> y y y y y> y* yy yy 1,2,3A 15 y y< y« y^ y y/ ft y* yy y* ->y> y y y x y* y y* X* V* y< y* 1,4/2 18 /3 Xi y y* x> y> Xs / 3 x y 3/2,4 19 X x % "X y X x y X\ V\ y y y* y 1/2,3 20 ■y y 'A X v* X y* X y* y i/i. CR. XL- a 70 6f 7" /o /^ /o 7c Af 7-i- ?o n AEC-427 (6/68) :CM 3201 U.S. ATOMIC ENERGY COMMISSION UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR DISPOSITION OF SCIENTIFfC AND TECHNICAL DOCUMENT I Se# Instructions on R Sid*) AEC REPORT NO. Report No. U90 COO-2118-0027 2. TITLE A PROBLEM IN FORM PERCEPTION: ODD SHAPE DETECTION rYPE OF DOCUMENT (Cheek one): Qa. Scientific and technical report ~^\ b. Conference paper not to be published in a journal: Title of conference Date of conference Exact location of conference Sponsoring organization □ c. Other (Specify) RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): J3 a. AEC's normal announcement and distribution procedures may be followed. ~2 b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. ^ c. Make no announcement or distribution. REASON FOR RECOMMENDED RESTRICTIONS: SUBMITTED BY: NAME AND POSITION (Please print or type) iyoshi Maruyama, Research Assistant Organization Department of Computer Science University of Illinois jignature ^rf.to'fl,,,.^ Date February 7, 1972 FOR AEC USE ONLY »EC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY, ON ABOVE ANNOUNCEMENT AND DISTRIBUTION IEC0MMENDATION: ATENT CLEARANCE: U a. AEC patent clearance has been granted by responsible AEC patent group. U b. Report has been sent to responsible AEC patent group for clearance. U c. Patent clearance not required. I LIOGRAPHIC DATA Ibt 1. Report No. UIUCDCS-R-71-1+90 3. Recipient's Accession No. itle and Subtitle A PROBLEM IN FORM PERCEPTION: ODD SHAPE DETECTION 5. Report Date December 1971 uthor(s) Kiyoshi Maruyama 8. Performing Organization Rept. No. erforming Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 6l801 10. Project/Task/Work Unit No. Illiac III 11. Contract /Grant No. AT (ll-l) -2118 ; sponsoring Organization Name and Address U. S. Atomic Energy Commission 13. Type of Report & Period Covered Sept. 1971-Dec. 1971 14. i Supplementary Notes Abstracts A procedure to determine an odd shape or form in a given set of shapes , each ; consisting of more than two shapes, is described. The "oddness" of a shape in a J; of shapes on the basis of a specified feature is defined and then the "most odd" lipe on the basis of many detected features is defined. To make the system response :e an average human response, modification of an assumed weight vector by means of an tention mechanism" is considered. Some of the results attained for various criteria ; the system simulation are as follows: (l) A comparison of Linear Separation and -tlidean Distance Separation methods indicates that the Linear Separation model ade- tely models psychological form perception. (2) The attention mechanism described in Is paper gives better system responses than those given without it. (3) The gestalt usure (emphasized in the psychology literature) which is defined as the function of j: perimeter and area of a given shape becomes less important. (k) The shape repre- ftation used in this paper, called the pattern sequence, S r (x ), has many advantages : the analysis of geometric shapes. ' iey Words and Document Analysis. 17a. Descriptors shape, form, oddness, feature, pattern sequence, linear separation, Euclidean distance separation, attention mechanism, system response, human response 'I Identifiers/Open-Ended Terms '•COSATI Field/Group '•variability Statement Releasable to the public " NTIS-35 (I0-7C 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 115 22. Price USCOMM-DC 40329-P7I ^ \$l V 0ND «T m JBS900BI 88BH M ■ *;% #w*