Biological shape and visual science (part I)

I

.I; rheor. Bioi. (1973) 38,205-287

Biological Shape and Visual Science (Part I) HARRY BLUM Division

of Computer Research and Technology,

National Institutes of Health, Bethesda, Maryland 20014, U.S.A. (Received 10 August 1971) “To my wife and children who bore the brunt of the turning inward that this work required.” Appreciatively, Harry Blum A new geometry based on the primitive notions of a point and a growth is explored. Growth from a boundary generates a description of an object that is centered on the space it includes. Growth from this centered or core description generates the boundary by an inverse growth. This leads to new properties and descriptions which are particularly suitable for many biological objects. Some implications for mathematics and biology are discussed. Part II explores the use of this description to the understanding of the visual process. Some implications for a revised view of nervous system structure and function are discussed.

Part I Biological

Shape . . .we.. . bypass the dogmatism of modem mathematics with its popular tendency to display geometry as a chapter of formal logic. Lanczos, Space Through the Ages

1. Introduction Part I of this paper introduces a new method of shape description, one that is of particular relevance to biology. Shape is an unsolved problem of biology which crops up in three fundamental ways. First, there is the morphological and taxonomic problem. How do we describe the shape of a cell, organ or 205 14 T.B.

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organism in natural, yet mathematical terms? Second, there is the neurophysiological and psychological problem. How do organisms describe and characterize other organisms’ shapes? Third, there is the developmental problem. How is shape coded by developing organisms (and their environments) so that they can assumethe shapesthat they in fact do assume?In vision, we perform such biological shape operations naively. Our theory is implicit. Consequently, we are without a formal statement for a fundamental aspect of biology. We are in the position of a physicist trying to develop mechanicswithout Euclidean geometry. Biology has no set of statements for its everyday spatial relations. It is small wonder then that scientific biology appears largely as epi-physics. Part II of this paper applies the new mathematics to our understanding of vision, a piece of the second of the three problems, to show that it is not vacuous. Obviously, a mathematics for vision is likely to be applicable to other spatially distributed sensory systems-auditory, tactile, somatic. These extensions are not explored here, although application to them and to other aspectsof nervous system understanding is suggested.The range of applications goes beyond the second problem, perhaps beyond biology and into physics. But the application to visual science is a reasonably self-contained topic and will suffice to give an idea of the profound changes that must be accomplished if we are to see in biology those aspects of it which are not simple extensions of our ways of seeingin physics. Further comments on the application to vision and biology are deferred until the introduction to Part II. Shapes are normally described by their boundaries. Here we shift the description to the interior by using as primitive, a growth or a disc (growth of a point). Two-dimensional, filled-in shapesare then described using the discs which just fit inside. The description consists of two parts: the symmetric axis or locus of centers of these di.scsand the radii of these discsalong the symmetric axis:1 A number of important new shape properties of an object become apparent in this new description. (The extension to higher dimensions is reasonably straightforward, but is only suggested in this paper.) I commend to the reader a brief review of the illustrations of the paper before undertaking a detailed readin g. In this way he will get an overt In introducing this concept (Blum, 1967) and in a variety of related work done subsequently (see Supplementary Bibliography), other names have been used-the medial axis, the skeleton and the stick figure. I submit that the name used here reflects the crucial role of symmetry better than the medial axis. “Skeleton” is confusing both to the biologist and to the mathematician, who has already used it in topology for an almost diametrically opposite notion-for those elements of a simplicial complex that are boundaries of it. “Stick figure” is too specific and dots not carry with it the accompanying notion of space width.

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view of the altered content of this geometry, and have a feeling for the direction toward which the foundation material is headed. For example, a simple local distance property identifies a pinch in the object. Curvature of the locus of centers identifies how the object veers. When curvature of the distance function is introduced, simple shapes are no longer the rectilinear triangle and polygon, but curved and flexural: the worm, wedge, cup and flare of section 10. New notions of length and width applicable to amorphous objects appear. That these new properties are naively recognizable represents to me the strongest confirmation of their importance to vision. A great variety of new shape formalisms particularly suited to amorphous forms are possible via this new description. Two are discussed at length, symmetric disc coordinates and A-morphology. The first is a precise coordinate description based on this circular width measure of an object. The second is a quasitopology, lying between the traditional “hard” geometries and the newer topology; providing a rich new tool for describing biologically important objects-ones which can flex and/or are not projections of planar bounded objects. A wide variety of theoretical and applied disciplines have a legitimate interest in shape. It is impossible to cover the topic in a way that would satisfy them all. The path taken here is aimed at illuminating the breadth of material opened up. Rather than becoming involved with the mathematical formalisms for their own sake, I have aimed at providing the most robust conceptual base for rethinking shape in our experimental sciences that involve processes in matter. For it is my sincere belief that we are at the stage in biology where we must properly come to grips with some basic unknowns. Process is the domain of operation there, not number or symbol manipulation. The kinetic orientation of this paper is no accident for another reason. It is the simple result of my naivete with many of the niceties of modern mathematics. Fortunately, a number of people without this defect have at least partially filled the gap: Calabi and Hartnett, for continuous mechanisms, and Mott-Smith, Rosenfeld and Montanari for discrete mechanisms. Their work has influenced this paper both explicitly and implicitly. A supplementary bibliography is supplied for those interested, although many of the formal details will be obvious to those with the necessary background. The existence of good physical models for the geometry indicates that a proper formal structure exists. Lapses in the formalisms are then research problems and may generate new mathematics which is still close to important unknowns of the immediately sensible world. Since Part I is aimed at providing a wide base for shape in biology, some of the material may be unnecessary for Part II, which is concerned primarily with shape in vision. Those who are essentially interested in this more 14*

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restricted aspect, and who wish to avoid undue involvement with the geometry, per se, should scan sections 3 and 11 and the illustrations of Part I; then go directly to Part II. They can return to Part I selectively when clarification is needed via the Glossary at the end, which contains page references to the definitions. The italicized terms are defined there. 2. Preliiinaries

We start by modifying the notion of symmetry. Conventional symmetry is a congruence property of shapes under a motion. Figure 1 shows the three types of symmetry in the planar case. Line or mirror symmetry results from

(b)

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FIG. 1. The three types of conventional symmetry. (a) Mirror symmetry. (b) Rotational symmetry. (c) Translational symmetry, (d) An exquisite combination taken from a Moorish panel.

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a congruence under a “Sipping over” motion; point or rotational symmetry, from a rotation; translational symmetry, from a linear motion. Conventional symmetry is an important area of mathematical physics and has been used there extensively. An elegant introduction to conventional symmetry can be had from Weyl’s (1952) book on the subject. Of the conventional types, only line symmetry will be directly meaningful here. Symmetry, as used here, is a property of points in space generated by shapes, rather than a property of the shapes themselves. (This concept of symmetry has a strong analogy with the Relativity Theory concept of simultaneity, in which events separated in space are seen as simultaneous only at certain special points in the space. This relationship will be explored more closely later.) Shapes generate symmetric points in space in either of two equivalent ways: by an equidistance criterion or by a maximal disc criterion. While the latter view is more elegant as a pure geometry, the former is closer to today’s thinking about shape. For tutorial reasons, therefore, the concepts are introduced using the equidistance criterion. The maximal disc criterion is also introduced because it supplies new mathematical and experimental insights needed later. A number of steps are taken now to keep the development as uncomplicated as possible. Initially, shapes are two-dimensional and planar. To avoid retracing well-worn steps of geometry, terms are used in their conventional geometric sense unless otherwise defined. Pathological curves, which are of no shape interest, are avoided by limiting the development to specially defined objects. An object is a point or a closed collection of connected points whose boundary is smooth (has curvature defined) everywhere, except at a finite number of points (line ends, corners, cusps and branches), at which it must have one-sided tangents. Figure 2 shows some allowed and non-allowed objects under this definition. To avoid involvement with open sets, a new definition of boundary of an object is introduced. It consists of two types of points. Exterior boundary points are those object points which are “next to” non-object points. Thus, they form a zero- and/or one-dimensional set. Interior boundary points are specially designated non-exterior points which otherwise fullill the criterion of a boundary: they are “next to” nonboundary object points and curvature exists everywhere except at a finite collection of points. Touching objects may be treated in a variety of arbitrary ways. We use here a rule forbidding the interior boundary from partitioning the remaining interior points. Therefore, partitioned interior points of an object may be connected only by an exterior boundary. Objects may touch each other as shown in Fig. 2(b); their interior boundary is common to both objects and it partitions the objects’ interior points. A finite collection of objects is a scene.

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Scene

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(b) FIG. 2. Some (a) allowed, and (b) non-allowed objects. Heavy lines represent boundaries; crosshatched areas, interior points. (Interior points abutting directly on outside space represent open sets in that neighborhood.)

The object ground (Unlike objects

of an object or scene is the collection of points not in the or scene together with the boundary of the object or scene. The of the ground is an identity operation. Note that all sets are closed. the concept of “closure of the complement”, the ground maintains of lesser dimension than the space.) Figure 3 shows the ground of

ground

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the objects and scenes of Fig. 2. Note that in the ground as well as in the object, a boundary can be entirely “next to” object or ground. For example, the ground of a point or line segment is the entire plane with the same point or line as boundary. A ground may not be an object or a scene since it is unbounded, but it may contain them. The boundary is a distinct category of points that belongs both to object and ground. (Let me relieve those who feel the “ground” notion abhorent or unnatural since points belong to both object and ground. The difference between “ground” and “complement”

FIG. 3. Grounds of objects and scenes in previous figure.

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is in principle unmeasurable since it is infinitesmal. The complement is used in set theory because of the importance of disjointness there.) The distance from a point in an object to that object shall be the shortest Euclidean distance to any point of its ground; from a point outside the object, to the object itself. Points inside the object shall be assigned a positive value; points outside, a negative value. Where the object consists of boundary points entirely (lines), all distances shall be zero or negative. Thus, distance

FIG. 4. Pannormals, lines along which distance between a point and object is measured. Normals issue from smooth points, radials from non-smooth points. Solid lines show nearest pannormals, ones using nearest distance constraint. Dotted lines show global pannormal extensions. These have no such constraint.

to a boundary can apply to either object or ground if we ignore sign and do not move domains in the discussion. Figure 4 shows some examples of distance measured from an arbitrary point to an object or boundary. Note that the line along which the shortest distance is measured is a normal to the boundary when the nearest point is on a smooth part on the curve. For other points it is along a radial. For discrete points, the radials form completely around the point; for line ends, they form along the half circle left empty by the normals; for corners and branches, along the angle left empty by the normals. These shortest paths or point-object geodesics, we shall call pannormals. Some geometries define distance from a curve, rather than to

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a curve. These ignore the nearest distance requirement, simply measuring along infinite normals to the curve. Other variations are also possible. We may use full (doubly infinite) lines or rays (singly infinite lines), and either terminate them or not. Note that in all these definitions, distance with regard to a line segment is defined only on its normal. These differences are important since we start with one definition and generalize the process later using another one. The differences between lines and rays are not significant enough to explore here in general. These differences exist only for radials. It suffices to state that we use rays when using nearest distance termination and lines when not. Geometric reasons for this will become more apparent later. Since distance to a boundary may be defined in different ways, so may parallels, as shown in Fig. 5. We may step off a given distance along a ray pannormal on each side of the boundary and call each such locus a parallel. Or we may only allow points on the parallel whose given distance is also the nearest distance. Both of these definitions are used in different domains of geometry. Differential Geometry ignores distance constraints and generates the global entities, G-pannormals and G-parallels. Convex Geometry invokes the nearest distance constraint and generates N-pannormals and N-parallels. The difference is apparent in the two point scene of Fig. 5(a). Physically, the G-parallels are in the succession of wavefronts that result from dropping two pebbles in a pond, the N-parallels are the wavefronts resulting from a fire started at two points in a field of grass. The first has “flow-through”; the second, not. (We will return to the grassfire analogy shortly.) Figure 5(b) shows the situation for a simple object, an ellipse with interior. In the ground parallel and nearer object parallel, N-parallels and G-parallels are identical. They differ only when they have crossed a point on the pannormal which is equidistant to another point on the boundary via a different pannormal, as occurs for the non-major axis points of the ellipse; or when they have reached the center of curvature of the boundary, as occurs for the two major axis points of the ellipse. These conditions are disjoint, the latter case being a special situation which we examine in a later section. For all but this latter case, the length of the pannormals at these transition points is equal. The N-parallel is the boundary of the union of discs centered on the originating curve; the G-parallel, the envelope of discs. Physics, and this is most evident in optics, has been concerned with wavefronts beyond centers of curvature and has resorted to g-parallels and Fermat’s Principle. Until section 11, on generalized symmetry, we are interested exclusively in the b-parallels and b-pannormals. Thus, the logically simpler definition, union of discs, is applicable. By observing the curves in Fig. 5(c), it becomes clear that under the “nearest” definition, “A parallel to B” does not imply “B parallel to A”.

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Observing Figs 4 and 6, it can be seen that in going from a boundary into an object along a pannormal, one passes a symmetric point (sym-point), a point beyond which another pannormal becomes shorter. (Pannormals that coincide are considered distinct if they issue from distinct boundary points-for example, those that issue from parallel object boundaries.) Such points always occur in traverse into the object, since it is bounded. In the

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FIG. 5. Nearest and global parallels. (a) shows parallels for two paints. Note similarity of global parallels with waves in a pond, of blocked parallels with a grassfire. (b) shows parallels for ellipse. Note that parallel is boundary of the union of discs on ellipse to the center of curvature (shown dashed) only. Beyond that, envelope of circles create boundary. (c) shows parallels for object of previous figure. (Blocked parallels are shown in solid lines, global parallels in solid plusdotted lines.)

Fur. 6. The sym-transform. The sym-function consists of the locus of sym-points and their associated sym-dist. Note that the separated angles of the boundary at the pannormal feet become real angles in the parallel at this sym-point. Note also that the touching disc at each sym-point touches only at the feet of its pannormals. The remainder of the disc lies entirely in the object or ground. This property makes the sym-function a descriptor of the object that is equivalent to the boundary.

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ground, such points occur only for pannormals issuing from points which do not also lie on the boundary of the object’s c~nuex lzull (the minimum convex object containing the object). This is a simple consequence of Motzkin’s Theorem, which states that all circles touching a convex set from the outside, touch at only one point (Yaglom & Boltyanskii, 1961). For simplicity, we consider initially only points inside the object; there sympoints always occur. The sym-point is the fundamental unit of interest here. We define a number of terms relating to it. The length of the pannormal from the boundary to the sym-point is the symmetric point distance (sym-dist); the locus of sympoints, the symmetric axis (sym-ax); the sym-ax with associated sym-dist at each point, the symmetric axis function (sym-function); and the sym-function is the symmetric axis transform (sym-transform) of the object. Note that the sym-ax can exist in the ground also. Thus we may have object sym- properties, ground sym- properties and total sym- properties. Our primary concern will be the object sym-and that will be understood unless otherwise indicated. The absence of any ground sym-ax indicates a convex object. Objects or object parts which have a sym-dist of zero (line segments, arcs, points, corners) are their own sym-ax. The fundamental property of the sym-function, the sym-ax and associated sym-dist, is that the object sym-function (defined only in the object) is a complete description of the object and fully equivalent to the boundary for defining it. Although these two descriptors are completely interchangeable, they bring out very different properties of the object. We may show this equivalence most easily with a specific reconstruction process, which generates objects from the sym-function.7 The object is the union of discs of sym-dist radius on each object sym-point of the sym-ax. Returning to Fig. 6, we first note that every boundary point has at least one object sym-point associated with it. Secondly, each sym-dist represents the shortest distance from the point to the boundary. Thus, the disc of sym-dist radius touches those boundary points only for which it is the sym-ax. All other boundary points are further away; so that it lies entirely in the object. This generation of the boundary from the sym-function is the inverse transform. A number of other methods can be used to perform this inverse transform. The processes for generating the transform and the inverse transform give important insights into the nature of the sym-function and into potential mechanisms for implementing them by both organismic and synthetic processes. 7 This follows from the Calabi-Hartnett Theorem (1968). It states that the sym-function of a set defines its convex deficit (the difference between the set’s convex hill and itself). A boundary whose sym-dist >O, includes this interior as deficit. This result is developed differently here to bring out the object description directly.

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3. Physical Analogies and Heuristic Aids This section makes a digression in the development to give the reader a more intuitive grasp of the material. This will keep the later development grounded in physical reality and also allow the reader to assimilate the new shape descriptors in a variety of ways. These aids are still geared at a development using pannormals and parallels. The grass$re, mentioned earlier, is probably the intuitively most appealing physical implementation. We define a grassfire as a process in a plane with the following three properties: (i) it has value either one or zero at each

FIG. 7. A grassfire using “magic marker” type ink and blotting paper. The sym-ax is deposited by the transported dye when the motion of the fluid is stopped by contact.

point (all or none excitation): (ii) a point may be excited from an external source, or by an adjacent point with a delay proportional to the distance between (external excitation and uniform isotropic propagation); (iii) an excited point cannot be re-excited for some interval of time (refractory property). An arbitrarily short refractory period will prevent the waves from flowing through. Figure 7 shows a simple implementation using “Magic Marker” type fluid and blotting paper. Note the depositing of additional dye at the sym-points (Nichols, 1966). Figure 8 shows a successionof wavefronts for some simple lines, convex objects and a point scene. Each has a particular point of interest. The corner shows an asymmetry of the ensuing sym-ax. It exists “inside” the corner only. The velocity of the sym-point is determined by the angle generating it. Tt is always greater than the space propagation velocity-a 1‘3. I?

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FIG. 8. A succession of grassfire wavefronts for some simple inputs. At the top, the grassfire is started along open contours. The sym-ax (shown dotted) occurs on the inside of the angle only, starting at the center of curvature and starting at a pinch in the space--ending at the center. The center panel shows some closed contours combining the above features. The sym-ax disappears at the largest inscribed circle. Note that the boundaries are convex and have no outside sym-ax. The bottom panel shows the sym-ax for a parallel boundary and for a set of points on a circle. In the parallel oval, the grassfire disappears all at once. The points on a circle give an example whereby the object is in the ground and discrete points can be treated as equivalent to a contour in generating an object.

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concept analogous to “phase velocity” in wave physics. The circular arc shows the sudden appearance of the sym-ax when the wavefront reaches a contour’s center of curvature. These open line inputs are extended to form closed convex object boundaries. No sym-ax appears outside the object. The internal wavefront disappears at the largest sym-dist point, the center of the largest inscribed circle. The circle shows an isolated sym-point, and the parallel oval, a sym-ax with constant sym-dist. The last input is a scene of dots arranged in a circle. It shows that the sym-ax may branch in the ground as well as in the object. An interesting figure completion property also appears. The two point scene of Fig. 5 showed the sym-ax issuing along the bisector of the two points; starting with infinite velocity, and decreasing

FIG. 9. Grassfire and inverse grassfire for boundary, object and ground. Wavefront times are shown for boundary. When the object is known, grassfire can be separated. Object times (or distances) are positive; ground, negative. The inverse grassfire is generated by exciting the internal sym-ax in reverse times. It generates the wavefronts in reverse order. At time zero, the original object is generated. Note that new inverse grassfire points are always excited outside the wavefront generated by previous points since the sym-ax velocity is faster.

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asymptotically to the space velocity. It is seen here that the velocity is ir measure of the smoothness of the wavefront. A threshold on velocity can be used to make the sym-ax disappear as the wavefront becomes smooth enough. This permits points on a circle to be identified as a circle by its isolated point sym-ax. We shall return to this non-topological property that permits figure completion later. The previous illustration used the boundary of objects as inputs, so that sym-axes generated inside and outside are both negative and indistinguishable by sym-dist sign. If we know u priori (becauseof color or boundednessof the space or for other reasons) which is the inside and which is the outside. we may assignpositive values to the object and negative values to the ground. To implement this with a grassfire, we break it into two parts : a total ground excitation yielding the object sym-ax and a total object excitation yielding the ground sym-ax (see Fig. 9). The first is assigned positive values: the second, negative values. Actually, one need not excite the entire object 01 ground to restrict excitation if it is possible to inject a refractory state. By such an injection on the side of the boundary to be inhibited. even of arbitrarily thin width, one can inhibit propagation through the entire object or ground. Stated even more tersely, a boundary and one “side” defning point, define the object (Mott-Smith. 1970). The processis shown for some non-convex shapesin Fig. IO. Parallels are now omitted and arrows indicating direction of flow on the sym-ax have been added at points where there is a change in direction of how. Note here the external sym-ax and the interesting internal sym-ax points representing global shape features of the object. For example, the appearance of two corners that issuefrom a single point representsa pinch in a shape. (Contrast this with the disappearance of two corners in the ellipse of Fig. 8.) The three-branching sym-ax of the other objects shown representsthe three-sided character of the objects, as does the triangle of Fig. 8. The grassfire can also be used to create the object from its sym-function that is, to perform the inverse transformation. We have seenearlier that the object is the union of discs of sym-dist size, centered on each sym-point of the object sym-ax. We may generate such a single disc by exciting a point and letting it grow until its sym-dist is reached. This may be done simultaneously for all the sym-points by changing the sign of their sym-dist-making them negative-and running the clock forward from some negative time which is algebraically less than the greatest negative sym-dist. Each sym-point excites the spaceat the time when its negative sym-dist is reached. (Equivalently, we may simply run the clock backwards.) Refer particularly to Fig. 9. At time zero, each point has had a chance to grow into the right size disc and the grassfire produces precisely the union processif we remember

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FIG. 10. Sym-axes of the boundaries of more complex objects. Arrows have to indicate direction of sym-ax flow, or increasing sym-dist, at places where Observe (i) the appearance of double sym-axes going in opposite directions object, (ii) the three-sided convergence of the sym-ax that results from the objects and (iii) the existence of ground sym-axes for the non-convex objects.

been added it changes. at pinch in three-sided

and include all those points that have been “burned”. This can occur because the sym-ax velocity is always greater than the space velocity, so that new points are ignited outside the wave generated by the old points. The wavefronts are now generated in reverse order. If the process is allowed to continue, it generates the ground sym-ax. Note that while the grassfire process is the samefor the direct and inverse transform, the inverse grassfire must be ignited as a function of time. The inversion processis related to the “matched filter” process in electrical engineering. In that process, the delay of a network to signal parts is determined by its responseto an “impulse” stimulation. which contains all signal parts. When the time inverse of that response is passed through the network, each part is pre-excited by the delay of the network and arrives at the output in synchrony. The requirement for synchrony of input is not always acceptable. particularly in some biological systems. It is possible to eliminate this require-

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ment by complicating the grassfire, coding distance in a way other than absolute time. In the unsynchronized case, the inputs at a point may arrive at different times and flowthrough must be permitted. Each point is assumed to contribute an independently increasing circularly symmetric function. Each point of the space must respond to the first value of the function from each point of the excitation, even though it arrives at a later time. Distance may be coded by amplitude change, rise-time change, variance spread (in a noisy velocity grassfire), velocity change, waveshape change, frequency change, or a host of other distance dependent processes in a plane. An extremely interesting system can be set up using a two velocity propagation in one or two planes. A point of excitation would lead to two waves. the time difference between which would be a code of distance. The two plane version is particularly suited to doing the direct and inverse transform. since the first excitation would be inserted in the slow plane and the second in the fast-both arriving in synchrony at the object boundary. Fortunately, in all of these schemes each point must know only that it has received two or more excitations at minimal distance. It need only remember the minimal distance that has passed it. An awareness of the variety of implementations possible for the grassfire is essential in a search for the process in biological systems and the synthesis in artificial systems. However, it is only diverting tutorially with respect to shape properties. Therefore, the development continues with the simple grassfire. We should remember. however. that it can take a much wider variety of forms than the simple one we are using to explore the geometry. The inverse grassfire excitation may be thought of as a wavefront generator as well as a disc initiator. Consider the previous figure again. The sym-ax at the non-end points has been generated in a continuous way, the sympoint always moving faster than the fire velocity. When this process is inverted. new points are started outside the expansion of earlier ignited points since the sym-ax velocity exceeds this fire velocity. The rate and direction is precisely that needed to generate an identical and oppositely moving wavefront to the one that generated the sym-ax. Thus, such smoothly moving points generate the same pannormals at the point as those that created the point in the first place. We may look at the sym-ax as an antenna surface receiving a wave at the sym-dist times. Again, this reciprocity between time inversion and the receiving-transmitting relationship of antennas is well known. The main innovation introduced here is the non-linear propagating space of the grassfire and the definition of the receiving surface by the input itself rather than by a predetermined boundary condition. The receiver antenna is now generated by the signal. In addition, sine waves and associated phase angles have been avoided by the use of impulse signals.

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Another view of the transform can be obtained by substituting a threedimensional static process for the two-dimensional dynamic grassfire (see Fig. 11). If a space-time plot of the grassfire is made with the input plane as the x, y surface, and time as an orthogonal z coordinate, a surface will be generated. Each plane at time t will contain the firefront at that time. Of course, for time we may substitute a parallel distance coordinate and get a purely geometric surface.? Since each parallel curve is the union of discs of

FIG. 11. Static three-dimensional representation of a grassfire. The vertical coordinate represents time, or distance between parallels. The slope of this surface, which is the reciprocal of the grassfire velocity, is everywhere one except at the sym-ax, where it lies between zero and one. The surface is the boundary of the union of right angle cones with apices on the object boundary.

increasing radius, the surface generated is the boundary of the union of right cones whose apex sits on the input excitation and whose axis is perpendicular to the object plane (the x, y plane, in which the input excitation occurs). Figure 12 shows the surface for a humanish boundary generated in precisely such a way with a conical cutter on an engraving machine. The surface is smooth and has a maximum directional slope of one everywhere except at the sym-ax points, where it lies between zero (for parallel curves) and one. When the “outside” surface is made negative and reflected across the input t Such a surface appears in analytic function theory and is called a “distance surface” there (see, for example, Goodman, 1964). It appears as a pursuit strategy surface in the Theory of Games and is called a “sandpile function” there (Isaacs, 1965). Neither of these fields develop the geometry we do here.

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plane, the surface becomes smooth there also, except for sym-ax points a~ zero sym-dist. This surface, either in space-time coordinates or in space distance coordinates, gives different insights from the equivalent kinematic grassfire, since it can evoke our knowledge of solid geometry. For example. the velocity of a grassfire generated by two points can clearly be seen to be hyperbolic by simply noting it is the reciprocal of the hyperbola of intersection of the two cones. This purely geometric surface is used later to show the equivalence of the pannormal and maximal disc formulation of the sym-ax. Still later, the space-time formulation is used to show the relationship between the sym-ax and simultaneity in relativity physics. Before leaving this topic, it should be noted that the equivalent to the inverse grassfire is obtained by placing right cones on the object sym-ax. The boundary of their union is also the desired surface. Its intersection with the input plane is the object boundary. Its sym-ax beyond the input plane is the ground sym-ax. Field theory has been used in a number of ways in both psychology and automatic recognition devices. Except for “nearest distance classifiers”, which we make reference to in section 11, these have generally been modelled after the fields of physics. The sym-transform can be thought of from a field theory viewpoint, but it departs sharply in character from those fields. It is a nearest distance field. Whereas the conventional fields of physics have the

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value at a point depending on the total of the input, the value here is determined only by the nearest point. Such fields have a sharp barrier effect, a property we consider more carefully later. It generates a description of an object which is not perturbed by anything put in the field that lies outside the object. Thus descriptions learned under isolated conditions hold under complex conditions (unless the object is occluded). “Flow through” fields do not have this property and can make the reading of a complex scene intractable. A great variety of implementations are possible, a small number of which have been explored, at least cursorily: optical defocusing, creepage in photoluminescent storage surfaces, image intensifier tubes, flying spot scanners, and many others. They indicate the possibility of doing useful intellectual work by active devices with extremely simple neighborhood properties that can be carried out by granular materials. Iterative logical networks are not needed for implementation. Indeed, lattice organized devices can in some important ways be a deterrent to a good grassfire. For example, good circularity requires the use of a large number of element-to-element connections at all distances, or exceedingly complex communications and computation at each point. An interesting property of the grassfire as a shape processor is brought out in an implementation in which it is observed through a set of circular apertures. Normally, one scans a pattern to observe it. In this case, the pattern, via the grassfrre, scans the apertures. A simple property in the aperture extracts sym-axes. Non-sym-ax points spend full time in the aperture, sym-ax points, less than full time. Another interesting viewpoint was used (Mott-Smith, 1967) to determine sym-ax points in a computer implementation. A parallel to a wavefront was generated by a union of discs on it. The original wavefront was then reconstructed by a similar process from the new one. Original points which were not included in this reconstruction were sym-ax points. Thus the asymmetry of parallelism in blocked symmetry was used as a sym-ax detector. 4. Maximal

Disc Formulation

In the previous section, the input started a process which generated the sym-points by laying off parallel displacements to its boundary. Thus, starting with the given object, successive operations performed on it led to a new description. In this section, a complete collection of the new descriptive elements is used directly. Logical operations are then performed to select the applicable subset of descriptors. Whereas the earlier method used operations traditionally developed for working with line and arc elements of conventional geometry, this new method goes directly to discs as descriptors, leaving the line as a derived element. Although this second viewpoint is T.B. 16

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important to both theory and application, the resulting description is identical with the earlier one. Hence, this section may be skipped initially for those interested in going directly to the shape properties that result. These two bases are required however before going into generalized symmetry and visual application. The primitive notions consist of an object and a disc, and the relation of inclusion, which is used as normally understood-one set includes another if all the points of the included set are in the including set. A maximal disc is one which is included in the object, but not included in any other disc in the object (see Fig. 13). We can pick a set of increasing discs, each including

FIG. 13. Object and ground maximal discs (heavy circles). An object maximal disc is a disc which fits entirely inside the object but in no other disc inside the object. The right set of discs shows these in arbitrary relationship. The left set shows these touching a common boundary point. A ground maximal disc is also shown.

a boundary point and the smaller discs, by going out along the pannormals for the discs’ centers. Then, every boundary point is included in some maximal disc. We show below that the maximal disc description is exactly equivalent to the sym-ax description of the last section. The notion that space could be treated as a collection of elements other than points goes back over 100 years (Plucker, 1865). While developing in detail only the notion of the plane as a two-dimensional collection of lines, Plucker also suggested that it could also be considered as a three-dimensional collection of circles or a five-dimensional collection of conic sections-that is, 2, 3 or 5 independent parameters, respectively, could uniquely specify each element. He proposed a three-dimensional half-space (a plane and the space on one side of it) for representing by points all the circles in the plane. We shall use it for discs instead and call it the PJucker disc coordinates (see Fig. 14). The center of each circle is represented by its location in the base or object plane, the radius by the height of the point above the plane. Thus,

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Fro. 14. Section through a disc inclusion cone in Plucker disc coordinates. Each point in the space above the plane represents a disc in the plane centered below it, of radius equal to its height. The apex point represents the heavy disc in the object plane. (a) represents discs included in the apex disc, (b) represents discs including the apex disc, (c) represents discs lying entirely outside of the apex disc and (d) represents discs whose boundary intersects the apex disc. If we allow the cone to cut through the object plane, we obtain the ground space. (~3 then represents the discs lying in the ground of the apex disc. The apex reflection is the representation of the apex circle. Applying the inclusion cone to an object generates the same three-dimensional surface as the grassfire. See Figs 11 & 12 and text.

each point represents a unique disc and every disc is represented. The right 90” cone whose apex is a particular disc point, bounds the inclusion relation for that disc with respect to all the other discs and is called the inclusion cone. This full double cone, reflected by the base plane, divides the upper half space into four parts. Above the apex are those points whose represented discs include the apex disc. Between the apex and the base plane are those discs which are included in the apex disc. Below the reflected cone are those points whose discs lie entirely outside (are disjoint with) the apex disc. The remaining space consists of points representing discs whose boundaries intersect the apex disc. Note that “lying outside of” is equivalent to “included in the ground of”. If we permit the cone to continue into the negative half-space instead of reflecting it, we may interpret it as the ground spaceas opposed to the object spacefor the positive half-space. The boundary of a disc (a circle) is represented by a reflection of the disc point into ground

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space. Boundaries or other zero- or one-dimensional objects have no included discs in the object space except of zero size. They are consequently their own object sym-ax. A point or straight line has no other ground sym-ax, a fact which will be of importance later on. In this geometry, a straight line is a derived, rather than a primitive, object. The disc inclusion properties with regard to an object is shown in Fig. 11. The sequence of nested discs that include a boundary point lie on a line in the disc coordinate space which is normal to the boundary if it is a differentiable point, or along a radial if it is an “outside corner” point. That line rises from the object plane at 45”. As in the earlier development, two different conditions can terminate lines. One occurs if the nested tangent discs reach the radius of curvature first. The other occurs if the nested boundary discs touch another boundary point first. We can then reach this maximal disc point by traversing along a second pannormal. Looked at this way, we see that the inclusion surface in disc coordinate space is precisely the same surface as the nearest distance surface of Figs 11 and 12, and these two definitions are equivalent. This allows us to use disc concepts when they are more illuminating or convenient. We introduce, therefore, the notion of sym-disc and sym-circle as the disc and circle, respectively, of a sym-point or boundary point. The description of shape by discs instead of lines has some interesting implications for an entirely new geometry that includes Euclidean as a special case. Section 15 discusses this further. 5. Types of Symmetric Points Sym-points are now classified by considering three properties defined either on its pannormals or on the object boundary touching points of its sym-disc: (i) discreteness or continuity, (ii) the number of intervals between and (iii) the character of the sym-dist in the neighborhood of the sym-point. We are interested here in points whose sym-dist is not zero. In that case, a number of degenerate situations are possible which deflect the development and are therefore saved until later. Figure 15 shows a variety of discrete and continuous pannormal sympoints along with the pannormals and boundary parallels at the point. Pannormals for a sym-point exist where its sym-disc touches the boundary. The shape of the boundary outside these points is irrelevant provided only that it not touch the disc anywhere else. Contact with the boundary may be discrete if there is an isolated touching point, or it may be continuous if the boundary has a circular arc centered on the sym-point. A sym-point may have both types of boundary contact. A sym-point all of whose pannormals

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FIG. 15. Types of sym-points. The left column shows continuous pannormal or finite contact types. The contact angle is shown cross hatched. The right-hand column shows the isolated pannormal, or zero contact, sym-point types. Pannormals, parallels and interval angles (the angles between the pannormals) are shown. Each interval angle has a sym-ax associated with it. The n-sym-point designates the number of sym-axes in the neighborhood of the point. Sym-points can be further categorized by how the sym-dist is changing near the point. An a,b-sym-point designates a sym-axes have decreasing sym-dist as one leaves the point, and b have increasing sym-dist. Only discrete pannormal 2-sym-points can exist over an interval. All others are discrete sym-points. The angle associated with the smooth 2-sym-point is the object angle and the sym-ax tangent, the object direction.

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are isolated lines is an isolated pannormal sym-point or zero contact sympoint. If the sym-point contains any continuous pannormals, it is a continuous pannormal sym-point or jinite contact sym-point. Each continuous pannormal has its contact angle. Their sum at a sym-point is its totaE contact. The angle between pannormals or disc touching points is the interval angle. These are also shown in the figure. The number of intervals associated with a point make it an n-sym-point. The interval angles and contact angles about a point sum to 360”. Figure 15 also shows parallels to the boundary at a sym-point. Note that the parallels are continuous in the interval even if the boundary generating it is not. The angle formed by the tangents to the parallels is the sym-ax angle and it is 180” less than the interval angle. Thus, it is positive for an interval angle greater than 180” ; it is negative for an angle less than 180”. The sym-dist increases and decreases, respectively, as one goes away from the point. For a small interval around the point the geometry is representable by a rectilinear approximation, and the angle bisector of the sym-angle is equidistant from the boundary. (This approximation is improved in the next section by using the circles of curvature as boundary approximations.) Thus, each pannormal interval has a sym-ax associated with it and the n-sym-point also refers to the number of sym-axes which enter the point. We now examine shape properties in more detail starting with the isolated pannormal sym-points. The contact angles are now zero and the sym-ax angles sum to (2-n) x 180”. Consider first the 2-sym-points, an example of which is the non-end points of Fig. 9. The parallels to the boundary are now smooth through the sym-point and one sym-ax angle is the negative of the other. We define the positive sym-ax angle as the object angle at that point. There is now a unique sym-ax tangent, the object angle bisector, which we call the object tangent at the point. The sym-points at which exactly two pannormals intersect form a differentiable curve whose tangent is the object tangent. The smooth sym-ax and sym-dist generate a smooth object angle and object tangent; consequently we call 2-sym-points, smooth points also. They are the only type of sym-points that can exist over an interval, all others being isolated discrete points. We will further classify these points when we introduce sym-dist criteria in the neighborhood. Meanwhile, all sym-points that follow are non-smooth or discrete sym-points. The “open” and “closed” polygons at the bottom of Fig. 15 lead to branch points or n-sym-points, where n is three or greater. These need not be rectilinear and more general curves generating such points are also shown. At the top is a l-point or end point. It occurs at the small end of a curve where the radius of curvature is reached along a single pannormal, its object angle being 180”. The parabola and ellipse are two common curves with such

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sym-ax end points. Simple geometric reasoning shows that a smooth end point cannot occur at the large end of the sym-ax. This isolated pannormal 1-sym-point is the unique case in which the sym-ax is not definable as an equidistance point-only by virtue of the pannormal changing as one proceeds beyond that point. In this limiting case, the sym-disc has more than tangent contact but less than finite disc contact with the boundary. Note that the existence of the end sym-point forces the property stated earlier, that the center of curvature of a boundary cannot lie between itself and its sym-point. We go next to the continuous pannormal sym-points. They must lie at the center of some circular arc of the object boundary. The large and small end points of Fig. 9 show these. Figure 15 also shows the continuous pannormal sym-points. The disc is a unique case, a O-sym-point, having no pannormal intervals. Next are shown end sym-points. They may now be either at the large or small end of a sym-ax. The continuous pannormal 2-sym-point has no branches, but its two sym-ax angles are not the negative of each other. Unlike the smooth 2-sym-point, no object angle exists here. Consider the action at these points in a grasstie. Isolated pannormal I-sym-points accelerate smoothly from the space velocity, and continuous pannormal ones with an impulse. The same is true of 2-sym-points if they are not local maxima or minima of the sym-dist, a case we cover below. Isolated pannormal 2-sym-points have a smooth acceleration of the grasstire, continuous pannormal ones have an impulse given to the velocity at that point. We may think of the boundary pushing the grassflre corner differentially when a differential length impinges on it and impulsively when a finite boundary length impinges on it. Continuous pannormal 23-sym-points are like their equivalent isolated pannormal ones with the exception that the sym-disc has a finite arc contact with the boundary. Let me stress that the object angle is a real angle (corner) at the sym-ax but only a pseudo-angle on the boundary since no actual corners exist at the boundary points of tangency. These pseudo-angles are not easily defined on the object boundary directly since the points to be related are not obvious there. We now apply the sym-dist properties of the neighboring sym-ax to the sym-points by observing whether their sym-dist is a strict maximum, a strict minimum, constant, or none of these in their neighborhood. Pictured in the grassfire, this notes whether the fire burns into the point, away from the point, burns at the same time in the neighborhood (as occurs for a parallel boundary) or flows across the point. To the sym-points that are already discrete, this merely gives an additional descriptor. For smooth points, it distinguishes a new set. We define these without regard to isolated or continuous pannormal points. Sym-points that are a strict maximum in their

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neighborhood are bulb points. Two types of strict minimum points exist: I-sym-points are sprout points and 2-sym-points are pinch points. (No minimum >3-sym-points can exist. To have an increasing sym-dist along a sym-ax, the interval angle must be equal to or greater than 180”. Clearly, no sym-point can have more than two such intervals.) Sym-ax branch points at which the sym-dist increases along some axes and decreases along others are fork points. (Again, the sym-dist can be increasing along one sym-ax at most.) Points at which the sym-ax is constant must be 2-sym-points and are called worm points. These are obviously smooth points and exist in an interval called the worm interval. Their end points we call worm ends. The directed sym-ax is defined in the direction of increasing sym-dist. We shall also expand the sym-point designation. An “a, b-sym-point” shall mean that a sym-widths decrease from the point and b increase. In the event we wish to designate worm neighborhoods, we shall use a triplet indicating decreasing, constant and increasing sym-axes in the neighborhood. Thus, a pinch point is a 0, 2-sym-point and in the in-center of a triangle is a 3, 0-sym-point. The non-smooth points plus other points specified by curvature criteria that are later invoked become distinguished points, which later become bases for breaking objects into pieces. 6. Width and Length In this section, we continue defining shape properties on the sym-ax. These are compared and contrasted with similar properties now in use. The sym-dist is a width measure of the object or ground at the boundary touching points of the maximal disc or its center on the sym-ax. Geometers have defined width of a convex object at some orientation as the distance between two parallel lines that just touch it from the outside-exterior parallel lines of support (Yaglom & Boltyanskii, 1961). See Fig. 16. (The same result is achieved by defining it as the length of the projection on a line perpendicular to the parallels.) A small body of interesting work has been associated with the notion, especially with regard to curves of constant width. From a shape point of view, its insight is limited, although it has been adopted implicitly in some important visual applications. Its major failing is that no proper description appears for non-convex objects, each having the same description as its convex hull. But the projection view has some subtler deficiencies as well. Using this view, one would conceive of the axes of an ellipse as being determined by the extrema of its width. While mathematically proper, the variation of width with angle is too “soft” (its values vary very little near the defining points) to permit obtaining precise axes in practical systems. We return to this problem again in the application of the

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theory. The sym-ax provides a straight line in such a case. To distinguish these two notions of width, I will call the one defined on “parallel lines of outer support”, the outer width at an angle, and the other using the “maximal disc of inner support”, the inner width at a point. In this paper, width will mean inner unless otherwise specified. Note that width = 2 x sym-dist. The maximum inner width we call the object width. Maximum and minimum exterior width we call projected length and projected width, respectively. Note both here and below, that “length” does not have a universally applicable

FIG. 16. Inner and outer width. Projected or outer width is the conventionally used one Inner width is the maximal disc diameter. Some other useful definitions are also shown. The ground sym-ax defines the convex deficit (shown cross hatched)-the difference between the object and its convex hull (the smallest convex set including it).

definition. For a rectangle, it is considered as the larger distance between parallels, for a sharp rhombus, it is the diagonal. Picture coders have also used a notion of width in describing black and white line drawings. Rather than code point by point and obtain long strings of “0”s occasionally interspersed by “13, they simply code the distance or “run length” of “0”s between “1”s along a prescribed direction of scan (usually a simple TV type raster). See Fig. 17. A more natural (independent of external coordinate system) way to consider width would be to measure distance orthogonal to the contour. But, as can be seen from the figure, this leads to an order dependent definition of space width. The width at point A is measured to B if it is the starting point, and to C if it is the terminal point. Thus an order independent space width must be defined on the space, as

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occurs with the sym-ax, rather than on the boundary. Some examples of width are shown in Fig. 18. Length on the sym-ax is more complex than width. For sym-axes with no loops (we shall see later that this means the object has no holes), sym-lengthor where it is unambiguous, length-is the largest value of the sum of path lengths plus sym-dist of the end points. A number of such lengths are also shown in Fig. 18. A careful examination of this figure is worth many pages of text, which is consequently omitted. Note that the sym-length is larger than the projected length, except when the sym-ax path involved is a straight line. In that case, they are equal. The only figure whose sym-length, symwidth, projected length and projected width are equal, is the circle. Note the

TV SCOI-

FIG. 17. Coding of space width. Run length codes give the distance between contour along a prescribed scan. Length of the normal from one contour to another is more natural but it depends on the starting point. Distance from B is to A. Distance to B is from Cs The sym-width, defined on the space, is independent of order.

intuitively satisfying definitions for length of the coiled and wiggly figures. Note also the unintuitive (from our present views) length and width for the rectilinear figures shown. The relative constancy of sym-length/sym-width ratio with variation in boundary ripples makes it a far better slenderness measure than the conventionally used perimeter squared/area ratio. The octopus-like object shows a length need not concern the same part of an object as its width. A number of intuitively reasonable definitions of length are possible for objects with holes in them. We present an example here for its simplicity, not efficacy. Length is defined the same way as before with the proviso that the sym-ax path may not repeat any arc. It may cross itself only at a point, as shown in the snake of Fig. 18. It is premature to expand this definition before expanding our view. Obviously, length and width are only a small subset of attributes for classifying objects. Shape must have many other measures for incisively dealing with the variety of tasks imposed on it. Those measures remain a central focus,

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FIG. 18. Sym-width and sym-length for a number of objects. The sym-width is shown as the largest inscribed circle. The sym-length is the largest sum of path length plus symdist value at the path’s end points. See text for details on sym-length for objects with sym-ax loops.

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7. Curvature, Plexure and Symmetrization Further shape properties of an object as seen along smooth sym-ax points are now developed. These are extensions of the object width and object angle. The approach taken is still “constructive” since it reflects process and mechanism more directly. In addition, constructive methods are much older and bear out the important point that a major part of this geometry could have been developed with pre-Cartesian methods if the problem of biological shape had been appreciated. The next section reworks this material from an “analytic” point of view and ties it in with coordinate and differential methods. For those who prefer analytic methods, I suggest the following section precede this one. Curvature is traditionally defined by change in tangent or normal angle with respect to an arc length and we use the sym-ax as that arc. The reciprocal of curvature is the radius of curvature. The center of curvature lies on the normal at a distance equal to the radius of curvature. The circle of curvature is the touching circle at the center of curvature. In Fig. 19, the boundary is

FIG. 19. Geometry of object curvatures. The boundaries are approximated by their circles of curvature. The sym-ax of two circles is a conic section in general, here an ellipse. Asymmetric or axis curvature is the average curvature of the boundary parallels in the direction of the sym-ax normal. Symmetric or width curvature is the rate of change of the object angle with regard to the sym-ax arc length. The construction shown above works with the radii of curvature, the reciprocals of the curvature.

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locally approximated by its circles of curvature. Circles parallel to these are drawn through the sym-ax point. Positive direction is defined as that of increasing object width. (For pinch points and smooth bulb points, the methods hold even though the direction changes in the point’s neighborhood. For worm points, direction may be arbitrarily chosen and may depend on aspects of the curve outside the worm interval.) The sym-ax of the two circles is a conic section or a line in the degenerate case. In the above figure, it is an ellipse. The center of curvature of the sym-ax is the center of curvature of this conic section. It lies on the angle bisector of the focal radii and is constructible by the simple ruler and compass method shown (Yates, 1947). Also shown are the projections of the boundary centers of curvature on the sym-ax normal. The projected boundary curvuture is the angular rate of change of boundary normal with respect to sym-ax arc length. The sym-ax curvature is the average of the projected boundary curvatures along its normal. (This procedure, using the property of an ellipse as the sym-ax of two intersecting circles gives a new, albeit more cumbersome, way of obtaining the center of curvature of a conic section-using the average curvature of the focal circles at that point.) We call this curvature the axis curvature or asymmetric curvature of the object at that sym-ax or boundary point since it is the average amount the projection of both boundaries curve in the same direction. We take the difference between both projected boundary curvatures as the width curvature or symmetric curvature since this represents the curvature of the object width or the rate at which the object angle is changing symmetrically about the sym-ax. Width curvature is positive then, if the width angle increases with respect to increasing object width, that is it flares out; and negative if the width angle decreases with respect to increasing object width, that is it bows in. We have need later on for half object angle properties, and we shall call them the sym-angle and the angle curvature. We may take the reciprocal of the width curvature and get an equivalent width radius of curvature. It is the projected center of curvature of the boundary when the sym-ax is straight. Otherwise, the physical meaning is complex and unnecessary here. A symmetric neighborhood of the sym-ax or object exists if its asymmetric curvature is zero there. A symmetric object is one whose asymmetric curvature is zero at all smooth points. Flexure of an object is an operation which changes the axis or asymmetric curvature while maintaining the sym-dist, and consequently the width or symmetric curvature. It has the interesting and important properties that area, perimeter and integral of boundary curvature are invariants of the operation as long as the boundary caused by flexing the sym-ax does not overlap itself. Locally, this means that the radius of axis curvature cannot be less than the sym-dist anywhere and the sym-ax topology remains the same.

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Fro. 20. Geometry of flexural invariance. Flexure consists of changing axis curvature while maintaining the sym-dist and thus the width curvature also. A symmetric boundary (asymmetric curvature = 0) is shown in solid lines and a flexing of it by dashed lines. Note, the triangle added at the top and subtracted at the bottom rapidly approach congruence when the sym-ax interval approaches zero. Since perimeter and area of the figure depend on the sum of these, they remain constant under flexure.

Figure 20 shows the geometric reasoning for this. The object represented by a segment of sym-ax can be broken into two parts, one on each side of the sym-ax. The quadrilaterals are congruent when axis curvature is zero. When not zero, the axis curvature angle adds a triangular piece to one side and subtracts one from the other. As these points are taken closer together and the continuous case is approached, the difference between the remaining quadrilaterals becomes negligible rapidly. Thus, any property that depends on the algebraic sum of these two counter-changing quadrilaterals will remain independent of flexure of the sym-ax. Some examples of flexure are

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shown in Fig. 21. Note that this operation also leaves length and width unchanged. A variety of symmetrization procedures have been devised in geometry. They use the notion of taking chords through the object with respect to a line or point and building a symmetrized object by splitting these chords symmetrically about that line or point (Polya & Szego, 1951; Yaglom 8z Boltyanskii, 1961). A typical example is the Steiner symmetrization shown in Fig. 22. An object is symmetrized with respect to a line by taking the

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FIG. 21. Flexure of a number of objects. A careful examination of these figures will help considerably in understanding the process. The eccentric annulus, which must be pasted to form a cigar band, is a good example of the unexpected results.

intercepts of all perpendiculars to the line and splitting them symmetrically about the line. This process, as well as other types of symmetrization, circularizes (maintains area but reduces perimeter). Sym-ax symmetrization, or simply symmetrization, where no ambiguity exists, shall consist of making the asymmetric or axis curvature zero everywhere, that is straightening the sym-ax. This procedure is always doable for a linear object, one whose

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FIG. 22. Steiner symmetrization and sym-ax symmetrization. Conventional symmetrization consists of splitting chords through an object about a line (Steiner) or a point. Steiner symmetrization about three lines is shown. Its result is not intuitively satisfying for the above. Conventional symmetrization circularizes-maintains area but reduces perimeter. Sym-ax symmetrization maintains both.

sym-ax is not closed and has no branches. We showed some examples earlier in Fig. 21, including non-linear objects, but do not pursue the topic further. Symmetrization and other flexures for a number of “plane geometry” objects have been included in Fig. 21. Because flexing does not always give the expected results for these, they are illuminating examples. It reminds us that the transform is not designed for these kinds of objects. They are the special cases. We now define con$exity, the property of an object whose symmetrization is convex. Thus, the kidney bean of Fig. 21 is conflex, while the tadpole of that figure is not. It is apparent that an object possesses this property only if its width curvature is everywhere non-positive-it does not curve away from its sym-ax. 8. Symmetric Disc Coordinates It is not the intent of this shape mathematics to become involved with the calculus and analysis. But they are so insightful that one cannot bypass them. T.B. 17

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The relationship is two-sided, however. This section proposes a new coordinate system and shows its use as an augment to the extremely rich collection that has been amassed already. It has a number of distinct properties however that distinguish it from those coordinates in use. We replace here the more purely geometric arguments of the last section with analytical ones. Before doing this, it is worth re-emphasizing that the tools of analysis are of little help in determining the sym-ax. Its power comes in describing, by well understood techniques, the representation of boundary and other object properties along smooth sym-ax intervals. We are so used to it that we are considerably unaware of what it does poorly. But the simple problem of determining the locus of equidistance points between two curves, even simple polynomials, is one such example. One would have to repeat the whole geometric procedure described in the last section with analytical tools, which for the sym-ax determining purpose are inappropriate and extremely unwieldy. (One may assessthe difficulty of the problem by trying to compute analytically the sym-function of a simple cubic parabola.) Thus, simple properties which depend on separated points, such as equidistance, can be much harder to obtain via analysis than apparently more complex single neighborhood properties, such as curvature and still higher order derivatives. Therefore, we assume in the following that the sym-ax has been determined a priori and we are concerned only with the character of the new descriptions that occur there. Consider the pair of boundary contours and sym-ax of Fig. 23. All are smooth. If the boundary center of curvature lies on the object side, it must lie

Fk. 23. Differential geometry for sym-ax analysis. Note particularly to axis bending is added on top and subtracted on bottom. See text.

that the angle due

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beyond the sym-ax. The sym-function consists of two parts when described in intrinsic Gaussian coordinates, k,(s) and w(s) (we use w in this section for d to avoid confusion with the derivative symbol. Width W = 2w = 2d), where k, is the curvature of the sym-ax, w is the distance from the boundary to the sym-ax along the normal to the boundary, and s is the sym-ax arc length measured from some distinguished point in the direction of increasing w. The angle between the boundary normal and the sym-ax tangent is the width angle, a. Cos a or dw/ds is the width rate. The width curvature, k,(s), is da/ds. Its reciprocal is r,,,(s). Substituting, we obtain k&)

= !!f = - d2 wlds2 ds Jl - (dw/ds)*’

Thus, width curvature is an explicit function of dw/d.r. Four componentsw, u, k,, k,-are the basic elements of boundary and object properties on the sym-ax. Each has clear physical meaning. Width curvature represents the degree that space is curving toward the sym-ax. Positive values represent curving toward the axis (this property is later called a cup). Negative values represent curving away (later called a flare). Without getting too enmeshed in the details, we show below key points in deriving geometric properties from these coordinates. Using the geometry of Fig. 23, we first take into account the curvature rate difference between the boundary parallel at the sym-ax and the sym-ax to obtain the equation of curvature at that parallel, kp. k, =

kkk cos u .

Taking into account that the boundary and sym-point radii of curvature differ by the sym-dist 1 cos a rb=-=kb

k,&k,

+ w’

where kb is the curvature at the boundary and r, is the radius of curvature at the boundary. Both boundary curvatures have the symmetric curvature term contributing to it in the same way, and the asymmetric curvature contributing in opposite ways. Note also, the sym-dist subtracts from the curvature at the boundary parallel when it is positive and toward the ground, and adds when it is negative and towards the object. Thus, two values are given for boundary curvature, one for each side of the sym-ax. Using the same approach, we may derive equations for area and perimeter. d(perimeter) = {cos r + w[k,(s) f k,(s)]) ds, d(area) = {cos c(+ w[k, 4 k,(s)])(w/2) ds.

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We can now see an important invariant of the symmetric axis description. Two contributions to area and perimeter appear for each differential sym-ax element, one on each side. Note that the axis curvature adds and subtracts linearly on opposite sides, making its contribution zero. Again, we see through analytic methods that area and perimeter for continuous axes are independent of axis curvature as long as basic smooth sym-ax conditions are not violated. Invariance to flexure of the integral of curvature about both sides of the sym-ax is a corollary of the perimeter invariance, since the total change of angle over the interval represented also remains invariant to flexure. By applying the above equations relating axis and boundary curvature, a number of interesting relationships can be obtained. Many of these can be derived from the simpler principles of the last section. (i) I,(S) 2 w(s). The axis curvature at a point cannot be greater than the disc width at that point. This occurs for an “outside” corner, the limiting case of boundary curvature. (ii) dw/ds < 1. The width rate must be less than unity. This occurs because one maximal disc may not include another. (iii) Iw2 - wi 1 < J(x, -x1)’ + (yl -yJ2. The width values at any two points must be less than the distance between the points. Else, one maximal disc would include the other. (iv) For a straight boundary, k,(s) = k,(s). Axis curvature must be equal to width curvature. (v) For convex figures, k,(s) 2 0 and k,(s) 2 k,(s). Width curvature must be non-negative and not less than axis curvature. (vi) Every convex polygon is symmetric (has only straight sym-axes). (vii) A non-convex polygon has only straight and parabolic symaxes. (viii) An object or scene composed of points, lines and circular arcs has only points, lines and conic sections as sym-axes. Some familiar curves are now redescribed in symmetric disc coordinates. Conic sections take the form on a straight sym-ax: w(s) = JAs2 +Bs + C.

The conic section is not the simple quadratic curve in these coordinates. Parallels to curves are now of the same order of analytical complexity as the curves themselves. One need only add a constant to the width function to obtain a parallel. Cycloids, which are awkward to describe explicitly in conventional coordinates have an extremely simple form here, W(s) = A sin 2s. By flexing the axes of cycloids into circular arcs, the outside one

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FIG. 24. An ellipse among other doubly symmetric ovals. Reading left to right and downward are: an ellipse, a parallel to an ellipse, a true sym-ax quadratic, a parallel to a sym-ax stretched cycloid, a four arc “ellipse” and a six arc “ellipse”. What is a visually adequate tilted circle? What are the rules in and the forms generated by biological growth processes?

becomes an epicycloid and the inside one, a hypocycloid. Figure 24 shows some doubly symmetric ovals of the same proportions. These forms include the ellipse, a parallel to the ellipse, a parallel to a sym-ax stretched cycloid, W(s) = A sin B(s) (note, this is not an “X coordinate” stretch though that may be used also-the sym-ax stretched object collapses to a circle, the X coordinate stretched one to a vertical line segment), a true quadratic in sym-ax coordinates, IV@) = As2 +Bs+ C, and a curve of four circular arcs. Note the ease with which a great variety of such curves can be defined. Some interesting new problems arise. What is the class of all curves whose sym-ax is a major diameter, such as the cycloid? Which contains all others? Which is contained by all others ? Let us now flex a simple wedge, W(s) = As + B. In Fig. 25(a), the sym-ax of a wedge is flexed into a circular arc. What is the bounding curve? Because

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FIG. 25. Some flexings of a wedge. At the upper left, the sym-ax is deformed into a circle. The boundary is the involute of the interior circle. At the upper right, the boundary is deformed into a circle. The sym-ax is an equiangular spiral; the opposite boundary is an equiangular involute of the interior circle (see text). Below is shown an equiangular spiral, the “maximal wind” of a wedge (Thompson, 1942). These talon and shell shapes occur later as examples of sym-ax rules for organismic development.

the pannormals all intersect the sym-ax circle at a constant angle, they are all tangent to an interior circle. The curve whose normal is tangent everywhere to a circle is the involute of that circle. The boundary of the wedge is made circular in Fig. 25(b). The sym-ax now makes a constant angle with the radius vector and is an equiangular spiral. A more interesting case occurs with the second boundary. The base of the sym-angle makes a constant angle with the other boundary and is thus tangent to an interior circle. This new curve, the equimgzdar involute, has similarities with both the circular involute and equiangular spiral. [Such curves have a long, though sporadic history (Reaumur, 1709).] It is asymtotic to the equiangular spiral. The

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equiangular spiral has been applied successfully to the description of many sea shells, but must start at zero size (see, for example, Thompson, 1942). The new curve allows for the description of equiangular growth that starts about a circular nucleus. Also shown is an equiangular spiral with its sym-ax and pannormals. Because an enlargement of this spiral is equivalent to a rotation and angles are invariant to both of the operations, the sym-ax angle is the same along its whole length. The equiangular spiral is then the maximal wind of a wedge. We return to this in section 13. Similar flexural questions could be asked about curves obtained by flexing about other boundaries or flexing with different width functions. Certainly, this would be interesting for the quadratic width curve which is the next order of descriptive complexity. The symmetric axis transformation also leads to new ways of obtaining derivative curves from given curves. One such example is the equiangular involute, described above. Perhaps a more interesting procedure leads to the bisectrix of a curve. Consider the half elliptical boundary of Fig. 26. The sym-point for a particular boundary point is generated where its normal intersects the bisector line of the nearest maximum curvature point and itself. This will always occur before the center of curvature of the point itself. (In

FIO. 26. The bisect& the sym-ax of a spiral arc. The circular wavefront generated by the high curvature endpoint precedes the center of curvature generated by a lower curvature point. The sym-ax is created at the bisector line as shown.

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FIG. 27. Symmetric chord coordinates. The sym-ax is modified by moving to the base midpoint and the sym-dist by using the half base of the sym-ax interval angle. It gives a coordinate system perpendicular to its axis, making equal angles with the boundary. Note from the middle figure, that when the sym-ax is straight, Cartesian coordinates result. The bottom figure shows the complexity caused by finite contact points and the disruption caused by sym-ax branches.

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an earlier paper, I had mistakenly said that the sym-ax of a semi-elliptical boundary is its evolute. I wish to thank Dr. Montanari twice for pointing out the error in personal conversation.) Whereas the involute point for a curve is defined in the neighborhood of that point, the bisectrix is defined globally. An interesting conjecture is that the bisectrix is the envelope of the bisectors in the same way that the evolute is the envelope of the normals. The relationship of an object boundary and its sym-ax is such that its sym-ax point is equidistant along the boundary pannormals, perpendiculars from the boundary. They make equal angles with the sym-ax but are not perpendicular to it. If we substitute for each smooth sym-ax point a new point midway between its boundary touching points, and a modified symdist equal to half the distance between boundary points, we obtain symmetric chord coordinates (see Fig. 27). The new measurements are normal to the coordinate axis and make equal angles with the boundary. This new coordinate system leads to a number of difhculties at the discrete sym-ax points. These are not insurmountable, but detract from the much cleaner geometric meaning of the conventional sym-ax. What is particularly interesting is that this transformation leads to conventional Cartesian coordinates when the sym-ax is straight. 9. A-Morphology-Geometries

on the Sym-ax

The preceding section treated the symmetric axis description as a coordinate system which still concerned itself with traditional geometric properties-congruence, area, perimeter, etc. But many biological problems, including vision, concern themselves with much “softer” properties. When confronted with an ellipse no one really imagines (at least, I cannot imagine anyone imagining it) that the nervous system computes the sum of the distance to two non-existant (in the stimulus, at least) foci or the precise coordinate equations or Gauss’s intrinsic coordinates or the precise sym-function. Thus the question arises as to how one loosens up shape description so that slop is allowed. This can be done by setting up equivalence classes on whole objects directly, or by tist breaking them up into parts. The first view taken in this section for simple objects, such as the tadpole and kidney bean, can be fruitful. Before continuing, we compare these with other geometries. Topology gives us a relief from congruence and other hard properties of the earlier geometry. But it is too unconstrained, its main concern being connectivity. Any system in which all one-pieced objects (speaking twodimensionally) without holes in them are equivalent is hardly enough for shape. A square, a circle, and a silhouette of a fish and a man are hardly alike. Can we have a mathematics between these extremes which is both “natural”

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and serviceable? Morphology is used in a number of applied fields to deal with shape and the term is adapted for our purpose here. Such a geometry I call axis morphology, or A-morphology to distinguish it from the older boundary morphology or B-morphology. A “geometry”, according to Klein’s now famous Erlangen proposal-a generally accepted view-is the study of those invariances maintained over some group of operations. (Group is used in the mathematical sense, here.) A-morphology, as well as B-morphology, is then a collection of geometries. A whole new category arises from the symmetric axis transformation. We introduce a few of the possibilities here, using an increasing set of constraints which converge on the particular shape properties of interest to us. Many others are possible; some obtained by extending further the constraints applied here, others obtained by taking alternate paths at various places along the way. Consider a scene whose objects are not identifiable by color, location, intensity or other object properties (see Fig. 28). Each space enclosed by a contour is a potential object. (The contour as object is ignored unless other identification for it exists.) All potential objects are ground objects and described by a non-positive sym-dist. Between smooth boundary points,

FIG. 28. The sym-ax graph that results from a field of contours when distinguished points are made nodes. The connected spaces result in a connected sym-ax (except for the “outside” space). The sym-axes of objects without holes are trees (graphs with no loops).

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not on the convex hull of the scene, and smooth sym-ax points, each boundary point contributes to two sym-ax points, one on each side of it, and each sym-ax point is the result of two boundary points. On the convex hull, only one sym-ax point is generated by a boundary point. The non-smooth sympoints show other boundary to sym-ax point transformation ratios, both finite and infinite. Boundary points generating radials have a one-to-infinity point generating ratio. Boundary points of finite disc contact have an infinityto-one ratio. The number of such special transformations is finite, however, if we consider ones bound to a single point on the boundary or sym-ax as a single transformation. By distinguishing non-smooth points on the sym-ax and disconnecting the sym-ax at discrete W = 0 points we obtain an undirected graph-a set of nodes and arcs. (This W = 0 disruption prevents the combinatoric concatenation of minimal objects.) This graph already contains important geometric information. Finite connected graph pieces (this rules out external convex deficits whose sym-ax goes to infinity) represent finite connected spaces or minimal potential objects. Loops in the graph represent holes in the spaces; a tree (a graph without loops) represents an object without holes. Thus the topology of the scene is given in sym-ax connectivity.? The number of connected intinite pieces tell us the number of intervals at which the scene does not touch its convex hull. A-morphologym, n represents the geometry with those numbers of internal and external (their sym-ax goes to infinity) pieces, respectively. A-morphology-l, X is the class of single objects. A-morphology-l, 0 is the class of convex objects. Although we may continue putting constraints on scenes, it is more interesting to go to single objects. The exciting thing now is that non-topological properties of the object or its boundary become topological properties of the sym-ax (see Fig. 10, for example). Thus, we may count the number of sym-ax loops, endpoints and branchpoints, both internal and external. This gives us a “second order” A-morphology in which sym-ax topology as well as connectivity enters. We may designate it A-morphology-l, X*p, q, r, . . . ** s, t, 24, . . . , where 1,X are earlier parameters, and the new numbers specify numbers of special points of each type-internal and external. Again we may specify some or all of these terms, or some combination, such as the sum of all branches. Thus amorphology-l,X*O,2,O**X is the class of linear objects, objects with no loops, two end points, no branches and arbitrary t Note that this is not precisely the topology conventionally detined because the object ground distinction is made rather than the set-complement distinction of regular topology. In addition, edging has seriously perturbed conventional topological notions, since solid interiors for closed curves are implicitly formed, Both the edging process and the rules of disruption may be changed to accommodate a variety of detailed object segmentation schemes, including the one of conventional topology. But this is a peripheral issue here and consequently passed by.

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.:.

FIG. 29. Equivalent objects in some simple A-morphologies. In the upper half, the object sym-axes have the same topology. In the lower half, the object and ground have the same directed graph.

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ground sym-ax. Note that sym-width information has not been used and the ellipse and a figure with a pinch in it are as yet equivalent objects. A directed graph is now defined using minimal width data. Arrows in the direction of increasing W are now inserted. Double arrows, head-on, are used over an interval where W is constant. New nodes are thus defined. The result tells us how the expanding and constant spaces abut on each other. Again, we may use this property on the internal and/or external sym-axes, Figure 29 shows us some examples of the equivalence classes that are now defined. Using the matrix of the graph or a number string derived from it, we can extend the previous nomenclature to higher order systems. New shape properties are defined, but they are still far from adequate. We may wish to separate as well as identify one specific wiggly-tailed tadpole from another although they may still be flexurally equivalent. Gross properties of object curvature, both symmetric and asymmetric, must be obtained from the sym-ax. To do this, we go to one more level. We will stop there except for some length measures and relational properties, although we could carry the process on indefinitely since the fundamental sym-ax theorem guarantees completely imaging the original object if it is not bounded by a pathological curve. But a reasonable shape language should have a reasonably small set of primitives. We continue, then, using object parts for description. 10. A-morph

and their Shape Primitives

This section explores in more depth the amorphology which is required in the visual shape theory of Part II. The capability of the visual system to make objects of parts that are topologically separate, to juxtapose parts of objects, to apply context and semantics to objects implies the existence of a visual language with at least some of the properties of spoken language. If the analogy holds, the language requires a set of primitives which are operated on at the higher, relational (perhaps, even linguistic) level. These primitives are the shape morphs. Those based on sym-ax properties are A-morphs and those on boundary properties B-morphs. The A-morphs concern us here. Use is made of both kinds of morphs in the next section. Let us distinguish all points that are extrema and zeroes of symmetric and asymmetric curvature, and the end points of those intervals over which the curvatures are constant. The resulting graph consists of node A-morphs, defined at a point, and arc A-morphs, defined on an interval over which sym-ax properties are increasing, decreasing or constant. Arc A-morphs are of two types-width A-morphs and axis A-morphs. Again, the direction of the A-morph is taken as that of increasing W. Obviously, increasing W has no meaning for constant W A-morphs and is either irrelevant or determined

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by criteria outside the interval. The catalog of A-morphs that follows dwells on the object space primarily. A-morphs can exist in ground space also and are used in the next section. The width A-morphs are shown in Fig. 30. The simplest is the worm, an object of constant width. The worm can be open or closed but can have no branches; if open, it must have end discs to form a complete object. Any arc A-morph must be joined to another A-morph if its end width is not zero. The minimum such A-morph is the end disc shown in the figure. For uniformity, we define for A-morph ends of W = 0, an end disc of that size. Note that any line, straight or curved, is a worm of W = 0 on itself as sym-ax. For closed boundaries, this represents the always possible case of boundary as object. For the wedge, width increases linearly with arc length; W = ks

FIG. 30. The width A-morphs shown on symmetric sym-axes. The top shows the worm, open and closed (the closed worm cannot be symmetric). Next is the wedge with and without non-zero end widths. Next are shown cups. At bottom are flares. Note that the A-morphs do not include the end discs.

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or equivalently, d W/ds = constant. If the wedge starts at W = 0, the A-morph has a comer. Curvature toward the axis, d2 W/ds2 < 0, is a cup; curvature away, d2 W/ds2 > 0, a flare. When a cup and a flare are joined smoothly without a wedge, the object goes through a point of width inflection. We go on to make similar distinctions for axis A-morphs by terminating arcs at those discrete points where the axis curvature is zero or an extrenum, and end points of intervals over which it is constant. These are shown in Fig. 31 and consist of the following types. Symmetric: The sym-ax is a straight line. Veer right or veer left: The sym-ax curves right or left as seen in direction of increasing W. When a right veer and a left veer are smoothly joined, the object goes through a point of axis inflection. Circular: The sym-ax has constant curvature over this interval. Spiral: The sym-ax has decreasing curvature over this interval. Respal: The sym-ax has increasing curvature over this interval.

.. . . . . . . . . . . . . . . . . . . . . . . . , . . . . t

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FIG. 31. The axis A-morphs. Shown are symmetric, circular, spiral (decreasing curvatum with increasing sym-width) and respal (increasing curvature with increasing sym-width) A-morphs.

We now go on to the node A-morphs. They exist at a point and with one exception, either terminate or join arc A-morphs. The disc is the only case of an isolated point sym-ax. It is the simplest A-morph and presents no problems. Joining discrete discs however are a special case since they generate new arc A-morphs. We catalog them here, although they are not, strictly speaking, node A-morphs. Figure 32 shows the two possible disc cases. The two arc and one node A-morph is generated in one case; one arc A-morph is generated in the other. (These “joins” generate ground sym-axes also.) We call these A-morphs ligatures since they are gratuitous contributions that image no new boundary points. These are limiting cases of boundary curvature away from the object. When the sym-ax is formed by radials

256

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.....). ... If i \

; .... ........ .... ..... ...... ... ... ..... Y I FIG. 32. The ligature and semi-ligature-finite length sym-axes generated from single boundary points. The top shows ligatures formed by the joining of two discs. The middle shows a ligature formed by the joining of two wedges. The bottom shows a semi-ligature, a boundary point interacting with a smooth boundary. Note that a ground sym-ax is associated with each ligature boundary point(s). issuing from one side only, as in the case of a parabolic sym-ax generated by a point and a line, we call it a semi-ligature. Note from the figure that these ligatures may be generated by other A-morphs which are connected with an

outside corner, as in the two wedges put together at the small ends. We return to proper node A-morphs. Note first that, as defined here, the axis and width A-morphs do not necessariIy terminate at the same points on the sym-ax. For simplicity, we treat them here as two independent systems with separate break points. We define first those cases which use only the

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properties of the directed graph. We assign to each node A-morph its sympoint number pair-sym-axes entering and leaving the point, respectively. Figure 33 shows a chart of these A-morphs. It is interesting that each of these falls into a plausible visual category. 2,0 refers to an A-morph with two branches entering and none leaving, an ellipse or similar two-sided oval. A pinch in the space is equivalent to a 0,2 A-morph. An opening space is always a 0,l A-morph. A three-sided closing is a 3,0 A-morph. Note that the sum of the numbers describes the branching of the space. An X,0 A-morph always refers to a maximal disc in the neighborhood along the sym-ax. Note that a normal looking point on the sym-ax graph is distinguished if

FIG. 33. Node A-morphs based on the directed sym-ax graph. Each type is associated with an a&-sym-point. Columns are a, the number of sym-axes whose sym-dist decreases

from point. Rowsare 6, numberof sym-axeswhosesym-distincreases from sym-point. Crosshatchedformsare thosethat musthavea finite contactdiscassociated with them. Distinguishing

width and axis curvature inflection points adds only zero contact l,l-sym-

points.Note, all theseA-morphsfall into simply recognizable categories. T.8.

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Fk. 34. Use of A- and B-morphs to describe simple objects. At the top right is a triangle.

It now consists of three symmetric wedges joined by a 3,0 node A-morph. To the right of it are shown the differences that occur when the wedges do not properly meet. In one case,

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the disc associated with it has a defect, making it a I,1 A-morph. The forbidden A-morphs have also been discussedearlier under sym-point types. Directed graph node A-morphs are given the following “natural” names: 0,O 0. I 0.2 X.0 I,0

disc, sprout, pinch, bulb (includes O,O), one bulb,

2,0 N,O I, I x, I N, I

twvobulb. N bulb. chain. fi/rk, x > I ( N,fork.

Curvature A-morphology introduces new points of the chain type only. Whereas the aforementioned 1,l point had to have a disc of finite contact, it can now be a point at the end of an arc A-morph. a smooth chain A-morph or zero contact chain A-morph.

11. Object Descriptions Using A-morphs The simplest object of linear B-morphology is the triangle (see Fig. 34). In A-morphology, it consists of three symmetric wedges with a 3,0 node A-morph joining them. Also shown are the additional or modified A-morphs that would exist if these wedges did not “properly” meet. Either ligatures or a finite contact node A-morph would be present. As noted before, all convex polygons are constructed of symmetric wedgesand zero contact angle node A-morphs; all polygons, of symmetric wedges, ligatures, parabolic semiligatures and node A-morphs. Note that “concave” angles are more easily represented in ground wedges than in object ligatures. Flexed variants of these are easily derived, as seen earlier in the flexed triangle, the starfish and the “curved” rectangle of Figs 18 and 21. Not unexpectedly, the descriptions are awkward for the simple rectilinear figures since they are not the purpose for which A-morphs are concocted. But important descriptive advantages can accrue none-the-less. Rectilinear objects, such as the square of Fig. 34. become part of a much larger shape continuum. In addition. A-morphs and a finite contact node A-morph is created; in the other case. ligatures and ground sym-axes. The center panel shows some objects that have a 4.0 node A-morph. Rectilinear objects are now part of a much larger shape continuum-and are transition cases there. The next panel shows the flexural, as well as width, continuum of rectilinear shapes. The bottom panel shows use of A-morphs and B-morphs together. In the perspective triangle, the sym-ax is the cue to the vertical, the base is the cue to the horizontal. (Note that for wide angles of perspective, the vertical cue can be off considerably. This becomes part of an experimental procedure in Part II.) In the perspective rectangle, the vertical boundary is the cue to the vertical. The central object angle, representing the convergence angle of the longer sides, is the cue to the plane of the rectangle.

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B-morphs may be combined. For example, the isosceles triangle in perspective can be described by the vertical sym-ax to define the vertical and by the base line to define the horizontal. Note that there is an error in this vertical that is an increasing function of the perspective angle of the figure, suggesting some testable visual experiments. When the trapezoid of the figure it

FIG. 35. Description of some simple curved objects by A-morphs. Where a finite contact point exists, a finite length of arc is described by a node A-morph. Note in the “pear” that the central section can be described by both object and ground A-morphs. The curvature property is simpler in the ground, since it exists as a location. The representation in the object requires precise velocity and acceleration information. The bottom shows two objects with the same sym-ax. Note that as A-morphs are arbitrarily defined here, the width and axes A-morphs need not terminate at the same places.

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considered to be a rectangle in perspective, the central wedge angle is a good cue for the perspective plane. We go on to curvilinear objects in Fig. 35. Shown there are an egg, a pear and a pointed object-all single symmetric arc A-morphs with end node A-morphs. The ellipse shown in Fig. 24 consists of two cups of finite entry width and zero contact entry angle, joined by a 2,0 chain, with no disc contact there. As discussed earlier, this defines a whole range of curves which need not be mathematical ellipses at all. The “three-sided” objects of Fig. 10 have a common feature in a three-bulb. The difference between the tapered and bulbous ends are represented by a number of A-morphs: a pinch. the presence or absence of other bulbs, and the presence of flares in the “arms”. Figure 36 shows a dog, an early scheme for obtaining the computer derived sym-ax and its computer reconstruction, using simple node locations and straight linear wedges connecting them (Philbrick, 1968). The tail has been straightened out by this simplistic procedure, but it already gives a good reconstruction. The objects of Fig. 37 make the important point that the equivalence of exact boundary and sym-ax descriptions, no longer holds when objects are caricatured by A-morphs and B-morphs. The A-morphs introduced thus far extract only the width data at distinguished points and the consistency of curvature changes between these points. This is obviously inadequate for many important shape distinctions. For example, all triangles have the same A-morphs. It is necessary to add some quantitative descriptors or at least some relations among them. Fortunately the zeros and extrema of many quantities exist at distinguished points. At the points, we may extract sym-width, sym-angle, object angle. object orientation, contact angle(s), symmetric and asymmetric curvature. Over intervals between these points, we may wish to extract distance and sym-length. The simplest way of using them is to specify the desired quantities and some tolerance limits on them. Unfortunately, that does not even allow for the identification of similar (size changed) objects. To accomplish this it is necessary to introduce global properties of the object. Normalization may be done by setting the object width to unity. We then divide lengths and widths, and multiply curvature by the object width. Other object parameters may also be used. A more compact scheme would simply order these parameters by size. Such a scheme has been proposed for intensity and color estimation models in human vision (Land, 1964). A less global method would represent the increase, decrease or sameness of the parameter between adjacent A-morphs. Some of this information is inherent already in the width and curvature A-morphs. It could be extended to include A-morph symlength, for example. The advantage of such an “adjacent A-morph” scheme is in its minimizing the level at which parameter comparisons must be made.

FIG. 36. -econstruction

Early

automatic using node

extraction A-morphs

of sym-function by computer. Beneath is show1 and wedges as connections (Philbrick. 1966).

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FIG. 37. Some un-nice geometric properties. Top shows a boundary curve and a sym-ax point for it. Whereas a smooth perturbation of the boundary will generally lead to a smooth perturbation of the sym-ax, the insertion of a corner, no matter how small, will generate a new sym-ax from that point discontinuously. To avoid this discontinuous change, the velocity of the sym-ax must be considered part of its description. The middle and bottom figures show the luck of invariance of the sym-ax description with affine transformation. The middle shows the skewing of a worm. The bottom shows the compression of a “fingered” object.

Other pairwise statements seem extremely useful also. One may wish to make a statement about the curvature of a tadpole boundary, one that is most simply imaged in a ground A-morph, in relation to the tadpole size. This would define the maximum boundary curving of the normalized tadpole. One may wish instead to make such a maximum bending condition

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depend on the sym-width of the tadpole at that point. Unfortunately for till\ case, they are not part of the normal object sym-ax and a special identilicatior of related object and ground sym-axes must be sought. How does one now represent this collection of information? Cleari!., there are many ways. A canonical description for biological taxononl> would go about it in quite a different way from one that would serve the purpose of visual science. Figure 38 shows an object, its aym-ax and it triangular matrix in which the above features are encoded. Node features go onto the diagonal, arc (or point pair) features on the off-diagonal. We ma\ use the graph, its matrix or a symbol string derived from either of these as ;I description. A variety of such string conversions have been developed fat-

*8,4,7(3,10)(5,9),l,m,6,00 FIG.

W*

38. Description

of an object by a graph, a matrix and a symbol string.

computer use. Such strings unfortunately take some properties that are local on the graph and disperse them. Careful syntactic methods are required to recognize features that are so disrupted, but it can be done. It is required however that the graph be a single piece and that a proper object exist. Part II concerns shape features as input to a perceptual and cognitive system that I cannot conceive as following the above rules. While formulating that process is not part of this work (it is the next level of operation), we would be left without a proper senseof completenesswithout an indication of the inputs to this next step and its general character. The domain of this higher process is the abstraction of shape by an adequate but not oppressive number of features from visual objects, necessary for the meaningful distinctions we must make. Rather than a graph or its equivalent, the perceptual memory stores and retrieves on parallel lists of features and feature combinations. For an example let us take the letter A of Fig. 39. “A-ness” can consist

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of a 3,0 point above and smaller than a 2,l point. Note that these properties are maintained over a wide range of modifications that include topology, abjectness and flexure. Note also that it is not a universal cure-all. The symax topology of the serif figure is disturbed, and requires a different description. Clearly, the “a” is an entirely different shape from the “A” and is associated through an entirely different process. Note that the problem of orientation has been brought in by an external coordinate system. The vertical and/or horizontal may depend on complex cognitive, non-visual processes. At this level cognitive and semantic processes depending heavily on context are brought in. The use of B-morph alone may be thought of as treating the boundary as object, with W = 0 everywhere. No width properties are allowed and a purely intrinsic coordinate system based on axis curvature is defined. Some

FIG. 39. An archetype “A” and some variations. The letter consists of two ground A-morphs, a 3,0 with the upper arc approximately vertical, above and with smaller sym-dist than a 2,l with the lower arc approximately vertical. (Further properties could be stated if one wanted a more precise letter.) Note the existence of these properties even under the distortion of the “written” A. Note also that the letter need not even be in one piece. The serifed letter does not work properly and required a modification of rules. Obviously, the lower case letter is not the same shape at all and must be solved by a different category of operation. T.B. 19

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quasi-topologies with such properties have been developed for computer recognition purposes (Eden, 1961; Shaw, 1968; Freeman, 1961, and others). These have on the whole been applied to topics in which the boundary is the object, handwriting, cloud chamber pictures, jig-saw puzzles (under the rules there used) and others. For object recognition, B-morphology has been useful for projections of linear bounded objects in which coordinate geometry is very satisfactory (Guzman-Arenas, 1967 and others including industrial design departments). But even here, the problem of hidden line elimination has been serious since the B-morphology does not code object-ground directly. The use of both A- and B-morphology is also shown for two extremely simple cases. A-morphology has been shown to be more convenient for set theoretic concatenation of shapes in certain cases such as performing “set theoretic” operations on shapes (Pfaltz & Rosenfeld, 1967). An adequate set of distinctive features of non-rectilinear objects cannot be extracted via B-morphs in a reasonable way. We leave out subcategorizations of objects that are bigger than A-morphs (bulb forms, necks, limbs, bumps, etc.-here left undefined) although they are likely to be useful stages in the hierarchy of shape description. Some may be genetically determined but most are likely to be part of a learned language of vision. Object descriptions which lie in the ground, such as the well-known implicit diamond among four triangles, the dots in a circle of Fig. 8 and other boundaries approximated by points, can be well treated by a feature space. For “closure figures”, objects defined by amorphous blobs, the same holds true except that we may require the global symmetry principle that follows for some features. 12. Global Symmetry and Blocked Contour Description Thus far, our concern has been with objects that are solid, intact silhouettes, or boundaries of such objects. Unfortunately, that is not always the case. What happens if we draw a line through a circle, or a dog has a shadow line, or an object has an internal textural pattern, or an internal occluding object(s), or additive point noise interferes? The features which are necessary for the proper description of the object may be barred from interacting because they are no longer on adjacent contours. An extension is now made to permit interaction of non-adjacent contours. However, this is done without giving up the barrier effect with its strong segmentation properties. An index is introduced which identifies the number of boundaries that have intervened in the propagation of the wave. This extension of the earlier process we shall call global symmetry or pond symmetry (in contrast to grassfire symmetry). A complete unquenched interaction now takes place. The notion is intro-

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duced in one dimension by examining its operation on a set of points. Some interesting and novel descriptive capabilities for statistical and other data analysis appear. Nevertheless, they are ignored here. The topic becomes important for understanding the visual system, organization and function. Other potential applications, such as to contour extraction and visual texture analysis, are passed over here. We start with global symmetry in one dimension. Figure 40 shows three events or pulses. The sym-width gives a measure of adjacent intervals only, whereas the autocorrelation function, for example, also gives a measure of the interval between the outside pulses too. If we extend the notion of a distance field to include more than the nearest distance, the third point, g appears. (Note the similarity of these propagation lines to the “cones of simultaneity” in relativity physics.) But we now have an opportunity not available to the autocorrelation function. We define the order of the sym-ax as the number of

FIG. 40. Global symmetry in one dimension. The top shows a simple three-point input and identifies the additional point that is blocked under the nearest constraint. Bottom shows a more complex input and its global sym-axes indexed by the number of intervening events. The downward projection along the dashed line gives the events organized by location and width of symmetric pairs.

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pulses or points between those that are interacting. Thus the blocked sym-ax is of order zero. From a propagation viewpoint, we can think of the meeting of waves as no longer quenching them, but raising their order and the symaxes that follow, by one. Figure 40(b) shows this occurring in a more complex but still one-dimensional situation. The extended nearest distance concept is no longer a single valued field. A point may have several W values associated with it. All pulse interval relationships are given, but attached to each is the order, that is the number of points or events that intervene. For example, if one traces from the central peak down along the quadrilateral diagonals, as shown by the dotted lines, the distribution of points is characterized by the width and displacement of nested symmetric pairs. If one were to count the number of times a width parameter appeared in such a collection of measurements (quantized by some interval), the result would look like a pulse interval histogram, but would have the additional information on the number of intervening events. Thus, we preserve the object or event segmentation property that is characteristic of blocked symmetry. Such a property may be very important for analyzing pulse trains in which change of interval properties is directly related to an intervening event or eventsas, for example, is likely to be the case in neural or automata pulse trains. Further, this new parameter can be of considerable use in pattern recognition and classification procedures which now use distance functions alone, and leave order out as a parameter. These methods have already been explored for fixed orders and have been found to be of considerable use (Nilsson. 1965; Cover & Hart, 1967; Patrick & Fischer III, 1970). We go on to objects in two dimensions. Figure 41 shows operation of the global sym-ax for some simple examples. Distance is not measured between all points on the input pattern but only along pannormals. The sym-axes are now ordered and codes not only the nearest interactions, but all, indexed by the number of contours that interfere. Note shortly that interfering need not mean between, but only closer. Parts (a) and (b) show two blocked symmetry cases in which descriptive features of contours have been blocked by an interfering inner contour. They now appear as higher order sym-axes. Beyond that, figure properties that did not exist in the earlier blocked symmetry now have good representations. Parts (c) and (d) show two examples. The parallel short sides of the rectangle now have a proper straight worm representation. And symmetry along the long axis of the ellipse is now a proper feature. But a new dithculty has been added. The maximal disc notion based on the “union” principle is no longer valid. The isodistance contours, could be generated by the union of discs on the boundary only to the sym-ax of order zero. They cannot, in general, be used beyond the sym-ax of order zero, since a center of curvature may occur or other contours may

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. ..‘.0.A,.=.-..*. . . .+-.r . t 0’ .. .c. .aI. =* FIG. 41. Global sym-axes in two dimensions. The top figures show objects with extraneous interference. The global process guarantees interaction of non-nearest contours whose features may otherwise be blocked. Thus, the circle has a proper representation as a first order sym-ax but not as a zeroth order sym-ax. The bottom shows global symmetry for two simple objects without interference. Note that features which were not apparent in the nearest sym-ax, now are so. Symmetry of ellipse about the vertical axis is now represented. The evolute is now present, giving the maximum as well as minimum radius of curvature. The rectangle has a worm representation of its parallels over their full length. Very unfortunately, however, global symmetry requires that convex geometry notions be abandoned and differential geometry notions using full line normals be used. The simple “union of discs” for parallels no longer holds. Fortunately. unlike the nurelv differential geometric views, we are able to m&tain the order of -the sym-ax and iis notion of “abjectness”-zeroth-order sym-ax, first-order sym-ax, second-order sym-ax, etc.

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interfere. For wavefronts beyond centers of curvature, physicists have gone to Huygen’s and Fermat’s principles. [For the restricted case of points as inputs, the “union of discs” can still be used for extended symmetry (Blum, 1967, 1968).] This required the injection of a new basic way of doing business. Vertebrate visual systems have augmented the circular organization in the lower vertebrates with a linear organization in the higher ones. In addition, the higher ones have a translation process normal to the contour which can form the base of such a global parallel system (Hubel & Wiesel, 1962). To my mind, no proper explanation has been offered for this drastic shift in neurophysiological organization, although it could be argued, I suppose, that the acuity for linearity of object boundaries became important for survival, or perhaps, that lines are better edge detectors than discs. The extended symmetry hypothesis suggests that this change came primarily for obtaining descriptive features of objects under more complex recognition situations such as partial occlusion, shadows and animal camouflage. In summary, the extended transform permits the extraction of a wider range of shape features and allows extraction of features under more adverse conditions. It suggests types of experiments which are concerned not with object descriptions alone, but interference conditions under which descriptions can be extracted. 13. Three Dimensions The use of the symmetric axis transformation for our world of objects certainly demands its extension to three dimensions and perhaps even higher. It is an extensive undertaking and beyond our capabilities here. We indicate an important part of it that has been done and review some work in organismic development which is relevant. Attention is drawn to the visual problem created in trying to construct three-dimensional objects from two-dimensional projections. Figure 42 shows sym-axes for some simple solids. It can now be a surface as well as an arc. As before, the problem breaks up into smooth and discrete sym-points. For the smooth points it is possible to build a symmetric disc coordinate system. Fortunately, this has been done with great generality. Nadenik (1966, 1967, 1968) has developed equations for the description of n-dimensional surfaces as the envelope of n-balls whose centers lie on both arcs and surfaces. I have only recently come across abstracts of the work (Math. Rev. 33, 5619; 35, 7218; 38, 6417.) The extreme generality does not, in the abstracts at least, bring out the full implications for small dimensioned spaces. The following results from my own independent work in the far more simplistic situation. Some of the interesting properties in two dimensions

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FIG. 42. Sym-axes of some three-dimensional objects. The ellipsoids at the top show that the sym-ax can now be both arcs and surfaces. The bottom shows the sym-ax for a rectangular solid and for a general spherical envelope.

do not hold in three. The invariance of area and perimeter to the sym-ax curvature is one such property. They still hold for object parts described by arc and cylindrical sym-axes whose flexing keeps the cylindrical form since there is still a congruence of the elements added and subtracted during flexure. The area-perimeter invariance for arc sym-axes is similar in concept to Pappus’ Theorem concerning the volume and area of solids of revolution. Some recent work extended it to solids constructed by moving closed planar shapes orthogonal to an arbitrary curve (Goodman & Goodman, 1969;

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Pursell, 1970; Flanders, 1970). The work showed that the perimeter invariance does not hold in general. An interesting and potentially important question concerns the most general condition of flexure for maintaining perimeter-area invariance. Such answers would give insight into geometric configurations whose internal volume and surface area must remain constant -such as a cell. Note in the long and flat ellipsoids of Fig. 42, the transformation can take boundary descriptions that are quite close and put them in entirely different sym-ax categories. In symmetric disc coordinates, both earlier and in the above paragraph, it is the inverse sym-transform that is being stated. The determination of the sym-ax or sym-surface from the boundary is still not amenable to a reasonable analytic formulation. We are thus led back to the fundamentally unsolved higher dimensional problem, building a catalog of A-morphologies and a catalog of A-morphs. In addition, descriptive schemes for describing objects from the A-morphs must be generalized. For it is clear that we have no properly defined names for three-dimensional amorphous shapes or their parts. As simple a form as a filament of varying diameter has not been given a proper name. (Canal surface is sometimes used for this. But even among mathematicians, its use is not uniform.) Although there is no necessary relation between visual description and rules of biological growth, it should not startle anyone if such a relation did in fact occur. Given that biological tissue can operate in certain ways and that both processes have need for such efficient coding of the same types of shapes, it has some real plausibility. That these processes are the inverse of each other, one need only look at tree rings to see an inverse grassfire taking place. It is interesting that even there the A- and B-morphological processes take place at the same time. The basic structure is A-morphic and the fillets between trunk and branches, B-morphic. Note the talon shapes and equiangular spiral of Fig. 25. These require only that the generating point of the inverse grassfire grow faster than the space velocity. The talons have an axis curvature on the growth point or on the boundary, a case representing a surface constraint or definition. The spiral can be generated by either type of constraint. When this is considered as the generator of a three-dimensional shape, such as a Nautilus shell, the result is a three-dimensional wound up cone. Note that if contact definition holds, and the wedge angle and spiral angle are not matched, there may be an interrupted part of circular crosssection that occurs. In the case of such growth it is normally assumed that the new part is added at the large end, not that the whole shell continues expanding as new points are added at the small end. A paradox for sym-ax growth described here results. But the ability of the sym-ax geometry to describe such shapes suggests that it may have a relationship to such large

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end growth also. The spiral has certain of these surface growing problems in common with the egg of Fig. 35, which is simpler in most ways. It has width curvature and hence growth point acceleration. Note that a chicken egg is a one arc figure since the large end is spherical. Such a growth coding is obviously simpler than a two-arc A-morph oval. An interesting difference occurs between the tree or talon and the seashell. The first two shapes are solid three-dimensional objects, the third is basically a surface. Consequently, we must consider at least two types of growth, which in the planar case would be a disc and a circle. But the boundary growth can be still more complicated. We may have to consider its growth as a two-dimensional manifold. This problem will become clearer, not solved, in the next section. Some recent experimental and theoretical work in organismic development has taken a view with a certain similarity to sym-ax geometry. Wolpert (1968) has proposed principles of shape coding which depend on a spatial coordinate system broadcast through a developing organism. Goodwin & Cohen (1969) have proposed a biochemical implementation of such a code which actually includes a grassfire. These have important differences with the methods we used here, though. We consider the organization of a distributed stimulus. Goodwin and Cohen are concerned with a coordinate system set up from a single point. Consequently, they ingeniously interweave a grasstire with a biochemical resonance to generate a maximal pannormal in their system. Thorn (1969, 1970), for example, has developed a geometric view for organismic growth which applies to morphological description also. It uses “germs” of growth and “catastrophies” in space. These have an intimate relation with this work. Their temper is of differential geometry and classical topology. Consequently, they emphasize different properties. An important three-dimensional problem for vision concerns the building up of object representations from two-dimensional projections. Two types occur-a simultaneous reconstruction from a binocular view and a sequential one obtained from viewer or object motion. If the sym-ax is important in vision, then there arises the practical as well as theoretical question concerning how one takes the cues from two-dimensional sym-axes and converts them into statements about three-dimensional objects. In binocular situations, the sym-width will be different for both eyes and will thereby contribute information. If there is a priori information, then one eye can be used. For example, the plane of a rectangle seen in perspective was treated earlier. This extends simply to the case of ripply or twisted surfaces that arise as changes in width between lines of the flexed surface of parallel curves (angle cues to depth could also be used here) or the twisted ribbon. See Fig. 43. But other cases will be no means be so simple, such as describing the amorphous sculpture of the figure as it rotates with respect to the viewer.

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The Miracle

FIG. 43. Some problems with objects in visual and motor behavior. The upper left shows the potential of sym-width cues to add depth to a surface. (Angle may also be used.) The lower left shows a twisted ribbon using the sym-ax as the centerline cue and sym-width as the twisting cue. The upper right poses the problem of interpreting an “abstract” object from a succession of outlines. The lower right shows the potential of sym-function coding for the motor system. When reaching for an edge, one reaches for the stimulus. But when reaching for an object such as a glass, the musculature requires two signals-one for centering the hand, the other for coding the width of the fingers. Such a coding serves biting, eating, handling, moving through openings, etc.

14. Manifolds and Cut Sets Sym-ax geometry has an intimate, albeit contrasting, relation with one of the extensive and fruitful geometric developments of modern times. Starting with Gauss and Riemann, there was a major development of

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geometry that described lines, surfaces, spaces, etc. in a way that is independent of an external coordinate system. Thus Gauss proposed an “intrinsic coordinate system” for curves and surfaces. In the simple two-dimensional case, curves are described by defining the curvature as a function of arc length from some distinguished point on it. (We actually use such a coordinate system but only after the sym-ax transformation. Here we wish to show its limitations when used unaugmented directly on the boundary.) Riemann extended this notion to higher dimensional objects (manifolds) by the use of local metrics within the space. There is both a great aesthetic appeal and practical use to such a notion. Thus, for example, the surface of a sphere can be described entirely by its curvature within the surface, leaving aside entirely the questions of its description in its embedding space. Generally, embedding constraints require complex integral formulations. We present here a counter-Riemann geometry and argue, theoretically in Part I and experimentally in Part II, that an essential problem of geometry for the understanding of vision concerns the description of imbedded objects in embedding objects-the boundary-object relationship. It is shocking that geometry to date could have missed the difference between an object and its boundary. Vision collects boundary data and must produce object descriptions. Paradoxically, within the development of the Gauss-Riemann viewpoint, modern manifold theory, a sym-ax type transformation (without sym-dist, however) has been defined. But it has been developed for other purposes. It has been used to explore the character of complex manifolds by considering equidistance properties from single points (degenerate objects) in them. We have used it here on complex objects in a trivial manifold to explore the objects, not the manifolds. We illustrate with some simple examples. Consider geodesics from a point in the two-dimensional manifolds of Fig. 44, a sphere and a torus (the bounding surfaces, not the solids). The cut set for that point is defined by considering all the geodesics traversing it. Since geodesics are infinitely extendable by local criteria, a dilhculty arises. These can be extended beyond the length for which they are the shortest path between their end points. This is shown by P and P’ in the figure. To keep the notion of a geodesic as a minimum path between its end points, one must introduce the cut set. A cut set is the locus of points such that as one traverses along a geodesic beyond them, a second geodesic path becomes shorter. As would be expected, a number of related concepts and theorems exist for the cut set and sym-axcenters of curvature (conjugate points), connectedness of the cut set, etc. (Kobayashi, 1967). For the sphere, the cut set is the opposite pole from that point. The important aspect of the cut set is that it is a subset of points of

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FIG. 44. Sym-axes and cut sets. To insure that geodesics on a manifold are shortest paths, it is necessary to cut the manifold at loci equidistant from the starting point. These are the equivalent of sym-axes for point objects on manifolds and are shown for sphere, ellipsoid and torus. Shown also on the ellipsoid is the conjugate point locus, the centers of curvature that can occur in a convex manifold for a point object. In manifolds also, “evolutes” occur on or after the cut set or sym-ax.

the manifold from which topological properties of the manifold can be ascertained. The cut set of the torus is the two connected circles shown, neither of which can be shrunk to a point within the space. The ellipsoid entails a more complex situation in which centers of curvature and cut sets only partially coincide (Struik, 1950). Its topology is nonetheless determined by the lack of loops in its cut set. An extension of the cut set to objects in the Euclidean plane has been proposed for the purpose of defining a coordinate system based on an arbitrary closed smooth curve therein (Hartman, 1964). Thus, the motion of cut set within a manifold can be extended to the imbedding space to describe the manifold itself. This extension is then a counter-Riemann geometry. It is possible, and is likely to be both interesting and important to natural and artificial vision, to consider complex objects in complex manifolds-the retina is spherical, the cortex who knows what and topologically disrupted at that, and the normal computer display a flat

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torus not imbeddable in Euclidean three-space. We do not attempt that step here though. Since the topologist’s concern has been connectedness primarily, he has not elicited the properties of concern to us. A-morphology of section 10 considers only constancies and direction of changing of metric properties, and obtains a variety of descriptors that can include the topological ones, as in the directed graph representations, or cuts across topology and allows identification of objects with entirely different topologies as the various A’s of Fig. 37. This gives us a set of non-topological shape features which lead to “gestalts” of shape. Manifold theory has been extremely fruitful in cosmological physics and has profoundly influenced our notions of the universe. An integral part of that application has been the notion of simultaneity in relativity physics. (We are not concerned with the measuring scale changes that occur with increasing velocity.) An event propagates radially and leads to an infinite four-dimensional space-time signal cone that is similar to our formalism. A major difference occurs for the recognition of objects using our cone of simultaneity. It does not extend to infinity when there is more than a point input, but terminates, as in the blocked symmetry case, or raises an order index in the global symmetry case. The use of active propagation spaces rather than the passive space of interstellar physics allows a much wider variety of actions to take place in space, actions that are much richer than a passive linear space could support. Further, in Part II we are not dealing with a single localized observer, but with a collection of them everywhere. Thus, we can guarantee that special events will be observed. These may be re-collected at a new level of the brain to allow a new cognitive synthesis without being subject to a new simultaneity principle, by using a different kind of brain organizing principle beyond that first level. Thus, despite the similarities that occur because both systems are measuring space by propagation processes, there are profound differences possible also. 15. Axiomatics

of Growth

Geometry

We are so proud of the accomplishments of our mathematical culture that we are blinded to its abject failures. Understanding biological shape is one. The fatal flaw of traditional geometry goes to its choice of point and line as primitive. Here we use a point and a disc (or growth) as primitive notions. While the Euclidean line is defined by two separated points, requiring information over a distance, the disc is defined entirely by a single point. It is in that sense simpler. The implications of this change are quite profound. Mathematicians exploring the foundations of Euclidean geometry early in this century, became aware that the circle as well as the line could be used

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as its basis (Pieri, 1908; Huntington, 1913). More recently, symmetry has been shown to form an adequate base for Euclidean, hyperbolic and elliptical geometries (Tarski, 1959; Robinson, 1959; Bachmann, 1959; Scott, 1956; and many others). However, they did not explore other possibilities such a new base would allow. Questions about what shape mathematics was “natural” for these new primitives were not asked. Since they were reconstructing Euclidean geometry, they still defined the line between two points as its basis (Pieri, 1908; Huntington, 1913). More recently, symmetry has in the bisector line. From our view here, these attempts were indeed confined. It is apparent that our shape preconceptions are a conceptual tyranny, even for some of our best mathematicians. We do not develop an axiomatic approach here, but wish to show its possibility and changed character. It would start with the notion of a point and a growth. A disc would be the growth of a point or a disc. The first and most simple line defined by two points is their sym-ax, the bisector line. (Busemann, 1959, has explored such lines in considering geodesics defined on symmetry relations in a variety of spaces.) It would take an additional step to define the joining line. This second step also allows for the definition of perpendicularity and colinearity. A line segment can then be defined as a terminated growth of two points-terminated where the boundary has a perpendicular discontinuity. It could alternatively be defined by a symmetry criterion-those sym-ax points whose segment returns the same points. The segment operation performed twice is the identity operator on the segment’s end points, then. The following table lists the anticipated new sequence of complexity for conventional geometric items. primitive Point Growth primitive Disc point and a growth Boundury all growths are inside and outside Circle disc boundary Concentricity disc grown from another Parallel for curve, boundary of growth from it growth of two points (synchronous) Line (Bisector) Joining Line growth on parallel points of (bisector) line Perpendicular (bisector) line and joining line relation Line Segment line stopped by boundary perpendicular or iterated segment returns line ends growth of two points (asynchronous) or two unequal Hyperbola disjoint discs Ellipse growth between circle and non-central inside point Parabola growth between point and line

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Beyond this, however, shape complexity does not follow the “normal” geometric hierarchy based on polynomial description. Euclidean geometry would emerge as a subclass. Note again that as in our shape description earlier, dimensional hierarchies are not maintained. A point in n-space grows directly into an n-dimensional ball. A consequence of this is the simplicity with which objects follow primitives. In Euclidean geometry, it takes three primitives, lines, to define the simplest object boundary. This must be augmented with special concepts to obtain the simplest “filled in” object, a triangle with interior. In “growth geometry”, the simplest filled in object, a disc, follows immediately from a point. It becomes possible to specify filled in two-dimensional objects, and even full scenes, by the growth of appropriately placed points. Note that the scheme described here is not necessarily the same as the inverse grass&e. Thus, one interior point of a triangle (any point) and three symmetrically placed exterior points (with respect to the bounding lines) will grow into such a triangle. Thus we may paint the triangle with bacterial cultures. If the central point is a red culture and the other three white, we paint a red triangle. This can be extended to more complicated painting from points, or in the limit, from the symfunction via the inverse grassfire. 16. Additional Discussion

I have taken the view that the symmetric axis transformation is a fundamentally new geometric development. In this section, however, I would like to relate it to a variety of other mathematics and show that this leads to a more general view of the geometries we have. Some of these relations have been suggested earlier. But it is here that the aesthetic and philosophical view is explicitly stated. Shape is traditionally the concern of geometry. As its name bears witness, geometry is f?rmly rooted in surveying. Its early evolution took place in collaboration with mechanics and it still bears the imprint of that historyconcern with the “hard” properties of figures: congruence, similarity, linearity, equivalence under planar projection, etc. With the advent of coordinate geometry and the description of shape by mathematical functions, the possibilities became vastly greater, but the absolute precision of description remained. Not until recent times, with topology, have we seriously departed from the notion of shape as rigid and precise. Both notions are dealt with here. But topology has been too “soft” for capturing the essence of biological shape. It is concerned primarily with connectivity. Thus, for example, the silhouettes of a circle, a square, a fish and a man are equivalent in topology. The ancient Greeks already knew of polyhedra and conic sections,

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and we have yet to define a kidney bean simply or a tadpole shape at all. (The tadpole is not a single shape, of course, since it can wiggle in many ways and still retain its identity.) Some startling consequences result from this neglect. Biometry, which properly includes shape, is largely involved in exploring statistical variation of measurement, rather than finding the incisive measurements to take. Yet, it has been shown that even when dealing with the statistical variations of organismic measurement alone, only one set of measurements can be optimum (Mosimann, 1970). To an overwhelming degree, we have used in our biology the shapes suggested by our conventional geometry. So towering is this intellectual edifice that it seems a universal base. The shape insights we need certainly should reside in this large and impressive body of mathematics, or in some simple extension or specialization of it. Our failings seem more likely to lie in the experimental shape sciences; so recent in comparison, and so tedious and capricious in execution. So many constraints and preassumptions appear in experiment and data organization that it seems preposterous to think that the failing is in our mathematics. Yet, science must be borne of experiment. The view we take here is that we cannot expand science properly without rebuilding our shape mathematics. It is clear that our geometries have failed biological researchers as few researchers feel compelled to learn or apply much of it and few modern geometers are involved in biology, geometry having gone beyond the immediately sensible world into metagalaxies and even into pure poetry. The geometry visual researchers use comes implicitly from traditional Euclidean concepts and their progeny. To correct this failing, I have gone back to the initial Euclidean primitives and revised them. It is indeed startling that after roughly two and a half millenia of development, this can still be done. Unlike the common non-Euclidean geometries [Gauss had used the more appropriate word, “anti-Euclidean” (Jammer, 1954)] that have used the same primitives while rejecting one axiom, the shape mathematics presented here is truly un-Euclidean. I feel that a “re-structuring” of geometry is necessary for biology. Pythagoras hypothesized that all is number. The development of mathematics, particularly the advent of coordinate geometry, has gone a long way toward making that a reality. Geometry as an operation in space has all but disappeared. It has become an exercise in symbols. It starts with the natural numbers and building thereon, converts our geometry into algebra and analysis. But in using these tools, the list of axioms is not enough. It does not tell us how to build a comprehensible intellectual structure that we can cognitively operate with. It is necessary to design a system of symbols whose interaction is matched to a perceptual and motor apparatus that has evolved to deal with our natural world of objects and events. Thus, the Roman numeral

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system is inadequate even with the proper axioms of arithmetic. The digital computer operation on pictures is a strong case in point. In converting geometry to numbers, we are forced to put it on a lattice and end up with a strongly anisotropic process. Thus, the number of points in a line along the lattice direction (the x and y axis) are different from the number of points along the diagonal. Correcting such a process is by no means an easy computational job. The conversion of space to numbers is a highly unnatural way of representing it. Should we not represent space in space ? To what degree can we represent other intellectual problems as spatial ones? The Venn Diagram is a case of set theory converted to geometry. In the complementary work to this, I have proposed just such a spatial process as the base of a perceptual system. This work can then be considered as a piece of such an approach. Its purpose is to explore the foundations of a theory of mind and brain, and devices also, organized on spatial principles. In this way, my hope is that we may see ourselves properly, not as poor logicians; and we may construct our environment and our technical aids along a set of operations that are amenable to us. This geometry is still too “mechanical” for that purpose and I apologize for the philosophical inconsistancy. My complementary work exploring higher level intellectual structure and function based on geometric principles, corrects that defect. But its formulation is as yet too amorphous and the experimental possibilities not properly formed. The present problem gives us the opportunity to explore geometric principles on a more concrete problem. The discussion following Part II considers further the implication of spatial processing for biological and engineered visual systems. It proposes a conceptual alternative (or augment, at least) to the “receptive field” or “McColluch-Pitts net” view that treats connectivity of “neurons” as the sole important interneural element, almost totally ignoring distance properties. Thus, one must assume, for example, that the change from point or circular to linear receptive fields in mammalian vision must take place via a very intricate connection network in which each lower element is very precisely connected to a two-dimensional expansion of higher elements-translation along a tied orientation and the collection of orientations to make all proper connections. Using the growth viewpoint, the linear fields can be envisioned as a two neuron property-for example, one which was sensitive to the first neuron at which a signal arrived and so mapped out their bisector line. But beyond questions of structural simplicity, the contemporary view has failed to illuminate the shape problem. Indeed in a beautiful elaboration of the geometry implicit in that view (Minsky & Papert, 1969), it has been shown that this view is singularly unproductive. The authors make a plea for the development of a proper “computational geometry”, one precisely specified. Their assumption has been that this T.B. 20

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would occur within the computer framework. I submit this geometry as just such a development-rich in possibilities, yet largely overlooked. Although it has been used in computer applications, the methods are more cumbersome there. The proper formulation seems to me to be on a continuous space. Another, and very beautiful, approach to the understanding of shape in vision centered about the notion of Lie Groups (Hoffman, 1966). It is the most serious attempt to carry conventional B-morphology and analytic function concepts into the shape problem. A comparison of that approach and this one will bear out the fundamental differences in principles despite some superf%al similarities-such as explaining the eye’s sensitivity to parallelism. That work does not lead to the insights concerning soft descriptive features, object-ground problems and the exciting inversion relationship between organismic development and vision. This paper proposes a geometry which is an extension of convex geometry (it is the degenerate case), an area which has not properly fitted in with the analytical insights we have been developing. This is a specific example in which our failure in a biological problem results from lack of naivete, not sophistication. It is to be expected, as our sophistication comes from physical science, which is an early and strongly indoctrinated lore. It gives us a deep, albeit implicit, cultural bias. It has selected problems for biology not on the basis of fundamental importance alone, but overwhelmingly in favor of their susceptibility to its “universal” methods and tools. Clearly, this would not be possible if our current scientic cultural biases were not as productive as they are. Molecular biology, for example, is probably the most exciting area of contemporary biology. But it is precisely this productivity that has seduced us from other fundamental biological problems that these insights do not attack. Illuminating as such areas are, they cannot by themselves put biology on a proper footing. Without a proper shape mathematics for biology, we are in the position that physics would have been in trying to develop mechanics without geometry. For, if we have no efficient statements for describing biological shapes, how can we iind relationships and causes for them? We will forever be in the position of feeling that science requires the fulfilling of the thesis that life does not exist. I am grateful to Dr. A. W. Pratt, without whose confidence, patience and tolerance this work could not have been done. REZERE?NCES (1967). In Models for the Perception of Speech and Visual Form (W. Wathen-Dunn, ed.). Cambridge, Mass.: M.I.T. Press. A transformation for extracting new descriptors of shape. BLUM, H. (1967). Perspect. Biol. Med. 10, 381. A new model of global brain function.

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t Parke Mathematical Laboratories, Inc., Carlisle, Mass. 2 Air Force Cambridge Research Laboratories, Bedford, Mass. (Reports obtainable from the U.S. Gov’t Printing Office, Washington, D.C.).

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Glossary T0rm A-morph A-morphology anglecurvature arc A-morph asymmetriccurvature axis A-morph axis curvature axis intlection axis morphology

Page 253 250 237 253 237 253 237 255 250

Term G-pannormal G-parallel global symmetry ground ground space ground-sym grassfire

Page 213 213 266 210 227 216 217

B-morph B-morphology bisectrix boundary boundary morphology branch point bulb A-morph

inclusioncone inner width 253 interval angle 250 inversetransform 247 isolatedpannormal sym-point 209 250 length 230 ligature 259 linear object

227 233 230 216 230

center of curvature chain A-morph circle of curvature circular A-morph conflexity contact angle continuous pannormal sym-point conventional symmetry convex deficit convex hull cup curvature

236 259 236 255 241 230 230 208 216 216 255 236

maximal disc medial axis morph

226 206 253

N-pannormal N-parallel n-sym-point nearestdistance node A-morph normal

213 213 230 213 253 212

234 255 240

209 object 230 object angle directed sym-ax 232 object plane 223, 226 disccoordinatespace-seePlucker object space 227 disc coordinates 226 object sym 216 discretesym-points 230 230 object tangent distance 212 object width 233 order of the sym-ax 267 end point 230 outer width 233 envelopeof discs 213 equiangularinvolute 212 246 pannormal parallel 213 finite contact sym-point 259 230 pinch A-morph flare 232 255 pinch points flexure 226 237 Plucker disccoordinates fork A-morph 259 projectedboundary curvature 237 fork points 232 projectedlength 233

BIOLOGICAL

Term projected width point pond symmetry pseudo-angle radial radius of curvature respal A-morph scene semi-ligature shape morph skeleton smooth chain A-morph smooth points spiral A-morph sprout A-morph sprout points Steiner symmetrization stick figure sym-angle sym-ax sym-ax angle sym-dist sym-function sym-length sym-point sym-transform symmetric A-morph symmetricaxis symmetric axis function

Page 233 seen-point 266 231

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Term Page symmetricaxis transform 216 symmetricchord coordinates 249 symmetriccurvature 237 symmetric neighborhood 237 symmetric object 237 212 symmetric point 209, 214 236 symmetricpoint distance 216 255 symmetrization 240 symmetry 209 209 256 total contact (angle) 230 253 total sym 216 206 259 union of discs 213 230 255 veer 255 259 232 wedge 254 239 width 233 206 width A-morph 253 237 width angle 243 216 width curvature 237, 243 230 width inflection 255 216 width rate 243 216 worm 254 234 worm end 232 214 worm interval 232 216 worm point 232 255 202, 216 zero contact chain A-morph 259 216 zero contact sym-point 230