Computational Topology for Shape Modeling John C. Hart School of EECS Washington State University
[email protected]
Abstract
two components: topological interrogation and topological control.
This paper expands the role of the new field of computational topology by surveying methods for incorporating connectedness in shape modeling. Two geometric representations in particular, recurrent models and implicit surfaces, can (often unpredictably) become connected or disconnected based on typical changes in modeling parameters. Two methodologies for controlling connectedness are identified: connectedness loci and Morse theory. The survey concludes by identifying several open problems remaining in shape modeling for computational topology to solve.
Dey et al. [1998] overview a number of examples of computational topology, but focus nearly exclusively in the domain of computational geometry. These topological interrogation examples focus on the topological analysis of existing shapes. Their algorithms return values such as the Euler characteristic and the more informative Betti numbers, and classify shapes based on their genus. This paper expounds on the birth of a new area of computational topology that focuses on controlling connectedness during the modeling process. The topological type of some geometric representations are dependent on the proximity of their primitives. Two such representations are recurrent (fractal) models and implicit surfaces1 . The computational tools used for exercising topological control during the modeling process are the connectedness locus and Morse theory.
1 Introduction One might ask “what is topology?” and receive the proper definition: “the study of open sets.” While this answer may enlighten some, the novice puzzles “what is an open set?” at about the same time the definition continues on to add that open sets are whatever a topology defines them to be (so long as they pass a few conditions regarding closure under union and intersection, etc.). This might be the main reason students are discouraged from learning topology. The intuition behind the study of topology is the study of the properties of a shape that do not change under deformation. Any deformation is allowed so long as it is oneto-one and bicontinuous, which is sufficient to prevent the deformation from cutting or joining segments, and piercing or sealing holes. In computer graphics, connectedness is the most fundamental of the topological properties. A definition less rigorous but more palatable to the computer graphicist might be “the study of connectedness.” As there are various sub-disciplines within computer graphics (e.g. modeling, rendering, animation), there are also several sub-disciplines of topology (point-set, algebraic, differential). As computational topology develops into an area of computer science, it also can be divided into
2 Connectedness Loci for Shape Modeling Fractal geometry has been used at the grade school level to increase interest in mathematics. That the iteration of a simple function can yield such intricate and unexpected shapes intrigues young minds. Likewise, fractal geometry can also help computer scientists understand topology. Fractals were present at the birth of topology, and their use as counter-examples disturbed many at the time. The Cantor set demonstrated that a compact uncountable set could be totally disconnected. 1 Other geometric representations such as Bezier patches, B-splines and NURBS, typically do not present problems with connectedness during the modeling process. When the control points of such representations are perturbed, the topology of the surface remains constant. Their only topologyrelated problem was due to the shape of patches at corners and ”suitcase handles” which yielded three- and five-sided patches for which S-patches and subdivision surfaces have adequately solved.
1
2.1 The Mandelbrot Set The Mandelbrot set (Figure 1) is a connectedness locus that maps the single complex parameter c corresponding to the Julia set of the iterated function f (z ) = z 2 + c: The connectedness of a Julia sets is very easily determined. The Julia set of an iterated function f is connected if and only if
f n (ci ) 6! 1 nlim !1
(1)
where ci is the finite set of critical points which are solutions to f 0 (ci ) = 0: For the Mandelbrot set in Figure 1, the function f (z ) = z 2 + c has a single critical point c0 = 0 for all parameters c: This particular Mandelbrot set is more than just a connectedness locus because its buds and dendrites completely organize all of the topological properties of the Julia sets [Norton, 1989]. While their are many ways people have been inspired by the Mandelbrot set, it is perhaps that the Mandelbrot set maps out all of the topological properties of the Julia sets that leads one to the investigation of similar maps for other families of shapes.
Figure 2. A connectedness locus for Julia sets in the quaternions. (Image courtesy Greg Turk.)
and their variants, which we call recurrent models have been shown to model a wider variety of rough shapes. An iterated function system uses a finite set of transformations fi Ni=1 to represent the shape A as the unique solution to
f g
A=
[N fi(A):
i=1
(2)
In fractal function form [Barnsley, 1986], they have been used to model images and terrain. When not constrained to this fractal function form, the topological type of the shape they model is affected by their transformation parameters. When modeling, this geometric representation can easily collapse into a totally disconnected dust. Figures 3 and 4 illustrate a shape transformation (morph) between a dragon curve and Sierpinski’s gasket. All of the shapes are described by three transformations, but the parameters controlling these self-transformations are continuously interpolated.
Figure 1. A connectedness locus for Julia sets in the complex plane.
2.2 Quaternion Julia Sets These same Julia sets can be extended into the fourdimensional quaternions, and drawn using a variety of algorithms [Norton, 1982; Hart et al., 1989; Hart et al., 1990]. While all of the quadratic Julia sets in the complex plane are indexed by a single complex parameter, all of the quadratic Julia sets in the quaternions are spanned by six quaternion parameters because quaternion multiplication is noncommutative. Moreover, connected quaternion Julia sets may be simply or multiply connected. Figure 2 shows only a portion of the full 24-dimensional parameter space of these shapes.
Figure 3. A “straight line” morph between a dragon curve and Sierpinski’s gasket.
2.3 Recurrent Models Julia sets in either the complex plane or the quaternions model little else than themselves. Iterated function systems
Direct interpolation of these parameters yields intermediate shapes that collapse into the totally disconnected dusts 2
3 Morse Theory for Shape Modeling Morse theory is the study of the relationships between a function’s critical points and the topology of its domain [Milnor, 1963]. This can be demonstrated using a classic example where the function is height and the domain is a torus. Figure 6 illustrates this in modern form by dunking a donut in coffee. Let p = (x; y; z ) be a point on the torus, and let f (p) = z return the height of point p on the torus. As we dunk the donut, the portion immersed consists of the points p such that f (p) < T for some threshold T: Note that at certain times and points, the level of the coffee is tangent to the surface of the donut. Such points are called critical points and occur when f (p) = 0: The topological type of the immersed portion of the torus changes after these critical points are dunked. Morse theory not only indicates when the topological type changes, but what kind of change takes place. Such changes are determined by the type of the critical point, which is given by the number of negative eigenvalues of its Hessian2 . Intuitively, this is the number of axes of a surface coordinate frame one can move along to make the function more negative. Hence, the critical point at the base of the torus is type 0, those at the bottom and top of the torus hole are type 1, and the one at the top of the torus is type 2. A new surface component is created when a type 0 critical point is encountered, opposing surfaces are connected when a type 1 critical point is encountered, and a hole in the surface is sealed when a type 2 critical point is encountered. Critical points and Morse theory have appeared in various modeling applications in computer graphics. For example, Cheng [1989] used critical points to find surfacesurface intersection curves for constructive solid geometry, and Shinagawa et al. [1991] used Morse theory to reconstruct anatomical surfaces from cross sections. Morse theory also applies to the standard geometric representations of implicit surfaces, polygonal meshes, and discrete volume and terrain data.
Figure 4. A connected morph between a dragon curve and Sierpinski’s gasket.
r
shown in Figure 3. A 2-D slice of the connectedness locus of this parameterized family of iterated function systems was generated and appears in Figure 5. The sequence of shapes in Figure 4 remains connected because the parameters used followed the path illustrated in Figure 5.
Figure 5. A connectedness locus for the iterated function systems in the morph. White indicates the parameters of connected shapes.
There are no critical points in iterated function system models corresponding to the critical points used to compute the Mandelbrot set. The connectedness locus in Figure 5 was instead computed using a new algorithm to determine if a recurrent model is connected [Burch & Hart, 1997]. The technique is particularly well suited for computer graphics in that it determines if the rendering of the recurrent model appears connected, even if the actual recurrent model is theoretically not connected.
3.1 Implicit Surfaces
!
Morse theory can also be extended for application on imR be a differentiable function plicit surfaces. Let f : R3 defining an implicit surface. While f could be considered a height function on the 3-manifold R3 ; the topology of a 3-manifold with boundary is not highly intuitive. Rather, in computer graphics, the topology of its 2-manifold boundary is desired. A recent set of theorems [Hart, 1998] shows that the Morse theorems on the 3-manifold with boundary also apply to its boundary, which is the implicit surface of f at some threshold.
It is currently an open problem whether such connected paths exist in the parameter space of iterated function systems. The Mandelbrot set was proven to be connected such that a sequence of connected intermediate Julia sets could be found to interpolate any two given connected Julia sets [Douady & Hubbard, 1982]. No such proof exists for the connectedness loci of families of iterated function systems.
2 If the Hessian has an eigenvalue of zero, then the critical point is degenerate, and Morse theory can not be directly used.
3
Since we are now examining critical points in a three dimensional solid, there are three kinds of critical points. Assume the implicit surface is constructed using an inverted blobby model [Blinn, 1982] such that the Gaussian functions form negative bumps3 . Then the type 0 critical points occur at or near the key points of the blobs. The type 1 critical points occur where blobs join or could join, and type 2 critical points occur where there is, or could be, a donut hole. There are also type 3 critical points where an air bubble has occurred or could occur. The critical points exist independent of the implicit surface threshold. By varying this threshold, each of the critical points’ events can be made to occur. This result was used to guarantee that the topology of an implicit surface matched that of its polygonization [Stander & Hart, 1997]. This result ensured that connections and disconnections between polygonized components were not artifacts of polygonization resolution, that all disjoint components of an implicit surface are found and polygonized, and provided the basis for a polygonization and localized real-time mesh maintenance algorithm. The organization of the critical points bears mention. We have found that a type 1 critical point is typically found between two type 0 critical points. A type 2 critical point is commonly found in the plane spanning three or more type 1 critical points. The type 3 critical point usually lies in the volume contained within the planar boundaries defined by the type 2 critical points. This is perhaps most clearly illustrated by the family of implicit surfaces defined by eight blob placed at the vertices of a cube in Figure 7. The resulting points, edges, faces and volumes form a graph that is known in algebraic topology as the CW-complex. A CW-complex is a generalization of a simplicial complex that allows faces to be polygonal instead of trianglular, and volumes to be polyhedral instead of tetrahedral. In Figure 7, the CW-complex consists of the eight vertices, twelve edges, six faces and single volume corresponding to the eight type 0, twelve type 1, six type 2 and one type 3 critical points.
3.2 Triangular Meshes Axen & Edelsbrunner [1998] applies Morse theory to triangular meshes in the form of compact piecewise linear 2-manifolds, using a wave traversal technique. The wave traversal is similar to the height function. First an initial mesh vertex is chosen and assigned a height of zero. Then all of the vertices sharing an edge with that vertex are selected and assigned a height of one. This process continues until all of the vertices are assigned their “distance” from the initial vertex. Examining all vertices at a given distance
Figure 6. The topological type of the dunked surface of a donut changes when the donut surface is tangent to the coffee level.
3 The bumps are inverted such that the type of the critical points agrees with the Morse theory conventions.
4
4 Shape Modeling Challenges for Computational Topology Examining current problems and tools in shape modeling reveals several open problems involving the connectedness of the geometry during the various stages of the modeling process. This list of future work serves to conclude this survey of computational topology applications in shape modeling.
4.1 Morphing Figure 7. A blobby cube at various threshold levels.
Often models are interpolated between keyframes, or morphed for transition effects. The interpolation between two shapes, particularly in implicit form, can produce forms covering a variety of topological types. For example, transforming the implicit model of a coffee cup into that of a donut might produce implicit models of one or more components, each of genus zero, one or even greater.
as the distance increases yields a wave traversing through the triangular mesh similar to the coffee rising up the donut surface. As this wave traverses the mesh, some vertices are found to be critical. The wave begins at the initial vertex, which is a type 0 critical point since all edge-neighbor vertices surrounding it have larger distance values. A wave terminates at type 2 critical points, which are represented by edgeneighbor vertices with smaller distance values. The wave divides or is reconnected at type 1 critical points, whose edge neighbors have distances that vary with periodicity of two when evaluated in rotation order. Morse theory requires all critical points to be nondegenerate. This form of wave traversal can yield two vertices that share an edge and the same distance value, or three vertices that share a face and the same distance values. A theorem shows that such cases are removed by performing a barycentric subdivision of all of the triangles followed by a perturbation. Barycentric subdivision replaces each triangle with six triangles by adding a new vertex in the middle of each triangle, and a new vertex in the middle of each edge.
As shown earlier, Burch & Hart [1997] investigated maintaining the connectedness of intermediate shapes when performing a shape transformation on recurrent models. The method developed a fast algorithm for determining the connectedness of the recurrent model, plotted a connectedness locus on the parameter space of the model indicating which parameters yielded connected results, then provided an interactive environment for specifying a path through this connectedness locus between the start and goal shape parameters. For implicit surfaces, we want to maintain the topological type of the shape from start to finish. This could be accomplished with an interval search of all critical points across the embedding space of the implicit surface along with its parameter space. This space would consist of regions where the implicit surface maintains the same topological type. These regions would be bounded by areas where the value at one of the critical points is zero, indicating a change in topological type of the implicit surface. Maintaining the topological type of the implicit surface during a shape transformation is then a matter of determining if the start and goal shape parameters are in the same connected region of this space.
3.3 Discrete Data Helman & Hesselink [1991], Delmarcelle & Hesselink [1994] and others use critical points of vector and tensor fields to delineate topologically-distinct regions in the visualization of flow fields. While the critical points are not used to determine changes in topology, the techniques used to detect critical points can be used to support Morse theory applications. That critical points can be defined on discrete data sets means that Morse theory could be applied to discern the topological types of terrain data (e.g. which islands are connected at a given sea level) and of volume data.
Useful shape models have numerous degrees of freedom. Hence their parameter spaces span a large number of dimensions. Visualizing such spaces directly is out of the question. Projections or slices of such spaces run the risk of obscuring possible paths of connectedness through them. Proving such a path exists would be a critical first step toward formulating an automatic method for navigating along it. 5
4.2 Texture Mapping
4.4 Simplification Often polygonal objects are stored at a variety of levels of detail, to facilitate their efficient display without sacrificing their rendered appearance. Numerous techniques have been developed to simplify polygonal objects. These techniques progressively remove geometric detail, and commonly change the topological type of the polygonal object. For example, holes are filled and disjoint objects are connected. Often much of the qualitative nature of the object is contained in its topological type. In such cases a typechanging simplification is not desired. The simplification of a coffee cup might close the handle, which is an otherwise important feature even if only a single pixel in its image indicates the handle exists. Bajaj & Schikore [1998] solved a similar problem by using Morse theory to simplify terrain data while ensuring that the terrain retained the structure of its “peaks, pits and passes.” The simplification would not allow the insertion or collapse of a vertex if it caused a change in the critical point structure of the terrain.
During the modeling of a shape, surface coordinates are usually assigned for texturing. If the model then changes its topological type, these texture coordinates need to be maintained in a consistent manner. An implicit surface changes topological type when a critical value of the function is zero (a critical point intersects the implicit surface). If the critical point is Morse, then the function can be locally approximated by a quadric function [Milnor, 1963]. In 3-D, this quadric function is either that of a sphere or a cone. Defining the change in texture coordinates when a sphere is created or disappears, or a cone becomes connected or disconnected suffices to define the texture coordinate changes for any topological type change in an implicit surface (up to a perturbation to eliminate any non-Morse critical points).
4.3 Compression
4.5 Constraints
Geometric models are often compressed for efficient storage or transmission. Depending on the application, the topological type of the geometry may need to be maintained. Schr¨oder & Sweldens [1995] demonstrated a problem of topology in compression. Their application was extending the wavelet basis to operate on the sphere and other spaces. They demonstrated their algorithm by compressing a map of the earth on the sphere. At various levels of compression, some land masses became connected and others separated due to compression errors, as shown in Figure 8. While errors obscuring geometric details may be forgivable, errors that change the topological type of the data should be avoided.
Numerous constraint methods have been applied to aid the geometric modeling process. One method for example shows that a structure’s components may be scattered, and the structure built automatically by constraining the appropriate ends of the materials to attach to each other [Barzel & Barr, 1988]. The notion of a topological constraint seems to remain open. Consider the ability to model an implicit surface with the constraints that some components could not connect with other components, or such that some components remained connected to other components. Indeed a significant portion of the implicit modeling process seems to focus on keeping the appropriate components connected, and disconnected from all else. Such topological constraints at first seem difficult to even specify. However, the language of Morse theory provides the ability to articulate such a specification. Such Morse constraints in a modeling system would identify each critical point as it appears, and would allow the user to specify whether the critical point should remain above or below the current implicit surface threshold. The impact on the controls due to this constraint could then be solved for using the same tools as physically-based constraints, as they were in another implicit surface modeling system [Witkin & Heckbert, 1994].
Figure 8. Data compression can cause unwelcome changes in topology. (Image courtesy Peter Schroeder.)
4.6 Blending Perhaps the holy grail of implicit surface modeling systems is to be able to control blending. Unwanted blend6
affect implicit surface topology, but didn’t discover Morse theory until much later, and then taught it to each other. I should not forget to thank Patty who is home alone with the kids tonight because I’ve again waited until the last minute to finish writing a paper.
ing causes webbed fingers and arms to fuse to legs. Techniques for avoiding unwanted blending have been developed. Gascuel [1993] warped the field function to simulate precise contact instead of unwanted blending. KacicAlesic & Wyvill [1991] and Guy & Wyvill [1996] used a blend graph to control blending, but the result was not always smooth. Using the language of Morse theory, unwanted blending is prevented by maintaining the critical points on the appropriate side of the threshold. However, a different modeling method may be a better alternative. Consider instead the modeling of an implicit surface by directly controlling its critical points organized in a CW-complex.
References [Axen & Edelsbrunner, 1998] Axen, U. and Edelsbrunner, H. Auditory morse analysis of triangulated manifolds. In Hege, H.-C. and Polthier, K., eds., Mathematical Visualization, pp. 223–236. Springer-Verlag, Heidelberg, 1998.
Acknowledgments
[Bajaj & Schikore, 1998] Bajaj, C. L. and Schikore, D. R. Topology preserving data simplification with error bounds. Computers and Graphics 22(1), 1998, pp. 3– 12.
My education in topology was quite circuitous as my chosen field is computer science, and I would like to take this opportunity to trace it while acknowledging those involved in it. Lou Kauffman and Dan Sandin originally hooked me on rendering quaternion Julia sets in the Electronic Visualization Laboratory at the University of Illinois at Chicago. Alan Norton’s guidance and insight during an internship at IBM’s T.J. Watson Research Center left me with a deeper understanding of their dynamics and structure. My advisor Tom DeFanti deserves thanks for allowing me to receive a Ph.D. in computer science even though most of my graduate courses were in math. Martin Tangora patiently suffered through my homework proofs while providing me a formal education in point-set topology. George Francis constantly reminds me that I don’t know the slightest thing about topology, but with such panache that I don’t feel too badly about it, and all the while mentoring me through its more subtle passages. He also introduced me to Ulrike Axen at the University of Illinois at UrbanaChampaign. I was thrilled when Ulrike joined the CS faculty at Washington State University, allowing us to collaborate more closely as we simultaneously attack computational topology from independent fronts. Her knowledge of computational geometry and formal training in Morse theory have been an invaluable resource. Thanks to Greg Turk and Peter Schroeder for getting me pictures at the last minute. Greg’s picture is the result of a single day we spent together at the University of North Carolina achieving the shared goal to recreate Alan Norton’s pictures with Jules Bloomenthal’s implicit surface polygonizer. I appreciate Bart Stander’s faith in my direction of his Ph.D. dissertation research on polygonizing implicit surfaces. We started with the idea that critical points somehow
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