Convergence of Geometric Algorithms for Graphics & Animation

Transcription

1 Convergence of Geometric Algorithms for Graphics & Animation E Antonov, J. Bisceglio, I. Borovikov, N. Noble, T. J. Peters October 28, 2008 Abstract Many graphics and animation applications depend upon convergence of geometric algorithms, which is the unifying theme for the two innovations presented here. The first extends ray-tracing in graphics over a richer family of input surfaces. The second improves temporal anti-aliasing in animation through topological considerations. Results of numerical experiments on convergence are provided for both algorithms to indicate the performance measures used by these authors in developing commercial applications for graphics and animation. Keywords: G.1.2: Numerical Analysis; Approximation, Special function approximations. I.3.5: Computer Graphics: Computational Geometry and Object Modeling, Splines. Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA, me@eugeneantonov.com Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA, justin.bisceglio@gmail.com Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA, igor.borovikov@gmail.com Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA, norm@kerner.com Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA and Department of Computer Science & Engineering, University of Connecticut, Storrs, CT , tpeters@cse.uconn.edu. 1

2 1 Introduction Algorithmic theory can often be refined by experimentation and testing [11], as will be demonstrated for two geometric algorithms: the determination of plane/surface intersection and finding line segments whose end points are both normal to a given surface. The first is used for ray-tracing and the second is used for temporal anti-aliasing. Algorithms for ray/surface intersections for Bézier surfaces have appeared [8], relying upon uniform parameterization. In the industrial practice of aircraft manufacture, it is necessary to compute ray/surface intersection with more general NURBS surfaces. Applications include the projection of curves onto surfaces along a particular vector and surface gridding. For this more general problem, no applicable research was found in the literature, leading to the generalizations presented here in Section 2. Tubular neighborhoods from differential topology [3] have been shown to be a useful topological construct to prevent temporal aliasing [4]. An algorithm to determine the appropriate tubular radius for NURBS curves has appeared [9]. It is based upon finding all pairs of points on the curve such that the line segment joining them is normal to the curve at both end points. These will be called doubly normal segments. The natural extension to surfaces is presented here. Furthermore, dynamic updates of these computations are desirable for temporal anti-aliasing in animation, but all known algorithms involve root finding techniques that are not easily adapted to dynamic situations. Aggressive convergence methods are presented in Section 3 that approach the performance needed for dynamic updating. In the study of each of these problems, topological analysis has proven valuable for extending known theoretical foundations to wider practical applications. 2 Topological Analysis for Ray Tracing The consideration of intersections between sets is a fundamental, unifying concern between topology and geometry. The integration of a topological analysis with a surface intersector provides a sound basis for extending ray/surface intersections beyond Bézier surfaces. The integration begins with the planar cut method, as explained, below. 2

3 2.1 Planar Cut Method The state of the art for efficient ray/surface intersection algorithms is limited to Bézier surfaces [8]. The successful extension here to NURBS surfaces relies upon rapid convergence obtained from adopting the topological analysis technique used in the planar cutting method. This extension also delivers the accuracy required by CAD/CAM applications. We first review the planar cutting method and then describe how we use it to compute ray/surface intersections. The planar cut method computes the intersection between a parametric surface and a plane. The intersection is represented as contours along the parametric surface. Developed by Grandine and Klein [2], planar cutting is summarized by the following steps: 1. subdivide the parametric region into panels to find all turning and critical points that lie on the intersection contours; 2. find all intersections of the surface boundaries and the plane; 3. order all points found in steps 1 and 2 according to their distance from the origin; initialize an ordered list of contours; 4. iterate through each point in the list; if the point is a boundary point go to step 5, otherwise go to step 6; 5. decide if the point corresponds to a contour which is starting or ending as determined by the panel direction and, accordingly, either add a new contour to the list or remove a completed contour from the list; return to step 4; 6. decide if the point is a turning or critical point; for turning points, test if two contours start or two contours end; return to step 4; 7. use spline collocation to trace the contours. Both critical and turning points are roots along the intersection contour where the gradient vanishes. The distinction is that turning points lie on the intersection of two contour branches. The spline collocation method used for approximating the contours can achieve arbitrary accuracy and can be adjusted to accommodate the requirements of any particular computer graphics application. The topology resolution scheme is designed to determine the number of contour branches as well as the endpoints for each of those branches. This helps guarantee the robustness of the method. The next section explains how the planar cutting algorithm is used in computing ray/surface intersections. 3

4 2.2 Root Finding with Intersecting Contours Consider an intersecting ray and a surface, such as shown in Figure 1. The ray is reformulated as two planes and two planar cuts are performed. Each planar cut contour represents an intersection between a plane and the surface, and these two contours themselves intersect at the same point(s) where the ray intersects the surface. Boundary and turning points are found in the same manner as the planar cut method by using the projected polyhedron algorithm (PPA) [9]. Once these Figure 1: Surface and ray. roots are ordered, topologically correct contours may be traced and intersections of the two planar cut contours queried. This intersection problem may easily be solved in the two dimensional parametric space of the surface. Much of the collocation work to trace the contours to an arbitrary degree of accuracy can be eliminated since only the intersection regions are of importance. Because the ordered roots guarantee a topologically correct contour, a piecewise linear approximation to the contour that connects the points will suffice to quickly identify and narrow the intersection regions. 2.3 Numerical Examples An experiment was conducted with a quadratic, non-rational, Bézier surface, depicted in Figure 1. A ray was drawn by selecting a position on the surface at random and drawing a line to the origin. A solution within tolerance to the selected point on the surface was required of the intersection algorithm. A tolerance condition of was used throughout the experiment. Generally, tolerance conditions are measured in parametric space. Because the solution for this experiment is known, a tolerance measure in model space was possible, as recorded here. Isoparametric curves were extracted from the surface boundaries and from 4

5 the interior of the patch where minimum and maximum features are defined by convex hull spans [10]. This creates a bound around the features and guarantees a topologically correct contour. For this example, twelve curves are produced, as shown in Figure 2(a). The isoparametric curves were used to find roots between the surface and each of the two planes that represent the ray. This was done with PPA. Transforming the ray and the twelve curves such that the ray became parallel with the x-axis allowed the control points of the twelve curves to be projected onto the XY-plane and the XZ-plane. Creating a graph with the parametric axis yielded the twenty-four curves shown in Figure 2(b). In this example, there were eight intersections with the two planes - four with each plane. These eight roots appear in Figure 2(c). Since none of the roots were turning points, the roots were ordered sequentially across the parametric domain and connected by piecewise-linear segments to form an approximation of the intersection contours. This is shown in parametric space in Figure 3(a) and in model space in Figure 3(b). The two dimensional line segments were then intersected: in this step the segments that intersect can be tracked to any desired level of accuracy before they are intersected, and can be subdivided to increase accuracy after intersection, until a solution within a desired tolerance is reached. Iterative refinements under convergence are shown in Figure 3(c), the final intersection result is illustrated in Figure 3(d), and Table 1 demonstrates the convergence to the solution through each iteration. In this experiment, the root ordering was a simple step, but ordering roots can sometimes be tricky. Detailed explanations and illustrated examples have been published for this task [2]. Iteration Model error Parametric error e e e e e e e e e e-16 Table 1: Convergence for first example. Three additional experiments, illustrated in Figure 4, were performed to examine the robustness of the algorithm. In the second experiment, the 5

6 (a) (b) (c) Figure 2: (a) Contours. (b) Projections. (c) Roots. 6

7 (a) (b) (c) (d) Figure 3: (a) Parameter space. (b) Approximation. (c) Refinement (d) Solution. 7

8 ray intersected a discontinuous region along the crease of the example surface. The third experiment was a classic tangency case. The last experiment demonstrates the flexibility of the algorithm with a rational quadric NURBS surface. The internal knots on this surface make it more general then a Bézier surface. Furthermore, the ray intersects this surface in two locations, both of which are reported by the algorithm. These experiments had similar convergence behavior to the first experiment. 3 Temporal Anti-aliasing & Topology Animation displays slightly differing images (called frames) at rates that produce an illusion of motion. That illusion can be destroyed by unexpected changes between frames a problem known as temporal aliasing. A topological approach for preventing temporal aliasing [4] relies upon ambient isotopy [3]. Following previous work on curves (1-manifolds) [9], algorithms for identifying ambient isotopies rely upon finding the doubly normal segment of minimal length. This minimal length is known as the minimum separation distance (MSD). The PPA was previously used [9] to find the MSD on 1-manifolds. As an alternative, Newton s method 1 was used elsewhere [7]. While both the PPA and Newton s methods are effective in the case of 1-manifolds, the extension of Newton s method to 2-manifolds is more natural [5] and is pursued here. The formulation of the partial differential equations for this bivariate Newton s method is a straightforward adaptation from the previously published univariate case [6, 7]. As with the previous demonstration on a trefoil knot [6, 7], the imperative for optimal performance is the efficient creation of seeds for Newton s method. This use of Newton s method over multiple seeds can result in many local minima being returned. From these local minima, the length of the shortest segment is chosen as MSD. Since seeds are easily created and Newton s method is called for each seed, the focus becomes to effectively cull to a small set of good seeds in order to reduce both: the number of times Newton s method is called, and the convergence time for each invocation of Newton s method. 1 Newton s method uses basic tools of calculus to iteratively improve upon an initial estimate to converge to an acceptable approximation of the roots of a given equation. 8

9 (a) (b) (c) Figure 4: (a) Discontinuous. (b) Tangential. (c) Multiple. 9

10 Only manifolds without boundary are considered. Consistent with the previous approach [6, 7], each 2-manifold is partitioned into sub-patches, with computations proceeding between each pair of sub-patches. For this initial research, the typical restriction to Bézier surfaces was adopted. 3.1 A Naïve Seeding Approach The most straight-forward approach to generating seeds is to select N points along each side of the unit square (in parametric coordinates), providing a uniformly distributed set of points over both manifolds, and to consider all N 4 pairs as initial seeds. Of course, most of the pairs can be culled since the connecting line segment is far from satisfying the normality conditions, but still the fourth power growth of data raises concerns about scalability. With this method, taking N = 10 was sufficient for some initial experiments. A representative example of two sub-patches is shown in Figure 5, where all 28 seeds were retained (Figure 5(a)) as being within sufficient tolerance of satisfying the normality threshold. Coded in C# and running on a 3 GHz computer, our prototype found three segements within tolerance of the double normality condition described in Section 3.3, below. The time expended was less than 1 second (Figure 5(b)), approaching the speeds needed to satisfy realistic requirements. Further optimization is expected by even more aggressive culling of seeds. 3.2 A Continuum of Seeds Despite this initial success, some problems have appeared when there is a high degree of symmetry, as often happens in objects used in animation, as well as in engineering design. Under those circumstances, there can be a continuum of double normals, as shown in Figure 6. For this continuum case, the culling comparison described in Section 3.3, below, fails to satisfactorily reduce the sample set of seeds for Newton s method. This is in contrast to the good performance seen on cases where such symmetry does not prevail, such as in examples previously published [6], shown here in Figure 7. One generally thinks of symmetries as reducing the difficulty of a problem, but, here, this essentially amounts to much redundant information that needs culling. One culling alternative under consideration is prompted by other graphics developments, specifically the gaming technique for shadow volume rendering using stencils [1]. 10

11 (a) (b) Figure 5: (a) Seeds. (b) Result. 3.3 Numerical Experiments for Seeding The results of Newton s method will, of course, vary according to the number of seeds permitted by the user and by the thresholds used for culling out seeds that are unlikely to be helpful in the convergence process. The results of numerical experiments to assess those trade-offs are reported in Table 2. Samples Thresholds ; 2 ; 0.015s * * * ; 3 ; 0.280s 16 ; 3 ; 0.050s * * ; 3 ; 4.22s 432 ; 3 ; 0.72s 44 ; 3 ; 0.41s * ; 3 ; 70.6s 7540 ; 3 ; 11.1s 784 ; 3 ; 5.81s 64 ; 3 ; 5.28s Table 2: Trade-offs between Samples and Thresholds The explanations for this table are as follows: The left hand column indicates the number of seed pairs initially cho- 11

12 Figure 6: A continuum of normals. sen. Each value is of the form N 4, where N is the number of samples chosen along each axis in the parametric domain. The variable D used for culling is defined by D = max{d 1, D 2, D 3 }. Let P 1 and P 2 be sample points defined in the previous item, and let N 1 and N 2 be the unit normals at each of P 1 and P 2, respectively. Let L be the normalized vector between P 1 and P 2 and let D 1 (N 1, N 2 ) = min( N 1 N 2, N 1 + N 2 ), D 2 (N 1, L) = min( N 1 L, N 1 + L ), D 3 (L, N 2 ) = min( L N 2, L + N 2 ). If D is greater than a user-supplied threshold value, then the pair of points P 1, P 2 was culled. An asterisk indicates that no solutions were found. Time spent for such failed runs was always less than or around 1 second, except for the last row where it was a few seconds (C# code, 3 GHz CPU). In each column, consider the numbers separated by semi-colons. The first number denotes the number of seeds left after culling, the second number is the number of solutions found and the third number is the 12

13 Figure 7: Many doubly normal segments efficiently calculated. time required to terminate. Only one case produced an incorrect number of solution as 2 the correct number of solutions for all cases was 3. When the threshold was lowered to no solutions were found for any of the numbers of initial seeds indicated. Increasing the initial sample size further was not explored, nor were smaller thresholds examined. The expectation is that for smaller thresholds, the sample size needed would become prohibitive regarding performance expectations. The conclusion draw from these experiments is that finding good parameters is a non-trivial endeavor: the sweet spot is located along the diagonal; moving far to the right gives no solutions (failure), while moving too far down leads to a very steep increase in calculation time while all solutions are produced. Moving to the left also results in increased calculation time. Determination of an optimal combination of parameters remains an object of our investigations. Concurrent with these investigations, contemporary graphics cards are shifting to support massively parallel computation. The analysis of the numerous seeds for Newton s method is thought to be a good candidate to 13

14 leverage the massively parallel capabilities of today s conventional graphics card. It is expected that future experiments will show that parallelization will nicely complement the algorithms presented here. 4 Conclusion Careful attention to convergence properties of geometric algorithms led to the innovations in graphics and animation presented here. For a graphics application, topological analysis helped to solve a geometric problem in computing ray/surface intersections robustly and efficiently, allowing extensions beyond Bézier surfaces to NURBS surfaces. Regarding animation, culling by geometric methods produce convergence of Newton s method with sufficient performance to obtain topological limits for temporal anti-aliasing. In both cases, a wider viewpoint on convergence has proven to be beneficial in developing software for commercial applications in graphics and animation. Acknowledgements: All authors were partially supported by NSF SBIR grant IIP and the third author was also partially supported by NSF grant CCR All statements here are the responsibility of the authors, not of the National Science Foundation. The authors express gratitude to Fritz W. Klein IV for many helpful conversations. References [1] [2] T. A. Grandine and F. W. Klein. A new approach to the surface intersection problem. Computer Aided Geometric Design, 14(2), , [3] M. W. Hirsch. Differential Topology. Springer-Verlag, New York, [4] K. E. Jordan, L. E. Miller, E. L. F. Moore, T. J. Peters, A. C. Russell, Modeling time and topology for animation and visualization with examples on parametric geometry, Theoretical Computer Science 405, (1-2), Computational Structures for Modelling Space, Time and Causality, 41 49,

15 [5] J. H. Mathews, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, [6] L. E. Miller, E. L. F. Moore, T. J. Peters, A. C. Russell, Topological neighborhoods for spline curves : practice & theory, 5045, , [7] E. L. F. Moore. Computational Topology of Spline Curves for Geometric and Molecular Approximations. PhD Thesis, The University of Connecticut, [8] T. Nishita and T. W. Sederberg and M. Kakimoto. Ray tracing trimmed rational surface patches. Siggraph 90: Proceedings of the 17th annual conference on Computer graphics and interactive techniques, , 1990, ACM Press. [9] Nicholas M. Patrikalakis and Takashi Maekawa. Shape Interrogation for Computer Aided Design and Manufacturing, 2002, Springer-Verlag. [10] Les Piegl and Wayne Tiller. The NURBS Book, 1995, Springer-Verlag. [11] N. F. Stewart, Science and computer science. ACM Computing Surveys (27), No. 1, 39-41,