Polygonization of Implicit Surfaces Kenneth E. Hoff III Computational Geometry Presentation

1. Polygonization of Implicit Surfaces Kenneth E. Hoff III Computational Geometry Presentation Sept ‘98

2. Overview • Define implicit surfaces • Why polygonize? • Why are they more difficult to render than “explicit” surfaces? • Why use them? • General framework for polygonization: • 1D implicit “surfaces” (pts) : root-finding • 2D implicit “surfaces” (contour lines) • Potential Problems • Application of framework to 3D

3. What is an implicit surface? • The set of points X that satisfies the implicit equation: f(X)=0 • Points that satisfy the “property” defined by the implicit function f: • sphere : pts at a specific distance from a point • map height contour : pts at a specific height on a topographical map • Also called iso-surfaces or contour surfaces • The sign of the implicit function indicates whether a point is inside or outside the surface (we use + to indicate inside)

4. Polygonization : Explicit vs. Implicit Definitions • Why polygonize? • to obtain an efficient representation for rendering • Explicit functions • Simple, direct polygonization • Parametric surfaces: • Vector-valued function so output is a surface point • Just iterate through domain points to tesselate the surface • Implicit functions • More difficult to polygonize • Reduces to solving the implicit equation - multiple-root finding • Scalar-valued function so just gives a single value for any point in space. • Zero-set (or root) defines the surface

5. Implicit Surfaces are difficult to render. Why use them? • Simpler collision queries • Simpler CSG operations (union, difference, intersection) • Allows simple blending between primitive surfaces • only changes f (polygonization method need not be modified) • Easy to apply free-form deformations • only transforms input point to f (polygonization method need not be modified) • have to invert the deformation M: f(M-1*x)=0 • “analytically” deformed • “Blobby” models, MetaBalls, etc.

6. Given a function f, we wish to find the set of x values (1D points) that satisfy f(x)=0 From calculus, the Intermediate Value Theorem states: “as x varies from a to b, the continuous function f takes on every value between f(a) and f(b)” If f(a) and f(b) have opposite signs, then the root is said to be “bracketed” in the interval [a,b] BUT, how did we find a and b? 1D Implicit “Surfaces” : 1D root-finding

7. 1D : How to bracket the root(s) • Uniformly subdivide 1D point domain into cells. • Evaluate f for each cell boundary point • Cells enclosed by two grid pts that evaluate to opposite signs contain at least one root • Is a non-uniform sampling better? Maybe, but more difficult in higher dimensions. • How much of the domain do we subdivide?

8. 1D : Finding the point domain subdivision boundaries Small cells may bracket a root, but how much of the domain must be subdivided? • Already know the extents of the implicit surface defined by f • Conservatively guess the boundaries - inefficient and unreliable • Use method that does not require domain subdivision

9. 1D : Finding the bracketed root • If cell size is small enough, perhaps either boundary value may be close enough to the root • OR use bisection search : depends on the intermediate value theorem • Recursively subdivide intervals bounded by grid pts evaluating to opposite signs • Continue until cell is below some threshold size • Boundaries or center value is considered “close enough” to the root of f • Why not start the uniform subdivision at this threshold size? Excessive evaluation of grid pts.

10. 2D Implicit “Surfaces” : Finding the boundary curve • Given a surface defined by f(X)=0 where X = (x,y) • Bracket roots by subdividing 2D point domain into rectangular cells • Again, how much of the space should be subdivided • Evaluate f at all grid pts • Roots are bracketed in cells that have grid pt evaluations of opposite sign • The surface passes through cells containing bracketed roots

11. 2D : Find the surface passing through a bracketing cell • Similar to 1D, if cell subdivision is considered small enough, simply find where the surface intersects the edges of the cell and connect the intersection points with a line. • Only search for crossings along edges whose endpoints evaluate to opposite signs • OR perform adaptive subdivision

12. 2D : Find the surface edge crossing • Given an edge whose endpoints evaluate to opposite signs • Use bisection search along the edge to find the intersection point

13. 2D : Adaptive-subdivision • Recursively subdivide cells containing a surface crossing down to a threshold cell size • Find surface crossings through smallest cells : bisection search of crossed edges, connect intersections • Since the adaptive subdivision is converged to a threshold cell size for all areas containing the boundary, adjacent cells containing the surface are all the same size - no cracks and no adjacency info is required; just dump the line segments! • Again, we perform adaptive subdivision rather than uniformly subdivide down to the threshold size to avoid excessive evaluations of the implicit function • Can we do better? Evaluate only for the smallest, surface-crossing cells.

14. 2D : Continuation methods • Assume the the domain space is uniformly subdivided into cells at the threshold size, but do not perform any function evaluations • Find a single starting “seed” point that lies on the surface (use ray-casting, etc) • Find cell containing the seed point • “Grow” the set of cells across the surface • Evaluate adjacent grid pts to find next surface crossing cells (or use constrained dynamics) • Do we even need a uniform subdivision (domain mesh)?

15. 2D : Using particle systems to avoid meshing Relaxation (Turk91): • Distribute pts fairly evenly over surface • Repeatedly iterate over all pts maintaining a certain neighbor distance Adaptive Repulsion (Witkin94): • Find any pt on the surface (seed pt) • Give pt a large “sphere of influence” • Pts with spheres of influence over a threshold size are split - fissioning • Split pts have smaller spheres of influence • Pts within another pt’s sphere of influence are repelled across surface • Pts with sphere of influence below another threshold size are destroyed

16. Potential Problems • Bounding the Domain Space • Discretization Error • Disconnected Components • Ambiguous Surface-Crossing Cells

17. Potential Problem #1 : Bounding the Domain Space • May not completely contain the object • May miss disconnected components • Will result in “clipping”

18. Potential Problem #2 : Discretization Error • Too large uniform cell size: • May not be able to converge (entirely miss the surface) • Too large adaptive threshold cell size: • Misses small, completely-contained features • Coarse resolution model • Incorrect topology: • ambiguous cells • connects or breaks components • Too small cells sizes are inefficient and more susceptible to numerical error • Particle systems are an attempt to avoid this problem - no meshing!

19. Potential Problem #3 : Disconnected Components • Uniform cell subdivision may find them, but inefficiently • Particle system and continuation methods may miss them • must have a seed pt on each component • Related to discretization error • Sometimes results from ambiguous cell polarity configurations

20. Potential Problem #4 : Ambiguous Surface-Crossing Cells • Many cell vertex polarity configurations are ambiguous • Possible to polygonize in different arrangements • May even result in disjoint polygons • Solutions: • Detect and recursively subdivide • Choose a consistent convention (e.g. always join positive pts) • Use simplices (triangulate) • Also, related to discretization error

21. Review before going to 3D. . . Method 1: Uniform Spatial Subdivision • Uniformly subdivide domain space into cells • Evaluate implicit function at grid pts to find surface crossing cells • Polygonize surface crossing cells Method 2: Adaptive Spatial Subdivision • Start with Method 1 • Adaptively subdivide surface crossing cells to threshold size • Polygonize surface crossing “leaf” cells Method 3: Continuation • Find any pt on the surface (seed pt) • Find cell containing seed pt at threshold size • Grow surface crossing cells in directions determined by crossed cell boundaries Method 4: Particle Systems • Distribute pts evenly and perform relaxation iterations • OR perform adaptive repulsion and fissioning

22. 3D Implicit Surface Polygonization using Spatial Subdivision Conceptually identical to 1D and 2D version: • Given a surface defined by f(X)=0 where X = (x,y,z) • Subdivide 3D space into uniform cells • Evaluate implicit function for all grid points • Find surface-crossing cells (check polarity of cell vertices) • Subdivide surface-crossing cells to threshold size • Search for surface-crossing along edges of opposite polarity • Polygonize the cell based on the edge intersections - much more difficult in 3D

23. 3D : Polygonizing a Surface-Crossing Cell The most difficult part. . . • Finding cell polygon vertices • Algorithmic method • Table lookup (based on vertex polarities) • Resolving ambiguities • Triangulating the cell polygon

24. 3D : Finding cell polygon face vertices Algorithmic method: • Begin with any edge-surface intersection point (1) • Proceed to negative corner (white open pts) and then clockwise about the cube face (w.r.t outside) until another intersection point is found (2) • Repeat for each subsequent face (34, 45, 51)

25. 3D : Finding cell polygon face vertices Table Lookup: • Cube table lookup: 8 verts, 256 entries • Tetrahedron table lookup: 4 verts, 16 entries • Requires tetrahedral decomposition of the cube • Tetrahedral subdivision is more difficult

26. 3D : Finding cell polygon face vertices Tetrahedral decomposition of the cube: • Different ways to decompose: 5, 6, or 12 tetrahedra • Adjacency problems: 5 must alternate, 6 and 12 is ok, but more expensive • Even 24 is possible to avoid bias

27. 3D : Finding cell polygon face vertices Resolving ambiguities: • many ambiguous polarity configurations • resolve by subdividing, using tetrahedral decomposition, or by being consistent (e.g. always join positive pts)

28. 3D : Triangulating the cell polygon • non-coplanar • preserve aspect ratio • may insert new points

29. 3D : Continuation Methods Same as in 2D: • Start with a seed pt and its containing cell • Visit adjacent cells through surface crossing cube faces (faces that have opposite vertex polarities) • Grow until surface is covered • Polygonize each cell

30. 3D : Particle Systems Similar to 2D: • Use relaxation if we have a uniform distribution • more expensive starting conditions • Use adaptive repulsion if we wish to “grow” across the surface • starts with a single point • better for interactive applications

31. ASIDE: Ray Intersection with Implicit Surface • Particle system methods require ability to push particles around on the surface • Usually a random direction is chosen in the tangent plane • Requires gradient of implicit function • Jumping in random direction causes new position to be off of the surface • We know the gradient at the new position, so we have a ray (start is at new pt, dir is along gradient towards the surface) • RAY CASTING! • Ray defined by r(t) = S + Dt where S is ray start, D is ray direction, t is step in units of length of D along ray • Implicit equation is defined as f(X)=0 where X is a point on the surface if f(X)=0 • Point along ray is defined by r(t), so substitute into f(X): f(r(t))=f(S+Dt)=0 • This reduces to the 1D implicit equation f(t)=0 • Solution(s) t of this equation correspond to the distance along the ray to a point that lies on the surface. . .

32. ASIDE: Ray Intersection with Implicit Surface • We need only find the smallest positive root t of the 1D implicit equation f(t)=0 • We have the gradient vector-valued function: [ fx(X), fy(X), … ] that gives the direction (in the domain) of maximum change in value of the implicit function for a particular point (in the domain) • The domain of f(t) is [0,] since we are only concerned with intersections in front of the ray start pos • We can try a variety of methods to find the root: uniform step size starting at t=0, adaptive step size, Newton iterations (use directional derivative) • To use Newton’s methods, we need the derivative at particular t values: f’(t), but we only have the gradient for pts in the implicit function domain: use the directional derivative in the ray direction to get f’(t): • Given a t value for which we want f’(t): • Find pt in implicit function domain: r(t)=S+Dt • Evaluate gradient at r(t): G(t) = [ fx(r(t)), fy(r(t)), … ] • f’(t) = “directional derivative in ray dir w.r.t. G(t)” = DG(t)

33. References Bloomenthal, Jules. Introduction to Implicit Surfaces. Morgan Kaufmann Publishers, Inc. San Francisco, CA. 1997. Bloomenthal, Jules. Polygonization of Implicit Surfaces. Computer Aided Geometric Design, Nov. 1988, vol 5, p 341-355. Witken, Andrew P. and Paul S. Heckbert. Using Particles to Sample and Control Implicit Surfaces. SIGGRAPH ‘94, 1994, p 269-277. Bloomenthal, Jules. Interactive Techniques for Implicit Modeling. Computer Graphics 24, 2 (Mar. 1990), p 109-116. Sclaroff, Stan and Alex Pentland. Generalized Implicit Functions for Computer Graphics. SIGGRAPH ‘91, July 1991, p 247-250. Stander Barton T. and John C. Hart. Guaranteeing the Topology of Implicit Surface Polygonization for Interactive Modeling. SIGGRAPH ‘97, 1997, p 279-286. Lorenson, William E. and Harvey E. Cline. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. SIGGRAPH ‘87, July 1987, p 163-169. Ning, Paul and Jules Bloomenthal. An Evaluation of Implicit Surface Tilers. IEEE Computer Graphics and Applications. November 1993, p 33-41. Turk, Greg. Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion. SIGGRAPH ‘91, July 1991, p 289-298. Szeliski, Richard and David Tonnesen. Surface Modeling with Oriented Particle Systems. SIGGRAPH ‘92, July 1992, p 185-194.