1. 1 Bounding Iterated Function Systems Orion Sky Lawlor olawlor@uiuc.edu CS 497jch November 14, 2002

2. 2 Roadmap Introduction to IFS Rice’s Bounding Spheres Lawlor’s Polyhedral Bounds

3. 3 Iterated Function Systems

4. 4 Iterated Function Systems--IFS A finite set of “maps”— distortions of some space Apply the maps in random order Converges to unique “attractor” Equivalent to L-systems, others E.g., Mandelbrot set is just convergence diagram for a one- map 2D IFS: complex squaring

5. 5 Example IFS—Sierpinski Gasket Shape is 3 copies of itself, so we use 3 maps:

6. 6 Example IFS—Sierpinski Gasket Shape is 3 copies of itself, so we use 3 maps: Map to top

7. 7 Example IFS—Sierpinski Gasket Shape is 3 copies of itself, so we use 3 maps: Map to top Map down right

8. 8 Example IFS—Sierpinski Gasket Shape is 3 copies of itself, so we use 3 maps: Map to top Map down right Map down left

9. 9 Example IFS—Sierpinski Gasket Many other, equivalent options

10. 10 IFS Gallery: Menger’s Sponge

11. 11 IFS Gallery: Spirals

12. 12 IFS Gallery: Five Non-Platonic Non-Solids Reproduced from Hart and DeFanti, SIGGRAPH 1991

13. 13 IFS Gallery: Fractal Forest Reproduced from Hart and DeFanti, SIGGRAPH 1991

14. 14 IFS Conclusions An IFS is just a set of maps of space Pastes shape onto copies of itself IFS are useful tool for representing fractal shapes Wide variation in results Arbitrary number of dimensions Beautiful, natural look Easy to produce/manipulate

15. 15 Sphere Bounds for IFS

16. 16 Why bother bounding an IFS? For display, processing, etc. Raytracing [Hart, DeFanti ‘91] Intersect rays with bounds Replace nearest intersecting bound with a set of smaller bounds Repeat until miss or ‘close enough’

17. 17 Why Bound IFS with Spheres? Spheres are a commonly used bounding volume for raytracing Very fast intersection test—a few multiplies and adds Invariant under rotation Rotate a sphere, nothing happens Closed under scaling Scale a sphere, get a sphere Easy to represent and work with

18. 18 Recursive Bounds for IFS Each map of the bound must lie completely within the bound B contains map(B) Now we just recurse to the attractor B contains map(B) contains map(map(B)) contains map(map(map(B)))...

19. 19 Sphere Bound for IFS Each map of the sphere must lie completely within the sphere This is our “recursive bound” Knowns w i Map number i s i Scaling factor of w i Unknowns r Radius of big sphere x Center of big sphere

20. 20 Sphere Bound for IFS [Rice, 1996] We require dist(x, w i (x)) + s i r < r Equivalently r > dist(x, w i (x))/(1 - s i ) We must pick x to minimize r Nonlinear optimization problem (!) dist(x, w i (x)) s i r r x

21. 21 Sphere Bound Conclusions Spheres are nice bounding volumes Especially for raytracing Hart gives a heuristic for sphere bounds Rice shows how to find optimal (recursive) sphere bound Requires nonlinear optimization Complex, slow (?)

22. 22 Polyhedral Bounds for IFS

23. 23 Why Bound IFS with Polyhedra? Includes many common shapes Box, tetrahedron, octahedron,... Bounding boxes are the other commonly used bounding volume for raytracing A better fit for elongated objects Computers don’t like curves (nonlinear optimization); a polyhedron has no curves

24. 24 Why not Bound IFS with Polyhedra? Polyhedra have corners, which might stick out under rotation Can always fix by adding sides Not so bad in practice !

25. 25 Polyhedral Bound for IFS Each map of the polyhedron should lie completely within the original polyhedron Again, a “recursive bound” Knowns w m (x) Map number m n s Normal of side s Unknowns d s Displacement of side s

26. 26 Polyhedron Bounding, in Words We will require Each corner of the polyhedron Under each map To satisfy all polyhedron halfspaces

27. 27 Point-in-Polyhedron Test Points inside polyhedron must lie inside all halfspaces Point lies in a halfspace if

28. 28 Polyhedron corners (2D Version) The corner of sides i and j is where both halfspaces meet or, if we define

29. 29 Polyhedron Bounding, in Equations We require: Each corner of the polyhedron (linear) Under each map (linear) To satisfy all the halfspaces (linear) These are linear constraints ( I M S of them)

30. 30 Linear Optimization [Lawlor 2002] We’ve reduced IFS bounding to a problem in linear optimization Constraints: Just shown Unknowns: Displacements d s Objective: Minimize sum of displacements? (Probably want to minimize area or length instead) Guaranteed to find the optimal bound if it exists (for some definition of “optimal”)

31. 31 2D Implementation Used open-source linear solver package lp_solve 3.2 Written in C++ Generating constraints take about 40 lines (with comments) Would be even shorter with a better matrix class Welded to a GUI

32. 32 Time vs. Number of Sides O(s 4.6 ) time; all in solver

33. 33 Area vs. Number of Sides Little benefit to using more than 12 sides

34. 34 IFS Gallery: Spirals, with Bounds

35. 35 Convex IFS Bounds: Conclusions Optimal polyhedron bounding using linear optimization Off-the-shelf solvers Piles of nice theory (optimality!) Fast enough for interactive use Future directions RIFS Bounding (solve for attractorlet bounds) Implement in 3D