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Bounding Iterated Function Systems

Orion Sky Lawlorolawlor@uiuc.edu

CS 497jchNovember 14, 2002

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

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Iterated Function Systems

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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

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Example IFS—Sierpinski GasketShape is 3 copies

of itself, so we use 3 maps:

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Example IFS—Sierpinski GasketShape is 3 copies

of itself, so we use 3 maps:

Map to top

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Example IFS—Sierpinski GasketShape is 3 copies

of itself, so we use 3 maps:

Map to top Map down right

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Example IFS—Sierpinski GasketShape is 3 copies

of itself, so we use 3 maps:

Map to top Map down right Map down left

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Example IFS—Sierpinski GasketMany other,

equivalent options

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IFS Gallery: Menger’s Sponge

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IFS Gallery: Spirals

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IFS Gallery: Five Non-Platonic Non-Solids

Reproduced from Hart and DeFanti, SIGGRAPH 1991

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IFS Gallery: Fractal Forest

Reproduced from Hart and DeFanti, SIGGRAPH 1991

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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

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Sphere Bounds for IFS

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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’

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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

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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)))...

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Sphere Bound for IFS Each map of the sphere must lie

completely within the sphere This is our “recursive bound”

Knowns

wi Map number i

si Scaling factor of wi

Unknowns

r Radius of big sphere

x Center of big sphere

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Sphere Bound for IFS [Rice, 1996]

We require dist(x, wi (x)) + si r < r

Equivalently r > dist(x, wi (x))/(1 - si )

We must pick x to minimize r Nonlinear optimization problem (!)

dist(x, wi (x)) si r

rx

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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 (?)

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Polyhedral Bounds for IFS

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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

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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

!

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Polyhedral Bound for IFS Each map of the polyhedron

should lie completely within the original polyhedron Again, a “recursive bound”

Knowns

wm (x) Map number m

ns Normal of side s

Unknowns

ds Displacement of side s

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Polyhedron Bounding, in Words We will require

Each corner of the polyhedron Under each map To satisfy all polyhedron halfspaces

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Point-in-Polyhedron Test Points inside polyhedron must

lie inside all halfspaces

Point lies in a halfspace if

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Polyhedron corners (2D Version) The corner of sides i and j is

where both halfspaces meet

or, if we define

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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)

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Linear Optimization [Lawlor 2002] We’ve reduced IFS bounding to

a problem in linear optimization Constraints: Just shown Unknowns: Displacements ds 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”)

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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

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Time vs. Number of Sides

O(s4.6) time; allin solver

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Area vs. Number of Sides

Little benefit to using morethan 12 sides

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IFS Gallery: Spirals, with Bounds

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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