Bounding Iterated Function Systems

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Bounding Iterated Function Systems

• Orion Sky Lawlor
• olawlor_at_uiuc.edu
• CS 497jch
• November 14, 2002

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Roadmap

• Introduction to IFS
• Rices Bounding Spheres
• Lawlors Polyhedral Bounds

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• Iterated Function Systems

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Iterated Function Systems--IFS

• A finite set of mapsdistortions 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 IFSSierpinski Gasket

• Shape is 3 copies of itself, so we use 3 maps

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Example IFSSierpinski Gasket

• Shape is 3 copies of itself, so we use 3 maps
• Map to top

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Example IFSSierpinski Gasket

• Shape is 3 copies of itself, so we use 3 maps
• Map to top
• Map down right

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Example IFSSierpinski Gasket

• Shape is 3 copies of itself, so we use 3 maps
• Map to top
• Map down right
• Map down left

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Example IFSSierpinski Gasket

• Many other, equivalent options

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IFS Gallery Mengers 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 testa 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 lt r
• Equivalently r gt dist(x, wi (x))/(1 - si )
• We must pick x to minimize r
• Nonlinear optimization problem (!)

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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 dont 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

• Weve 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 all
• in solver

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Area vs. Number of Sides

• Little benefit
• to using more
• than 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