John C. Hart

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Computational TopologyforComputer Graphics

• John C. Hart
• School of EECS
• Washington State University

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What is Topology?

• The topology of a space is the definition of a
  collection of sets (called the open sets) that
  include
• the space and the empty set
• the union of any of the sets
• the finite intersection of any of the sets

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No, Really.What is Topology?

• The study of properties of a shape that do not
  change under deformation
• Rules of deformation
• Onto (all of A ? all of B)
• 1-1 correspondence (no overlap)
• bicontinuous, (continuous both ways)
• Cant tear, join, poke/seal holes
• A is homeomorphic to B

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Why Do We Care?

• The study of connectedness
• Understanding
• How connectivity happens?
• Analysis
• How to determine connectivity?
• Articulation
• How to describe connectivity?
• Control
• How to enforce connectivity?

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For Example

• How does connectedness affect
• Morphing
• Texturing
• Compression
• Simplification

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Geometric Modeling

• The connectedness of some geometric
  representations depends on the proximity of their
  primitives
• Recurrent models
• Implicit surfaces

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Implicit Surfaces
Manifold
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Interactive Implicit Surface Modeling

• Particles constrained to surface
• Surface constrained to particles

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Goal Meshing Implicit Surfaces

• Constrains vertices to surface
• As surface moves, so do vertices
• Reconnect skinny triangles
• Problems

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Problem Mesh Aliasing

• Determines shape from point samples
• Different coordinates, different shapes

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Problem Interloping Components

• Blob functions accumulate away from blob centers
• Incremental meshes can miss such components

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Problem Mesh Reconnection

• How do we reconnect polygons when an implicit
  surfaces topology changes?
• How do we even detect when an implicit
  surfacestopologychanges?

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Solution Morse Theory

• Determines the connectedness of a space from the
  critical points of a real function
• Critical point occur where the gradient ??f
  (??f/?x, ?f/?y,) vanishes
• Index of a critical point is of principal
  directions where f decreases
• Perturb to remove degeneracy

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Example Dunking a Donut

• Space is torus
• Function f is height
• Elevation gradient
• Connectedness of f ?? h
• Four critical points
• Index 0 minimum
• Index 1 saddle
• Index 1 saddle
• Index 2 maximum

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How Does It Work?

• Need more serious topology
• Homotopy equivalence
• Deformation retraction
• Cells

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Homotopy Equivalence

• A B
• Same of components
• Same of holes
• Not necessarily same dimension
• Not same as homeomorphic

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Deformation Retraction

• Function that continuously reduces a set onto a
  subset
• Any shape is homotopic to any of its deformation
  retracts
• Skeleton is a deformation retract of the solids
  it defines

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Cells

• Cells are dimensional primitives
• Cells are attached at their boundary

0-Cell
1-Cell
2-Cell
3-Cell
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How It Works

• Homotopy type changes only when height crosses a
  critical point
• Let A before critical point
• Let B after critical point
• Let l index of critical point
• Then B A with a l-cell attached (!)

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Building a Torus
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Oh Yeah,Implicit Surfaces

• Space is now 3-D
• Homotopy type of solid f(x,y,z) ?? 0
• Critical points
• Index 0 center of sphere
• Index 1 wasps waist
• Index 2 dimple
• Index 3 air bubble

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Finding Critical Points

• Search all of 3-D space
• Simple interval search
• Interval Newtons method
• Tracking
• Constrain particles to follow critical point
• Problem critical points appear out of nowhere

Degenerate
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Implicit Surface Topology Changes
Createð Minimum - ïDestroy
Attachð 1-Saddle - ïCut
Burstð Maximum - ïBubble
Spackleð 2-Saddle - ïPierce
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Cut/Spackle Example
Cut
Spackle
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Blending

• Automatic, based on proximity
• Uncontrolled, based on proximity

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Controlled Blending

• Precise Contact Modeling
• Cani (Gascuel) SIGGRAPH 93
• Uses a graph to indicate blending
• Primitives mutually distorted if not blended
• Blending Graph
• Guy Wyvill IS96
• Uses a graph to indicate blending
• Primitives unioned (CSG) if not blended
• Results in C1 discontinuities

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Blend Graph Example
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CW-Complex

• Created by attaching cells
• Generalizes graph to higher dimensions
• Vertices
• Edges
• Faces
• Volumes
• Represents the homotopy typeof the shape

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Example Torus
a
a
b
a
a
b
b
b
FundamentalDomain
Geometry
CW-Complex
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Morse CW-Complex

• A surface has the homotopy type of a CW-complex
  with one l-cell for each critical point of index
  l
• But how do we construct this CW-complex for a
  given shape?

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Follow the Separatrices
(a)
(a)
(b)
c
d
(c)
a
b
(a)
b
a
d
c
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Computing the CW-Complex of a Surface

• 0-cells (vertices)
• Surface search for all minima
• 1-cells (edges)
• Integrate ODE x ?v0(x) from saddles to minima
• 2-cells (faces)
• Seed particle system at 2-saddles
• Bound domain by 1-cell edges

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CW-Complex of a Solid

• Use cells to describe shape topology
• Vertex component
• Edge connection between components
• Ring of edges handle
• Face no hole
• Enclosure of faces hollow shell
• Volume solid
• CW-complex ?? shape

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Example Cube
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Computing the CW-Complex of a Solid

• 0-cells (vertices) minima
• 1-cells (edges)
• Integrate x ?v0(x) from 1-saddle to minima
• 2-cells (faces)
• Surface particles
• Additional constraint x? v2(x) 0
• 3-cells (volumes)
• Voxel region growing seeded at maximum

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Future Work

• Topological skeleton
• MATCW-complex of distance
• Removes geometric noise
• Crystallography
• Carroll Johnson salt molecule
• Critical net graph of CW-complex
• Meshing moving interface surfaces
• Fluid dynamics - droplets
• Level sets ODE surfaces

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The End, For Now

• Birth of a new field
• Morse theory is key
• Solves holy grail problems in implicits
• Interactive modeling
• Controlled blending
• Thanks NSF, Ulrike Axen, Bart Stander, Brian
  Wyvill,Peter Schroeder