Computational Topology for Computer Graphics

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Computational Topology for Computer Graphics
Klein bottle
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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
• Topological space is a set with the least
  structure necessary to define the concepts of
  nearness and continuity

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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 Topology?

• What is the boundary of an object?
• Are there holes in the object?
• Is the object hollow?
• If the object is transformed in some way, are the
  changes continuous or abrupt?
• Is the object bounded, or does it extend
  infinitely far?

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Why Do We (CG) 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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Problem Mesh Reconstruction

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

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Topological Properties

• To uniquely determine the type of homeomorphism
  we need to know
• Surface is open or closed
• Surface is orientable or not
• Genus (number of holes)
• Boundary components

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Surfaces

• How to define surface?
• Surface is a space which locally looks like a
  plane
• the set of zeroes of a polynomial equation in
  three variables in R3 is a 2D surface x2y2z21

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Surfaces and Manifolds

• An n-manifold is a topological space that
  locally looks like the Euclidian space Rn
• Topological space set properties
• Euclidian space geometric/coordinates
• A sphere is a 2-manifold
• A circle is a 1-manifold

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Open vs. Closed Surfaces

• The points x having a neighborhood homeomorphic
  to R2 form Int(S) (interior)
• The points y for which every neighborhood is
  homeomorphic to R2?0 form ?S (boundary)
• A surface S is said to be closed if its boundary
  is empty

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Orientability

• A surface in R3 is called orientable, if it is
  possible to distinguish between its two sides
  (inside/outside above/below)
• A non-orientable surface has a path which brings
  a traveler back to his starting point
  mirror-reversed (inverse normal)

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Orientation by Triangulation

• Any surface has a triangulation
• Orient all triangles CW or CCW
• Orientability any two triangles sharing an edge
  have opposite directions on that edge.

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Genus and holes

• Genus of a surface is the maximal number of
  nonintersecting simple closed curves that can be
  drawn on the surface without separating it
• The genus is equivalent to the number of holes or
  handles on the surface
• Example
• Genus 0 point, line, sphere
• Genus 1 torus, coffee cup
• Genus 2 the symbols 8 and B

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Euler characteristic function

• Polyhedral decomposition of a surface
• (V vertices, E edges, F faces)
• ?(M) V E F
• If M has g holes and h boundary components
  then ?(M) 2 2g h
• ?(M) is independent of the polygonization

? 1
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Summary equivalence in R3

• Any orientable closed surface is topologically
  equivalent to a sphere with g handles attached to
  it
• torus is equivalent to a sphere with one handle
  (? 0, g1)
• double torus is equivalent to a sphere with two
  handles (? -2 , g2)

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Hard Problems Dunking a Donut

• Dunk the donut in the coffee!
• Investigate the change in topology of the portion
  of the donut immersed in the coffee

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Solution Morse Theory

• Investigates the topology of a surface by the
  critical points of a real function on the surface
• Critical point occur where the gradient ??f
  (??f/?x, ?f/?y,) 0
• Index of a critical point is of principal
  directions where f decreases

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Example Dunking a Donut

• Surface is a torus
• Function f is height
• Investigate topology of f ?? h
• Four critical points
• Index 0 minimum
• Index 1 saddle
• Index 1 saddle
• Index 2 maximum
• Example sphere has a function with only critical
  points as maximum and a minimum

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How does it work? Algebraic Topology

• Homotopy equivalence
• topological spaces are varied, homeomorphisms
  give much too fine a classification to be useful
• Deformation retraction
• Cells

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Homotopy equivalence

• A B ? There is a continuous map between A and
  B
• Same number of components
• Same number of holes
• Not necessarily the same dimension
• Homeomorphism Homotopy

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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
• We attach cells at their boundaries

0-cell
1-cell
2-cell
3-cell
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Morse function

• f doesnt have to be height any Morse function
  would do
• f is a Morse function on M if
• f is smooth
• All critical points are isolated
• All critical points are non-degenerate
• det(Hessian(p)) ! 0

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Critical Point Index

• The index of a critical point is the number of
  negative eigenvalues of the Hessian
• 0 ? minimum
• 1 ? saddle point
• 2 ? maximum
• Intuition the number
  of independent
  directions in which
  f
  decreases

ind2
ind1
ind1
ind0
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If sweep doesnt pass critical pointMilnor 1963

• Denote Ma p ? M f(p) ? a (the sweep region
  up to value a of f )
• Suppose f ?1a, b is compact and doesnt contain
  critical points of f. Then Ma is homeomorphic to
  Mb.

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Sweep passes critical pointMilnor 1963

• p is critical point of f with index ?, ? is
  sufficiently small. Then Mc? has the same
  homotopy type as Mc?? with ?-cell attached.

Mc?
Mc??

Mc?
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This is what happened to the doughnut
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What we learned so far

• Topology describes properties of shape that are
  invariant under deformations
• We can investigate topology by investigating
  critical points of Morse functions
• And vice versa looking at the topology of level
  sets (sweeps) of a Morse function, we can learn
  about its critical points

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Reeb graphs

• Schematic way to present a Morse function
• Vertices of the graph are critical points
• Arcs of the graph are connected components of the
  level sets of f, contracted to points

2
1
1
1
1
1
0
0
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Reeb graphs and genus

• The number of loops in the Reeb graph is equal to
  the surface genus
• To count the loops, simplify the graph by
  contracting degree-1 vertices and removing
  degree-2 vertices

degree-2
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Another Reeb graph example
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Discretized Reeb graph

• Take the critical points and samples in between
• Robust because we know that nothing happens
  between consecutive critical points

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Reeb graphs for Shape Matching

• Reeb graph encodes the behavior of a Morse
  function on the shape
• Also tells us about the topology of the shape
• Take a meaningful function and use its Reeb graph
  to compare between shapes!

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Choose the right Morse function

• The height function f (p) z is not good enough
  not rotational invariant
• Not always a Morse function

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Average geodesic distance

• The idea of Hilaga et al. 01 use geodesic
  distance for the Morse function!

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Multi-res Reeb graphs

• Hilaga et al. use multiresolutional Reeb graphs
  to compare between shapes
• Multiresolution hierarchy by gradual
  contraction of vertices

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Mesh Partitioning

• Now we get to Zhang et al. 03
• They use almost the same f as Hilaga et al. 01
• Want to find features long protrusions
• Find local maxima of f !

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Region growing

• Start the sweep from global minimum (central
  point of the shape)
• Add one triangle at a time the one with
  smallest f
• Record topology changes in the boundary of the
  sweep front these are critical points

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Critical points genus-0 surface

• Splitting saddle when the front splits into two
• Maximum when one front boundary component
  vanishes

min
splitting saddle
max
max