1. Computational Topology for Computer Graphics
Klein bottle

2. 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”
If you open a book on topology, you get very obtuse definitions involving open sets and such. Topology is really much simpler than that conceptually.

3. 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)
Can’t tear, join, poke/seal holes
A is homeomorphic to B
Say we want to deform a shape A into a shape B. The deformation is onto, which means every point of A is used and every point of B is used. The deformation is also one-to-one, which means that each point of A maps to a single point on B; that there is no overlap. The deformation is also continuous so that we can’t tear A, join sections of A together, poke holes into A, or seal up holes in A. When all these are true, then we say A is homeomorphic to B, which means that they are topologically equivalent. For example, the donut is topologically equivalent to the coffee cup, because we can deform the torus into the cup. We pull out one end of the torus and form it into a cup. The hole in the torus becomes the handle of the cup.

4. 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?

5. Why Do We (CG) Care? The study of connectedness Understanding Analysis
How connectivity happens?
Analysis
How to determine connectivity?
Articulation
How to describe connectivity?
Control
How to enforce connectivity?
Topology studies the properties of a set that don’t change under a homeomorphism (well-behaved deformation). These properties include completeness and compactness. However, in computer graphics, there is really only one property we are interested in and that is connectedness. Specifically how many disjoint components there are, and how many holes do they have. We are also interested in the geometric configuration of these properties, such as which components have how many holes, and where are the holes relative to each other.

6. For Example How does connectedness affect… Morphing Texturing
Compression
Simplification
If we morph a donut into a coffee cup, we might want to make sure the donut hole becomes the handle, instead of the hole sealing up and a new hole poked through the handle. We are also interested on how to make a texture consistent when an object changes topological type, or how we can cut up an object to lay it flat and place a texture on it. When compressing the globe, Peter Schroeder at CalTech discovered that his compression technique had only small errors in geometry, but these small errors changed the geography of the European countries! Finally, when we simplify an object to describe it with fewer primitives, we want to know its topology so we can retain important properties such as holes, or remove these properties when they become imperceptible, insignificant or unnecessary.

7. Problem: Mesh Reconstruction
Determines shape from point samples
Different coordinates, different shapes
One problem with meshing an implicit surface is that the location of the samples can infer a different topology that that of the underlying surface. Note that if we translate the implicit surface one half unit up or down, the topology of the approximating mesh changes.

8. 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

9. 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: x2+y2+z2=1

10. 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

11. 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

12. 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)

13. 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.

14. 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

15. 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
 = 2
 = 0
 = 1

16. 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, g=1)
double torus is equivalent to a sphere with two handles ( =-2 , g=2)

17. 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
Here we have dunked the donut in the coffee. We are interested in the change in topology of the portion of the donut immersed in the coffee. In this case, we started with no donut and then the bottom critical point was immersed and the topology changed to a surface that is topologically a disk.

18. Here we have dunked the donut in the coffee
Here we have dunked the donut in the coffee. We are interested in the change in topology of the portion of the donut immersed in the coffee. In this case, we started with no donut and then the bottom critical point was immersed and the topology changed to a surface that is topologically a disk.

19. Next we pass the second critical point and the dunked portion changes topology, from a disk…

20. To a cylinder.

21. Passing the third critical point, we see that the topology is now changing from a cylinder…

22. To a punctured torus (a 2-manifold with boundary of genus 2)
To a punctured torus (a 2-manifold with boundary of genus 2). If we let go of the donut, the entire surface is immersed. We have passed the final critical point, and the punctured torus changes its topology to that of a torus.

23. 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
Morse theory is many decades old. Our work is based on the book Morse Theory by John Milnor. Morse theory draws a connection between the differential properties of a surface and its topological type. It was coined by Marsden Morse. It is based on critical points, where the partial derivatives of a function are zero, such that the gradient no longer has a specific direction.

24. 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 2 : maximum
Example: sphere has a function with only critical points as maximum and a minimum
Assume we live on a torus and our function is altitude. Then the derivative of the altitude has the slope for its magnitude and the direction of steepest ascent for its direction. Notice that the four star points denote the four “level” areas on the torus, where there is no single direction of steepest ascent. We will label these points with an index. The index is the number of “principal” (maximal/minimal) directions of descent (including the opposite direction). So the point on the bottom has no direction of descent and its index is zero. The two points on the hole each have one principal direction of descent (and one principal direction of ascent), so their indices are both one. The point at the top has two principal directions of descent so its index is two.

25. 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

26. 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
We say two shapes are homotopic when they have the same number of components and the same number of holes. This is not necessarily the same as homeomorphic because the shapes need not be the same dimension. (Dimension is a topological property and so is preserved by homeomorphism.) For example, the point is homotopic to the solid ball (one component, no hole). Also the symbol for female is homotopic to the symbol for male (one component, one hole). ~ ~

27. 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
The easiest way to show homotopic is to use the deformation retract. A deformation retract is a continuous erosion of material, leaving a subset of the original shape. For example, the solid torus can be eroded into a flat anulus, and the anulus can be eroded to a circle. Notice we can’t shrink the circle to a point at its center because that point is not a subset of the circle. We also can’t erode the circle into a point on the circle because it would require us to break the circle, and a deformation retract is continuous. Notice we can erode the plus in the female sign to its base on the circle and we can likewise erode the plus in the male symbol. Hence all these shapes are homotopic because they can all be deformation retracted to a circle. ~

28. Cells Cells are dimensional primitives
We attach cells at their boundaries
0-cell
1-cell
2-cell
3-cell

29. Morse function
f doesn’t 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

30. 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
ind=2
ind=1
ind=1
ind=0

31. If sweep doesn’t pass critical point [Milnor 1963]
Denote Ma = {p  M | f(p)  a} (the sweep region up to value a of f )
Suppose f 1[a, b] is compact and doesn’t contain critical points of f. Then Ma is homeomorphic to Mb.

32. Sweep passes critical point [Milnor 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
with -cell
attached
Mc ~
Mc+

33. This is what happened to the doughnut…

34. 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

35. 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

36. 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

37. Another Reeb graph example

38. Discretized Reeb graph
Take the critical points and “samples” in between
Robust because we know that nothing happens between consecutive critical points

39. 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!

40. Choose the right Morse function
The height function f (p) = z is not good enough – not rotational invariant
Not always a Morse function

41. Average geodesic distance
The idea of [Hilaga et al. 01]: use geodesic distance for the Morse function!

42. Multi-res Reeb graphs
Hilaga et al. use multiresolutional Reeb graphs to compare between shapes
Multiresolution hierarchy – by gradual contraction of vertices

43. 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 !

44. 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

45. 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