Download presentation

Presentation is loading. Please wait.

Published byShaylee Barter Modified over 2 years ago

1
Matthew L. WrightMatthew L. Wright Institute for Mathematics and its ApplicationsInstitute for Mathematics and its Applications University of MinnesotaUniversity of Minnesota

2
4 For a polyhedron, ? – 6+ 4= 2 verticesedgesfaces (# vertices) – (# edges) + (# faces) =

3
For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

4
For a polyhedron, ? (# vertices) – (# edges) + (# faces) = 8– 12+ 6= 2 verticesedgesfaces

5
For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

6
For a polyhedron, ? (# vertices) – (# edges) + (# faces) = The Euler characteristic of a polyhedron is 2.

7
Euler Characteristic, more generally: Finite set Finite graph, no loops Finite graph, with loops Compact planar region = 8= 8 – 5 = 3= 8 – 6 = 2= 8 – 12 + 3 = -1 (number of connected components) – (number of holes)

9
Nota Bene Euler Characteristic is independent of decomposition: = 1 …but is sensitive to open/closed sets: = 0 = 1 = -1

10
Key Property This property is called additivity, or the inclusion-exclusion principle.

12
Integral with respect to Euler Characteristic

13
1 2 3

14
Preferred in practice! Proof: telescoping sums Now integrate.

15
2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

16
2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

17
2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4 compute: There are 7 sets! sum 2 1 3 4 5 6 7

18
Proof: Therefore,

19
Application: Euler integration is useful for enumeration problems (Baryshnikov and Ghrist).

20
Can we count by sensors? I count one. sensors network I count none.

21
Can we count by sensors? ??? ? ? ?

22
How can we compute Euler integrals over networks?

23
Computation:

26
Example: Start with a grayscale digital image.

27
246 635 52 25 164 236 31 56 42155

28
246 635 52 25 164 236 31 56 42155

29
246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

30
246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

31
246 635 52 25 164 236 31 56 42155 threshold Euler characteristic Euler characteristic graph

32
Richardson and Werman used the Euler characteristic graph to classify 3D objects. a catcurvature function on the cat curvature above threshold 0.05

33
Richardson and Werman used the Euler characteristic graph to classify 3D objects.

34
The Euler characteristic curve helps distinguish between similar artifacts from two sites.

36
Euler characteristic is a simple topological invariant. Homology is a much more sophisticated topological invariant. Betti numbers Connection:

37
Persistent Homology e.g. components, holes, graph structure e.g. set of discrete points, with a metric Persistent homology is an algebraic method for discerning topological features of data.

38
Example: What topological features does the following data exhibit? Problem: Discrete points have trivial topology.

39
Idea: Connect nearby points.

40
Idea: Connect nearby points, build a simplicial complex. 3. Fill in complete simplices.

42
…then we detect noise.

46
A collection of bars is a barcode.

49
?

Similar presentations

OK

Chap. 11 Graph Theory and Applications 1. Directed Graph 2.

Chap. 11 Graph Theory and Applications 1. Directed Graph 2.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on wildlife and natural vegetation of india Ppt on water pollution free download Neurological anatomy and physiology ppt on cells Ppt on computer graphics applications Ppt on computer malware scanner Ppt on duty rosters Ppt on causes of 1857 revolt ppt Ppt on environmental degradation vs industrial revolution By appt only movie john Seminar ppt on power electronics