1. Oct 16, 2013: Zigzag Persistence and installing Dionysus part I.
MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Oct 16, 2013: Zigzag Persistence
and installing Dionysus part I.
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

3. http://geometrica. saclay. inria

4. Lee-Mumford-Pedersen [LMP] study only high contrast patches.
Collection: 4.5 x 106 high contrast patches from a
collection of images obtained by van Hateren and van der Schaaf
Recall from Sept 20 lecture

7. M(100, 10) U Q
where |Q| = 30
On the Local Behavior of Spaces of Natural Images, Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, International Journal of Computer Vision 2008, pp 1-12.

9. The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian
"Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams,
multidimensional persistence, Gröbner bases, Gunnar Carlsson 

10. Computing Multidimensional Persistence,
Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian

12. S(1, 1) Z2[x1, x2]/x2

17. Witness Complexes

18. Witness Complexes

19. Time varying data
X[t0, t1] = data points existing at time t for t in [t0, t1]
X[t1, t2] X[t2, t3]
X[t0, t2] X[t1, t3] X[t2, t4]

21. Time varying data
X[t0, t1] = data points existing at time t for t in [t0, t1]
X[t1, t2] X[t2, t3]
X[t0, t2] X[t1, t3] X[t2, t4]
VR(X[t1, t2], ε) VR(X[t2, t3], ε)
VR(X[t0, t2], ε) VR(X[t1, t3], ε) VR(X[t2, t4], ε)

22. Time varying data
X[t0, t1] = data points existing at time t for t in [t0, t1]
X[t1, t2] X[t2, t3]
X[t0, t2] X[t1, t3] X[t2, t4]
VR(X[t1, t2], ε) VR(X[t2, t3], ε)
VR(X[t0, t2], ε) VR(X[t1, t3], ε) VR(X[t2, t4], ε)
C0  C1  C2  C3  C4

23. C C3
C C C4
H H3
H H H4

24. Hki, p = Zki /(Bki+p Zki) = L(i, i+p)( Hki)
Persistent Homology:
C0  C1  C2  C3  C4
H0  H1  H2  H3  H4
Hki, p = Zki /(Bki+p Zki) = L(i, i+p)( Hki) U
Zigzag Homology:
C0  C1  C2  C3  C4
H0  H1  H2  H3  H4

25. H0  H1  H2  H3  H4

26. H0  H1  H2  H3  H4
Z2  Z2    0

27. H0  H1  H2  H3  H4
Z2  Z2    0
0  Z2  Z2  Z2  0

28. H0  H1  H2  H3  H4
Z2  Z2    0
0  Z2  Z2  Z2  0
Z2  Z2 × Z2  Z2  Z2  0 x

29. Gabriel (1972) For Dynkin-Coxeter graphs:
H0  H1  H2  H3  H4
F  F    0
0  F  F  F  0
F  F × F  F  F  0 x

30. F  F    0
0  F  F  F  0
F  F × F  F  F  0 x
I(0, 2) I(1, 4)

35. idarcy$ hg clone http://hg.mrzv.org/Dionysus/
destination directory: Dionysus
requesting all changes
adding changesets
adding manifests
adding file changes
added 278 changesets with 1396 changes to 388 files
updating to branch default
82 files updated, 0 files merged, 0 files removed, 0 files unresolved
idarcy$ cd Dionysus
:Dionysus idarcy$ hg up tip
277 files updated, 0 files merged, 18 files removed, 0 files unresolved

36. idarcy$ man hg type q to quit HG(1) Mercurial Manual HG(1) NAME
hg - Mercurial source code management system
SYNOPSIS
hg command [option]... [argument]...
DESCRIPTION
The hg command provides a command line interface
to the Mercurial system.
COMMAND ELEMENTS
files...
indicates one or more filename or relative
type q to quit

38. Need C-compiler first
For example:

40.
go to app store for free app

41. 2. From within xcode preferences
(or download directly from web per next slide)

42. https://developer.apple.com/downloads/
Log in with Free apple ID

46. Clicking on downloads should install
Depending on your security settings, you may need to control click to install

47.
See Oct 18 lecture for easier way to download and install Boost

48. http://www. mrzv. org/software/dionysus/python/zigzag-persistence
See Oct 18
& 21 lecture.