Seminar on Topological Data Analysis for the UC Davis Plant Science Symposium, April 16, 2018. #UCDPSS18

1. Topological Data Analysis
What is it?
What is it good for?
How can it be used to study
plant morphology?
Dan Chitwood
UC Davis Plant Science Symposium
April 16, 2018
Mitchell Eithun, Liz Munch
Jacob Landis, Dan Koenig

2. Liz Munch
Computational Mathematics,
Science & Engineering
Michigan State University
Topologists
Mao Li
Donald Danforth
Plant Science Center
St. Louis, Missouri USA

3. Why topology?
A way to measure global features quantitatively
from complicated geometric structures

4. A way to measure global features quantitatively
from complicated geometric structures
There are ways to do this statistically,
without topology . . .
Why topology?

5. Measuring shape without topology
Landmarks
Chitwood et al. 2016 Plant Physiol
Climate and developmental plasticity:
Interannual variability in grapevine leaf morphology

6. Chitwood et al. 2016 Plant Physiol
Climate and developmental plasticity:
Interannual variability in grapevine leaf morphology
Measuring shape without topology
Landmarks

7. Chitwood 2014 PLOS One
Imitation, genetic lineages, and time influenced the
morphological evolution of the violin
https://en.wikipedia.org/wiki/Fourier_transform#/media/
File:Fourier_transform_time_and_frequency_domains_(small).gif
Wikipedia
Measuring shape without topology
Elliptical Fourier Descriptors

8. New York Times, International Art
Stephen Heyman:
How Stradivari came to dictate violin design
Measuring shape without topology
Elliptical Fourier Descriptors

9. Topological Data Analysis
What is it?
What is it good for?
How can it be used to study
plant morphology?

10. Betti #
Blobs Holes Voids
Up to
N dimensions
What are topological features?
Blobs, holes, and voids
“Properties of space preserved
under continuous
deformations, such as
stretching, crumpling and
bending, but not tearing or
gluing” –Topology, wikipedia
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

11. https://en.wikipedia.org/wiki/Homotopy#/media/File:Mug_and_Torus_morph.gif
Wikipedia
What are topological features?
Blobs, holes, and voids
“Properties of space preserved
under continuous
deformations, such as
stretching, crumpling and
bending, but not tearing or
gluing” –Topology, wikipedia

12. What is a simplicial complex?
A collection of simplices
0-simplex = 1 vertex
1-simplex = 2 vertices, an edge
2-simplex = 3 vertices, a triangle
3-simplex = 4 vertices, a tetrahedron
n-simplex = n + 1 vertices
Simplicial complex = a network!!!
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

13. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data (a metric set)
But pick a value t so if distance between two
vertices <=t, then an edge
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

14. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data (a metric set)
But pick a value t so if distance between two
vertices <=t, then an edge
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

15. Persistent homology
A continuum of values to create a simplicial complex
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

16. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data (a metric set)
Huang et al., 2018 arXiv
Demonstration of Topological Data Analysis on a Quantum Processor

17. Huang et al., 2018 arXiv
Demonstration of Topological Data Analysis on a Quantum Processor
Persistent homology
A continuum of values to create a simplicial complex

18. Persistence diagrams
The birth and death of features across a function
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

19. Bottleneck distance
The distance between two persistence diagrams
GUDHI
http://gudhi.gforge.inria.fr/doc/latest/group__bottleneck__distance.html

20. Topological Data Analysis
What is it?
What is it good for?
How can it be used to study
plant morphology?

21. How is topology useful for plants?
Complex plant morphologies
Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353

22. How is topology useful for plants?
Complex plant morphologies

23. How is topology useful for plants?
Complex plant morphologies
Mary Lu Arpaia, Eric Focht
UC Riverside

24. How is topology useful for plants?
Complex plant morphologies
Jacob Landis, Daniel Koenig
UC Riverside

25. How is topology useful for plants?
Complex plant morphologies
Amy Litt
UC Riverside

26. How is topology useful for plants?
Complex plant morphologies
Carolyn Rasmussen
UC Riverside

27. How is topology useful for plants?
Complex plant morphologies
Peter Cousins (Gallo), Keith Duncan

28. Topological Data Analysis
What is it?
What is it good for?
How can it be used to study
plant morphology?

29. Are there applications to
plant morphology?
2D
• Shapes
• Local features: leaf serrations
• First order homology: loops
3D
Branching architectures
• Shoots and roots
Mapper
• Converting morphology
to graphs

30. 16 annuli Density estimator
A tool: Subset and smooth Side view
A persistent
homology
morphometric
method:
Blind to size,
position, and
orientation
2D point cloud
Mao Li

31. plane height
(level value)
connectedcomponent
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function Mao Li

32. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

33. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

34. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

35. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

36. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

37. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

38. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

39. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

40. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

41. plane height
(level value)
connectedcomponent
Mao Li
The function is pixel density
subsetted by a ring
Persistent homology
measures topology, or
connected components,
across the scale of the
function

42. Morphometrics with persistent homology
Compares simple, deeply lobed, and compound leaf shapes

44. Where do the leaves come from?
“Transect” and Leafsnap data
Transect data
Dana Royer, Wesleyan University
Daniel Peppe, Baylor University
Peter Wilf, Penn State
Huff PM, Wilf P, Azumah EJ. 2003. Digital future for
paleoclimate estimation from fossil leaves? Preliminary
results. Palaios 18: 266-274.
Royer DL, Wilf P, Janesko DA, Kowalski EA, Dilcher DL.
2005. Correlations of climate and plant ecology to leaf size
and shape: potential proxies for the fossil record.
American Journal of Botany 92: 1141-1151.
Peppe DJ, Royer DL, Cariglino B, Oliver SY, Newman S,
Leight E, Enikolopov G, Fernandez-Burgos M, Herrera F,
Adams JM, Correa E, Currano ED, Erickson JM, Hinojosa LF,
Iglesias A, Jaramillo CA, Johnson KR, Jordan GJ, Kraft N,
Lovelock EC, Lusk CH, Niinemets U, Penuelas J, Rapson G,
Wing SL, Wright IJ. 2011. Sensitivity of leaf size and shape
to climate: global patterns and paleoclimatic applications.
New Phytologist, 190: 724-739.
Leafsnap: A Computer Vision System for
Automatic Plant Species Identification
Neeraj Kumar, Peter N. Belhumeur, Arijit
Biswas, David W. Jacobs, W. John Kress, Ida
C. Lopez, João V. B. Soares
Proceedings of the 12th European
Conference on Computer Vision (ECCV),
October 2012

45. Analysis
Mao Li, Danforth Center
Isolation
Rebekah Mohn, Miami University
Potato
Shelley Jansky, USDA, Wisconsin-
Madison
Diego Fajardo, National Center for
Genome Resources
Pepper
Allen van Deynze, UC Davis
Theresa Hill, UC Davis
Tomato
Viktoriya Coneva, Danforth Center
Margaret Frank, Danforth Center
Chris Topp, Danforth Center
Arabidopsis
Ruthie Angelovici, University of Missouri,
Columbia
Batushansky Albert, University of Missouri,
Columbia
Clement Bagaza, University of Missouri,
Columbia
Edmond Riffer, University of Missouri,
Columbia
Braden Zink, University of Missouri,
Columbia
Brassica
J. Chris Pires, University of Missouri,
Columbia
Hong An, University of Missouri, Columbia
Sarah Gebken, University of Missouri,
Columbia
Cotton
Vasu Kuraparthy, North Carolina State
University
Grape
Allison Miller, Saint Louis University
Jason Londo, USDA/ARS, Geneva, NY
Laura Klein, Saint Louis University
Passiflora
Wagner Otoni, Universidade Federal de Vicosa
Viburnum
Erika Edwards, Brown University
Elizabeth Spriggs, Yale University
Michael Donoghue, Yale University
Sam Schmerler, American Museum of Natural
History
Grasses
Lynn Clark, Iowa State
Timothy Gallaher, Iowa State
Phillip Klahs, Iowa State
Where do the leaves come from?
Specific plant taxa

46. Morphometrics with persistent homology
A morphospace for leaf shape

47. Mao Li, Margaret Frank, Viktoriya Coneva,
Washington Mio, Chris Topp, Dan Chitwood
Persistent homology: a tool to universall measure
plant morphologies across organs and scales
bioRxiv, 2018
How is topology useful for plants?
Local features: serrations

48. Mao Li, Margaret Frank, Viktoriya Coneva,
Washington Mio, Chris Topp, Dan Chitwood
Persistent homology: a tool to universall measure
plant morphologies across organs and scales
bioRxiv, 2018
How is topology useful for plants?
First order homology: loops

49. How is topology useful for plants?
Genetics and persistent homology

50. Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353
How is topology useful for plants?
Branching architectures

51. Mao Li

52. Mao Li

53. Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353
How is topology useful for plants?
Branching architectures

54. Bottleneck distances
Overall differences in morphology
Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353 Mao Li

55. Bottleneck distances
Overall differences in morphology
Mao Li

56. Mapper
Converting structure to a graph
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017

57. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

58. Jacob Landis, Daniel Koenig
UC Riverside
Mapper
Converting structure to a graph

59. Mitchell Eithun, Liz Munch
Michigan State University
Mapper
Converting structure to a graph

60. Mitchell Eithun, Liz Munch
Michigan State University
Mapper
Converting structure to a graph

61. Reeb graphs
The limit of cover intervals . . .
Wikipedia
https://en.wikipedia.org/wiki/Reeb_graph#/media/File:3D-Leveltorus-Reebgraph.png

62. Scan all the things!!!
MSU Museum and the Broad

63. Scan all the things!!!
MSU Museum and the Broad

64. Scan all the things!!!
MSU Museum and the Broad