Topological data analysis is a technique that can be used to study plant morphology. It involves using tools from topology and algebraic geometry to analyze shapes and structures. Persistent homology in particular allows researchers to quantify topological features like blobs, holes, and voids that remain consistent under deformations. These techniques have been applied to study plant branching architectures, leaf shapes and serrations, and can provide a way to universally measure plant morphology across scales.
3. What are topological features?
A way to measure global qualitative features
from complicated geometric structures
4. What are topological features?
A way to measure global qualitative features
from complicated geometric structures
There are ways to do this statistically,
without topology . . .
5. What are topological features?
Not spatial positions
Chitwood et al. 2016 Plant Physiol
Climate and developmental plasticity:
Interannual variability in grapevine leaf morphology
6. What are topological features?
Not spatial positions
Chitwood et al. 2016 Plant Physiol
Climate and developmental plasticity:
Interannual variability in grapevine leaf morphology
7. What are topological features?
Not a Fourier transform
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
8. What are topological features?
Not a Fourier transform
New York Times, International Arts, Stephen Heyman
How Stradivari came to dictate violin design
9. 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. 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
12. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data
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
13. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data
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. 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
15. Vietoris-Rips complex (Rips complex)
A simplicial complex of your data
Huang et al., 2018 arXiv
Demonstration of Topological Data Analysis on a Quantum Processor
16. 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
17. 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
18. Bottleneck distance
The distance between two persistence diagrams
GUDHI
http://gudhi.gforge.inria.fr/doc/latest/group__bottleneck__distance.html
19. Mapper
Converting structure to a graph
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017
20. Mapper
Converting structure to a graph
Elizabeth Munch
A User’s Guide to Topological Data Analysis
Journal of Learning Analytics, 2017
22. 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
23. Daniel Schachtman, Keith Duncan,
Ni Jiang, Mao Li
How is topology useful for plants?
Complex plant morphologies
24. How is topology useful for plants?
Complex plant morphologies
Mary Lu Arpaia, Eric Focht
UC Riverside
25. How is topology useful for plants?
Complex plant morphologies
Jacob Landis, Daniel Koenig
UC Riverside
26. How is topology useful for plants?
Complex plant morphologies
Mitchell Eithun
Liz Munch
27. How is topology useful for plants?
Complex plant morphologies
Amy Litt
UC Riverside
28. How is topology useful for plants?
Complex plant morphologies
Carolyn Rasmussen
UC Riverside
29. How is topology useful for plants?
Complex plant morphologies
Peter Cousins (Gallo), Keith Duncan
30. Chopping down the cherry tree
Isolating the inner tree
Jimmy Larson
Mitchell Eithun
Liz Munch
Greg Lang
37. Are there applications to
plant morphology?
2D
• Shapes
• Local features: leaf serrations
• First order homology: loops
Branching architectures
• Shoots and roots
38. 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
39. 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
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. 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
43. 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
44. 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
45. 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
46. 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
47. 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
48. 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
49. 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
52. 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
53. 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
55. 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
56. 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
57. How is topology useful for plants?
Genetics and persistent homology
58. 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
61. 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
62. 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
