Presentation to be given at the Japanese Society of Developmental Biologists, on May 17th 2019 (Osaka, Japan).

1. Topological Data Analysis:
What is it?
What is it good for?
How can it be used to study
developmental biology?
Dan Chitwood
Dept. Horticulture
Dept. Computational Mathematics,
Science & Engineering
Michigan State University
@EndlessForms

2. Shape is information
• How do we measure shape
comprehensively?
• How do we measure non-traditional
shapes? Like branching in plants?
• How do we measure across scales
and emergent properties?
• When we achieve the above, how
do we analyze? New statistics?
… and information has shape
An unlikely answer:
Topology!

3. Main goal of TDA:
Provide
quantifiable,
comparable,
consise
summaries of the shape of data

4. Introduction to Topological Data Analysis
Grapes: modeling functional data
using topological signatures
Barley: example of applying topology and
geodesic distance
Citrus: example of applying topology and
geodesic distance
Overview

5. wikipedia/topology, LucasVB
Topological features exist in every day objects
Topological features include:
• Blobs (connected components)
• Loops
• Voids
• Higher dimensional features

6. A User’s Guide to Topological Data Analysis
Journal of Learning Analytics (2017)
Elizabeth Munch
How do we use topology to measure shape?
Topological Data Analysis (TDA)
• Collect data (points)
• Pick filter function (radius)
• Use function to assign a real number
value to each data point
• Apply function across level sets…all
values
• Monitor when topological features
arise, disappear

7. Slides made by:
Matthew Wright
St. Olaf College
Topology exists in everyday objects
… but data is noisy !
How do we use topology to measure shape?

8. Slides made by:
Matthew Wright
St. Olaf College
Topology exists in everyday objects
… but data is noisy !
How do we use topology to measure shape?

9. Slides made by:
Matthew Wright
St. Olaf College
Topology exists in everyday objects
… but data is noisy !
How do we use topology to measure shape?

10. Slides made by:
Matthew Wright
St. Olaf College

11. If 𝑑 is too small…
…then we detect noise.
Slides made by:
Matthew Wright
St. Olaf College

12. Slides made by:
Matthew Wright
St. Olaf College

13. If 𝑑 is too large…
…then we get a giant simplex (trivial homology).
Slides made by:
Matthew Wright
St. Olaf College

14. 𝑑
Problem: How do we choose distance 𝑑?
This 𝑑
looks good.
Idea: Consider all distances 𝑑.
How can we
say this hole is
a feature,
rather than
noise?
Slides made by:
Matthew Wright
St. Olaf College

15. 𝑑: 0 1 2 3
Example:
Record the barcode:
4

16. Short bars
represent
noise.
Long bars
represent
features.
𝑑: 0 1 2 3 4
Example:
Record the barcode:

17. Introduction to Topological Data Analysis
Grapes: modeling functional data
using topological signatures
Barley: example of applying topology and
geodesic distance
Citrus: example of applying topology and
geodesic distance
Overview

18. Mao Li
Topological Data Analysis (TDA)
• Collect data (voxels)
• Pick filter function (geodesic
distance to bottom)
• Use function to assign a real
number value to each data
point
• Apply function across level
sets…all values
• Monitor when blobs arise,
disappear
• Create barcode

19. Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353
Bottleneck distance
• Compare overall similarity
of any two barcodes to
each other
• Create a pairwise distance
matrix
• Do statistics

20. Mao Li, Keith Duncan, Chris Topp, Dan Chitwood
Persistent homology and the branching topologies of plants
Am J Bot, 104(3):349-353
Bottleneck distance
• Compare overall similarity
of any two barcodes to
each other
• Create a pairwise distance
matrix
• Do statistics

21. Characterizing grapevine (Vitis spp.) inflorescence architecture
using X-ray imaging: implications for understanding cluster
density. bioRxiv (2019) Mao Li
Model traits as function of topology
• Interpret topology using
traditional measures

22. Characterizing grapevine (Vitis spp.) inflorescence architecture
using X-ray imaging: implications for understanding cluster
density. bioRxiv (2019) Mao Li
Model traits as function of topology
• Interpret topology using
traditional measures
• Model functional traits using
comprehensive topological
features
• Correlation, prediction,
classification

23. Characterizing grapevine (Vitis spp.) inflorescence architecture
using X-ray imaging: implications for understanding cluster
density. bioRxiv (2019) Mao Li
Model traits as function of topology

24. Characterizing grapevine (Vitis spp.) inflorescence architecture
using X-ray imaging: implications for understanding cluster
density. bioRxiv (2019) Mao Li
Model traits as function of topology

25. Introduction to Topological Data Analysis
Grapes: modeling functional data
using topological signatures
Barley: example of applying topology and
geodesic distance
Citrus: example of applying topology and
geodesic distance
Overview

26. Diversification of floral morphology in barley
Hordeum
spontaneum
Hordeum
vulgare
Wild Domesticated
Jacob Landis
Dan Koenig
UC Riverside

27. The Composite Cross II - Parental Diversity
Jacob Landis
Dan Koenig
UC Riverside

28. The Composite Cross II - Half Diallele Design
Jacob Landis
Dan Koenig
UC Riverside

29. ● 4 spikes per reconstruction
● Seeds higher X-ray absorption
● Awns, rachis, and floral organs
lower absorption
X-ray CT: Dr. Michelle Quigley
Barley: X-ray CT reconstruction

30. Barley: Weighted geodesic distance (VIDEO)
● Geodesic distance to the
base
● High densities weighted to
provide less “resistance”
● Highlights the branches of
the spike, through the
seeds.
Geodesic distance:
Dr. Tim Ophelders
Mitchell Eithun

31. Barley: Weighted geodesic distance
● Geodesic distance to the
base
● High densities weighted to
provide less “resistance”
● Highlights the branches of
the spike, through the
seeds.
Geodesic distance:
Dr. Tim Ophelders
Mitchell Eithun

32. Barley: Weighted geodesic distance
● Geodesic distance to the
base
● High densities weighted to
provide less “resistance”
● Highlights the branches of
the spike, through the
seeds.
Geodesic distance:
Dr. Tim Ophelders
Mitchell Eithun
How many paths through each voxel?

33. Introduction to Topological Data Analysis
Grapes: modeling functional data
using topological signatures
Barley: example of applying topology and
geodesic distance
Citrus: example of applying topology and
geodesic distance
Overview

34. Citrus: complex hybridization and domestication
Genomics of the origin and evolution of Citrus. Nature 554, 311-316 (2018)

35. Citrus: slicing the segments
Citrus work: Danelle Seymour (UC Riverside), Mitchell Eithun (MSU)

36. Citrus: Isolating anatomical features
● Exocarp (skin)
● Mesocarp (rind)
● Endocarp (pulp)
● Measure volumes
● Allometry?
Citrus work: Danelle Seymour (UC Riverside), Mitchell Eithun (MSU)

37. Citrus: Isolating anatomical features
● Oil glands
● Vasculature
Citrus work: Danelle Seymour (UC Riverside), Mitchell Eithun (MSU)

38. Citrus: Weighted geodesic distance
● Geodesic distance to the
base
● High densities weighted to
provide less “resistance”
● Highlights the branches of
the citrus, through the
fruit.
Citrus work: Danelle Seymour (UC Riverside), Mitchell Eithun (MSU)
Geodesic distance: Tim Ophelders (MSU)

39. Citrus: Weighted geodesic distance
● Geodesic distance to the
base
● High densities weighted to
provide less “resistance”
● Highlights the branches of
the citrus, through the
fruit.
Kleinhans et al. Computing Representative
Networks for Braided Rivers

40. Where to from here?
• We have a method to measure shape comprehensively
• A major focus is interpreting so much information
• Traditional measurements?
• Predicition, classification?
• The inverse problem
• A statistical genetic and phylogenetic framework
• Dealing with time series and development
• Molecular biology: a focus on networks
• Unifying analysis across emergent levels?

41. Thank you!
@EndlessForms
Michigan State University
Department of Horticulture
Department of Computational
Mathematics, Science & Engineering
Department of Mathematics
Erik AmézquitaDr. Michelle QuigleyDr. Liz Munch
Dr. Tim Ophelders Mitchell Eithun

42. Thank you!
@EndlessForms
Michigan State University
Department of Horticulture
Department of Computational
Mathematics, Science & Engineering
Department of Mathematics