Formation of low mass protostars and their circumstellar disks
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Topological Data Analysis (TDA) for volumetric X-ray CT data
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
โข How do we measure shape
comprehensively?
โข How do we measure non-traditional
shapes? Like branching in plants?
โข How do we measure across scales
and emergent properties?
โข When we achieve the above, how
do we analyze? New statistics?
โฆ 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
6. A Userโs Guide to Topological Data Analysis
Journal of Learning Analytics (2017)
Elizabeth Munch
How do we use topology to measure shape?
Topological Data Analysis (TDA)
โข Collect data (points)
โข Pick filter function (radius)
โข Use function to assign a real number
value to each data point
โข Apply function across level setsโฆall
values
โข Monitor when topological features
arise, disappear
7. Slides made by:
Matthew Wright
St. Olaf College
Topology exists in everyday objects
โฆ 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
โฆ 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
โฆ but data is noisy !
How do we use topology to measure shape?
13. If ๐ is too largeโฆ
โฆthen we get a giant simplex (trivial homology).
Slides made by:
Matthew Wright
St. Olaf College
14. ๐
Problem: How do we choose distance ๐?
This ๐
looks good.
Idea: Consider all distances ๐.
How can we
say this hole is
a feature,
rather than
noise?
Slides made by:
Matthew Wright
St. Olaf College
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)
โข Collect data (voxels)
โข Pick filter function (geodesic
distance to bottom)
โข Use function to assign a real
number value to each data
point
โข Apply function across level
setsโฆall values
โข Monitor when blobs arise,
disappear
โข 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
โข Compare overall similarity
of any two barcodes to
each other
โข Create a pairwise distance
matrix
โข 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
โข Compare overall similarity
of any two barcodes to
each other
โข Create a pairwise distance
matrix
โข 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
โข 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
โข Interpret topology using
traditional measures
โข Model functional traits using
comprehensive topological
features
โข 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. โ 4 spikes per reconstruction
โ Seeds higher X-ray absorption
โ Awns, rachis, and floral organs
lower absorption
X-ray CT: Dr. Michelle Quigley
Barley: X-ray CT reconstruction
30. Barley: Weighted geodesic distance (VIDEO)
โ Geodesic distance to the
base
โ High densities weighted to
provide less โresistanceโ
โ Highlights the branches of
the spike, through the
seeds.
Geodesic distance:
Dr. Tim Ophelders
Mitchell Eithun
31. Barley: Weighted geodesic distance
โ Geodesic distance to the
base
โ High densities weighted to
provide less โresistanceโ
โ Highlights the branches of
the spike, through the
seeds.
Geodesic distance:
Dr. Tim Ophelders
Mitchell Eithun
32. Barley: Weighted geodesic distance
โ Geodesic distance to the
base
โ High densities weighted to
provide less โresistanceโ
โ 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)
38. Citrus: Weighted geodesic distance
โ Geodesic distance to the
base
โ High densities weighted to
provide less โresistanceโ
โ 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
โ Geodesic distance to the
base
โ High densities weighted to
provide less โresistanceโ
โ Highlights the branches of
the citrus, through the
fruit.
Kleinhans et al. Computing Representative
Networks for Braided Rivers
40. Where to from here?
โข We have a method to measure shape comprehensively
โข A major focus is interpreting so much information
โข Traditional measurements?
โข Predicition, classification?
โข The inverse problem
โข A statistical genetic and phylogenetic framework
โข Dealing with time series and development
โข Molecular biology: a focus on networks
โข 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 AmรฉzquitaDr. 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