The topological shape of plants

.title[
# The topological shape of plants
]
.subtitle[
## A <strong>handwavy</strong> introduction to Topological Data Analysis
]
.author[
### <strong>Erik Amézquita</strong> <br> — <br> Computational Mathematics, Science, and Engineering <br> Michigan State University <br> —
]
.date[
### 2023-03-19
]

---

background-image: url("../../img/endlessforms.png")
background-size: 150px
background-position: 89% 7%

# Plant morphology

<div class="row">
  <div class="column" style="max-width:50%">
    <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/oM9kAq0PBvw?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
    <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/V39K58evWlU?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
  </div>
  <div class="column" style="max-width:50%">
    <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/efF5PSvFQ2A?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
    <iframe width="375" height="210" src="https://www.youtube-nocookie.com/embed/qkOjHHuoUhA?controls=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
  </div>
</div>
<p style="font-size: 24px; text-align: right; font-family: 'Yanone Kaffeesatz'">Check out more 3D X-ray CT scans at <a href="https://www.youtube.com/@endlessforms6756">youtube.com/@endlessforms6756</a></p>

---

# Topological Data Analysis (TDA)

<div class="row" style="font-family: 'Yanone Kaffeesatz'">
  <div class="column" style="max-width:25%; font-size: 20px;">
    <img style="padding: 25px 0 35px 0;" src="../figs/S019_L0_1.gif">
    <p style="font-size: 30px; text-align: center; color: DarkRed;"> Raw Data </p>
    <ul>
      <li> X-ray CT </li>
      <li> Point clouds </li>
      <li> Time series </li>
    <ul>
  </div>
  <div class="column" style="max-width:40%; padding: 0 25px 0 25px; font-size: 20px;">
    <img src="../figs/ecc_X.gif">
    <p style="font-size: 28px; text-align: center; color: DarkRed;"> Topological Summary </p>
    <ul>
      <li> Euler Characteristic </li>
      <li> Persistence diagrams </li>
      <li> Mapper/Reeb graphs </li>
    <ul>
  </div>
  <div class="column" style="max-width:35%; font-size: 20px;">
    <img src="../figs/svm_mds_ect.gif">
    <p style="font-size: 30px; text-align: center; color: DarkRed;"> Analysis </p>
    <ul>
      <li> Statistics </li>
      <li> Machine learning </li>
      <li> Classification/prediction </li>
    <ul>
  </div>
</div>

---

background-image: url("../../img/cmse_logo.svg")
background-size: 150px
background-position: 1% 50%

# My research: Crossing and merging bridges

<div class="row">
  <div class="column" style="width:15%">
  <p style="font-size:5px">&mdash;</p>
  </div>
  <div class="column" style="max-width:28%">
    <a href="https://plantsandpython.github.io/PlantsAndPython/00_Opening_page.html" target="_blank"><img style="padding: 0 0 0 0;" src="../../tutorials/figs/plants_and_python_group_photo.jpg"></a>
  </div>
  <div class="column" style="max-width:50%">
    <a href="https://doi.org/10.1073/pnas.2217564120" target="_blank"><img style="padding: 0 0 0 0;" src="../../journal_diversity/figs/frequencies_globe_inv.jpg"></a>
  </div>
</div>

<div class="list" style="font-size: 10px; text-align: left;">
    <ul>
      <li>Marks <em>et al.</em> (2023) A critical analysis of plant science literature reveals ongoing inequities. DOI: <a href="https://doi.org/10.1073/pnas.2217564120" target="_blank">10.1101/2022.10.15.512190</a></li>
      <li><strong>A</strong> <em>et al.</em> (2022) The shape of aroma: measuring and modeling citrus oil gland distribution. DOI: <a href="https://doi.org/10.1002/ppp3.10333" target="_blank">10.1002/ppp3.10333</a></li>
      <li><strong>A</strong> <em>et al.</em> (2022) Genomics data analysis via spectral shape and topology. DOI: <a href="https://doi.org/10.48550/arXiv.2211.00938" target="_blank">10.48550/arXiv.2211.00938</a></li>
      <li>VanBuren<em>et al.</em> (2022) Teaching tools in plant biology. Plants and Python, coding from scratch in the plant sciences. DOI: <a href="https://doi.org/10.1093/plcell/koac187" target="_blank">10.1093/plcell/koac187</a>.</li>
      <li><strong>A</strong> <em>et al.</em> (2021) Measuring hidden phenotype: Quantifying the shape of barley seeds using the Euler Characteristic Transform. DOI: <a href="https://doi.org/10.1093/insilicoplants/diab033" target="_blank">10.1093/insilicoplants/diab033</a></li>
      <li><strong>A</strong> <em>et al.</em> (2020) The shape of things to come: Topological data analysis and biology, from molecules to organisms. DOI: <a href="https://doi.org/10.1002/dvdy.175" target="_blank">10.1002/dvdy.175</a></li>
    </ul>
</div>

---

# The shape of things to come

<div class="row" style="font-family: 'Yanone Kaffeesatz'; font-size:22px;">
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">The shape of data</p>
    <img style="padding: 0 22px;" src="https://raw.githubusercontent.com/sperciva/Backups/main/Tutorial/animation.gif"></img>
    <img style="padding: 0 10px;" src="https://raw.githubusercontent.com/sperciva/Backups/main/Tutorial/animation_diagram.gif"></img>
    <p style="text-align: center;">Ultra quick introduction</p>
    <p style="text-align: center;">TDA + Biology in literature</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;font-size:28px;text-align: center;">The shape of development</p>
    <img style="padding: 0 60px;" src="../../citrus/figs/SR01_L01_black_exocarp.gif"></img>
    <img style="padding: 0 26px;" src="../../citrus/figs/SW03_CRC1241-B_12B-4-3_L00_lambproj_v.jpg"></img>
    <p style="text-align: center;">Oil gland distributions</p>
    <p style="text-align: center;">Persistent Homology</p>
  </div>
  <div class="column" style="max-width:33%">
    <p style="line-height:0;text-align: center; font-size:28px">The shape of adaptability</p>
    <img style="padding: 0 70px;" src="../figs/S017_L3_1.gif"></img>
    <img style="padding: 0 55px;" src="../figs/ecc_X.gif"></img>
    <p style="text-align: center;">Seed phenotyping</p>
    <p style="text-align: center;">Euler Characteristic Transform</p>
  </div>
</div>

---

# 1. Shape is data and data is shape

## TDA meets biology in a number of ways

### The significance of detecting connected components and holes

---

# 1st TDA Ingredient: Complexes

- Think the data as a collection of elementary building blocks ( _cells_ )

Vertices | Edges | Faces | Cubes
---------|-------|-------|-------
  0-dim  | 1-dim | 2-dim | 3-dim

- A collection of cells is a _cubical complex_

- Count the number of topological features ( _holes_ ):

Connected components | Loops | Voids
---------------------|-------|-------
       0-dim         | 1-dim | 2-dim

- Example with 2 connected components, 1 loop, 0 voids

---

# 2nd TDA Ingredient: Filters

- Each cell is assigned a real value which defines how the complex is constructed.

- Observe how the number of topological features change as the complex grows.

.pull-left[
<img src="../figs/eigcurv_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../figs/gaussian_density_filter.gif" width="250px" style="display: block; margin: auto;" />
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.pull-right[
<img src="../figs/eccentricity_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../figs/vrips_ver2.gif" width="250px" style="display: block; margin: auto;" />
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---

# TDA + Biology in the literature

<div class="row">
  <div class="column" style="max-width:25%">
    <img style="padding: 0 0 0 0;" src="https://ars.els-cdn.com/content/image/1-s2.0-S1361841518302688-gr1.jpg">
    <img style="padding: 0 0 0 0;" src="https://ars.els-cdn.com/content/image/1-s2.0-S1361841518302688-gr2.jpg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1016/j.media.2019.03.014">Qaiser <em>et al.</em> (2019)</a></p>
  </div>
  <div class="column" style="max-width:25%">
    <img style="padding: 0 0 0 0;" src="https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/ce30e9df-d595-4520-a68a-74f3c6f151d1/assets/graphic/pnas.1313480110fig01.jpeg">
    <img style="padding: 0 0 0 0;" src="https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/b733c3a3-00f2-43d5-b372-9da25f3d33c9/assets/graphic/pnas.1313480110fig02.jpeg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1073/pnas.1313480110">Chan <em>et al.</em> (2013)</a></p>
  </div>
  <div class="column" style="max-width:22%">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_001.jpg">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_012.jpg">
    <img style="padding: 0 0 0 0;" src="https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_010.jpg">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1515/sagmb-2015-0057">Kovacev-Nikolic <em>et al.</em> (2016)</a></p>
  </div>
  <div class="column" style="max-width:28%">
    <img style="padding: 0 0 0 0;" src="https://media.springernature.com/original/springer-static/image/chp%3A10.1007%2F978-3-030-20867-7_7/MediaObjects/484957_1_En_7_Fig1_HTML.png">
    <p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1007/978-3-030-20867-7_7">Chitwood <em>et al.</em> (2019)</a></p>
  </div>
</div>

<div class="row" style="font-family: 'Yanone Kaffeesatz'; font-size:25px;">
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Cancerous tissue</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Horizontal Reassortment</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Holes</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Open/closed conformations</p>
  </div>
  <div class="column" style="width:25%;">
    <p style="text-align: center;">Components</p>
    <p style="text-align: center;">&darr;</p>
    <p style="text-align: center;">Panicle structure</p>
  </div>
</div>

---

## Persistent Homology: the Vietoris-Rips Complex

The Vietoris-Rips complex, `$VR(X, r)$`, is a simplicial complex where:

- If `$n$` points have pairwise distances `$\leq r$` , then the `$n − 1$` simplex formed by those points is added.
]

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![](https://raw.githubusercontent.com/sperciva/Backups/main/Tutorial/PH.gif)
<p style="font-size: 10px; text-align: right; color: Grey; line-height:0; margins:0;padding:0;"> Credits: Sarah Percival</p>
]

---

## Persistent Homology: Persistence Diagrams

![](https://raw.githubusercontent.com/sperciva/Backups/main/Tutorial/circle_example.gif)
<p style="font-size: 10px; text-align: right; color: Grey; line-height:0; margins:0;padding:0;"> Credits: Sarah Percival</p>

- Keep track of the birth and death of topological features in a death-vs-birth time diagram

- Handwavily: it is the multiset `$\text{dgm} = \bigcup_{d\geq0}\{(\text{birth},\,\text{death})_d\}\cup\text{diagonal}.$`

---

## Compare PDs via the Bottleneck distance

![](../../tda/figs/bottleneck_diagram.svg)

Given two persistence diagrams `$\text{dgm}_1$` and `$\text{dgm}_2$`, consider `$\Gamma$` the set of all bijections between these two multisets and

`$$d_B(\text{dgm}(X),\text{dgm}(Y)) = \inf_{\gamma\in\Gamma}\sup_{p\in\text{dgm}(X)}\| p - \gamma(p)\|_\infty$$`

---

## The motivation

- TL;DR: If the original point cloud wiggles a tiny bit, then the elements of its Vietoris-Rips-related persistence diagram will wiggle only a tiny bit as well.

<hr>

- **Theorem (Chazal, de Silva, Oudot, 2013)**: The bottleneck distance is stable with respect to the Gromov-Hausdorff distance. For `$X$` and `$Y$` totally bounded metric spaces,
`$$d_B(\text{dgm}(X),\text{dgm}(Y)) \leq 2d_{GH}(X, Y).$$`
Where `$d_{GH}$` is the Gromov-Hausdorff distance.

- The GH distance is a generalization of the Hausdorff distance: If `$X$`, `$Y$` are two subsets of the same metric space, then

`$$d_H(X,Y) = \inf\{\varepsilon > 0 | X\subset Y^\varepsilon, \; Y\subset X^\varepsilon\}$$`

- If `$X$`, `$Y$` are any two spaces, then

`$$d_{GH}(X,Y)  = \inf_{\text{isometric embeddings}\;X\hookrightarrow Z,\:Y\hookrightarrow Z}\{d_H^Z(X,Y)\}$$`

---

# Topology: The Euler characteristic

`$$\begin{align} \chi &= \#(\text{Vertices}) - \#(\text{Edges}) + \#(\text{Faces}) \\ &= \#(\text{Connected Components}) - \#(\text{Loops}) + \#(\text{Voids}). \end{align}$$`

---

background-image: url("../figs/ecc_ver2.gif")
background-size: 750px
background-position: 50% 90%

# Euler Characteristic Curve (ECC)

- Consider a cubical complex `$X\subset\mathbb{R}^d$`
- And a unit-length direction `$\nu\in S^{d-1}$`

- And the subcomplex containing all cubical cells below height `$h$` in the direction `$\nu$`
`$$X(\nu)_h =\{\Delta \in X\::\:\langle x,\nu\rangle\leq h\text{ for all }x\in\Delta\}$$`

- The Euler Characteristic Curve (ECC) of direction `$\nu$` is defined as the sequence `$$\{\chi(X(\nu)_h)\}_{h\in\mathbb{R}}$$`

---

background-image: url("../figs/ect_ver2.gif")
background-size: 800px
background-position: 50% 88%

## Euler Characteristic Transform (ECT)

- Repeat and concatenate for all possible directions.

$$
`\begin{split}
ECT(X):\; & S^{d-1} \to \mathbb{Z}^{\mathbb{R}}\\
&\nu\mapsto\{\chi(X(\nu)_h)\}_{h\in\mathbb{R}}.
\end{split}`
$$

- [**Theorem** (Turner, Mukherjee, Boyer 2014) & (Curry, Mukherjee, Turner, 2018)](http://arxiv.org/abs/1805.09782):  The ECT is injective with a finite bound of necessary directions.

---

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background-size: 280px
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# The motivation

- Easy to compute: a quick alternating sum.

- *Different* simplicial complexes correspond to *different* ECTs.

- [**Theorem _(ibid)_**](https://arxiv.org/abs/1310.1030): The ECT effectively summarizes all possible information related to shape.

There is elusive math research on computationally efficient reconstruction algorithms:
- [Turner, Mukherjee, Curry (2021)](https://arxiv.org/abs/1805.09782): Finite number of directions
- [Betthauser (2018)](https://people.clas.ufl.edu/peterbubenik/files/Betthauser_Thesis.pdf): 2D reconstruction
- [Fasy, Micka, Millman, Schenfisch, Williams (2022)](https://arxiv.org/abs/1912.12759): 3D reconstruction
- [Crawford, Monod, Chen, Mukherjee, Rabadan (2020)](https://doi.org/10.1080/01621459.2019.1671198): Smooth ECT
- [Jiang, Kurtek, Needham (2020)](https://openaccess.thecvf.com/content_CVPRW_2020/papers/w50/Jiang_The_Weighted_Euler_Curve_Transform_for_Shape_and_Image_Analysis_CVPRW_2020_paper.pdf): Weighted ECT

---

# Shameless plug

- [Review paper](https://doi.org/10.1002/dvdy.175) with more content, references, and examples

- [Jupyter notebook](https://github.com/amezqui3/demeter/blob/main/tutorial/nappn2022_shape_of_things_to_come.ipynb) with even more links and resources: `bit.ly/nappn22`

- This slides: `bit.ly/ustars23`

---

# 2. If life gives you lemons...

## Modeling citrus oil gland distribution

### Comparing limes to oranges with directional statistics

---

## Think of citrus as lego blocks

![](../../citrus/figs/citrus_family_tree_1.jpg)
<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p>

---

## X-rays &rarr; Image Processing &rarr; Oil glands

<div class="row">
  <div class="column" style="max-width:37%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../../citrus/crc_pics/crc_diversity.jpg">
    <p style="text-align: center;">UCR Collaboration</p>
  </div>
  <div class="column" style="max-width:37%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../../citrus/crc_pics/citrus_xrayct_scanning.jpeg">
    <p style="text-align: center;">3D X-Ray CT scan</p>
  </div>
  <div class="column" style="max-width:25%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/lemon_development.jpg">
    <p style="text-align: center;">Fruit development</p>
  </div>
</div>

<div class="row" style="color: black; font-size: 20px; font-family: 'Yanone Kaffeesatz'; margin: 0 auto;">
  <div class="column" style="max-width:20%;">
    <img src="../../citrus/figs/SR01_L01_black_endocarp.gif">
    <p style="text-align: center;"> Endocarp </p>
  </div>
  <div class="column" style="max-width:20%;">
    <img src="../../citrus/figs/SR01_L01_black_rind.gif">
    <p style="text-align: center; padding:5px 0;"> Rind </p>
  </div>
  <div class="column" style="max-width:20%;">
    <img src="../../citrus/figs/SR01_L01_black_exocarp.gif">
    <p style="text-align: center;"> Exocarp </p>
  </div>
  <div class="column" style="max-width:20%;">
    <img src="../../citrus/figs/SR01_L01_black_oil_glands.gif">
    <p style="text-align: center;"> Oil glands</p>
  </div>
  <div class="column" style="max-width:20%;padding: 10px 0 0 20px;">
    <p>
      <li>53 different citrus species</li><br>
      <li>All fundamental lego blocks and several relatives and hybrids of interest</li><br>
      <li>166 individual fruits</li>
    </p>
  </div>
</div>

---

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## Citrus modeling: Sour and sweet orange

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## Distortions: Kumquat and finger lime

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## Persistent Homology for Mandarins