Quantifying the shape of plants

.title[
# Quantifying the shape of plants
]
.subtitle[
## using Topological Data Analysis
]
.author[
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Elizabeth Munch, Dan Chitwood <br> Dan Koenig, Jacob Landis <br> -
]
.institute[
### Computational Mathematics, Science and Engineering <br> Michigan State University <br> -
]
.date[
### 2021-02-14<br> - <br> Slides available: <code>bit.ly/shape_gac2021</code>
]

---

# Plant morphology

---

# Topological Data Analysis

<div class="row">
  <div class="column" style="max-width:25%; font-size: 15px;">
    <img style="padding: 25px 0 35px 0;" src="../figs/S019_L0_1.gif">
    <p style="font-size: 25px; 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: 15px;">
    <img src="../figs/ecc_X.gif">
    <p style="font-size: 23px; 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: 15px;">
    <img src="../figs/svm_mds_ect.gif">
    <p style="font-size: 25px; text-align: center; color: DarkRed;"> Analysis </p>
    <ul>
      <li> Statistics </li>
      <li> Machine learning </li>
      <li> Classification/prediction </li>
    <ul>
  </div>
</div>

---

<div class="row">
  <div class="column" style="max-width:44%">
    <a href="https://kizilvest.ru/20150827-v-kizilskom-rajone-nachalas-uborochnaya-strada/" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_kizilskoye.jpg"></a>
    <a href="https://ipad.fas.usda.gov/highlights/2008/11/eth_25nov2008/" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_ethiopia.gif"></a>
    <a href="https://www.doi.org/10.1007/978-1-4419-0465-2_2168" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_historical_expansion.jpg"></a>
  </div>
  <div class="column" style="max-width:44%">
    <a href="https://www.resilience.org/stories/2020-03-09/the-last-crop-before-the-desert/" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_morocco.jpg"></a>
    <a href="https://www.tibettravel.org/tibetan-culture/highland-barley.html" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_seed_tibet.jpg"></a>
    <a href="https://www.nationalgeographic.co.uk/travel/2020/05/photo-story-from-barley-fields-to-whisky-barrels-in-rural-scotland" target="_blank"><img style="padding: 0 0 0 0;" src="../figs/barley_seed_scotland_cropped.jpg"></a>
  </div>
  <div class="column" style="max-width:8%; font-size: 15px;">
    <p style="text-align: center; font-size: 30px; line-height: 1em;"> <strong> Barley across the world </strong></p>
    <p>Kiliskoye (Chelyabinsk, Russia)</p>
    <p>Marchouch (Rabat, Morocco)</p>
    <p>Aksum (Tigray, Ethiopia)</p>
    <p>Salar (Tsetang, Tibet)</p>
    <p>Expansion of the barley. </p>
    <p>Turriff (Aberdeenshire, Scotland)</p>
    <p style="font-size:9px;line-height: 1em;">Click on any picture for more details and credits</p>
  </div>
</div>

---

background-image: url("../figs/composite_cross_v_05.svg")
background-size: 450px
background-position: 95% 90%

# Cross Composite II experiment

.pull-left[
- Cross all possilbe **28 parent** combinations (F1s)
- Self-fertilize the resulting 379 hybrids (F2s)
- Plant the progeny of hybrids in different plots and let nature do the rest
- Mostly self-fertilization was carried out throughout **58 generations**
- We have records of all plots throughout generations
    - Measure **morphology**
    - **DNA** sequencing
    - Which genes are selected for?
    - How did morphology change?
]

---

background-image: url("../figs/barley_lab_composition.jpg")
background-size: 750px
background-position: 99% 99%

# Raw data: X-ray CT scans

`$\sim30\mu m$` resolution

Batch scans - 4 per tray

2Gb+ per raw scan
]

---

## Image processing &rarr; Traditional descriptors

.pull-left[
- Length
- Width
- Height
- Surface area
- Volume
![](../figs/seed_orientation1.png)
![](../figs/seed_orientation3.png)
]

.pull-right[
![](../figs/boxplot_all_vol_h.png)
![](../figs/boxplot_all_length_h.png)
![](../figs/boxplot_all_area_h.png)
]

---

# Topology: The Euler characteristic `$\chi$`

`$$\chi = \#(\text{Vertices}) - \#(\text{Edges}) + \#(\text{Faces}).$$`
<img src="../../tda/figs/euler_characteristic_2.png" width="400" style="display: block; margin: auto;" />

--
- Summarize **topological features** with the Euler-Poincaré formula

`$$\chi = \#(\text{Connected Components}) - \#(\text{Loops}) + \#(\text{Voids}).$$`

--
- The Euler characteristic is a **topological invariant**.

---

background-image: url("../figs/ect.gif")
background-size: 700px
background-position: 50% 90%

# Euler Characteristic Transform (ECT)

- Slice the seed along a chosen direction.

- Record how `$\chi$` changes as we add slices.

- Repeat and concatenate for all possible directions.

---

# Why choose the ECT?

- Easy to compute: a quick alternating sum.

[**Theorem _(Turner, Mukherjee, Boyer 2014)_**](https://doi.org/10.1093/imaiai/iau011):

- Given all the (infinite) ECCs corresponding to all possible directions, *different* simplicial complexes correspond to *different* ECTs.

- The ECT effectively summarizes all possible information related to shape.

- 8 thresholds per direction

- Every seed is associated a `$158\times8=1264$`-dimensional vector

- Reduce dimensions with nonlinear PCA
]

.pull-right[
<img src="../figs/pole_directions_p7_m12_crop.jpg" width="300" style="display: block; margin: auto;" />

]

---

# Classification of 28 lines with SVM

---
class: right, bottom, inverse

background-image: url("../figs/acknowledgments.jpg")
background-size: 1000px
background-position: 50% 40%

Grab these slides at [`bit.ly/shape_gac2021`](http://bit.ly/shape_gac2021)