Quantifying the shape of barley

# Quantifying the shape of barley
## using the Euler Characteristic Transform
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Elizabeth Munch, Dan Chitwood <br> Dan Koenig, Jacob Landis <br> -
### Computational Mathematics, Science and Engineering <br> Michigan State University <br> -
### 2021-10-22 <br> - <br> To appear in <em>inSilico Plants</em>

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# Plant morphology

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# Topological Data Analysis (TDA)

<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>

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# Cross Composite II experiment

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background-image: url("../figs/composite_cross_v_01.png")
background-size: 425px
background-position: 95% 90%

# Cross Composite II experiment

![](../figs/barley_world.jpg)
- **28 parents/accessions** `$(F_0)$`
]

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background-image: url("../figs/composite_cross_v_02.png")
background-size: 425px
background-position: 95% 90%

# Cross Composite II experiment

![](../figs/barley_world.jpg)

- **28 parents/accessions** `$(F_0)$`

- Do `${28 \choose 2}$` **hybrids** `$(F_1)$`

]

---

background-image: url("../figs/composite_cross_v_03.png")
background-size: 425px
background-position: 95% 90%

# Cross Composite II experiment

![](../figs/barley_world.jpg)

- **28 parents/accessions** `$(F_0)$`

- Do `${28 \choose 2}$` **hybrids** `$(F_1)$`

- **Self-fertilize** the resulting 379 hybrids
]

---

background-image: url("../figs/composite_cross_v_04.png")
background-size: 425px
background-position: 95% 90%

# Cross Composite II experiment

![](../figs/barley_world.jpg)

- **28 parents/accessions** `$(F_0)$`

- Do `${28 \choose 2}$` **hybrids** `$(F_1)$`

- **Self-fertilize** the resulting 379 hybrids

- Each line grows in a different part of an open field
]

---

background-image: url("../figs/composite_cross_v_05.png")
background-size: 425px
background-position: 95% 90%

# Cross Composite II experiment

![](../figs/barley_world.jpg)

- **28 parents/accessions** `$(F_0)$`

- Do `${28 \choose 2}$` **hybrids** `$(F_1)$`

- **Self-fertilize** the resulting 379 hybrids

- Each line grows in a different part of an open field **for 58 generations**
]

---

# Raw Data: X-rays &rarr; Image Processing

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# Raw Data: X-rays &rarr; Image Processing

<div class="row">
  <div class="column" style="max-width:37.5%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../figs/barley_packing.jpg">
    <p style="text-align: center;">Barley shipped from California</p>
  </div>
  <div class="column" style="max-width:50%; color: Navy; font-size: 15px;">
    <img src="../figs/x3000_setup.jpg">
    <p style="text-align: center;">Proprietary X-Ray CT scan reconstruction</p>
  </div>
</div>

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# Raw Data: X-rays &rarr; Image Processing

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# Raw Data: X-rays &rarr; Image Processing

<div class="row">
  <div class="column" style="max-width:38%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../figs/barley_xray_setup.jpg">
    <p style="text-align: center;">Creative setup</p>
  </div>
  <div class="column" style="max-width:17.5%; color: Navy; font-size: 15px;">
    <img src="../figs/S019_L0_1.gif">
    <p style="text-align: center;"> 975 spikes </p>
  </div>
  <div class="column" style="max-width:17%; color: Navy; font-size: 15px;">
    <img src="../figs/S017_L3_1.gif">
    <p style="text-align: center;"> 3 generations </p>
  </div>
</div>

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# Raw Data: X-rays &rarr; Image Processing

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## 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="../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**.

---

# 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/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 `$$\{\chi(X(\nu)_h)\}_{h\in\mathbb{R}}$$`
---

# Euler Characteristic Transform (ECT)

- Repeat and concatenate for all possible directions.

- More formally, the ECT can be thought as a function

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

---

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.

- More formally, the ECT can be thought as a function

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

---

# Why choose the ECT?

- Easy to compute: a quick alternating sum.

[**Theorem _(Turner, Mukherjee, Boyer 2014)_**](https://doi.org/10.1093/imaiai/iau011): The ECT is injective for finite simplicial complexes in 3D.

[**Theorem _(ibid)_**](https://arxiv.org/abs/1310.1030): The ECT is a sufficient statistic for finite simplicial complexes in 3D.

*Translation:*

- 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.

Efficient reconstruction algorithms for 3D shapes ([Betthauser 2018](https://people.clas.ufl.edu/peterbubenik/files/Betthauser_Thesis.pdf); [Curry, Mukherjee, Turner 2018](https://arxiv.org/abs/1805.09782); [Fasy et al. 2019](https://arxiv.org/abs/1912.12759)) remains elusive.

---

background-image: url("../figs/S012_L2_Blue_33.png")
background-size: 150px
background-position: 99% 50%

# Game plan

- **Goal:** Classify 28 barley parental accessions using solely grain morphology information.

- **3121** grains in total

.pull-left[
<img src="../figs/pole_directions_p7_m12_crop.jpg" width="150" style="display: block; margin: auto;" />
]

.pull-right[
- 158 directions
- 16 thresholds per direction
- Every seed is associated a `$158\times16=2528$`-dim vector
- Dimensions reduced with UMAP
]

- Compare **3** sets of morphological descriptors

Descriptor | No. of descriptors
-----------|--------------------
Traditional | 11
Topological (ECT &rarr; UMAP) | ~~2528~~ &rarr; 12
Combined (Trad &oplus; Topo) | 23

- Sample randomly 75/25 training/testing sample for each accession.

- Repeat the sampling and SVM computation 100 and consider the average.

---

# Classification of 28 lines with SVM

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# Traditional shape descriptors

![](../figs/trad_pca_2_dims_founders_spikes.jpg)

---

# Topological shape descriptors + UMAP

![](../figs/umap_2_dims_founders_spikes.jpg)

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# Topological shape descriptors + KPCA

![](../figs/kpca_2_dims_founders_spikes.jpg)

---

# Hidden topological shape information

.pull-left[
- Analysis of variance to determine the most discerning directions and slices/thresholds.

- The top crease on the seed is highly descriptive!

<img src="../figs/kruskal_wallis_topo_summary.jpg" width="300" style="display: block; margin: auto;" />
]

.pull-right[
<img src="../figs/discerning_directions.png" width="225" style="display: block; margin: auto;" />

![](../figs/arrow_seed_09_0.gif)
]

---

# Into semi-supervised territory

- Train an SVM with 100% of the founders `$(F_0)$`

- Classify the progeny `$(F_{18}\text{ and }F_{58})$` to detect genotype enrichment

---
class: right, bottom, inverse

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

- [`egr.msu.edu/~amezqui3/barley/slides/fwcg_2021.html`](https://www.egr.msu.edu/~amezqui3/barley/slides/fwcg_2021.html)