A topologist and a plant biologist

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.title[
# A topologist and a plant biologist
]
.subtitle[
## go for a newly shaped beer
]
.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[
### 2021-10-14 <br> - <br> To appear in <em>inSilico Plants</em>
]

---

class: inverse

# Plant morphology

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# Topological Data Analysis (TDA)

---

# Topological Data Analysis (TDA)

---

# 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>

---

# Cross Composite II experiment

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

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

# Cross Composite II experiment

.pull-left[

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

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

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

# Cross Composite II experiment

.pull-left[

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

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

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

]

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

---

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

# Cross Composite II experiment

.pull-left[

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

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

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

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

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

---

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

# Cross Composite II experiment

.pull-left[

![](../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
]

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

---

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

# Cross Composite II experiment

.pull-left[

![](../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**
]

.pull-right[
![](../figs/composite_hybrid_mixture.jpg)
]

---

# 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:51%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../figs/x3000_setup.jpg">
    <p style="text-align: center;"> Proprietary X-Ray CT scan reconstruction </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>

---

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

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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:35%; color: Navy; font-size: 15px;">
    <img style="padding: 2px 0 2px 0;" src="../figs/seed_orientation1.png">
    <p style="text-align: center;"> Align all the seeds </p>
  </div>
  <div class="column" style="max-width:55%; color: Navy; font-size: 15px;">
    <img src="../figs/boxplot_all_traditional_h.gif">
    <p style="text-align: center;"> Traditional shape descriptors </p>
  </div>
</div>

---

# 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/ecc_ver2.gif")
background-size: 750px
background-position: 50% 90%

# Euler Characteristic Curve (ECC)

- Consider a voxel-based image `$X\subset\mathbb{R}^d$`.

- Slice it along a fixed direction.

- Keep track of how the Euler Characteristic changes as we keep adding slices.

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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.

- [**Theorem** (Curry, Mukherjee, Turner, 2018)](http://arxiv.org/abs/1805.09782):  The ECT is injective.

- [**Theorem** (Turner, Mukherjee, Boyer, 2014)](https://arxiv.org/abs/1310.1030):  The ECT is a sufficient statistic in 3D.

---

# Hidden topological shape information

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

- **3121** grains in total

.pull-left[
![](../figs/svm_f0_table_umap_results.png)
]

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

---

# Hidden topological shape information

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

- **3121** grains in total

.pull-left[
![](../figs/svm_f0_table_umap_results.png)

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

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

---

# Hidden topological shape information

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

- **3121** grains in total

.pull-left[
![](../figs/svm_f0_table_umap_results.png)

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

.pull-right[
<img src="../figs/discerning_directions.png" width="150" 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

.pull-left[
![](../figs/histogram_combined_gen1_d158_T16_topounscaled_vert.svg)
]

.pull-right[
![](../figs/histogram_combined_gen7_d158_T16_topounscaled_vert.svg)
]

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