Quantifying the shape of plants

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# Quantifying the shape of plants
## using Topological Data Analysis
### <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> -
### 2021-10-05

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# TDA applied to plant biology

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# TDA applied to plant biology

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# TDA applied to plant biology

<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>
    <p> X-Ray CT 3D voxel-based reconstructions </p>
    <p> Segmented seeds </p>
  </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>
    <p style="text-align:center;">Euler Characteristic Transform</p>
    <p style="text-align:center;"><a href="http://dx.doi.org/10.1093/imaiai/iau011" target="_blank">(Turner, Mukherjee, and Boyer)</a></p>
  </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> SVM classification </li>
      <li> Variance analysis </li>
      <li> Link phenotype with genotype </li>
    <ul>
  </div>
</div>

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# 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)
]

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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)
]

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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)
]

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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)
]

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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: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>

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# 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>

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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 filter it along the unit-length direction `$\nu\in S^{d-1}$`
`$$X(\nu)_h =\{\Delta \in X\::\:\langle x,\nu\rangle\leq h\text{ for all }x\in\Delta\}$$`

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

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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.

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

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

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# 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;" />
]

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# 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;" />
]

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# 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)
]

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# 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.png)
]

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

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background-image: url("../figs/acknowledgments.jpg")
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