Quantifying the shape of plants

.title[
# Quantifying the shape of plants
]
.subtitle[
## using the Euler Characteristic Transform
]
.author[
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br>Jacob Landis, Dan Koenig<br>Elizabeth Munch, Dan Chitwood<br> -
]
.institute[
### Computational Mathematics, Science and Engineering <br> Michigan State University <br> -
]
.date[
### 2022-04-08 <br> - <br> Published in <em>inSilico Plants</em> <strong>4</strong>(1) 2022
]

---

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

# Plant morphology

<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>

---

# Traditional Morphometry

<img src="https://nph.onlinelibrary.wiley.com/cms/asset/e0e8b362-efee-410a-a293-947c2d57acb9/nph16286-fig-0003-m.jpg" width="650" style="display: block; margin: auto;" />
<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href=" https://doi.org/10.1111/nph.16286">Gupta <em>et al.</em> (2019)</a></p>

---

# Modern Morphometric Methods might not be enough

.pull-left[
Elliptical Fourier Descriptor (EFD)
![](../figs/chitwood_sinha_2016_01_02.jpg)
<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1016/j.cub.2016.02.033">Chitwood and Sinha (2016)</a></p>
]

.pull-right[
Landmark-based morphometrics (GMM)
![](../figs/chitwood_2021_fig-004_1.jpg)
<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href=" https://doi.org/10.1002/ppp3.10157">Chitwood  (2020)</a></p>
]

---

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

# Something more robust is needed

---

# 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>

---

background-image: url("../figs/seed.png")
background-size: 325px
background-position: 99% 99%

# Roadmap for today

### Split into modules

1. The Euler Characteristic Transform

1. Quantifying barley morphology using TDA

1. Current work on modeling citrus and arabidopsis

---

# 1. Foucusing on the Euler Characteristic

## The Euler Characteristic Transform (ECT)

---

# Topology: The Euler characteristic

`$$\chi = \#(\text{Vertices}) - \#(\text{Edges}) + \#(\text{Faces}).$$`
<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**.

---

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

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.

---

# More on 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.

There is elusive math research on computationally efficient reconstruction algorithms:
- [Turner, Mukherjee, Curry (2021)](https://arxiv.org/abs/1805.09782): Finite no. 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

Alternative takes:
- [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

---

# 2. Quantifying barley morphology

## Using the Euler Characteristic Transform

---

<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_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_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

<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 class="column" style="max-width:20.5%; color: Navy; font-size: 15px;">
    <img src="../figs/S017_L0_seed_10_0.gif">
    <p style="text-align: center;"> 38,000 seeds </p>
  </div>
</div>

<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;">
    <div class="row">
      <div class="column" style="max-width:24%; color: Navy; font-size: 15px;">
        <img style="padding: 2px 0 2px 0;" src="../figs/seed_outlier1.png">
      </div>
      <div class="column" style="max-width:24%; color: Navy; font-size: 15px;">
        <img style="padding: 2px 0 2px 0;" src="../figs/seed_outlier2.png">
      </div>
      <div class="column" style="max-width:24%; color: Navy; font-size: 15px;">
        <img style="padding: 2px 0 2px 0;" src="../figs/seed_outlier3.png">
      </div>
      <div class="column" style="max-width:24%; color: Navy; font-size: 15px;">
        <img style="padding: 2px 0 2px 0;" src="../figs/seed_outlier4.png">
      </div>
    </div>
    <p style="text-align: center;"> Remove outliers </p>
  </div>
</div>

---

# Traditional shape descriptors

---

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 times and consider the average.

---

# Classification of 28 lines with SVM

---

# Traditional shape descriptors

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

---

# Topological shape descriptors + UMAP

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

---

# 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 `$F_{58}$`: DNA (genotype) &harr; shape (phenotype) enrichment

---

# 3. But wait, there is more!

## If life gives you lemons, make salad with the plant guinea pig

---

## 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="../../citrus/crc_pics/crc_diversity.jpg">
    <p style="text-align: center;"> UCR Collaboration </p>
  </div>
  <div class="column" style="max-width:38%; 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:23%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_CRC3289_12B-19-9_L01_raw.gif">
    <p style="text-align: center;"> Raw </p>
  </div>
</div>

<div class="row" style="margin: 0 auto;">
  <div class="column" style="max-width:20%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_L01_black_spine.gif">
    <p style="text-align: center;"> Spine </p>
  </div>
  <div class="column" style="max-width:20%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_L01_black_endocarp.gif">
    <p style="text-align: center;"> Endocarp </p>
  </div>
  <div class="column" style="max-width:20%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_L01_black_rind.gif">
    <p style="text-align: center;"> Rind </p>
  </div>
  <div class="column" style="max-width:20%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_L01_black_exocarp.gif">
    <p style="text-align: center;"> Exocarp </p>
  </div>
  <div class="column" style="max-width:20%; color: Navy; font-size: 15px;">
    <img src="../../citrus/figs/SR01_L01_black_oil_glands.gif">
    <p style="text-align: center;"> Oil glands</p>
  </div>
</div>

---

### Understanding the distribution of oil glands in citrus fruits

.pull-left[
![](../../citrus/figs/SW02_L00_center_oil_glands.gif)
- About 6K&rarr;20K individual oil glands detected

- Number in the right ballpark

- **We have a point cloud in `$\mathbb{R}^3$`.**

- Key to understand fruit development in citrus
]

.pull-right[
![](https://www.boredpanda.com/blog/wp-content/uploads/2020/06/5eec755907306_07f29bfovq551-png__700.jpg)
<p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://www.boredpanda.com/life-cycles-pics/">BoredPanda</a></p>

- Huge perfume and food industry behind essential oils

]

---

### Grand challenge: genotype (DNA) &harr; phenotype (shape)

Leaf development in Arabidopsis: the guinea pig of plant biology

.pull-left[
<img src="../../arabidopsis/figs/Day7_110921_Col_0_X_pot3_leaf_X.gif" width="225" style="display: block; margin: auto;" /><img src="../../arabidopsis/figs/Day7_110921_U112-3_pot0_leaf_X.gif" width="225" style="display: block; margin: auto;" />
]

.pull-right[
![](../../arabidopsis/figs/ecc_axis_X.gif)
![](../../arabidopsis/figs/ecc_axis_Y.gif)
![](../../arabidopsis/figs/ecc_axis_Z.gif)
]

---

background-image: url("../../img/phd_institutional_logos.jpg")
background-size: 500px
background-position: 95% 1%

## Thank you!

<div class="row" style="margin-top: -25px;">
  <div class="column" style="max-width:19.5%; font-size: 13px;">
    <img style="padding: 0 0 0 0;" src="https://i1.rgstatic.net/ii/profile.image/607374528233472-1521820780707_Q128/Elizabeth-Munch.jpg">
    <p style="text-align: center; color: White">Liz Munch<br>(MSU)</p>
    <img style="padding: 0 0 0 0;" src="https://alga.win.tue.nl/images/staff/tim.jpg">
    <p style="text-align: center; color: White">Tim Ophelders<br>(Utrecht)</p>
    <img style="padding: 0 0 0 0;" src="../figs/S017_L0_seed_10_0.gif">
  </div>
  <div class="column" style="max-width:19.5%; font-size: 13px;">
    <img style="padding: 0 0 0 0;" src="https://www.canr.msu.edu/contentAsset/image/9ae9777d-157c-46e6-9f12-d062ad35671e/fileAsset/filter/Jpeg,Resize,Crop/jpeg_q/80/resize_w/400/crop_x/0/crop_y/45/crop_w/400/crop_h/400">
    <p style="text-align: center; color: White">Dan Chitwood<br>(MSU)</p>
    <img style="padding: 0 0 0 0;" src="https://i1.rgstatic.net/ii/profile.image/926632407748609-1597937790454_Q128/Michelle-Quigley.jpg">
    <p style="text-align: center; color: White">Michelle Quigley<br>(MSU)</p>
    <img style="padding: 0 0 0 0;" src="https://publons.com/media/thumbs/academic/photos/252a7116-ee14-4a5e-b44c-7c62ac11e217.png.200x200_q95_crop_detail_upscale.png">
    <p style="text-align: center; color: White">Jacob Landis<br>(Cornell)</p>
  </div>
  <div class="column" style="max-width:14%; font-size: 13px;">
  <img style="padding: 0 0 0 0;" src="https://plantbiology.ucr.edu/sites/g/files/rcwecm1001/files/styles/scale_225/public/Koenig.jpg?itok=P92Ow61p">
  <p style="text-align: center; color: White">Dan Koenig<br>(UC Riverside)</p>
  <img style="padding: 0 0 0 0;" src="https://iigb.ucr.edu/sites/g/files/rcwecm5716/files/styles/scale_225/public/Seymour.jpg?itok=pnUK_orh">
  <p style="text-align: center; color: White">Danelle Seymour<br>(UC Riverside)</p>
  <img style="padding: 0 0 0 0;" src="../../citrus/figs/SR01_L01_black_exocarp.gif">
  </div>
  <div class="column" style="width:10%; font-size: 24px;">
  </div>
  <div class="column" style="max-width:30%; font-size: 24px; line-height:1.25; padding: 0 0 0 0;">
  <p style="text-align: center; color: White"><strong>Email</strong></p>
  <p style="text-align: center; color: Yellow">amezqui3@msu.edu</p>
  <p style="text-align: center; color: White"><strong>Website/slides</strong></p>
  <p style="text-align: center; color: Yellow"><a href="https://egr.msu.edu/~amezqui3/aboutme.html" target="_blank">egr.msu.edu/~amezqui3</a></p>
  <p style="text-align: center; color: White"><strong>References</strong></p>
  <p style="font-size: 15px; color: Yellow; line-height:1.1; font-family: 'Yanone Kaffeesatz'">
  <strong>E.J. Amézquita</strong>, M.Y. Quigley, T. Ophelders, J.B. Landis, D. Koenig, E. Munch, D.H. Chitwood (2022) <a href="https://doi.org/10.1093/insilicoplants/diab033" target="_blank">Measuring hidden phenotype: Quantifying the shape of barley seeds using the Euler Characteristic Transform"</a>. <em>in Silico Plants</em> 4(1); diab033
  </p>
  <p style="font-size: 15px; color: Yellow; line-height:1.1; font-family: 'Yanone Kaffeesatz'">
  <strong>E.J. Amézquita</strong>, M.Y. Quigley, T. Ophelders, D. Seymour, E. Munch, D.H. Chitwood <a href="" target="_blank">The shape of aroma: measuring and modeling citrus oil gland distribution"</a>. In preparation. 
  </p>
  <p style="font-size: 15px; color: Yellow; line-height:1.1; font-family: 'Yanone Kaffeesatz'">
      K. Turner, S. Mukherjee, D.M. Boyer (2014) <a href="https://doi.org/10.1093/imaiai/iau011" target="_blank">"Persistent homology transform for modeling shapes and surfaces,"</a> <em>Information and Inference</em>, 3(4) 310–344
  </p>
  <p style="text-align: center; color: White; font-size:10px; line-height:1.1">Slides made in xaringan and rmarkdown</p>
  
  </div>
</div>