Quantifying barley morphology

.title[
# Quantifying barley morphology
]
.subtitle[
## using Euler characteristic curves
]
.author[
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Elizabeth Munch, Dan Chitwood <br> Dan Koenig, Jacob Landis
]
.institute[
### Computational Mathematics, Science and Engineering <br> Michigan State University
]
.date[
### 2020-06-15
]

---

# 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_canberra_algerian_everest.png">
    <img src="../figs/svm_mds_canberra_multan_white-smyrna.png">
    <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>

---

# 1. Plant Biology: Barley

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background-image: url("../figs/barley_domestication.jpg")
background-size: 910px
background-position: 50% 70%

# Diversification of floral morphology

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background-image: url("../figs/composite_hybrid_mixture.jpg")
background-size: 400px
background-position: 95% 5%

# Composite Cross (CC II)

<div class="row">
  <div class="column" style="max-width:36%">
    <img src="../figs/BarleyEars_2_vs_6.jpg">
    <p style="font-size: 15px; text-align: center; color: DimGrey;"> 2-row vs 6-row barley </p>
  </div>
  <div class="column" style="max-width:64%">
    <img src="../figs/barley_world.jpg">
    <p style="font-size: 15px; text-align: center; color: DimGrey;"> 28 founder lines </p>
  </div>
</div>

- Experiment started in 1929, Aberdeen, Idaho
- Maintenance by a number of people at UC Riverside.

---

background-image: url("../figs/composite_cross_v_05.svg")
background-size: 450px
background-position: 95% 50%

# Experimental design

.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 almost 58 generations
- We have records of all plots throughout generations
    - Measure morphology
    - DNA sequencing
    - Which genes are selected for?
    - How did morphology change?
]

---

# 2. Image procesing

## X-ray CT scan 3D images

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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
]

---

# _Ad-hoc_ image processing

<div class="row">
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_0_original.gif">
    <p style="text-align: center"> Original </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_1_normal.gif">
    <p style="text-align: center"> Normalized </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_2_unair.gif">
    <p style="text-align: center"> Clean </p>
  </div>
  <div class="column" style="max-width:12%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S017_3_denoise.gif">
    <p style="text-align: center"> Prunned </p>
  </div>
  <div class="column" style="max-width:27%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S015_alignment.jpg">
    <p style="text-align: center"> Labeled </p>
  </div>
  <div class="column" style="max-width:21%; color: DimGrey; font-size: 15px;">
    <img src="../figs/S019_L0_1.gif">
    <p style="text-align: center"> Analysis! </p>
  </div>
</div>

- 224 raw scans in total
- 875 individual barley spikes

---

# Analyze seed distribution

<div class="row">
  <div class="column" style="max-width:63%; padding: 0px 50px;">
    <img src="../figs/seeds_batch1.png">
    <img src="../figs/seeds_batch2.png">
  </div>
  <div class="column" style="max-width:12%; background-color: DarkRed; padding: 5px; border-radius: 15px 0px 0px 30px;">
    <img src="../figs/seed_outlier1.png">
    <img src="../figs/seed_outlier2.png">
  </div>
  <div class="column" style="max-width:12%; color: Yellow; text-align: center; font-size: 15px; background-color: DarkRed; padding: 5px;">
    <img src="../figs/seed_outlier4.png">
    <img src="../figs/seed_outlier6.png">
    <p><strong> Outliers </strong></p>
  </div>
  <div class="column" style="max-width:12%; background-color: DarkRed; padding: 5px; border-radius: 0px 15px 30px 0px;">
    <img src="../figs/S102_1_44_0.png">
    <img src="../figs/S129_0_37_0.png">
  </div>
  
</div>

- `$\sim38,000$` clean seeds.

---

background-image: url("../figs/seed_mesh.png")
background-size: 250px
background-position: 80% 60%

# Traditional measures

All seeds are oblong: align them based on SVD

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

---

# Allometry and Morphological evolution

---

# 3. Topology

## Euler Characteristic Transform (ECT)

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background-image: url("../figs/euler_characteristic_2.png")
background-size: 450px
background-position: 97% 50%

# The Euler characteristic

`$$\chi = V_0 - V_1 + V_2$$`
**Betti numbers:** Number of homologically different holes.
- `$\beta_0$`: Number of component components
- `$\beta_1$`: Number of cycles
- `$\beta_2$`: Number of voids

`$\chi = \beta_0 - \beta_1 + \beta_2$`

In general:

`$\chi = \sum_{i=0}^n (-1)^iV_i = \sum_{i=0}^n (-1)^i\beta_i$`
]

---

background-image: url("../figs/ecc_X.gif")
background-size: 300px
background-position: 90% 60%

# Euler characteristic curve (ECC)

- A direction `$\nu\in S^{d-1}$`

- Height filtration
`$M(\nu)_r = \{x\in M: \langle x,\nu\rangle \leq r\}$` 
`$\simeq 	\{\Delta\in M : \langle x, \nu\rangle\leq r\:\forall	\,x\in\Delta\}$`

- Define the ECC as `$\chi(M,\nu):\mathbb{R}\to\mathbb{Z}$`

`$$\chi(M,\nu)(r) = \chi(M(\nu)_r).$$`
]

---

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

# Euler characteristic transform (ECT)

- Define `$ECT(M): S^{d-1}\to\mathbb{Z}^{\mathbb{R}}$` with `$\nu\mapsto\chi(M,\nu)$`
- Concatenate an infinite number of ECCs.

---

# 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.

---

# 4. Results on barley seeds

---

background-image: url("../figs/pole_directions_102.png")
background-size: 350px
background-position: 90% 50%

# Computing the ECT of seeds

- 74 directions

- 32 thresholds per direction

- Every seed is associated a `$74\times32=2368$`-dimensional vector.

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

---

# MDS + SVM

- Focus on two founders at a time

- Project each ECT to 2D with a metric MDS.
    - `$d(\mathbf x, \mathbf z) = \sum_i\frac{|x_i-y_i|}{|x_i|+|y_i|}$`

- Compute a linear SVM to gauge the separability of the ECT+MDS signals.

- Used 100% of the data set as training.
]

.pull-right[
![](../figs/svm_mds_canberra_algerian_everest.png)
![](../figs/svm_mds_canberra_multan_white-smyrna.png)
]

---

background-image: url("../figs/heatmap_svm_linear_C1_mds_canberra_k4.png")
background-size: 450px
background-position: 98% 50%

# Accuracy heatmap

Idea:

- If two founders are easy to separate, then their morphology must be significantly different.

- Thus, such founders ought to be distant in the phylogenetic tree.

- The reverse must be true as well.
]

---

background-image: url("../figs/hclust_ward.D2_svm_radial_C1_mds_manhattan_k4.png")
background-size: 450px
background-position: 99% 50%

# Hierarchical clustering

- The behavior is mantained as we modify:
    - Number of projected dimensions with MDS: `$k=2,4,8$` 
    - Canberra or Manhattan distance for metric MDS
    - Radial or linear SVM
    - Different hierarchical clustering criteria (complete, average, ward, mcquitty)
]

---

background-image: url("../figs/founders_rownums2_pca_d74_T32.png")
background-size: 450px
background-position: 99% 60%

# PCA suggests a slight asymmetry.

- Hypothesis: The lateral seeds in the 6-row barley present a natural asymmetry/handedness.

![](../figs/barley_inflorescence.png)
]

---

# Future work

- Do a Procrustes analysis to analyze seed handedness

- Figure out how to use the barley spike as a whole

- Find a more concrete link between morphology and genomics

- Many more plant morphology data sets to be analyzed!

---

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