Quantifying the shape of plants

.title[
# Quantifying the shape of plants
]
.subtitle[
## using the Euler Characteristic Transform
]
.author[
### <strong>Erik Amézquita</strong>
]
.date[
### 2022-07-18 <br> - <br> 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("../../img/cmse_logo.svg")
background-size: 150px
background-position: 5% 60%

# My research: Crossing and merging bridges

---

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

# Roadmap for today

### Split into modules

1. Survey of TDA in biology applications

1. The Euler Characteristic Transform

1. Quantifying barley morphology using TDA

1. Current work on modeling citrus (with TDA in the horizon)

---

# 1. TDA meets biology in a number of ways

## Shape is data and data is shape

### The significance of detecting connected components and holes

---

# 1st TDA Ingredient: Complexes

- Think the data as a collection of elementary building blocks ( _cells_ )

Vertices | Edges | Faces | Cubes
---------|-------|-------|-------
  0-dim  | 1-dim | 2-dim | 3-dim

- A collection of cells is a _cubical complex_

- Count the number of topological features ( _holes_ ):

Connected components | Loops | Voids
---------------------|-------|-------
       0-dim         | 1-dim | 2-dim

- Example with 2 connected components, 1 loop, 0 voids

---

# 2nd TDA Ingredient: Filters

- Each cell is assigned a real value which defines how the complex is constructed.

- Observe how the number of topological features change as the complex grows.

.pull-left[
<img src="../figs/eigcurv_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../figs/gaussian_density_filter.gif" width="250px" style="display: block; margin: auto;" />
]

.pull-right[
<img src="../figs/eccentricity_filter.gif" width="250px" style="display: block; margin: auto;" /><img src="../figs/vrips_ver2.gif" width="250px" style="display: block; margin: auto;" />
]

---

## Example 1

### Detecting holes &rarr; detect cancerous tissue

<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1016/j.media.2019.03.014">Qaiser <em>et al.</em> (2019)</a></p>

---

## Example 2

### Detect holes &rarr; detect reassortment and horizontal evolution

.pull-left[
![](https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/ce30e9df-d595-4520-a68a-74f3c6f151d1/assets/graphic/pnas.1313480110fig01.jpeg)
]

.pull-right[
![](https://www.pnas.org/cms/10.1073/pnas.1313480110/asset/b733c3a3-00f2-43d5-b372-9da25f3d33c9/assets/graphic/pnas.1313480110fig02.jpeg)
]

<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1073/pnas.1313480110">Chan <em>et al.</em> (2013)</a></p>

---

## Example 3

### Detect holes &rarr; detect open and closed conformations

.pull-left[
![](https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_001.jpg)
![](https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_012.jpg)
]

.pull-right[
![](https://www.degruyter.com/document/doi/10.1515/sagmb-2015-0057/asset/graphic/j_sagmb-2015-0057_fig_010.jpg)
]

<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1515/sagmb-2015-0057">Kovacev-Nikolic <em>et al.</em> (2016)</a></p>

---

## Example 4

### Detect connected components across slices &rarr; detect panicle structure

<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1007/978-3-030-20867-7_7">Chitwood <em>et al.</em> (2019)</a></p>

---

## Example 5

### Detect componets across the morphospace &rarr; detect leaf development

<p style="font-size: 10px; text-align: right; color: Grey;"> Credits: Percival <em>et al.</em> (in preparation)</p>

---

# Advertisement

- More content, references, and examples

> Amézquita _et al._ (2020) "The shape of things to come: Topological data analysis and biology, from molecules to organisms". _Developmental Dynamics_ **249**(7) pp. 816-833. DOI: [10.1002/dvdy.175](https://doi.org/10.1002/dvdy.175)

![](../../tda/figs/amezquita_etal_2020.png)

---

# 2. Foucusing on the Euler Characteristic

## The Euler Characteristic Transform (ECT)

---

# Topology: The Euler characteristic `$\chi$`

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

---

# 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.

---

background-image: url("../figs/amezquita_etal_2021.png")
background-size: 400px
background-position: 50% 95%

# 3. 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
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# 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>

---

# 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;">
    <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%

# Classification of 28 lines with SVM

.pull-left[
<img src="../figs/pole_directions_p7_m12_crop.jpg" width="250" style="display: block; margin: auto;" />
]

.pull-right[
- **3121** grains in total
- 158 directions
- 16 thresholds per direction
- Every seed is associated a `$158\times16=2528$`-dim vector
- Dimensions reduced with UMAP
]

---

# 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 `$(F_{18}\text{ and }F_{58})$` to detect genotype enrichment

---

background-image: url("../figs/comparison_lines_combined_topounscaled_d158_T16_horz.png")
background-size: 900px
background-position: 50% 50%

---

background-image: url("../../citrus/figs/biorxiv_preprint.png")
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# 5. If life gives you lemons...

## Looking to strike some oil

### Modeling citrus oil gland distribution

---

## Think of citrus as lego blocks

![](../../citrus/figs/citrus_family_tree_1.jpg)
<p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1038/nature25447">Wu <em>et al.</em> (2018)</a></p>

---

## Oil glands are closely linked to fruit development

.pull-left[
![](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>

Developing cycle of a lemon
]

.pull-right[
<iframe width="560" height="300" src="https://static-movie-usa.glencoesoftware.com/webm/10.1073/956/d916befc88029defb1ecef6c4a2fd83db89428d9/pnas.1720809115.sm02.webm" frameborder="0" allowfullscreen></iframe>
<p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1073/pnas.1720809115">Smith <em>et al.</em> (2018)</a></p>

- Cross-sectional view of a navel orange peel bending to the point of jetting.

- Huge perfume and food industry behind essential oils

]

---

## 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>

---

## Quantifying the shape of citrus and their oil glands

---

# Future directions

.pull-right[
- **Quantify oil gland distribution using persistence homology** (or any other TDA tools)

- Compare differences in oil gland density distribution
    - Between "species"
    - Between wild types and domesticated fruits
    
- Propose a distribution model

- Insights into fruit development
]

---

### 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">
  <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 and slides</strong></p>
  <p style="text-align: center; color: Yellow">egr.msu.edu/~amezqui3</p>
  <p style="text-align: center; color: White; font-size:12px; line-height:1.1">Slides made in xaringan and rmarkdown</p>
  </div>
</div>

---

# Appendix

---

# 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

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# Injectivity proof outline