The shape of the aroma

.title[
# The shape of the aroma
]
.subtitle[
## modeling the citrus oil gland distribution
]
.author[
### <strong>Erik Amézquita</strong>, Michelle Quigley, Tim Ophelders <br> Danelle Seymour, Elizabeth Munch, Dan Chitwood <br> -
]
.institute[
### Computational Mathematics, Science and Engineering <br> Michigan State University <br> -
]
.date[
### 2023-01-07<br> - <br> Published in <a href="https://doi.org/10.1002/ppp3.10333">Plants, People, Planet (2022)</a>
]

---

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

# When life gives you lemons...

<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>

---

## Think of citrus as lego blocks

![](../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="../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="../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="../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="../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="../figs/SR01_L01_black_oil_glands.gif">
    <p style="text-align: center;"> Oil glands</p>
  </div>
</div>

---

## The centers will be just fine

- 58 species and 163 raw scans

- About 6K&rarr;20K individual oil glands detected

- Number in the right ballpark

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

.pull-right[
![](../figs/oil_gland_number_vs_area.png)
<p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://doi.org/10.1006/anbo.2001.1546">Knight <em>et al.</em> (2001)</a></p>
]

---

# Looking to strike some oil

## Phenotyping all the citrus

---

# Allometry

+ Linear relationship between size-related measurements.

+ Total number of glands seems to be decoupled

---

## Average distance from each gland to its `$k$`-th nearest neighbor

![](../figs/panel_knn25_growth.svg)

+ `$\text{Avg.dist}(k) = \sqrt{Mk + B}$`
+ Oil glands might be distributing themselves following **normal diffusion** mechanics
+ The **outliers** for citrus groups usually correspond to hybrids.

---

# Citrus are quite spherical

## Ellipsoidal to be more precise

## Directional statistics to compare fruit surfaces

![](../figs/SR01_CRC3289_12B-19-9_L00_Geocentric.jpg)

---

background-image: url("../ellipsoids/Li_and_Griffiths_2004.png")
background-size: 250px
background-position: 95% 3%

## Best ellipsoid fitting &mdash; least squares

- A general quadratic surface is defined by the equation

`$$\eqalignno{ & ax^{2}+by^{2}+cz^{2}+2fxy+2gyz+2hzy\ \ \ \ \ \ \ \ \ &\hbox{(1)}\cr &+2px+2qy+2rz+d=0.}$$`

With our observations `$\{(x_i,y_i,z_i)\}_i$`,

$$
`\begin{pmatrix}
x_1^2 & y_1^2 & z_1^2 & 2x_1y_1 & 2y_1z_1 & 2x_1z_1 & x_1 & y_1 & z_1 & 1\\
x_2^2 & y_2^2 & z_2^2 & 2x_2y_2 & 2y_2z_2 & 2x_2z_2 & x_2 & y_2 & z_2 & 1\\
\vdots& \vdots& \vdots& \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
x_n^2 & y_n^2 & z_n^2 & 2x_ny_n & 2y_nz_n & 2x_nz_n & x_n & y_n & z_n & 1
\end{pmatrix}`
`\begin{pmatrix}
a \\ b \\ \vdots \\ d
\end{pmatrix}`
=
`\begin{pmatrix}
0 \\ 0 \\ \vdots \\ 0
\end{pmatrix}`
$$
or
$$
\mathbf{D}\mathbf{v} = 0
$$

The solution to the system above can be obtained via Lagrange multipliers

`$$\min_{\mathbf{v}\in\mathbb{R}^{10}}\left\|\mathbf{D}\mathbf{v}\right\|^2, \quad \mathrm{s.t.}\; \text{ some ellipsoid-specific conditions hold}$$`

---

## Model the whole fruit as an ellipsoid

.pull-left[
![](../figs/SW01_CRC3030_12B-8-5_L02_frontal_ell_projection.jpg)
![](../figs/SR01_CRC3289_12B-19-9_L02_frontal_ell_projection.jpg)
]

.pull-right[
![](../figs/C08C_CRC3546_18A-5-3_L00_frontal_ell_projection.jpg)
![](../figs/WR11_CR3228_12B-30-9_L02_frontal_ell_projection.jpg)
]

---

background-image: url("../figs/lambert_equal_area_N.gif")
background-size: 150px
background-position: 98% 1%

## Citrus modeling: Sour orange

![](../../citrus/figs/SR01_CRC3289_12B-19-9_L00_Geocentric.jpg)

---

background-image: url("../figs/lambert_equal_area_N.gif")
background-size: 150px
background-position: 98% 1%

## Citrus modeling: Pummelo

![](../../citrus/figs/P01A_CRC3781_12A-31-9_L00_Geocentric.jpg)

---

background-image: url("../figs/lambert_equal_area_N.gif")
background-size: 150px
background-position: 98% 1%

## Citrus modeling: Mandarin

![](../figs/M01_CRC3226_12B-27-1_L00_Geocentric.jpg)

---

background-image: url("../figs/lambert_equal_area_N.gif")
background-size: 150px
background-position: 98% 1%

## Citrus modeling: Kumquat

![](../figs/WR02_CRC3877_12B-44-13_L00_Geocentric.jpg)

<img src="../figs/WR02_CRC3877_12B-44-13_L00_geocentric_lazi.jpg" width="550" style="display: block; margin: auto;" />
---

background-image: url("../../citrus/figs/lambert_equal_area_N.gif")
background-size: 150px
background-position: 98% 1%

# Citrus modeling: Finger lime

![](../figs/WR08_CRC3661_18B-16-5_L00_Geocentric.jpg)

---

background-image: url("../ellipsoids/GarciaPortugues_etal_2020a.png")
background-size: 200px
background-position: 95% 3%

# Can we say something more?

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

---

# Uniform and rotationally symmetric spherical distributions

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

---

# But then, what distribution do oil glands follow?

## Spherical Kernel Density Estimates FTW

---

# Kernel Density Estimators

![](https://www.statsmodels.org/dev/_images/examples_notebooks_generated_kernel_density_16_0.png)
<p style="font-size: 8px; text-align: right; color: Grey;"> Credits: <a href="https://www.statsmodels.org/dev/examples/notebooks/generated/kernel_density.html">Statsmodels</a></p>

- The choice of bandwidth is crucial
- Large bandwidths tend to oversmooth
- Small bandwidths tend to overfit
- Plenty of literature on `$\mathbb{R}$` on how to choose an adequate value

---

background-image: url("../ellipsoids/GarciaPortugues_2013.png")
background-size: 300px
background-position: 95% 3%

# Spherical KDE

![](../figs/WR19_CRC3842_18B-18-5_L00_emi_kde_N.jpg)

- Oil glands of a papeda
- Red arrows indicate the most significant gradient values for the density function
- Concentration parameter `$k$` is chosen to be optimal

---

background-image: url("../ellipsoids/Vuollo_and_Holmstrom_2018.png")
background-size: 300px
background-position: 95% 3%

# More exploratory analysis

## SphereSiZer: Pummelo

![](../figs/P01C_L00_W.gif)

---

# Side view of a sweet orange

![](../figs/SW01_L00_W.gif)

---

# Side view of a mandarin

![](../figs/M01_L00_W.gif)

---

# Side view of a kumquat

![](../figs/WR02_L00_W.gif)

---

# Side view of a finger lime

![](../figs/WR08_L00_W.gif)

---

# Future topological directions

## Obligatory TDA plugin

---