Using **topology** to **analyze** the **shape** of barley

amezqui3@msu.edu

^{1} Computational Math, Science & Engineering, Michigan State University

^{2} Horticulture, Michigan State University

^{3} Mathematics and Computer Science, TU Eindhoven

^{4} Integrative Plant Science, Cornell University

^{5} Botany and Plant Sciences, University of California, Riverside

- 28 founders (land races). 58 generations.

- 3D X-ray CT scan data: 875 barley spikes.
- 38,000 seeds: generations F0, F18, and F58.
- Distribution of length, height, width, volume, etc.

Shape descriptors | # descr | F1 Score |
---|---|---|

Traditional | 11 | 0.55 ± 0.019 |

Topological (ECT + UMAP) | 12 | 0.74 ± 0.016 |

Combined (Trad + Topo) | 23 | 0.86 ± 0.010 |

- SVM to classify 3,000 seeds from the 28 founders
- (75% training vs 25% testing) \(\times\) 50 times
**Up to 84% classification accuracy**

**Acknowledgements:** This work is supported in part by Michigan State University and the National Science Foundation Research Traineeship Program (DGE-1828149).

\[\chi = \#(\text{Vertices}) - \#(\text{Edges}) + \#(\text{Faces})\]

- ECT is the record of how the EC changes as we reconstruct a given object in all possible directions.
- The ECT summarizes all shape information [2].

- Train with 100% of the founder seeds
- Classify 6000 unlabeled seeds from F58
- Three morphologies are enriched through time.
- Similar conclusion with genomic analysis!

[1] **EJA**, M. Quigley, T. Ophelders, J. Landis, D. Koenig, E. Munch, D. Chitwood (2022) "Measuring hidden phenotype: quantifying the shape of barley seeds using the Euler characteristic transform," *in silico Plants*, **4**(1) diab033

[2] K. Turner, S. Mukherjee, D. M. Boyer (2014) "Persistent homology transform for modeling shapes and surfaces," *Information and Inference*, **3**(4) 310–344.