Data-driven quasiconformal morphodynamic flows

Temporal imaging of biological epithelial structures yields shape data at discrete time points, leading to a natural question: how can we reconstruct the most likely path of growth patterns consistent with these discrete observations? We present a physically plausible framework to solve this inverse problem by creating a framework that generalises quasiconformal maps to quasiconformal flows. By allowing for the spatio-temporal variation of the shear and dilation fields during the growth process, subject to regulatory mechanisms, we are led to a type of generalised Ricci flow. When guided by observational data associated with surface shape as a function of time, this leads to a constrained optimization problem. Deploying our data-driven algorithmic approach to the shape of insect wings, leaves and even sculpted faces, we show how optimal quasiconformal flows allow us to characterise the morphogenesis of a range of surfaces. , , , # Data-driven quasiconformal morphodynamic flows

[https://doi.org/10.5061/dryad.v41ns1s6t](https://doi.org/10.5061/dryad.v41ns1s6t)

## Contributors

* Salem Mosleh: The University of Maryland Eastern Shore.
* Gary Choi: The Chinese University of Hong Kong
* L. Mahadevan: Harvard University

## Description of the data and file structure

In each dataset, there are 30 times steps arranged into subfolders containing the vertices, faces, landmark indices, and other derivative quantities defining the mesh at a given timestep. The units of measurement in the dataset are dimensionless since each dataset is normalized according to its total area in the analysis. 

There are two preprocessing Mathematica notebooks, one for planar and one for 3D surface data. After preprocessing, which generates extra files in the mesh folders, you can run the Python code for planar and surface meshes with and without landmarks (each case has a different script).

After this runs and saves the resulting files in...