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.