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 dilatation 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.

翻译：生物上皮结构的时间序列成像在离散时间点产生形状数据，这引出了一个自然问题：如何重建与这些离散观测结果一致的最可能的生长模式路径？我们提出了一个物理上合理的框架来解决这一逆问题，该框架将拟共形映射推广为拟共形流。通过允许生长过程中剪切场和膨胀场的时空变化，并受调控机制约束，我们得到了一类广义的里奇流。当以随时间变化的表面形状观测数据为指导时，这引出了一个约束优化问题。通过将我们的数据驱动算法应用于昆虫翅膀、叶片乃至雕塑面部的形状，我们展示了最优拟共形流如何使我们能够表征一系列表面的形态发生过程。