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 generalizes 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 generalized 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 characterize the morphogenesis of a range of surfaces.

翻译：生物上皮结构的时间成像在离散时间点产生形状数据，引出一个自然问题：如何重建与这些离散观测值最一致的生长模式可能路径？我们提出一个物理上合理的框架来解决这一逆问题，通过创建将拟共形映射推广为拟共形流的方法。通过在生长过程中允许剪切场和膨胀场的时空变化，并受调控机制约束，我们得到一种广义的里奇流。当以时间函数的表面形状相关观测数据为指导时，这导致一个约束优化问题。将我们的数据驱动算法方法应用于昆虫翅膀、叶片甚至雕塑面孔的形状，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。