Capturing data from dynamic processes through cross-sectional measurements is seen in many fields such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is instrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We rigorously evaluate our method by providing convergence guarantees and testing it on simulated cell data characterized by bifurcations and merges, comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories, but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.

翻译：通过横截面测量捕获动态过程数据在计算生物学等众多领域均有应用。轨迹推断旨在解决从这类观测数据中重建连续过程的挑战。本文提出基于B样条的逐次平均点云逼近与插值方法，该方法内蕴于Wasserstein空间。通过将细分格式与基于最优传输的测地线相结合，我们的方法能够在选定精度与平滑度下执行轨迹推断，并能自动处理粒子随时间分裂的场景。我们通过收敛性理论证明及在具有分岔合并特征的模拟细胞数据上的测试，将本方法与前沿轨迹推断及插值方法进行系统比较，从而严格评估其性能。实验结果不仅证实了本方法在轨迹推断方面的有效性，更凸显了遵循数据内在几何特性进行插值与逼近的显著优势。