A Numerical Framework for Efficient Motion Estimation on Evolving Sphere-Like Surfaces Based on Brightness and Mass Conservation Laws

Abstract

In this work we consider brightness and mass conservation laws for motion estimation on evolving Riemannian 2-manifolds that allow for a radial parametrization from the 2-sphere. While conservation of brightness constitutes the foundation for optical flow methods and has been generalized to said scenario, we formulate the principle of mass conservation for time-varying surfaces that are embedded in Euclidean 3-space and derive a generalized continuity equation. The main motivation for this work is efficient cell motion estimation in volumetric fluorescence microscopy images of a living zebrafish embryo. The increasing spatial and temporal resolution of modern microscopes requires efficient analysis of such data. With this application in mind we address this need and follow an emerging paradigm in this field: dimensional reduction. In light of the ill-posedness of considered conservation laws, we employ Tikhonov regularization and propose the use of spatially varying regularization functionals that recover motion only in regions with cells. For the efficient numerical solution, we devise a mesh-free Galerkin method based on compactly supported (tangent) vectorial basis functions. Furthermore, for the fast and accurate estimation of the evolving sphere-like surface from scattered data, we utilize surface interpolation with spatio-temporal regularization. We present numerical results based on the aforementioned data featuring fluorescently labeled cells.