  In this work we consider brightness and mass conservation laws for motion
estimation on evolving Riemannian 2-manifolds that allow for a radial
parametrisation from the 2-sphere. While conservation of brightness constitutes
the foundation for optical flow methods and has been generalised to said
scenario, we formulate in this article the principle of mass conservation for
time-varying surfaces which are embedded in Euclidean 3-space and derive a
generalised continuity equation. The main motivation for this work is efficient
cell motion estimation in time-lapse (4D) volumetric fluorescence microscopy
images of a living zebrafish embryo. Increasing spatial and temporal resolution
of modern microscopes require 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 regularisation and propose the use of
spatially varying regularisation functionals that recover motion only in
regions with cells. For the efficient numerical solution we devise a 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 utilise surface interpolation with
spatio-temporal regularisation. We present numerical results based on
aforementioned zebrafish microscopy data featuring fluorescently labelled
cells.
