by Lukas F. Lang (RICAM, Austrian Academy of Sciences, Austria)
Our main motivation is to estimate cell motion in time-lapse volumetric microscopy images depicting fluorescently labelled cells of a live zebrafish embryo. It has been only recently possible to obtain high-resolution observations of biological model organisms such as the zebrafish. Increasing spatial and temporal resolutions lead to vast amounts of data that make the extraction of information through visual inspection impracticable. Automated cell motion estimation therefore is key to large-scale analysis of such data.
Left: Volumetric zebrafish microscopy image. Center: Approximated sphere-like surface. Right: Colour-coded visualisation of the optical flow field. All dimensions are in micrometer (μm).
In image analysis and computer vision, motion estimation is a fundamental problem that can be treated by the optical flow computation. Optical flow delivers necessary quantitative methods and leads to insights into the underlying cellular mechanisms and the dynamic behavior of cells. We compute optical flow on evolving two-dimensional Riemannian manifolds which can be radially parametrised from the sphere , as they suit quite naturally to the given data. We give a variational formulation of optical flow and we solve the problem by a Galerkin method in a finite-dimensional subspace of an appropriate (vectorial) Sobolev space. We arrive at a minimisation problem over Rn where the optimality conditions can be written purely in terms of spherical quantities and can be solved on the 2-sphere.
Secondly, to obtain a smooth sphere-like surface from the observed microscopy data, we formulate the problem of finding its (spatially varying) radius as a variational problem on the sphere. Finally, we present numerical experiments on the basis of the mentioned cell microscopy data of a zebrafish.
Joint work with Otmar Scherzer (CSC, University of Vienna and RICAM, Austria).
Acknowledgments:This work has been supported by the Vienna Graduate School in Computational Science (IK I059-N) funded by the University of Vienna. In addition, we acknowledge the support by the Austrian Science Fund (FWF) within the national research network “Geometry + Simulation” (project S11704, Variational Methods for Imaging on Manifolds).