Optical flow on evolving sphere-like surfaces

Lang, Lukas F.; Scherzer, Otmar

doi:10.3934/ipi.2017015

Cited by 4 publications

(17 citation statements)

References 42 publications

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“…2.2), decrease in magnitude, which again is indicated by the radius R of the colour disk. These findings are in line with the results obtained in [34,37]. In Fig.…”

Section: Resultssupporting

confidence: 92%

“…which is a parametrised version of (37) and is referred to as parametrised optical flow equation. We highlight that the unknownŵ depends exclusively on the imposed surface velocityV.…”

Section: Conservation Of Brightnessmentioning

confidence: 99%

“…We refer to it as parametrised continuity equation. Let us mention that, with the help of (25) and (37), one can alternatively rewrite (41) and solve forŵ in…”

Section: Conservation Of Massmentioning

confidence: 99%

“…The main goal of this subsection is to estimate the time-evolving surface M together with a parametrisation y of the form (1). We extend the idea of surface interpolation from scattered data in [37] and seek a functionρ : I × S 2 → (0, ∞) which is sufficiently regular in time and in space. Given noisy dataρ δ : I × S 2 → (0, ∞), we seek a minimiser to the energy…”

Section: Surface Parametrisationmentioning

confidence: 99%

“…A major gain of this approach is that it can also reduce the computational effort during analysis of the recorded material, see e.g. [33,34,35,37,47]. In addition, introducing a geometric representation of the specimen allows to compute accurate measurements, such as distances, on curved surfaces rather than in-possibly distorting-projections.…”

Section: Introductionmentioning

confidence: 99%

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

Lang¹

2019

SIAM J. Imaging Sci.

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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.

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“…2.2), decrease in magnitude, which again is indicated by the radius R of the colour disk. These findings are in line with the results obtained in [34,37]. In Fig.…”

Section: Resultssupporting

confidence: 92%

“…which is a parametrised version of (37) and is referred to as parametrised optical flow equation. We highlight that the unknownŵ depends exclusively on the imposed surface velocityV.…”

Section: Conservation Of Brightnessmentioning

confidence: 99%

“…We refer to it as parametrised continuity equation. Let us mention that, with the help of (25) and (37), one can alternatively rewrite (41) and solve forŵ in…”

Section: Conservation Of Massmentioning

confidence: 99%

Section: Surface Parametrisationmentioning

confidence: 99%