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Joint Motion Estimation and Source Identification Using Convective Regularisation with an Application to the Analysis of Laser Nanoablations

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Time-dependent Problems in Imaging and Parameter Identification

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Abstract

We propose a variational method for joint motion estimation and source identification in one-dimensional image sequences. The problem is motivated by fluorescence microscopy data of laser nanoablations of cell membranes in live Drosophila embryos, which can be conveniently—and without loss of significant information—represented in space-time plots, so called kymographs. Based on mechanical models of tissue formation, we propose a variational formulation that is based on the nonhomogenous continuity equation and investigate the solution of this ill-posed inverse problem using convective regularisation. We show existence of a minimiser of the minimisation problem, derive the associated Euler–Lagrange equations, and numerically solve them using a finite element discretisation together with Newton’s method. Based on synthetic data, we demonstrate that source estimation can be crucial whenever signal variations can not be explained by advection alone. Furthermore, we perform an extensive evaluation and comparison of various models, including standard optical flow, based on manually annotated kymographs that measure velocities of visible features. Finally, we present results for data generated by a mechanical model of tissue formation and demonstrate that our approach reliably estimates both a velocity and a source.

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Notes

  1. 1.

    https://doi.org/10.5281/zenodo.3740696.

  2. 2.

    https://doi.org/10.5281/zenodo.3257654.

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Acknowledgements

LFL and CBS acknowledge support from the Leverhulme Trust project “Breaking the non-convexity barrier”, the EPSRC grant EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1, the RISE projects ChiPS and NoMADS, the Cantab Capital Institute for the Mathematics of Information, and the Alan Turing Institute. ND and JE were supported by ANR-11-LABX-0030 “Tec21”, by a CNRS Momentum grant, and by IRS “AnisoTiss” of Idex Univ. Grenoble Alpes. ND and JE are members of GDR 3570 MecaBio and GDR 3070 CellTiss of CNRS. Some of the computations were performed using the Cactus cluster of the CIMENT infrastructure, supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA) and the authors thank Philippe Beys, who manages the cluster. Overall laboratory work was supported by Wellcome Trust Investigator Awards to BS (099234/Z/12/Z and 207553/Z/17/Z). ES was also supported by a University of Cambridge Herchel Smith Fund Postdoctoral Fellowship. The authors also wish to thank Pierre Recho for fruitful discussions and the re-use of his numerical simulation code.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Lukas F. Lang & Carola-Bibiane Schönlieb

  2. Laboratoire Interdisciplinaire de Physique, Université Grenoble Alpes, Grenoble, France

    Nilankur Dutta & Jocelyn Étienne

  3. Department of Physiology, Development and Neuroscience, University of Cambridge, Cambridge, UK

    Elena Scarpa & Bénédicte Sanson

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  1. Lukas F. Lang
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  2. Nilankur Dutta
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  4. Bénédicte Sanson
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  5. Carola-Bibiane Schönlieb
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  6. Jocelyn Étienne
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Correspondence to Lukas F. Lang .

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Editors and Affiliations

  1. Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria

    Barbara Kaltenbacher

  2. Department of Mathematics, Saarland University, Saarbrücken, Saarland, Germany

    Thomas Schuster

  3. Department of Mathematics, Saarland University, Saarbrücken, Saarland, Germany

    Anne Wald

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Lang, L.F., Dutta, N., Scarpa, E., Sanson, B., Schönlieb, CB., Étienne, J. (2021). Joint Motion Estimation and Source Identification Using Convective Regularisation with an Application to the Analysis of Laser Nanoablations. In: Kaltenbacher, B., Schuster, T., Wald, A. (eds) Time-dependent Problems in Imaging and Parameter Identification. Springer, Cham. https://doi.org/10.1007/978-3-030-57784-1_7

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