Lukas Baumgärtner, Department of Mathematics, Humboldt University Berlin
and
PD Dr. Stephan Schmidt, Department of Mathematics, University of Trier

Title: Using Transport Equations for Image Processing and Mesh Generation in Medical Applications

Abstract:

We consider the use of transport equations for solving medical image registration problem by determining the best optical flow between different brain MRI scans. In a follow-up procedure, this registration data is used to make a default generic brain mesh patient specific.
In the ensuing optimal control problem, the transport equation is solved via an upwind scheme in a DG discretization, which usually introduces non-smoothness due to absolute values. To avoid this, we propose a smoothed version of the upwind scheme which is by construction consistent with the transport equation. Its L2-stability can be shown similarly to the regular upwind scheme by a von Neumann stability analysis, however, under a slightly different CFL-condition. For the linear transport equation, this yields a reasonable scheme by the Lax equivalence theorem.
Numerical results are presented for the image registration problem. The corresponding mesh deformations are compared to state-of-the-art brain meshing software FreeSurfer. Towards the end of this talk, non-smoothness of the regular upwind scheme is discussed in the context of the non-linear PDEs of fluid mechanics.

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