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