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doi: 10.3934/ipi.2022018
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## Using the Navier-Cauchy equation for motion estimation in dynamic imaging

 1 Department of Mathematics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 2 Department of Mathematics, University of Würzburg, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

*Corresponding author: Sandra Warnecke

Received  June 2021 Revised  December 2021 Early access April 2022

Fund Project: The first and second authors are supported by the Deutsche Forschungsgemeinschaft under grant HA 8176/1-1

Tomographic image reconstruction is well understood if the specimen being studied is stationary during data acquisition. However, if this specimen changes its position during the measuring process, standard reconstruction techniques can lead to severe motion artefacts in the computed images. Solving a dynamic reconstruction problem therefore requires to model and incorporate suitable information on the dynamics in the reconstruction step to compensate for the motion.

Many dynamic processes can be described by partial differential equations which thus could serve as additional information for the purpose of motion compensation. In this article, we consider the Navier-Cauchy equation which characterizes small elastic deformations and serves, for instance, as a simplified model for respiratory motion. Our goal is to provide a proof-of-concept that by incorporating the deformation fields provided by this PDE, one can reduce the respective motion artefacts in the reconstructed image. To this end, we solve the Navier-Cauchy equation prior to the image reconstruction step using suitable initial and boundary data. Then, the thus computed deformation fields are incorporated into an analytic dynamic reconstruction method to compute an image of the unknown interior structure. The feasibility is illustrated with numerical examples from computerized tomography.

Citation: Bernadette N. Hahn, Melina-Loren Kienle Garrido, Christian Klingenberg, Sandra Warnecke. Using the Navier-Cauchy equation for motion estimation in dynamic imaging. Inverse Problems and Imaging, doi: 10.3934/ipi.2022018
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##### References:
The mapping $\Phi^{-1}_t$ correlates the state $f_t$ at time $t$ to the reference state $f_0$ at the initial time
Initial state $f_0$ of a phantom (left) and its singularities (right)
Illustration of the boundary: The nodes 1 and 2 lie directly on the continuous boundary, and their behaviour is prescribed by the Dirichlet data $\psi$. For the node 0, the stencil for the update scheme only can be applied with the help of an interpolation since the values of the ghost node are not available. The average of the values of the nodes 1 and 2 are used to create an auxiliary node which corresponds to a slightly 'shifted' boundary
We illustrate the stencil for our numerical scheme. For the update of the values at node $x_{i, j}$ from $t_n\to t_{n+1}$, we have to provide information about the values at the other marked nodes
Cross-section of the numerical phantom during one cycling breath. The first image corresponds to the reference state, the second and third image correspond to the body after a quarter and after one half of a breathing cycle, respectively. The fourth image illustrates the body after one period when the initial configuration is reached again
Static and dynamic reconstruction results of the initial state function
Illustration of the numerical solution of the Navier-Cauchy equation with analytical boundary data. The initial density distribution used for solving the Navier-Cauchy equation is given in the first image. The second, third and fourth image correspond to the configurations after a quarter, after one half and after a full period of the breathing cycle
Dynamic reconstruction with motion information from solving the PDE with noisy boundary data
Dynamic reconstruction results with motion information from solving the PDE with only a small number of boundary nodes
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