Using the Navier-Cauchy equation for motion estimation in dynamic imaging

Figure 1.  The mapping $ \Phi^{-1}_t $ correlates the state $ f_t $ at time $ t $ to the reference state $ f_0 $ at the initial time

Figure 2.  Initial state $ f_0 $ of a phantom (left) and its singularities (right)

Figure 4.  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

Figure 3.  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

Figure 5.  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

Figure 6.  Static and dynamic reconstruction results of the initial state function

Figure 7.  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

Figure 8.  Dynamic reconstruction with motion information from solving the PDE with noisy boundary data

Figure 9.  Dynamic reconstruction results with motion information from solving the PDE with only a small number of boundary nodes