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Analytic ray propagation in FEM modelling of ultrasonic transmission tomography

Many problems are described by wave-type differential equations, which bring the notion of rays and their propagation in a natural way. The aim of this running work contribution is to investigate a non-approximative method of ray propagation, primarily in connection with the ultrasonic transmission tomography modelling. To keep the reconstruction efforts reasonable, it is necessary to use underlying physical models as simple as possible; in our case, a simple wave equation suffices. This allows us to bring forward the analytical formulas for ray trajectories within the finite elements framework. Consequently, we show how this result can be used to reduce the whole ray propagation into matching the boundary conditions between adjacent elements, while keeping the ray shape exact (up to the level introduced by the FEM itself). For clarity of the contribution, we discuss the 2D case and we give a comparison of an exact ray and its (usual) straight-ray approximation based on a speed of sound reconstruction from ultrasonic transmission tomography measurement.