A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only
This work provides a theoretically grounded, direct inversion technique for elasticity imaging, eliminating the need for boundary information and enabling recovery of discontinuous parameters, which is a known bottleneck in medical and geophysical imaging.
The paper presents a direct linear inversion method for recovering discontinuous elastic parameters from internal displacement measurements only, achieving L^2-stable reconstruction of shear modulus with a single measurement. The method also handles anisotropic stiffness tensors.
The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an $L^2$-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.