Abstract

We study the classical inverse problem to determine the shape of a three-dimensional scattering obstacle from measurements of scattered waves or their far-field patterns. Previous research on this subject has mostly assumed the object to be star-shaped and imposed a Sobolev penalty on the radial function or has defined the penalty term in some other ad-hoc manner which is not invariant under coordinate transformations.

For the case of curves in $\mathbb{R}^2$, Julian Eckardt suggests in his PhD thesis to use the bending energy as regularisation functional and proposes Tikhonov regularization and regularized Newton methods on a shape manifold. The case of surfaces in $\mathbb{R}^3$ is considerably more demanding. First, a suitable space (manifold) of shapes is not obvious. The second problem is to find a

stabilizing functional for generalised Tikhonov regularisation which on the

one hand should be bending-sensitive and on the other hand prevent the surface from self-intersections during the reconstruction.

The tangent-point energy is a parametrization-invariant and repulsive surface energy that is constructed as the double integral over a power of the tangent point radius with respect to two points on the surface, i.e. the smallest radius of a sphere being tangent to the first point and intersecting the other. The finiteness of this energy also provides $C^{1,\alpha}$ Hölder regularity of the surfaces. Using this energy as the stabilising functional, we choose general surfaces of Sobolev-Slobodeckij reguality, which are naturally connected to this energy.

The proposed approach works for surfaces of arbitrary (known) topology.

In numerical examples we demonstrate that the flexibility of our approach in handling the reconstruction of rather general shapes.

Authors: Jannik Rönsch, Henrik Schumacher, Max Wardetzky, and Thorsten Hohage