= Shape recovery from a Herglotz boundary identity
If measured <far-field patterns> determine a density $g$ and its <Herglotz wave function> equals a prescribed spherical wave on the obstacle boundary, that trace identity can locate the boundary. A star-shaped axisymmetric obstacle can be represented by one radial function $h(\theta)$. Numerically, expand $g$ and $h$ in suitable bases, match the far-field constraints, and minimize the complex boundary residual at angular collocation points, using <regularization of an inverse problem> for noise and small singular values. A single scalar pairing constraint alone does not determine an arbitrary density.
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