Recovering piecewise constant refractive indices by a single far-field pattern

Abstract

We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied for example in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.

15 pages, 1 figure

References in corpus (6)

• On vanishing near corners of transmission eigenfunctions
• Acoustic Scattering from Corners, Edges and Circular Cones
• Recovery of an embedded obstacle and its surrounding medium by formally-determined scattering data
• Addendum to: "On vanishing near corners of transmission eigenfunctions"
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Cited by in corpus (5)

• On an electromagnetic problem in a corner and its applications
• Calderón's Inverse Problem with a Finite Number of Measurements
• Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems
• Addendum to: "On vanishing near corners of transmission eigenfunctions"
• Reconstruction of piecewise constant functions from X-ray data