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We are concerned with the inverse scattering problem of recovering aninhomogeneous medium by the associated acoustic wave measurement. We prove thatunder certain assumptions, a single far-field pattern determines the values ofa perturbation to the refractive index on the corners of its support. Theseassumptions are satisfied for example in the low acoustic frequency regime. Asa consequence if the perturbation is piecewise constant with either apolyhedral nest geometry or a known polyhedral cell geometry, such as a pixelor voxel array, we establish the injectivity of the perturbation to far-fieldmap given a fixed incident wave. This is the first unique determinancy resultof its type in the literature, and all of the existing results essentially makeuse of infinitely many measurements.

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