The reconstruction of the full wavefield from spatially sparse samples is a fundamental challenge in several disciplines, including geophysical seismic imaging, structural health monitoring, and non-destructive-testing. For a band-limited signal with maximum spatial frequency B, perfect reconstruction from (particle) velocity measurements alone requires a sampling interval of at most Δx = 1/2B, as dictated by the Nyquist theorem. However, if first-order spatial derivatives of the (particle velocity) wavefield are measured alongside with the velocity at the same locations, the Nyquist limit can be relaxed proportionally to the new degrees of freedom. This enables perfect reconstruction using a sample spacing above the Nyquist limit. In this work, we design mechanical structures that are sensitive to spatial derivatives of the wave. Spatial derivatives are inherently difficult to isolate from velocity signals, and measurement noise further degrades reconstruction quality. The accuracy of the interpolation is linked to the condition number of the matrix formed by the interpolation basis functions, which in turn depends on the degree of orthogonality between measured quantities. We address this issue by using adjoint-based optimization to design mechanical structures that minimize the condition number of the matrix associated to the inverse problem.
Piciucco et al. (Wed,) studied this question.