Imaging from sparse measurements

We consider the problem of using scattered waves to recover an image of the medium in which the waves propagate. We address the case of scalar waves when the sources and receivers are sparse and irregularly spaced. Our approach is based on the single-scattering (Born) approximation and the generalized Radon transform. The key to handling sparse sources and receivers is the development of a data-weighting scheme that compensates for non-uniform sampling. To determine the appropriate weights, we formulate a criterion for measuring the optimality of the point-spread function, and solve the resulting optimization problem using regularized least squares. Once the weights are determined, they can be used to compute the point-spread function and thus determine resolution, and they can also be applied to the measured data to form an image. Tests of our minimization scheme with different regularization parameters show that, with appropriate weighting, individual scatterers can be resolved at subwavelength scales even when data is noisy and the locations of both sources and receivers are uncertain. We show an example in which the source-receiver geometry and frequency bandwidths correspond to seismic imaging from multiple local earthquakes (passive seismic imaging). The example shows that the weights determined by our method improve the resolution relative to reconstructions with constant weights.