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Wide-band pulse-echo imaging with distributed apertures in multi-path environments

We derive a new image reconstruction method for distributed apertures operating in complex environments. The aperture elements can be distributed spatially in an arbitrary fashion, can be several hundred wavelengths apart, and can involve transmission from multiple elements simultaneously. Moreover, the object to be imaged can be either in the near-field or far-field of the array. Our method is capable of exploiting information about multi-path scattering in the environment, statistics of the objects to be imaged and statistics of the additive (possibly non-stationary) noise. We formulate the image reconstruction problem as an inversion of a bilinear mapping that maps object reflectivity to an operator which in turn acts on the transmitted waveforms. We use transmitted waveforms to reveal the action of this bilinear mapping. We develop a minimum-norm inversion which takes the form of a family of linear operators applied to the pulse-echo measurements. This processing is implemented by means of inner products between the measurements and pre-computed quantities, separately for each receiving element. Our approach is therefore well suited for parallel implementation, and can be performed in a distributed manner.