Reconstruction of a reflectivity field from line integrals over circular paths

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Abstract
When a broadband acoustic pulse is emitted from an omnidirectional source into a two-dimensional reflecting medium and the resultant backscattered echoes are recorded as a function of time at a point coinciding with the source, measurements of line integrals of the acoustic reflectivity are obtained over concentric arcs centered at the source point. Sufficient line-integral data can be generated in this fashion, by translating the omnidirectional source–receiver point over a suitable aperture, to reconstruct the unknown reflectivity function. A closed-form solution to this image reconstruction problem is derived, and on the basis of this solution, computer reconstructions of a point reflecting object from simulated echo data are presented. Finally, the closed-form reconstruction formula is shown to be expressible as the sum of two terms, where the first term corresponds to a simple delay-and-sum operation applied to the echo data recorded over the aperture; the second term is new and represents a correction which is shown to provide a noticeable improvement in the temporal–spatial point spread function produced by conventional delay–sum processing alone.

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