Reconstruction of a reflectivity field from line integrals over circular paths

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 correcti...