Imaging via inversion of ellipsoidal projections of solutions to the linear acoustic wave equation (medical ultrasound)

Backscattered energy from a medium with nearly constant velocity measured by an omnidirectional source-sensor comes from spherical shells centered at the source-sensor location. If source and sensor locations differ, the energy comes from ellipsoidal shells. To image the scattering potential or reflectivity of the medium, backscattered energy must be projected onto ellipsoids with a weighting factor dependent on the eccentricity of the ellipsoid. Eccentricity is the ratio of the separation of source and sensor to the path length from source to scatterer to sensor. Because path length changes with time, this weighting factor is, in general, time-dependent. It is consistent only if the source and sensor location are the same, in which case its value is unity. To estimate scattering potential or reflectivity at a given site, the backscattered signal which could have arisen from that site as measured by a given source-sensor pair is weighted by that pair's time-dependent ellipsoidal factor, then the weighted backscattered signals are averaged over all available source-sensor pairs.<
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