Three dimensional fast factorized back projection for sub-sediment imaging sonars

Low frequency sonar has the capability of penetrating sediment and imaging buried objects, making it a candidate technology for the detection and classification of buried unexploded ordnance (UXO). To improve detection and classification, systems often use planar arrays: a real-aperture in the across-track dimension and a synthetic aperture in the along-track dimension. By using a synthetic aperture planar array three-dimensional volumetric imagery of the sub-sediment environment can be produced. Synthetic aperture processing improves resolution in the along-track dimension and also greatly improves the signal-to-noise level for structural acoustic response analysis. A difficulty introduced by synthetic aperture processing is the heavy computational burden associated with reconstructing a three dimensional image from a planar array. This is particularly true when applying a generalized beamforming algorithm, such as back projection, to make the signal processing robust to array distortions that may be caused by non-linear sonar platform movement. For the two-dimensional synthetic aperture beamforming case a variety of algorithms have been proposed that retain the attractive generality of back projection but leverage sub-aperture processing to reduce computational burden. Of these, the most well-known is fast-factorized back projection (FFBP). In this paper the FFBP algorithm is extended to the three-dimensional case by leveraging a series of transformations between sub-apertures represented in spherical coordinates. Additionally, an asymmetry correction is introduced to compensate for the fact that the source and receive arrays of the relevant sonars are rarely designed to be axisymmetric. The asymmetry correction allows the first sub-aperture stage to be constructed from two-dimensional real-aperture image frames that are very efficient to create.