Inversion of Parabolic and Paraboloidal Projections

摘要

The multidimensional inverse scattering problem for an acoustic medium is considered within the homogeneous background Born approximation. A constant density acoustic medium is probed by a wide-band plane wave source, and the scattered field is observed along a receiver array located outside the medium. The inversion problem is formulated as a generalized tomographic problem. It is shown that the observed scattered field can be appropriately filtered so as to obtain generalized projections of the scattering potential. For a 2-D experimental geometry, these projections are weighted integrals of the scattering potential over regions of parabolic support, whereas they become surface integrals over circular paraboloids for the 2-D case. The inversion problem is therefore similar to that of x-ray tomography, except that instead of being given projections of the object to be reconstructed along straight lines, parabolic or paraboloid projections are given. The inversion procedure that we propose is similar to the x-ray solution, in the sense that it consists of a backprojection operation followed by 2- or 3-D space invariant filtering. An alternative interpretation of the backprojection operation in terms of a backpropagated field is given. A Projection-Slice Theorem is also derived relating the generalized projections and the scattering potential in the Fourier transform domain.