An analytic examination of 3D holography under a 90° recording geometry was carried out earlier in which 2D spatial Laplace transforms were introduced in order to develop transfer functions for the scattered outputs under readout [1,2].
Thereby, the resulting reconstructed output was obtained in the 2D Laplace domain whence the spatial information would be found only by performing a 2D Laplace inversion. Laplace inversion in 2D was attempted by testing a prototype function for which the analytic result was known using two known inversion algorithms, viz., the Brancik and the Abate . The results indicated notable differences in the 3D plots between the algorithms and the analytic result, and hence were somewhat inconclusive. In this paper, we take a closer look at the Brancik algorithm in order to understand better the implications of the choices of key parameters such as the real and imaginary parts of the Bromwich contour and the grid sizes of the summation operations.
To assess the inversion findings, three prototype test cases are considered for which the analytic solutions are known. For specific choices of the algorithm parameters, optimal values are determined that minimize errors in general. It is found that even though errors accumulate near the edges of the grid, overall reasonably accurate inversions are possible to obtain with optimal parameter choices that are verifiable via cross-sectional views. Further work is ongoing whereby the optimized algorithm is to be applied to the 3D holography problem.
Available at: http://works.bepress.com/monish_chatterjee/64/