Recursive Linear Smoothing for the 2-D Helmholtz Equation

Résumé du livre

A fast algorithm for reconstructing images governed by a 2-D Helmholtz equation is presented. The computational complexity of the algorithm is O(NMlogM) or O(N(M-sq) depending on boundary conditions, where N and M are the number of spatial grid points in the x and y directions respectively. This problem arises when smoothing a large number of images governed by the 2-D wave equation, because a Fourier transform in time gives a new set of images governed by the Helmholz equation. When the images come from a scattering process, we show that a linear least-squares Born inversion of the wave field amplitudes can be performed during the smoothing procedure without changing the computational complexity. The smoothing algorithm is well posed, and the sample functions of the smoothed estimate possess smoothness properties consistent with the Helmholtz equation.