Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem

In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

Central and Periodic Multi-Scale Discrete Radon Transforms

The multi-scale discrete Radon transform (DRT) calculates, with linearithmic complexity, the summation of pixels, through a set of discrete lines, covering all possible slopes and intercepts in an image, exclusively with integer arithmetic operations. An inversion algorithm exists and is exact and fast, in spite of being iterative. In this work, the DRT forward and backward pair is evolved to propose two faster algorithms: central DRT, which computes only the central portion of intercepts; and periodic DRT, which computes the line integrals on the periodic extension of the input. Both have an output of size N×4N, instead of 3N×4N, as in the original algorithm. Periodic DRT is proven to have a fast inversion, whereas central DRT does not. An interesting application of periodic DRT is its use as building a block of discrete curvelet transform. Central DRT can provide almost a 2× speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.