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In this paper a one-dimensional inverse-scattering problem is considered. Two approximate inversion formulae are derived. Using these formulae, approximate permittivity and conductivity profiles of an inhomogeneous lossy medium are simultaneously generated from the partial knowledge of the reflection coefficient in the frequency domain. These formulae are deduced through the application of linear transformations to two nonlinear differential equations for the reflection coefficient in the frequency domain. Discrete versions of the inversion formulae are also developed. The transformed equations for the reflection coefficient in the frequency domain are applicable irrespective of whether the derivative of the unknown permittivity profile exists. An example to demonstrate these approximate inversion formulae is given.

Masks are used in optical image processing. They are used to generate gradient maps. These maps are applicable to the enhancement of feature extraction and edge detection. Lagrange mask is presented in this letter and criteria for the characterizations of mask performance are given. Through an illustration the performance of the presented mask is demonstrated where it is compared to that of Gabor mask. Results from the illustration support the applicability and suitability of Lagrange mask for the generation of gradient maps from a noise corrupted optical image.