This paper applies wavelets to the inverse scattering problem of reconstructing the permittivity of a one-dimensional dielectric medium from its reflection response to an electromagnetic plane wave. All multiple scattering effects, which can be significant in a sharply-varying medium, are included. The significance of this algorithm is that it works even in the presence of time and frequency support-limited noise or interference, as long as a single noise free wavelet component can be isolated (we can also switch from one scale to another). This differs from previous approaches, in which noise filtering is restricted to Fourier domain filtering. The added freedom of choice of the wavelet basis function makes it much easier to isolate a noise-free wavelet component; one such suffices to reconstruct the medium. Alternatively, if a noise-free wavelet component cannot be found, orthogonality of the wavelet components allows different noisy reconstructed permittivities to be combined to yield less-noisy reconstructed permittivities.
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