In this paper we present a non-linear edge-preserving solution to linear inverse scattering problems based on optimal basis-function decompositions. Optimality of the basis functions allow us to (i) reduce the dimensionality of the inverse problem; (ii) devise non-linear thresholding operators that approximate minimax and that improve the signal-to-noise ratio. We present a reformulation of the standard generalized least-squares/truncated-SVD formulation of the seismic inversion problem into a formulation based on thresholding, where the singular values, vectors and linear estimators are replaced by quasi-singular values, basis-functions and thresholding. To limit the computational burden we use a Monte-Carlo method to compute the quasi-singular values. With the proposed method we aim to significantly improve the signal-to-noise ratio in the model space and hence the resolution of the seismic image.
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