Sparse regularization of inverse problems by biorthogonal frame thresholding

09/20/2019

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by   Jürgen Frikel, et al.

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We analyze sparse frame based regularization of inverse problems by means of a biorthogonal frame decomposition (BFD) for the forward operator. The BFD allows to define a non-iterative (direct) frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse ℓ^1-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of ℓ^1-regularization as well as the BFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.