A general framework for soft-shrinkage with applications to blind deconvolution and wavelet denoising
- Justen, L.
- Ramlau, R.
DOIAbstract
AbstractWe consider the abstract problem of approximating a function ψ0∈L1(Rd)∩L2(Rd) given only noisy data ψδ∈L2(Rd). We recall that minimization of the corresponding Tikhonov functional leads to continuous soft-shrinkage and prove convergence results. If the noise-free data ψ0 belongs to the source space L1−u(Rd)∩L2(Rd) for some 0<u<1, we show convergence rates, which are order-optimal. We consider a priori parameter choice rules as well as the discrepancy principle, which is shown to be order-optimal as well. We then introduce a framework by combining soft-shrinkage with a linear invertible isometry and show that the results obtained for the abstract minimization problem can be transferred to applications such as blind deconvolution and ...
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