We present an iterative deconvolution algorithm that minimizes a functional with a non-quadratic wavelet-domain
regularization term. Our approach is to introduce subband-dependent parameters into the bound optimization
framework of Daubechies et al.; it is sufficiently general to cover arbitrary choices of wavelet bases
(non-orthonormal or redundant). The resulting procedure alternates between the following two steps:
1. a wavelet-domain Landweber iteration with subband-dependent step-sizes;
2. a denoising operation with subband-dependent thresholding functions.
The subband-dependent parameters allow for a substantial convergence acceleration compared to the existing
optimization method. Numerical experiments demonstrate a potential speed increase of more than one order of
magnitude. This makes our "fast thresholded Landweber algorithm" a viable alternative for the deconvolution
of large data sets. In particular, we present one of the first applications of wavelet-regularized deconvolution to
3D fluorescence microscopy.
We propose a generalization of the Cohen-Daubechies-Feauveau (CDF) and 9/7 biorthogonal wavelet families. This is done within the framework of <i>non-stationary</i> multiresolution analysis, which involves a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. We consider a dual pair of such multiresolutions, where the scaling functions at a given scale are mutually biorthogonal with respect to translation. Also, they must have the shortest-possible support while reproducing a given set of exponential polynomials. This constitutes a generalization of the standard polynomial reproduction property. The corresponding refinement filters are derived from the ones that were studied by Dyn et al. in the framework of non-stationary subdivision schemes. By using different factorizations of these filters, we obtain a general family of compactly supported dual wavelet bases of <i>L</i><sub>2</sub>. In particular, if the exponential parameters are all zero, one retrieves the standard CDF B-spline wavelets and the 9/7 wavelets. Our generalized description yields equivalent constructions for E-spline wavelets. A fast filterbank implementation of the corresponding wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. This new scheme offers high flexibility and is tunable to the spectral characteristics of a wide class of signals. In particular, it is possible to obtain symmetric basis functions that are well-suited for image processing.