Wavelet-based deconvolution using optimally regularized inversion for ill-conditioned systems

We propose a hybrid approach to wavelet-based deconvolution that comprises Fourier-domain system inversion followed by wavelet-domain noise suppression. In contrast to conventional wavelet-based deconvolution approaches, the algorithm employs a regularized inverse filter, which allows it to operate even when the system in non-invertible. Using a mean-square-error (MSE) metric, we strike an optimal balance between Fourier-domain regularization (matched to the system) and wavelet-domain regularization (matched to the signal/image). Theoretical analysis reveals that the optimal balance is determined by the economics of the signal representation in the wavelet domain and the operator structure. The resulting algorithm is fast (O(Nlog22N) complexity for signals/images of N samples) and is well-suited to data with spatially-localized phenomena such as edges. In addition to enjoying asymptotically optimal rates of error decay for certain systems, the algorithm also achieves excellent performance at fixed data lengths. In simulations with real data, the algorithm outperforms the conventional time-invariant Wiener filter and other wavelet- based deconvolution algorithms in terms of both MSE performance and visual quality.