Wavelet-domain regularized deconvolution for ill-conditioned systems

We propose a hybrid approach to wavelet-based image 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 is non-invertible. Using a mean-square-error metric, we strike an optimal balance between Fourier-domain regularization that is matched to the system and wavelet-domain regularization that is matched to the signal. Theoretical analysis reveals that the optimal balance is determined by economics of the input signal wavelet representation and the operator structure. The resultant algorithm is fast, O(N log₂ ² N) where N denotes the number of samples, and is well-suited to data with spatially-localized phenomena such as edges. In addition to enjoying asymptotically near-optimal rates of error decay for some systems, the algorithm also achieves excellent performance at fixed data lengths. In simulations with real data, the algorithm outperforms the conventional LTI Wiener filter and other wavelet-based deconvolution algorithms in terms of both visual quality and MSE performance.