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