The observed signal (e.g. image) is often degraded (e.g. blurred image) due to the constraints of the observation and other technical conditions. The aim of deconvolution is to restore the observed to the undistorted as exact as possible. Generally speaking, the process of degrading is a process of low-pass filtering, so the deconvolution is a classical ill-conditioned problem and must be regularized to make a stable estimation of the original signal. There have been many methods of deconvolution including based on frequency-domain, space-domain, wavelet-domain and other transformation-domain. Wavelet-domain deconvolution algorithms have same outstanding advantages over other non-wavelet-domain methods. The most important one is that the singularities of signal (e.g. the edges of image) can almost be preserved during the processing in wavelet domain thanks to the time-frequency locality and other characteristics of wavelet transformation. Two classes of methods can be summarized for deconvolution in wavelet domain, one is wavelet shrinkage and the other one is MAP estimation based on Bayesian deconvolution theory. The wavelet-domain deconvolution methods in the two classes are studied in this paper including the following. An image restoration algorithm based on Kalman filtering in multi-scale space is proposed for effectively restoring blurred image. We first build the observation equation of Kalman filtering in frequency domain through the de-blurring models in each scale and the state transformation equation from the wavelet reconstruction processes in frequency domain. The Fourier Transformation of the restoration result in the lowest resolution can be regarded as the original state vector for Kalman filtering, and then process predictions and rectifications on every low-frequency sub-band until to the scale (the finest resolution). Applying the inverse Fourier Transformation on the last result can get the final restored signal. The non-Gaussian bivariate distributions can represent accurately the dependencies between the coefficients and their parents, and work well in de-noising. In this paper, the bivariate distributions model is introduced into un-decimated wavelet domain image restoration. We discuss this restoration algorithm from the view of adaptive regularization. Because complex wavelets can provide both shift invariance and good directional selectivity, and the wavelet-doamin hidden Markov tree (HMT) model can represents exactly the distributions of the wavelet coefficients, a linear image restoration algorithm based on the wavelet-domain HMT model, and a feasible, fast method to estimate the model parameters adaptively are proposed. The restoration algorithm models the prior distribution of the real-world image by wavelet-domain HMT model and converts the restoration problem to a constrained optimization one which can be solved through the steepest descend method.
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