Although the Wiener filtering is the optimal tradeoff of
inverse filtering and noise smoothing, in the case when the blurring filter is
singular, the Wiener filtering actually amplify the noise. This suggests that a
denoising step is needed to remove the amplified noise. Wavelet-based denoising
scheme, a successful approach introduced recently by Donoho, provides a natural
technique for this purpose. Therefore, the image restoration contains two
separate steps: Fourier-domain inverse filtering and wavelet-domain image
denoising. The digram is shown as follows.

Donoho's approach for image restoration improves the
performance, however, in the case when the blurring function is not invertible,
the algorithm is not applicable. Furthermore, since the two steps are separate,
there is no control over the overall performance of the restoration. Recently,
R. Neelamani et al. proposed a wavelet-based deconvolution technique for
ill-conditioned systems. The idea is simple: employ both Fourier-domain
Wiener-like and wavelet-domain regularization. The regularized inverse filter
is introduced by modifying the Wiener filter with a new-introduced parameter:

The parameter can be optimally selected to
minimize the overall mean-square error. The diagram of the algorithm is
displayed as follows.

The implementation of the regularized inverse filter involves
the estimation of the power spectrum of the original image in the spatial
domain. Since wavelet transforms have good decorrelation property, the wavelet
coefficients of the image can be better modeled in a stochastic model, and the
power spectrum can be better estimated. This inspires a new approach: changing
the order of the regularized inverse filtering and the wavelet transform. (See
the following digram)

This way the both inverse filtering and noise smoothing can
be performed in wavelet domain. Specifically, the power spectrum of the image
in a same subband can be estimated under the assumption that the wavelet
coefficients are independent. Therefore, the power spectrum is just the
variance of the wavelet coefficients. We note that the exchange of the order of
inverse filtering and wavelet transform is valid only when undecimated wavelet
transform is used and the blurring function is separable. Therefore, for
interpretation we can exchange the order of the blurring operation and the
wavelet transform, which means that the inverse filtering cancels the blurring
in the wavelet domain. So, wavelet thresholding results in a reasonable
estimate. The above explanation can be visualized using the following figure.

**Simulation**

As usual
we corrupted the standard 256x256 lena test image by convolving wit the simple
4x4 square blurring filter

and adding zero-mean white Gaussian noise of variance 100.
The three introduced wavelet-based image restoration algorithms are applied to
the corrupted image, and the results are reported in the following table.
According to the visual performance and the mean square error, the algorithms
improve the restoration performance. However, the denoising step uses wavelet
thresholding to remove the noise, the images are blurred a little bit again,
although the MSE is improved.

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