So what can we say about image restoration? We can see that inverse filtering is a very easy and accurate way to restore an image provided that we know what the blurring filter is and that we have no noise. Because an inverse filter is a high pass filter, it will tend to amplify noise as was presented in our results. The second way of inverse filtering was through an iterative procedure. Since that is more or less an averaging method, it deals a little better with noise by averaging it out. But both methods do not deal well with noise. We must use some method of restoration which would trade off inverse filtering with de-noising.

Wiener filtering is the optimal tradeoff between the inverse filtering and noise smoothing. It can be interpreted as a inverse filtering step followed by a noise attenuation step. However, to implement the Wiener filter we have to estimate the power spectrum of the original image from the corrupted observation. This is the main task in making the Wiener filtering work well in practice.

Since the Wiener filtering contains the inverse filtering step, it amplifies the noise when the blurring filter is not invertible. More importantly, the amplified noise is unattended. To remove the amplified noise the best approach is to remove the noise using wavelet thresholding. Donoho's classical approach conatins two separate step, inverse filtering and wavelet denoising. It has not control over the overall restoration performance. One new approach, recently proposed at Rice University, comprises Fourier-domain system inversion followed by wavelet-domain noise suppression. It regularizes the inverse filtering by introducing a parameter into the Wiener filtering, and the optimal regularized parameter is chosen to minimize the overall MSE. The fact that the implementation of the Wiener filtering and regularized inverse filtering involve the estimation of the power spectrum of the original image inspires a new wavelet-based restoration approach. Since we believe the wavelet coefficients of the image are better modeled to estimate the power spectrum, we propose to exchange the order of the inverse filtering and wavelet tranform. This results in a subband restoration approach, meaning we do the inverse filtering and denoising for each subband, and the advantage of this algorithm is that we can perform different regularized inverse filtering for different subband.

We saw in the Blind Deconvolution section that it is possible to restore our image without having specific knowledge of our degradation filter, additive noise, or image spectral density. However, not knowing our degradation filter **h** imposes the strictest limitations on our restoration capabilities. The methods we examined all required zero phase degradation filters. There are methods that estimate phase, but they are very tricky and were not attempted in this project. We saw that our MSE was the worst by a small amount when we didn't know **h**, **Suu**, or **Snn** and used the Jain restoration approach Visually, however, the image was not very clear. We did not get our Stockham PSE approach to give us improvements on the degraded image. For the degradations we used, the PSE restoration was always inferior to Wiener restoration. This is because PSE restoration does not use inverse filtering, so the denominator in the PSE equation must be extremely accurate.

Finally, all of the homomorphic blind deconvolution techniques require iterations through either breaking up the degraded image or using multiple degraded images for estimation. For a given image size, we are limited in the number of blocks we can break our image into. For multiple degraded images, we may be limited by how many image snapshots we can obtain. So we are limited in both cases by how many iterations we can average over, and this profoundly affects our estimations. This is one of the main drawbacks that we found in homomorphic techniques.