1. Copyright, 1996 © Dale Carnegie & Associates, Inc. Image Restoration Hung-Ta Pai Laboratory for Image and Video Engineering Dept. of Electrical and Computer Engineering The University of Texas at Austin Austin, TX 78712-1084

4. Degradation Model n In noise-free cases, a blurred image can be modeled as n In the DFT domain, Y(u,v) = X(u,v) H(u,v)

5. Inverse Filtering n Assume h is known (low-pass filter) n Inverse filter G(u,v) = 1 / H(u,v) n

6. Implementing Inverse Filtering

7. Lost Information

8. Problems with Inverse Filtering n H(u,v) = 0, for some u, v n In noisy case,

9. n Least Mean Square Filter Wiener Filter Formulation n In practice

10. Wiener Filter Results

11. Maximum-Likelihood (ML) Estimation n h is unknown n Solution is difficult n Parametric set  is estimated by n Assume parametric models for the blur function, original image, and/or noise

12. Expectation-Maximization (EM) Algorithm n Find complete set  : for z  , f(z)=y n Expectation-step n Choose an initial guess of  n Maximization-step

13. Subspace Methods n Observe

14. Subspace Methods n Several blurred versions of original image are available n Construct a block Hankel matrix  of blurred images n  =  , where  is a block Toeplitz matrix of the blur functions and  is a block Hankel matrix of the original image

15. Subspace Methods Results

16. Conclusions n Noise-free case: inverse filtering n Multichannel blind case: subspace methods n Blind case: Maximum-Likelihood approach using the Expectation- Maximization algorithm n Noisy case: Weiner filter

17. Further Reading n M. R. Banham and A. K. Katsaggelos "Digital Image Restoration, " IEEE Signal Processing Magazine, vol. 14, no. 2, Mar. 1997, pp. 24-41. n D. Kundur and D. Hatzinakos, "Blind Image Deconvolution," IEEE Signal Processing Magazine, vol. 13, no. 3, May 1996, pp. 43-64.