&RS\ULJKW 'DOH &DUQHJLH $VVRFLDWHV ,QFImage RestorationHung-Ta PaiLaboratory for Image and Video EngineeringDept. of Electrical and Computer EngineeringThe University of Texas at AustinAustin, TX 78712-1084Degradation Model■ In noise-free cases, a blurred image canbe modeled asimage original:xfunction blur invariant-space linear : hh x y ∗=■ In the DFT domain, Y(u,v) = X(u,v) H(u,v)Inverse Filtering■ Assume h is known (low-pass filter)■ Inverse filter G(u,v) = 1 / H(u,v)■v)G(u, v)Y(u,v)(u,X =~Implementing Inverse FilteringLost InformationProblems with Inverse Filtering■ H(u,v) = 0, for some u, vnoise additive :nnhxy+∗=■ In noisy case,■ Least Mean Square Filter Wiener Filter Formulation[]v)(u,v)/S(u,Sv)H(u,v)(u,H v)G(u,xn2*+=■ In practice Kv)H(u,v)(u,H v)G(u,*+=2Wiener Filter ResultsMaximum-Likelihood (ML)Estimation■ h is unknown )}|p(y maxarg{ mlθθθ=■ Solution is difficult■ Parametric set θ is estimated by■ Assume parametric models for the blurfunction, original image, and/or noiseExpectation-Maximization (EM)Algorithm■ Find complete set Ζ: for z ∈ Ζ, f(z)=y■ Expectation-step■ Choose an initial guess of θ■ Maximization-step]y,|)|[p(z)|(kkθθθθEg=)|( max argkθθθθg=+1kSubspace Methods=43210210210210210432321210aaaaa bbb000bbb000bbbbbb aaaaaaaaa■ ObserveSubspace Methods■ Several blurred versions of originalimage are available■ Construct a block Hankel matrix Χ of blurredimages■Χ = Η Σ, where Η is a block Toeplitzmatrix of the blur functions and Σ is ablock Hankel matrix of the original imageSubspace Methods ResultsConclusions■ Noise-free case: inverse filtering■ Multichannel blind case: subspacemethods■ Blind case: Maximum-Likelihoodapproach using the Expectation-Maximization algorithm■ Noisy case: Weiner filterFurther Reading■ M. R. Banham and A. K. Katsaggelos"Digital Image Restoration, " IEEESignal Processing Magazine, vol. 14,no. 2, Mar. 1997, pp. 24-41.■ D. Kundur and D. Hatzinakos, "BlindImage Deconvolution," IEEE SignalProcessing Magazine, vol. 13, no. 3,May 1996, pp.
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