小波域图像复原,彭思龙 中科院自动化所 国家专用集成电路设计工程技术研究中心,Image Restoration,Image Restoration,References:,H.C. Andrews and B.R. Hunt , Digital Image Restoration , Englewood Cliffs, NJ:Prentice-Hall, 1977 邹谋炎 ， 反卷积和信号复原， 国防工业出版社， 2001.3 Mark R. Banham and A.K. Katsaggelos, Digital Image Restoration, IEEE Trans. Image Proc. , March 1997 R.L.Lagendijk,J.Biemond. Iterative identification and restoration of images. Kluwer Academic Publishers,1991 P.Mller, B.Vidakovic. Bayesian inference in wavelet-based models. 1999 Springer-Verlag New York, Inc,Image Restoration,Degradation model Ill-posed problem Regularization Bayes framework for image restoration Image Restoration Methods Frequency domain methods Spatial domain methods Wavelet domain methods,Image RestorationDegradation model,Degradation model (Continous form),h,+,f (u,v),g (u, v), (u, v),Linear vs. Non-linear,Many types of degradation can be approximated by linear, space invariant processes Non-linear and space variant models are more accurate Difficult to solve Unsolvable,Image RestorationDegradation model,Here, We only consider the linear, space invariant PSF !,Image RestorationDegradation model,Degradation model (Discrete form),Size:,f : N1N2 h : M1M2 g : (N1M1-1)(N2+M2-1),Image RestorationDegradation model,Matrix-Vector representation of image restoration problem: Stack f, g, row-by-row or column-by-column to form vector representations of these 2-D variables theoretic analysis more easily,Degradation model (Discrete form),Size:,f : N1N21 H : (N1M1-1)(N2+M2-1)N1N2 g : (N1M1-1)(N2+M2-1)1 H is a block toeplitz matrix,Image Restoration,Degradation model Ill-posed problem Regularization Bayes framework for image restoration Image Restoration Methods Frequency domain methods Spatial domain methods Wavelet domain methods,Image Restoration Ill-posed Problem,Inverse filtering solution H is ill-conditioned which makes image restoration problem an ill-posed problem Solution is not stable: not continuely depend on the observed data g,Restoration Problem: g, h and statistical properties of noise are Known, the task is to estimate the true image f,Another perspective Least square solution,Image Restoration Ill-posed Problem,Singular Value Decomposition of H :,U is MM orthornormal matrix V is NN orthornormal matrix,Another perspective Least square solution (Cont.),Image Restoration Ill-posed Problem,By simple computation:,It can be seen that if H have small singular values , then a small change in g or H will cause large change in the solution.,Noise-free Sinusoidal noise Noise-free Exact H Exact H not exact H,Image Restoration Ill-posed Problem,Examples:,Image Restoration,Degradation model Ill-posed problem Regularization Bayes framework for image restoration Image Restoration Methods Wavelet domain methods,Image RestorationRegularization,Generally speaking, any regularization method tries to analyze a related well-posed problem whose solution approximates the original ill-posed problem. The well-posedness is achieved by implementing one or more of the following basic ideas： restriction of the data; change of the solution space and/or topologies; modification of the operator itself; the concept of regularization operators; and well-posed stochastic extensions of ill-posed problems.,For g = Hf + h, the regularization method constructs the solution as u(f, g) describes how the real image data is related to the degraded data. In other words, this term models the characteristic of the imaging system. bv(f) is the regularization term with the regularization operator v operating on the original image f, and the regularization parameter b used to tune up the weight of the regularization term. By adding the regularization term, the original ill-posed problem turns into a well-posed one, that is, the insertion of the regularization operator puts some constraints on what f might be, which makes the solution more stable.,Image RestorationRegularization,Solution Formulation,Image RestorationRegularization,A case study,Consider,By SVD decomposition of H,we get,The introduction of reduced the affection of small singular values of H on the solution.,Image Restoration,Degradation model Ill-posed problem Regularization Bayes framework for image restoration Image Restoration Methods Wavelet domain methods,MAP (maximize a-posteriori probability),Formulate solution from statistical point of view: MAP approach tries to find an estimate of image f that maximizes the a-posteriori probability p(f|g) as According to Bayes rule, P(f) is the a-priori probability of the unknown image f. We call it the prior model P(g) is the probability of g which is a constant when g is given p(g|f) is the conditional probability density function (pdf) of g. We call it the sensor model, which is a description of the noisy or stochastic processes that relate the original unknown image f to the measured image g.,Image RestorationBayes Framework,MAP - Derivation,Bayes int