小波与滤波器组-中科院wavelets and filter

1、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， 2001.3Mark R. Banham and A.K. Katsaggelos, Digital Image Restoration, IEEE Trans. Image Proc.

2、 , March 1997 R.L.Lagendijk,J.Biemond. Iterative identification and restoration of images. Kluwer Academic Publishers,1991P.Mller, B.Vidakovic. Bayesian inference in wavelet-based models. 1999 Springer-Verlag New York, IncImage RestorationDegradation modelIll-posed problemRegularizationBayes framewo

3、rk for image restorationImage Restoration MethodsFrequency domain methods Spatial domain methodsWavelet domain methodsImage RestorationDegradation modelDegradation model (Continous form)h+f (u,v)g (u, v) (u, v): Point Spread Function, PSF or Blur Function : Pointwise nonlinear operation: Additive no

4、ise: True image: Observed image Linear vs. Non-linearMany types of degradation can be approximated by linear, space invariant processesNon-linear and space variant models are more accurateDifficult to solveUnsolvable Image RestorationDegradation modelHere, We only consider the linear, space invarian

5、t PSF ! Image RestorationDegradation modelDegradation model (Discrete form)Size:f : N1N2h : M1M2g : (N1M1-1)(N2+M2-1)Image RestorationDegradation modelMatrix-Vector representation of image restoration problem: Stack f, g, row-by-row or column-by-column to form vector representations of these 2-D var

6、iables theoretic analysis more easily Degradation model (Discrete form)Size:f : N1N21H : (N1M1-1)(N2+M2-1)N1N2 g : (N1M1-1)(N2+M2-1)1H is a block toeplitz matrixImage RestorationDegradation modelIll-posed problemRegularizationBayes framework for image restorationImage Restoration MethodsFrequency do

7、main methods Spatial domain methodsWavelet domain methodsImage Restoration Ill-posed ProblemInverse filtering solutionH is ill-conditioned which makes image restoration problem an ill-posed problemSolution is not stable: not continuely depend on the observed data gRestoration Problem: g, h and stati

8、stical properties of noise are Known, the task is to estimate the true image f Another perspective Least square solutionImage Restoration Ill-posed ProblemSingular Value position of H :U is MM orthornormal matrixV is NN orthornormal matrixAnother perspective Least square solution (Cont.)Image Restor

9、ation Ill-posed ProblemBy 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-freeExact H Exact H not exact HImage Restoration Ill-posed Problem Examples:Image RestorationDeg

10、radation modelIll-posed problemRegularizationBayes framework for image restorationImage Restoration MethodsWavelet domain methodsImage RestorationRegularization Generally speaking, any regularization method tries to analyze a related well-posed problem whose solution approximates the original ill-po

11、sed 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; andwell-posed stochastic extensions of ill-posed

12、 problems. For g = Hf + h, the regularization method constructs the solution asu(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 ope

13、rating 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

14、 f might be, which makes the solution more stable.Image RestorationRegularization Solution FormulationImage RestorationRegularization A case studyConsider By SVD position of H,we get The introduction of reduced the affection of small singular values of H on the solution. Image RestorationDegradation

15、 modelIll-posed problemRegularizationBayes framework for image restorationImage Restoration MethodsWavelet 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 probab

16、ility p(f|g) asAccording to Bayes rule, P(f) is the a-priori probability of the unknown image f. We call it the prior modelP(g) is the probability of g which is a constant when g is givenp(g|f) is the conditional probability density function (pdf) of g. We call it the sensor model, which is a descri

17、ption of the noisy or stochastic processes that relate the original unknown image f to the measured image g.Image RestorationBayes Framework MAP - DerivationBayes interpretation of regularization theory Noise termPrior termImage RestorationBayes Framework Noise TermAssume Gaussian noise of zero mean

18、, the standard deviation is MAP Derivation（Cont.）Image RestorationBayes Framework Prior TermThe prior knowledge of the original image refers to the a-priori belief that the state of a pixel is entirely determined by the states of its neighboring pixels. Specifically, it is expected that pixels close

19、 to each other tend to have the same or similar brightness values.A Markov Random Field (MRF) is a probabilistic process in which all interaction is local. It is an appropriate model to represent the local property in the image. However, MRF is difficult to estimate. There is an equivalence between

20、Gibbs distribution and MRF.Gibbs distribution allows the modeling of local structure through energies which describes the interactions of pixels within each clique of the neighborhood. MAP Derivation（Cont.）Image RestorationBayes FrameworkImage RestorationWavelet domain methodsWavelet domain represen

21、tation of image restoration problemImage RestorationWavelet domain methodsA practical wavelet domain restoration algorithm (M. Belge, 1999) position Strategy Image RestorationWavelet domain methodsA practical wavelet domain restoration algorithm (M. Belge, 1999)Prior Model: GGD: is a scale parameter

22、 similar to the standard deviation of a gaussian densityImage RestorationWavelet domain methodsA practical wavelet domain restoration algorithm (M. Belge, 1999)The cost functionalImage RestorationWavelet domain methodsA practical wavelet domain restoration algorithm (M. Belge, 1999)Taking the gradie

23、nt of the cost function we getA fixed point iteration to solve for f*We can solve this equation with conjugate gradient algorithm Image RestorationWavelet domain methodsA practical wavelet domain restoration algorithm (M. Belge, 1999)Practical implementation of the algorithmProblem: The size of is t

24、oo large to be implemented on nowadays computer, and cannot be approximated by circulate matrix !Solution: Transform the iteration problem back to spatial domainNow : the problem can be solved by convolution, wavelet position and reconstruction only. We need not generate the large scale matrix reall

25、y!Image RestorationWavelet domain methodsComparisionImage RestorationWavelet domain methodsOther wavelet domain methodsWavelet-based Regularized Deconvolution, WaRD Wavelet-Vaguelette position Multiscale kalman filtering Multiscale maximum entropy deconvolutionWavelet domain gaussian scale mixtureHi

26、dden Markov Tree Model based restorationLocal gaussian model based restoratoinWavelet domain EM algorithmImage RestorationWavelet domain methodsFurther reading (1):M.R. Banham and A.K. Katsaggelos, Spatially-Adaptive Wavelet-Based Multiscale Image Restoration , IEEE Trans. Image Proc. Vol. 5 , April

27、 1996 , 619634Matthew S. Crouse, Robert D. Nowak and Richard G. Baraniuk, Wavelet-Based Statistical Signal Processing Using Hidden Markov Models, IEEE Trans. Signal Proc. Vol. 46 , April 1998 , 886902J. Portilla and E.P. Simoncelli，Image restoration using Gaussian scale mixtures in the wavelet domain, Proc. 10th IEEE Intl Conf on Image Processing, Barcelona, Spain. Sep 2003J. Portilla, V. Strela, M. Wainwright and E.P. Simoncelli, Image denoising using scale mixtures o