Study On Variational Models And Efficient Algorithms With Application To Image Processing

Thestudyonvariationalmodelsandoptimizationtechniquesisanimportantresearchaspect in the field of digital image processing. This thesis mainly focuses on gray imagerestoration and segmentation problems, and proposes some efficient models and algo-rithms to overcome the shortcomings and difficulties of the previous works. The usedtools mainly include the local/non-local(NL) total variation(TV), augmented Lagrangianapproach, low-rank matrix restoration, etc. The main work and innovation are embodiedas follows.1. This thesis analyzes the limitations and shortcomings of the previous two-stepmethods in the field of image restoration, and proposes a TV-Stokes model for image de-convolution. In the new model, we first utilize the commuting property of the convolutionoperators and the Bayesian formula to establish the TV-Stokes model for the tangentialfield estimation of the deblurred image. Then a new defined regularization term which isonly orientation-matching is included in the TV model for image restoration, and it over-comes the defects of the variational models which utilize the estimated tangential vectorto reconstruct the image in the second step of previous two-step methods. The proposedmodel can be solved efficiently by the augmented Lagrangian approach.2. The classical TV restoration models always use scalar regularization parameters,but they are unable to meet the demand of removing the noise in the homogeneous regionsand preserving the details in the texture areas meanwhile with the uniform parameter val-ues. This thesis proposes two adapted TV models and corresponding algorithms. Firstly,some random variable with respect to Gamma noise is defined for the distinguishment ofthe homogeneous regions and texture areas. Then the local constrained TV model andthe corresponding iterative algorithm of adjusting the vector parameter of the TV modelare established for the Gamma noise removal. Next, the problem of poissonian image de-blurring is considered. The (Poisson) local discrepancy function is used to distinguish thehomogeneous and texture regions, and then the local constrained TV model and spatiallyadapted TV algorithm are proposed. Experimental results demonstrate that the proposedmodel can recognize the image regions exactly, and then adjust the regularization param-eters adaptively to remove the noise and retain the image details simultaneously. 3. Compared with the TV models, the nonlocal TV models are able to better preservethe texture and details of images. However, the computational amount of the nonlocalmodels is very large. This thesis proposes a fast alternative minimization algorithm basedon the the idea of variable splitting and penalty techniques in optimization. The proposedalgorithm uses Taylor series approximation and a continuation scheme to accelerate itsimplementation,andisprovedtobemoreefficientthanthePBOSalgorithm. ThepreviousNL-TV models are based on the assumption of Gaussian noise. This thesis further usesthe minimize mean-square error (MMSE) estimator to establish the nonlocal TV modeland corresponding iterative algorithm for Gamma noise removal. Experiments show thatthe proposed NL-TV model outperforms the corresponding TV model.4. This thesis researches the fast computation of the minimization problem in thefield of image processing, and proposes a fixed point algorithm and a subspace optimiza-tion accelerating augmented Lagrangian approach. On the one hand, we transform theminimization problem into a fixed point equation, and use the Picard iteration to obtainthe corresponding fixed point. Then the solution of the original problem can be obtainedby the fixed point; on the other hand, a subspace optimization technique is adopted tosolve the sub-minimization problem in each iteration of the augmented Lagrangian ap-proach, and then a subspace optimization accelerating augmented Lagrangian approach isproposed for solving the TV deblurring model. Numerical experiments demonstrate thatthe proposed algorithms are very efficient.5. The inpainting problems in the space and wavelet domains are investigated, andtwo inpainting models are proposed in this thesis. In the field of image inpainting, weconsider the PCA dictionaries and the unknown image information, and transform the in-painting problem into low-rank and joint-sparse matrix recovery based on matrix decom-position and PCA transform. Then a new inpainting model based on the PCA dictionariesis obtained. In the field of wavelet domain inpainting, the computational amount of thealgorithms corresponding to TV(NL-TV) wavelet inpainting models is very large. Thisthesisproposesanewwaveletinpaintingmethodbasedonimagedecomposition. Thecor-responding iterative algorithm has a simpler structure, and it can improve the efficiencyof wavelet inpainting dramatically.6. TheEWCVTmodelforimagesegmentationisverysimpleandefficient, however,it is unable to preserve irregular edges such as some thin and slight edges, and is not suit- able for the non-Gaussian noise. In order to overcome the shortcomings of the EWCVTmodel,thisthesisproposesanovellabel-matchingregularizationCVT(centroidalVoronoitessellation) model. Compared with the EWCVT model, the proposed methods can im-prove the image segmentation accuracy, and therefore obtain the superior segmentationperformance.