Variational Regularization Models Of Image Segmentation-Nonconvex, Sparse Theories And Algorithms

Recently, many researches are focused on the applications of variationalregularization method in image processing. Many classical models and methods werebuilt up. Some new notations, models and methods are proposed in this thesis. Imagesegmentation is mainly concernd. The main work of this thesis includes the followingaspects:Variational model for image segmentation consists of data term and regularizationterm. Usually data term is chosen as L2norm, and regularization term is determined bythe prior assumption. We present a novel model in the maximum a posterior probabilityframework. A new reweighted L2norm is used in the data term, which shares theadvantages of L2and mixed L21norm. An edge weighting function is addressed in theregularization term, which enforces the ability to reduce the outlier effects and preserveweak edges. An improved region-based graph cuts algorithm is proposed to solve thismodel efficiently. Numerical experiments show our method can get better segmentationresults, especially in terms of removing outliers and preserving weak edges.A general equivalent model is introduced based on the convex relaxation model ofa class of vector-valued minimization problems. The presented model can be solvedwith the split-Bregman algorithm. Therefore the computational efficiency is greatlyimproved. The method is applied to the Vese-Chan multi-phase segmentation model andMumford-Shah model. Numerical experiments show our method has fast computingspeed and good segmentation results, and is robust to the initial condition.Image superpixels segmentation is considered as the subspace clustering problem.A new constraint condition is presented to be equivalent to use the clean data asdictionary. The nonconvex proximal p-norm of the coefficients matrix is used forsparse constraint, and the nonconvex proximal p-norm of the singular values of thecoefficients matrix is used for low-rank constraint. Then a nonconvex minimizationmodel is proposed. The augmented Lagrangian method and the AM (alternatingminimization) method are applied for solving the unknown matrices. The numericalexperiments show that the presented constraint condition is better than using the originaldata as dictionary, and the nonconvex proximal p-norm has better segmentation resultthan the convex nuclear norm andl1norm. Two improved models of the TV (total variation) regularization are presented. Oneis the Mumford-Shah model based on weighted TGV (total generalized variation). Theweighted TGV is defined. The second order weighted TGV semi-norm of images isused as the regularization term. Besides, the second order weighted TGV semi-norm ofthe level set function is used for approximating length of the boundary. The numericalcalculation model for solving the unknown functions is presented by using alternatingsplit-Bregman method、 Fenchel dual method and FISTA (fast iterativeshrinkage-thresholding algorithm) separately. Simulation results show that the use ofsecond order weighted TGV semi-norm of images has better denoising effect than thecommon L2norm of gradient norm and the weighted TV semi-norm. And the result ofedge detection is better than the traditional TV semi-norm and weighted TV semi-normby using second order weighted TGV semi-norm of the level set function toapproximate length of the boundary. Another is the active contour model based on thenonlocal TV regularization. Based on the continuous global minimization approach forthe active contour model, the nonlocal TV is used for the regularization term ofboundary length. Numerical experiments show the new model can segment the mainstructures and the useful fine structures well.The nonlinear complex diffusion method based on topological optimization ispresented. Based on the topological optimization idea, the linear complex diffusioncoefficients at each pixel are perturbed. The diffusion coefficients corresponding to thesamllest topological derivative are optimal. Then the pixels having the enough smalltopological derivatives are chosen, and diffusion is applied to them using the optimaldiffusion coefficients. Stop criteria of the algorithm is introduced. Diffusion coefficientschosen here have the anisotropic property thus can remove noise along edges andpreserve edges well. Experiments show the real part shows better denoising effect andedges are well preserved in the imaginary part after the original noisy image isprocessed using our method. Besides, our method can reduce staircase effect effectively.