Research On Image Restoration Models And Algorithms By Combination X-let With Variational

X-let is a general definition of wavelets, including all the classical wavelets andtheir new progress, even those defined in future,which covers the high-dimensionalcontinuous wavelets, ridgelets, the second-generation curvelets, wave atoms and Log-Gabor wavelets et al related in this dissertation. The theory framework of X-let isrelatively mature and its application to image processing is the highlight of currentresearch. Another important method in mathematical image processing is variationalpartial differential equations, which has been applied to many fields in image process-ing and computer vision with good results.Image restoration is a kind of fundamental problem in image processing, which isan ill-posed inverse problem in the sense of Hadamard. The key point of solving thisproblem is how to use the priori knowledge to establish the corresponding mathemati-cal model, and obtain the well-posed solution.In this dissertation, we combine X-let with variational idea to explore and studyproblem of image restoration, and obtain a series of meaningful results. The main workcan be summarized as follows:1 Two theoretical results related to high-dimensional continuous wavelet trans-form and ridgelet frame are obtained. Firstly, the relationship of high-dimensionalcontinuous wavelet thresholding with pseudo-differential equation is theoretically ana-lyzed, which results in a new high-order nonlinear diffusion equation by using both theproperties of Fourier transform and operator representation theory in high-dimensionalcase. Secondly, a ridgelet-type structured admissible covering and associated framesusing Fourier basis are designed based on the ideology of constructing curvelet-typeframe. Moreover, we obtain three novel ridgelet-type frames by associating orthonor-mal ridgelet with the ridgelet-type covering.2 The novel variational models and algorithms for both image denoising and de-blurring are given with the second-generation curvelets. Firstly, we apply constraintof curvelet-type decomposition spaces as a regularizing term to variational model forimage denoising. Based on the equivalent modulus relationship between semi-norm ofcurvelet-type decomposition spaces and the weighted curvelet coefficients, solution ofthe proposed model equals to different curvelet shrinkages. Moreover, we prove that the decomposition model with constraint of semi-norm of curvelet-type decompositionspaces with negative degree of differentiability is equivalent to a denoising model withconstraint of L2 norm to the fidelity term on a given condition. Secondly, an iterativeregularization method and inverse scale space method of curvelets are introduced byapplying Bregman distance. Finally, a model for image deblurring is considered byemploying the sparse constraint of curvelet-type decomposition spaces. Especially, aniterative hard shrinkage algorithm and an adaptive stopping criterion are obtained bythe generalized conditional gradient method. These models and algorithms describedabove extend the wavelet method in Besov spaces. Moreover, experimental resultsshow that the use of curvelets in curvelet-type decomposition spaces is appropriate forcharacterizing image with rich structures.3 Several models and algorithms are presented with wave atoms. Firstly, a noveldenoising model for texture images is proposed by using Besov spaces, which results ina soft thresholding algorithm depending on both the scale of wave atoms and smoothingparameter. Secondly, considering the sparse representation of wave atoms for texturalimage, an iterative regularization moethod and an inverse scale spaces method for fullydiscrete wave atom coefficients are designed. Finally, as an extension of Perona-Malikmodel, we develop a nonlinear diffusion model by replacing Gaussian regularizationwith wave atoms regularization.4 A novel tensor diffusion model and a variational deblurring model arepresented with two tools of phase congruency and principal component analysis.Firstly,taking advantage of both the importance of phase and the merit of phase con-gruency for image feature detection, we design a new scatter matrix, then propose atensor diffusion model based on phase congruency. Secondly, a model for image de-blurring using principal component analysis is given. Then an iterative shrinkage al-gorithm is obtained in principal component analysis domain by employing the idea ofsurrogate functional. Due to the slow speed of convergence, we further offer a newaccelerated algorithm.