Application Research Of Variable Exponent Functional In Inverse Problems Of Image Processing

With the development of the theory of variable exponent spaces,such as variable exponent Lebesgue space and variable exponent Sobolev space,more and more physical objects which have the local properties of "pointwise dissimilarity" can be described by those functions in the variable exponent spaces.This phenomenon is helpful to open a new situation of applying variable exponent spaces.Especially it can be applied to modeling and solving for the variational problems.So it has not only made a breakthrough in the application of variable exponent spaces and variational problems,but also provided a new tool and theoretical basis for studying those objects with local properties.In this thesis,from the view of inverse problem,by using the regularization methods,several variational models based on the variable exponent Lebesgue space and variable exponent Sobolev space are made for image restoration,image denoising and image enhancement problems.By making full use of the theory of variable exponent space and optimization,the existence and uniqueness of solutions of those models can be proved theoretically.And this work is also contributed to discuss the feasibility of constructing local adaptive models in theory.In addition,this thesis also attempts to reflect and utilize the local characteristic and the priori information of images with the pointwise dissimilarity.In this thesis,the experiment results further support the premise that the image has the local characteristic and prove the validity and the advantage of variational models to deal with this kind of objects.The main research emphasises are shown as following:(1)An image restoration model based on the image manifold is proposed in the form of variable exponent functional.Firstly,the model is based on the related geometric theory of differential manifold and the relationship between manifold features and local geometric features of images.The whole image is restructured according to the local geometry similarity on the manifold.Secondly,the quantitative methods to distinguish geometric similarity of image manifold are given.Finally,the variable exponent functional combines the local properties by the manifold regularization method organically to form a local regularized variable exponent model.The existence and uniqueness of the solution of the model are proved by using the theory of variable exponent space and the convex optimization theory in this thesis.In addition,by means of the variational method,the "pointwise" restoration ability of this model is analyzed mathematically.In the end,the advantage and the validity of the proposed restoration model based on the image manifold are tested by the fast numerical algorithms.(2)A denoising model based on noise estimation and local feature is proposed.Firstly,using geometric features,an estimation method of noise variance is improved.And according to the solutions obtained from those optimization models with different energy functional and the corresponding residuals,the estimatied noise variance is utilized to define the first term of the variable exponent denoising model.In addition,the local features of the image are also used to select the form of the regularization term to let them more 'adaptive'.In this thesis,the proof of the existence of the solution is also finished.By using the proper iteration algorithm,the comparison of the experimental results can be made to other models.Those results show that this model has certain advantages in restoring images with big noise.(3)The Retinex image enhancement models in the variable exponent space is proposed.Based on the Retinex theory,obtaing the illumination component and reflection component of the degraded image is a kind of inverse problem.For the different local features of their own,two kinds of variable exponent functional Retinex models are proposed combing the local image information.The qualitative analysis on the enhancement model in the variable exponent space is also completed.The regularization method is used to overcome the illness of the original problem;the variable exponent functional and image features are useful to achieve more accurate illumination and reflection.The above two factors improve the effect of image enhancement.The split-Bregman algorithm is used in the numerical experiments for the proposed model.The results from artificial images,natural scenes images and special scene images show the advantages of the variable exponent Retinex enhancement model which is better than other Retinex enhancement models in the visual sense.