Image Processing Based On Cartoon And Texture

Image restoration is often formulated as an inverse problem:f=Au+ε, here,f∈Rl is the known observed image, u∈Rn is the unknown true iamge,εis a white Gaussian noise with varianceσ, A∈Rl×n is a linear operator.For the above questions,we done two part of the work in this paper,it is image denoising and image inpainting.Since the above problems are typically ill posed,it is standard to use a regularization technique to make them well-posed. In Chapter 2 Section 1 of this paper,first,we give the isotropic ROF model got by the regularization technique to solve the problem of image denoising.The model can often be formulated as the following minimization problem: here, (?)χu. (?)yu is the difference operator of u,||.||1 denotes the L1 norm. Second,we give the split Bregman iteration of the ROF model. At last,we provide several numerical simulation,from the simulation result of the ROF denoising model,we can see,the TV model can effectively remove noise and preserve edges,but the model does not recognize well texture from noise.Therefore part of texture information was removed in the process of removing noise.In Chapter 2 Section 2 of this paper,first,we give the denoising method which extract the detail of texture from the image got by the ROF model,it is the denoising method based on nonlocal total variation functional given by Xiaoqun Zhang in [20]. This method used similarities in images to synthesize textures and can often be formulated as the following minimization problem: here,||.|| stands for the L1 norm,ωis the nonnegative symmetric weight function fromΩ×Ωto R,second,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the nonlocal total variation functional denoising model,we can see,when this method synthesizes textures, the noise is brought back to the image.Since some wavelet have been proved or verified to represent the texture part well,we consider using wavelet to extract the detail of texture from the image got by the ROF model.In Chapter 2 Section 3 of this paper, first,we combine the denoising method based on wavelet shrinkage with the method based on total variation,give the denoising model based on cartoon and texture. This model can be formulated as the following minimization problem: here, uc is the cartoon part of image, uT is the texture part of image, uC+uT is the denoised image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely.Second,similar to the above two model,we use the split Bregman method to solve it.Third,we prove the convergence of the model.At last,we provide several numerical simulation.from the simulation result of the denoising model based on cartoon and texture,we can see,the denoising method provided in this paper can extract the detail of texture from the image got by the ROF model and get the better denoisng effect.Due to the above discussion about the image denoising model based on cartoon and texture and the regularization of image restoration problem,in chapter 3,first,we provide the image inpainting model based on cartoon and texture,this model can be formulated as the following minimization problem: subject to PΛ(uC+uT)=PΛf. here, uc is the cartoon part of image, uT is the texture part of image, uC+UT is the restored image, w is the framelet which can represent the texture part sparsely but not represent the cartoon part sparsely, PΛis projection.Second,similar to the denoising model,we use the split Bregman method to solve it.At last,we provide several numerical simulation,from the simulation result of the image inpainting model based on cartoon and texture,we can see,the image inpainting method provided in this paper get better effect.