Fractional-order Variational PDE Based Image Modeling And Denoising Algorithms

In image processing, it is important to preserving useful structures such as edges and textures. To preserve these important structures, we need to distinguish and deal with these different components of image such as cartoon, edge, texture and noise separately, and then we need modeling these components appropriately.In this paper, using wavelet analysis theory and methods, Function space image mod-eling methods and fractional calculus, new methods for modeling different components of image are proposed. Based on these works and using regularization theory and methods and variational methods, new fractional-order variational PDE models and algorithms are proposed for image denoising. The fractional-order variational PDE based image model-ing theory and methods can be applied in other field of image processing such as image segmentation, super-resolution reconstruction, have important theoretical significance and application values.The main results achieved and innovations include:(1)Based on multi-scale modeling the texture of image in negative Sobolev space and using the orthogonal wavelet transform based multi-scale norm in negative Sobolev space to characterize the norm of texture, a new multi-scale variational PDE model in negative Sobolev space is proposed. For solving the proposed model, this paper gives three kinds of numerical algorithms, and the sufficient conditions for convergence of these algorithms are theoretically analyzed and proved. Numerical experiments show that using the multi-scale modeling method in negative Sobolev space for the texture of image, we can distinguish and describe the texture and noise better at different scales. The proposed multi-scale variational PDE model can preserve texture efficiently while improve the PSNR of image, and the alternating projection algorithm based on the operator splitting method is is the most rapid and stable one of the three proposed algorithms.(2) Modeling the image by using fractional-order derivative, Fractional-order Varia-tion PDE models and algorithms are proposed for image denoising.First of all, using the fractional derivative defined in Fourier transform domain, a fractional-order variation model is established and the corresponding discrete gradient de-scent algorithm is proposed。Secondly, based on Grumwald-Letnikov fractional-order derivative, this paper pro-pose two models from the point of view continuous modeling and discrete modeling. From the point of view continuous modeling, using the relation between Griimwald-Letnikov fractional-order derivative and certain convolution integral, a convolution integral based fractional-order variation model and the corresponding algorithm are proposed. From the point of view Continuous modeling, based on the Grumwald-Letnikov fractional-order difference, this paper extend the discrete function space of bounded variation (BV space) to the discrete fractional-order function space of bounded variation BV_a at first and then using BV_a space for image modeling, a new fractional-order variation model and the corresponding projection algorithm are proposed. For the convergence of the projection algorithm, an efficient condition is proved.Numerical experiments show that using the fractional-order variation to model the image, we can preserve the finer scale details such as textures and edges, restrain the "blocky effect" and improve the PSNR effectively. In the proposed three kinds of algo-rithms, the the cost of computation of projection algorithm is minimum and the speed of taht is the fastest.(3) Coupling the multi-scale modeling method in negative Sobolev space and the fractional-order derivative based image modeling method, a uniform multi-scale fractional-order variational PDE model for image denoising and the corresponding alter-nating projection algorithm are propoesd. The convergence of the alternating projection algorithm is analyzed and an efficient condition is proved in this paper. Numerical experi-ments show that the coupled model combines the advantages of both models, it have better results in the improvement of the PSNR, preserving of textures, as well as restraining of the "blocky effect".Based on the uniform multi-scale fractional-order variational PDE model, by using image statistical information such as local variance of image to distinguish of "texture area" and "non-texture area" of image and the relationship between wavelet confidences and the regularity of function, the adaptive multi-scale fractional-order variational PDE model and the corresponding adaptive alternating projection algorithm for image denoising are proposed. Numerical results show the adaptive fractional-order model can not only remove noise and restrain "blocky effect" in the "non-texture area" effectively, but also preserve the finer scale details such as texture in the "texture area" efficiently and is a fast and effective method for image denoising.(4) Based on two kinds of image decomposition of iamge such as f= u+v+w and f= u+uv, making use of the fractional-order derivative based image modeling and multi-scale modeling method in negative Sobolev spaces, new fractional-order variational denoising models and algorithms are proposed for additive and multiplicative noise respectively.First of all, based on image decomposition form as f= u+v+w,using fractional-order derivative, this paper constructs a new space denote G(_μ~α), and then proposes two models: first, modeling cartoon, texture and noise of image with BV_a space, G(_μ~α) and Multi-scale negative Sobolev space respectively, and propose a new fractional-order variation PDE model and algorithm. Secondly, modeling cartoon, texture and noise with BV_a space, G(_μ~α) and Besov space B(_-1,∞~∞) respectively, and propose a new fractional-order variation model and algorithm. Numerical results show that the new models are effective for improving PSNR, preserving texture and restraining "blocky effect"Secondly, based on image decomposition form as f= u+uv, multiplicative noise re-moving problems whose noise follow the Gauss law and Gamma Law are considered. New multi-scale fractional-order multi-scale models are propose. This paper gives the adaptive parameter selection method and designs the corresponding numerical algorithms. Numer-ical experiments show that for multiplicative noise removing, fractional-order multi-scale models are effective for improving PSNR, preserving texture and restraining "blocky ef-fect" also.