Fractional-Order Regularization Method Based On Besov Space For Image Restoration

Image restoration is a process of establishing mathematical model through some prior knowledge in the image degradation process and the degraded image,so as to recover the original image.Image restoration includes image denoising,image deblurring and other issues.According to whether the information of blur kernel is known,the image deblurring can be divided into non-blind deblurring and blind deblurring.This paper focuses on image denoising and deblurring,including the establishment of models,the design of algorithms,the analysis of convergence theory and the experimental results.The main research contents of this paper are as follows:1.We propose a model for image deblurring,denoising and simultaneous decomposition based on fractional-order derivative regularization in Besov space.The model divides the restored image into cartoon and texture parts and reg-ularizes them separately.The fractional-order derivative Besov norm is used as the regularization term of the cartoon part,which can avoid the staircasing effect（blocky effect）caused by the integer-order regularization.Since wavelets provide unconditional bases for the Besov spaces,one can express whether the function1)belongs to a Besov space by a fairly simple and completely explicit require-ment on the absolute values of the wavelet coefficients of1).Compared with the model based on fractional-order total variation regularization,the model based on fractional-order Besov norm expands and solves the function in the wavelet space,which transform the partial differential equation（PDE）schemes that are numer-ically intensive into a simple wavelet soft-thresholding problem.The proposed minimization model is solved by the alternating direction method of multipliers（ADMM）.The experimental results show that our proposed fractional-order model has a good effect on dealing with different blur and Gaussian noise.2.We propose a model with fractional-order regularization and sparsity con-straint for blind deblurring.The model recovers the original image and estimates the blur kernel simultaneously,using an alternate iterative approach.The pro-posed model use fractional-order Besov norm as the regularization term of image,and use the sum of₂norm and the₁norm based on the wavelet frame as the regularization term of the blur kernel.We solve the minimization model by combining iterative thresholding algorithm and split Bregman algorithm.Exper-imental results show that our proposed method is able to maintain edges and smoothness,which have obvious advantages over other blind deblurring methods.3.Since the exponential-type（ET）function is very effective for removing impulse noise and the fractional-order Besov norm can avoid the blocky effect,we combine the ET function and Besov norm and propose a nonconvex model for removing blur and impulse noise.The data fitting term of the model is ET function,and the regularization term is the Besov norm with fractional-order derivative.For images degraded by blur and mixed Gaussian impulse noise,we propose a nonconvex model for removing blur and mixed Gaussian impulse noise.The data fitting term of the model is composed of ET function and₂norm,and the regularization term is still the fractional-order Besov norm.The proximal linear minimization（PLM）algorithm and the ADMM are used to solve these two nonconvex minimization models.Experiments show that the models based on fractional-order derivative regularization in Besov space have a good effect on dealing with different types of degraded images,and also shows obvious advantages when compared with related algorithms.4.We studied the inverse filtering method for blind deblurring of images,and improve the STAR-RIF method based on compressed sensing（CS）theory and surface-aware minimization method.Firstly,we propose a nonconvex inverse filtering method for blind deblurring of grayscale images.This method applies the nonconvex metric₁-₂（≥1）in the CS theory to the STAR-RIF mod-el.Experimental results show that the proposed nonconvex method can preserve the sparsity of the model and reduce disadvantageous structures in the recovered images.Secondly,we propose two surface-aware based inverse filtering estimation models.For the first model,we add a surface-aware regularization term to the STAR-RIF method to improve the quality of the restored image.The reason is that,geometrically,a recovered image with salient edges and fewer disadvanta-geous structures should have a smaller surface area.For the second model,we replace the₂norm of the nonnegative and the background constraint in model one with₁norm,which can make full use of the sparsity of text images.In this paper,two surface-aware based inverse filtering estimation models are applied to blind deblurring of color images.Since the color image has three channels,red（R）,green（G）,and blue（B）,in order to make full use of the different information of each channel,the inverse filter estimation model is applied to each channel of the color image respectively,and obtain a three-channel inverse filter with different numerical characteristics on different channels.The restored image is obtained by convolving the three-channel inverse filter and the degraded image on each channel respectively.The proposed inverse filter estimation models are all solved by the Chambolle-Pock algorithm.Numerical examples show that our proposed meth-ods have good effect on dealing with different types of blur,and shows obvious advantages compared with the existing related algorithms.