Fractional Order Total Variation For Image Reconstruction

Compressed sensing(CS)is a new processing method in the field of signal processing,which can reconstruct the original signal with a small number of observations.CS theory has three core problems: sparse representation,measurement matrix and reconstruction algorithm.Among them,reconstruction algorithm is the most important problem in CS theory.In this paper,the total variation(TV)model of reconstruction algorithm is studied,and the traditional algorithm is improved and discussed.The main tasks are as follows:1.Firstly,describing the three core technologies of compressed sensing theory sparse representation,measurement matrix and reconstruction algorithm,the influence of the selection of sparse basis on image sparsity is described in turn;the construction of Gaussian random matrix,random Bernoulli matrix and local Hadamard matrix;different reconstruction algorithms(OMP algorithm,BP algorithm,TV algorithm)are used to reconstruct the same image Performance impact.Finally,the mathematical models of two total variation algorithms(TVAL3,TVNLR)are analyzed.2.The influence of fractional order differentiation on signal and image detail texture is discussed.In this paper,three kinds of fractional differential models(G-L definition,R-L definition,Caputo definition)are selected,and their generalization process and relationship are analyzed.In the same model,different differential orders have different effects on the signal.Therefore,the different effects of different orders on the high-frequency component and the low-frequency component of the signal are expounded by taking multiple values of small step length between the first and second-order.The experimental results show that,when the order is between 0 and 1,the fractional differential has the attenuation effect on the high-frequency signal,but has a small improvement on the low-frequency signal When the order is between 1 and 2,the fractional differential can attenuate the low-frequency signal,but it can significantly improve the high-frequency signal.3.The fractional differential model is introduced into TV model,and the non local mean filter model is used to optimize the total variation algorithm.The algorithm can effectively improve the ladder effect in the traditional algorithm and improve the visual effect of the reconstructed image.The nonlocal mean filter is used to update the Lagrange operator in the iterative process,and the alternating direction multiplier(ADMM)method is used to solve the non differentiable problem in the reconstruction algorithm.Because the choice of parameters has a great impact on the quality of reconstruction,the influence of parameters is elaborated through experiments,and then the most appropriate parameter value is selected.Several images are simulated and reconstructed at different sampling rates(10%,15%,20%,25%,30%),and the peak signal-to-noise ratio(PSNR)and structural similarity are used to compare and analyze the experimental results.The results show that the reconstruction performance of the improved algorithm is better than the other four algorithms.Especially in the low sampling rate(10%),the proposed algorithm has a larger improvement value than the other two algorithms.Compared with the other four algorithms,the maximum PSNR gain of the improved algorithm can reach 11.51 db,7.59 db,5.28 db and 2.52 db,respectively.4.In the reconstruction,in order to make full use of the prior information,the high-frequency component and the low-frequency component of the image are separated,and then the high-frequency component is weighted.The gradient sparse prior information is added to the fractional total variation algorithm to protect the texture information of the image.At the same time,the nonlocal regularization term is added to the algorithm to increase the prior information of image structure,which makes the algorithm have better performance.The experimental results show that the improved algorithm has certain advantages in image reconstruction by comparing the four algorithms.In terms of preserving texture details,the improved algorithm is obviously better than other algorithms,especially in low sampling rate,it can recover more image information.Compared with the other four algorithms,the maximum gain of the improved algorithm is 12.28 db,9.63 db,4.83 db and 1.63 db,respectively.