Optimization Model And Algorithm Analysis For Limited-angle And Low SNR CT Image Reconstruction

X-ray Computed Tomography?CT?is one of the important non-destructive techniques,which utilizes the obtained projection data to reconstruct the attenuated coefficients of the object to be reconstructed.Since CT can reconstruct the scanned object for the internal structure of the object information,it has been widely used in medical,industrial nondestructive testing,biology,archeology and other fields.When the obtained projection data are complete and only contain low-level noise,the traditional algorithms,such as filtering back projection method?FBP?and simultaneous algebraic reconstruction technique?SART?,can reconstruct the high-quality images.However,when the obtained projection data are incomplete,it may lead to an internal,external,sparse-angle or limited-angle truncated data problem.At this time,the problem of the truncated data reconstruction is a seriously ill-posed and inverse problem.When the obtained projection data are low signal to noise ratio?SNR?,the structures and details of the image reconstructed by the traditional algorithm may be blurred and damaged.Based on the development and application of current CT technology,the research on the reconstruction algorithm of truncated projection data and low SNR projection data has not only academic value,but also important application value.CT is usually limited by the scanning environment?such as C-arm CT,dental CT,and breast detection,etc.?,radiation exposure?shorter scanning time,lower radiation dose,etc.?,and the incomplete rotation angle of the radiation source,which may lead to a limited-angle reconstruction problem.However,it is a challenging and ill-posed inverse problem to reconstruct an image of the object from the limited-angle projection data.At this time,the current popular algorithm,such as the total variation?TV?reconstruction algorithm,is used to reconstruct an image from the limited-angle projection data,which will suffer from the gray gradient artifacts?slope artifacts?near the strong edge of the reconstructed image.In order to relieve this ill-posedness,the regularization method provides an effective method to deal with it.In recent years,some researchers have studied l₁ regularization in order to solve such problem,but there exist some problems in restraining slope artifacts on the edges of the reconstructed image.To suppress the slope artifacts,a non-smooth and non-convex model based on l₀and l₁ regularizations is presented.On the one hand,data fidelity term is considered in this model;on the other hand,two regularization terms are introduced to alleviate the ill-posedness caused by the lack of projection data.Under certain conditions,the convergence analysis of the presented algorithm is given.Numerical experiments show that compared with SART and TV algorithms,the presented algorithm has more advantages in suppressing slope artifacts and preserving the details.Inspired by the classical image segmentation model?Mumford-Shah:MS?,Cai et al.proposed an image restoration model based on the evolution of the MS model,which maintains the edges and the details while restoring the image.Motivated by this,in order to settle the limited-angle reconstruction problem,this paper presents an l₀ and l₂regularization optimization model based on the image under the wavelet tight framelet transform.The artifacts of the reconstructed image not only exist in the high frequency coefficient under the wavelet tight framelet transform,but also in the low frequency.The coefficients of the slope artifacts under the wavelet tight framelet transform are changed to zero by using l₀ quasi-norm.However,if all the small coefficients are processed,some structural information with a little change in gray level will be damaged.Therefore,the presented model utilizes the l₂ norm to repair them.Experimental results and quantitative analysis show that compared with SART and SART+TV algorithms,the algorithm in this paper has certain advantages in restraining the slope artifacts and maintaining the details.When the rotation angle is severely limited,the limited-angle reconstruction problem is a severely ill-posed and inverse problem.At this point,for an object with the complex structures,the ordinary regularization method can not deal with the slope artifacts in the reconstructed result well.In order to reduce slope artifacts,an iterative reconstruction method combining the structural similarity between the reconstructed image and the prior image is proposed to settle the limited-angle CT reconstruction with severely limited rotation angle.This method utilizes the structural similarity between the prior image and the reconstructed image to compensate the distorted edges.l₀regularization and the wavelet tight framelet are used to suppress slope artifacts.This method includes the following four steps:the first step is to use SART to process data fidelity item;the second step is to use the prior image and the improved non-local mean?PNLM?to compensate the slope artifacts caused by the lack of the projection data;the third step is to use l₀ regularization to suppress the slope artifacts,and to use the iterative hard threshold algorithm to deal with the wavelet coefficients of the transformed images,noted as l₀W;step 4 is the inverse transform of the wavelet tight framelet to obtain the reconstructed image.The presented algorithm can be noted as l₀W-PNLM.Numerical experiments show that compared with TV algorithm and prior image constrained compressed sensing?PICCS?,the proposed l₀W-PNLM algorithm in this paper has better characteristics of suppressing the slope artifacts and preserving the edges.Quantitative evaluation verifies the effectiveness and superiority of our algorithm.Compared with the above algorithm,the presented l₀W-PNLM can obtain the highest peak signal-to-noise ratio?PSNR?,general image quality index?UQI?,structural similarity index measure?SSIM?and smaller relative mean square error?RMSE?.Limited by the factors?such as shorter X-ray exposure time,lower tube voltage,and faster scan speed?,the obtained projection data is usually low SNR.In this case,the traditional SART can not suppress the noise very well.The current popular TV algorithm will lead to the stair effect in the reconstructed result,what's more,the edges and details have suffered some ambiguity and damage.In order to improve this effect,this paper incorporated Gaussian kernel function and overlapping group sparsity?OGS?,short for GOGS,into the TV model to obtain a new reconstruction model,noted as GOGS-TV.This model not only considers the sparsity of the image under gradient transform,but also considers the sparsity of the local structure under gradient transform.Numerical experiments show that compared with the traditional SART algorithm and the popular TV algorithm,the presented algorithm has more advantages in suppressing noise and protecting edges.