基于Abel变换的图像重建自适应方法
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摘要
本文论述了利用轴对称物体的单幅投影信息进行密度重建的一种自适应正则化模型。所提模型基于全变分正则项与高阶全变分正则项的联合使用,主要的优点是在保持清晰的界面及恢复平稳变化区域的同时减弱了阶梯效应。并且使用自适应方法,提高了效果的同时简化了所使用的参数。对于其中涉及的最优化问题,我们采用增广拉格朗日方法来解。数值结果表明,这一模型提高了关于密度界面位置及密度值的准确度,具有较好的抗噪性。
In this paper, we discuss an adaptive regularization approach for density reconstruction of axially symmetric object whose tomography comes from a single X-ray projection. The method we proposed is based on the combination of total variation regularization and high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and recovering smoothly varying regions. Moreover, it simplifies the use of parameters. We apply the augmented Lagrangian method to solve the optimization involved. Numerical results show that the proposed method has improved the accuracy of density edges and values. Besides, the method is not sensitive to the measured data noise.
In this paper, we discuss an adaptive regularization approach for density reconstruction of axially symmetric object whose tomography comes from a single X-ray projection. The method we proposed is based on the combination of total variation regularization and high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and recovering smoothly varying regions. Moreover, it simplifies the use of parameters. We apply the augmented Lagrangian method to solve the optimization involved. Numerical results show that the proposed method has improved the accuracy of density edges and values. Besides, the method is not sensitive to the measured data noise.
引文
[1]牛轶杰,王子野,乔灵博.宽带全息重建算法应用于亚毫米波成像系统初探[J].CT理论与应用研究,2016,25(5):531-538.doi:10.15953/j.1004-4140.2016.26.05.04.Niu YJ,Wang ZY,Qiao LB.A primary study on wide-band holographic reconstruction algorithm applied to submillimeter-wave imaging system[J].CT Theory and Applications,2016,25(5):531-538.(in Chinese).doi:10.15953/j.1004-4140.2016.26.05.04.
[2]李高,郑旭辉,张宝君,等.基于弹性波CT技术的岩体破裂探测方法[J].CT理论与应用研究,2015,24(5):681-688.doi:10.15953/j.1004-4140.2015.24.05.05.Li G,Zheng XH,Zhang BJ,et al.A detecting method for rock mass fracture based on elastic wave CT technique[J].CT Theory and Applications,2015,24(5):681-688.(in Chinese).doi:10.15953/j.1004-4140.2015.24.05.05.
[3]吴胜利,潘瑞谊,文斌.锥束CT图像重建算法的快速实现[J].CT理论与应用研究,2007,16(4):31-37.Wu SL,Pan RY,Wen B.Fast accomplishment of reconstruction algorithm for cone beam CT[J].CT Theory and Applications,2007,16(4):31-37.(in Chinese).
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[8]Chan RH,Liang HX,Wei SH,et al.High-order total variation regularization approach for axially symmetric object tomography from a single radiograph[J].Inverse Problems&Imaging,2015,9(1):55-77.
[9]Rudin LI,Osher S,Fatemi E.Nonlinear total variation based noise removal algorithms[J].Physica D Nonlinear Phenomena,1992,60(1-4):259-268.
[10]Savage J,Chen K.On multigrids for solving a class of improved total variation based staircasing reduction models[M].Image Processing Based on Partial Differential Equations.Springer Berlin Heidelberg,2007:69-94.
[11]Wu CL,Tai XC.Augmented lagrangian method,dual methods,and split bregman iteration for ROF,vectorial TV,and high order models[J].Siam Journal on Imaging Sciences,2010,3(3):300-339.
[12]Dong FF,Chen YM.A fractional-order derivative based variational framework for image denoising[J].Inverse Problems&Imaging,2016,10(1):27-50.
[13]Tikhonvo AN,Arsenin VY.Solution of ill-posed problems[M].New York:Wiley,1997.
[14]Asaki TJ.Quantitative Abel tomography robust to noisy,corrupted and missing data[J].Optimization and Engineering,2010,11(3):381-393.
[15]Abraham R,Bergounioux M,Trélat E.A penalization approach for tomographic reconstruction of binary axially symmetric objects[J].Applied Mathematics&Optimization,2008,58(3):345-371.
[16]Lysaker M,Lundervold A,Tai XC.Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time[J].IEEE Transactions on Image Processing,2003,12(12):1579-90.
[17]Chambolle A.An algorithm for total variation minimization and applications[J].Journal of Mathematical Imaging and Vision,2004,20(1):89-97.
[18]Shi YY,Chang QS,Yang XZ.A robust and fast combination algorithm for deblurring and denoising[J].Signal,Image and Video Processing,2015,9(4):865-874.
[19]Wu CL,Zhang JY,Tai XC.Augmented Lagrangian method for total variation restoration with non-quadratic fidelity[J].Inverse Problems&Imaging,2011,1(1):237-261.
[20]Tai XC,Wu CL.Augmented Lagrangian method,dual methods and split bregman iteration for ROF model[C]//International Conference on Scale Space and Variational Methods in Computer Vision.Springer-Verlag,2009:502-513.
[21]Vetterli M,Kovacevic J,Venkatesh PK.Wavelets and Subband Coding[M].Prentice-Hall,Inc.1995.
[2]李高,郑旭辉,张宝君,等.基于弹性波CT技术的岩体破裂探测方法[J].CT理论与应用研究,2015,24(5):681-688.doi:10.15953/j.1004-4140.2015.24.05.05.Li G,Zheng XH,Zhang BJ,et al.A detecting method for rock mass fracture based on elastic wave CT technique[J].CT Theory and Applications,2015,24(5):681-688.(in Chinese).doi:10.15953/j.1004-4140.2015.24.05.05.
[3]吴胜利,潘瑞谊,文斌.锥束CT图像重建算法的快速实现[J].CT理论与应用研究,2007,16(4):31-37.Wu SL,Pan RY,Wen B.Fast accomplishment of reconstruction algorithm for cone beam CT[J].CT Theory and Applications,2007,16(4):31-37.(in Chinese).
[4]Smith LM,Keefer DR,Sudharsanan SI.Abel inversion using transform techniques[J].Journal of Quantitative Spectroscopy&Radiative Transfer,1988,39(5):367-373.
[5]Hanson KM.A Bayesian approach to nonlinear inversion:Abel inversion from X-ray attenuation data[J].1989,115:363-368.
[6]Asaki TJ,Campbell PR,Chartrand R,et al.Abel inversion using total variation regularization[J].Inverse Problem,2005,21(6):1895-1903.
[7]Asaki TJ,Campbell PR,Chartrand R,et al.Abel inversion using total variation regularization:Applications[J].Inverse Problems in Science and Engineering,2006,14(8):873-885.
[8]Chan RH,Liang HX,Wei SH,et al.High-order total variation regularization approach for axially symmetric object tomography from a single radiograph[J].Inverse Problems&Imaging,2015,9(1):55-77.
[9]Rudin LI,Osher S,Fatemi E.Nonlinear total variation based noise removal algorithms[J].Physica D Nonlinear Phenomena,1992,60(1-4):259-268.
[10]Savage J,Chen K.On multigrids for solving a class of improved total variation based staircasing reduction models[M].Image Processing Based on Partial Differential Equations.Springer Berlin Heidelberg,2007:69-94.
[11]Wu CL,Tai XC.Augmented lagrangian method,dual methods,and split bregman iteration for ROF,vectorial TV,and high order models[J].Siam Journal on Imaging Sciences,2010,3(3):300-339.
[12]Dong FF,Chen YM.A fractional-order derivative based variational framework for image denoising[J].Inverse Problems&Imaging,2016,10(1):27-50.
[13]Tikhonvo AN,Arsenin VY.Solution of ill-posed problems[M].New York:Wiley,1997.
[14]Asaki TJ.Quantitative Abel tomography robust to noisy,corrupted and missing data[J].Optimization and Engineering,2010,11(3):381-393.
[15]Abraham R,Bergounioux M,Trélat E.A penalization approach for tomographic reconstruction of binary axially symmetric objects[J].Applied Mathematics&Optimization,2008,58(3):345-371.
[16]Lysaker M,Lundervold A,Tai XC.Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time[J].IEEE Transactions on Image Processing,2003,12(12):1579-90.
[17]Chambolle A.An algorithm for total variation minimization and applications[J].Journal of Mathematical Imaging and Vision,2004,20(1):89-97.
[18]Shi YY,Chang QS,Yang XZ.A robust and fast combination algorithm for deblurring and denoising[J].Signal,Image and Video Processing,2015,9(4):865-874.
[19]Wu CL,Zhang JY,Tai XC.Augmented Lagrangian method for total variation restoration with non-quadratic fidelity[J].Inverse Problems&Imaging,2011,1(1):237-261.
[20]Tai XC,Wu CL.Augmented Lagrangian method,dual methods and split bregman iteration for ROF model[C]//International Conference on Scale Space and Variational Methods in Computer Vision.Springer-Verlag,2009:502-513.
[21]Vetterli M,Kovacevic J,Venkatesh PK.Wavelets and Subband Coding[M].Prentice-Hall,Inc.1995.