心脏磁共振扩散张量场正则化方法研究
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摘要
随着磁共振成像(Magnetic Resonance Imaging, MRI)技术的发展,扩散张量磁共振成像(Diffusion Tensor Magnetic Resonance Imaging, DTI)技术因其可以实现非侵入人体组织器官及多参数成像,引起了人们的广泛关注。由于扩散加权磁共振成像(Diffusion Weighted Magnetic Resonance Imaging, DWI)过程中引入大量噪声,被噪声污染的张量场偏离其真实走向,使得跟踪出来的纤维结构不够平滑,甚至可能得到错误的结果。张量场正则化以恢复其真实走向为目的,成为后续纤维跟踪工作的一个必要步骤。本文基于变分偏微分方程原理,对张量场正则化方法进行了系统深入的研究,并对其数值实现进行了探讨。
图像恢复问题是一类典型的不适定问题,解决不适定问题的方法,称为正则化方法。本文将全变分(Total Variation, TV)正则化方法引入到张量场的处理当中。TV正则化方法不仅在灰度图像恢复和处理方面取得了成功,在多值图像的正则化中也取得了良好的效果。
磁共振扩散张量场即为3×3的正定对称矩阵场,相对于一般的矢量,张量具有其特殊的性质(正定对称性)。因此,在处理过程中,必须要保证其正定对称性,以使其不脱离张量空间。把一般的TV正则化算法推广到磁共振扩散张量场,需要一个理想的坐标空间。本文在欧几里德坐标空间,把Cholesky分解引入到张量场,以保证张量的正定对称性。并采用耦合技术以更好的保护张量场的边缘特征。得到了较好的恢复效果,但是部分张量的各向异性降低了。由于欧几里德空间的局限性,黎曼空间成为张量场正则化一个更好的选择,本文采用对数欧几里德黎曼空间,在很好的恢复了张量场走向的同时,使恢复后的张量场更具有各向异性。
With the development of magnetic resonance imaging, Diffusion tensor magnetic resonance imaging, attracts wide attention for its non-invasive and multi-parameters. Diffusion weighted magnetic resonance imaging, introducing a large number of noise inevitably, leads to noisy tensor field with the messy and irregular direction. This results tracking by the lack of a smooth fiber structure, and may even get the wrong results. Regularization for tensor field in order to restore the real tensor field direction becomes a necessary step of the following fiber tracking work. Based on the variation partial differential equation principle, in this paper, we did a systematic and thorough study of the regularization method for tensor field, and its numerical realization was discussed.
Image restoration problem is a typical ill-posed problem. Regularization methods are known as the solution of ill-posed problem. TV regularization method, not only has successful restoration in gray-scale image, in the multi-valued image regularization also achieved good results.
Magnetic resonance diffusion tensor field is 3×3 field of the positive definite symmetric matrix. Compared with the normal vector, tensor has its unique prope-rty (symmetric positive-definite). Therefore, during the course of processing, it is necessary to ensure its positive-definite symmetry, to avoid going out of the tensor space. In order to generation the usual TV algorithm to diffusion tensor field, a consistent operational framework is needed. In order to ensure the positive-definite symmetry of tensor, in this paper, the Cholesky decomposition is introduced into tensor field on the Euclidean coordinate space, the experimental results is better, but the anisotropy of part tensors reduced. To remedy the limitation of the Euclidean space, Riemann space becomes a better choice for tensor field regularization. In this paper, Log-Euclidean Riemann space obtains a better effect of regularization, tensor is more anisotropic.
图像恢复问题是一类典型的不适定问题,解决不适定问题的方法,称为正则化方法。本文将全变分(Total Variation, TV)正则化方法引入到张量场的处理当中。TV正则化方法不仅在灰度图像恢复和处理方面取得了成功,在多值图像的正则化中也取得了良好的效果。
磁共振扩散张量场即为3×3的正定对称矩阵场,相对于一般的矢量,张量具有其特殊的性质(正定对称性)。因此,在处理过程中,必须要保证其正定对称性,以使其不脱离张量空间。把一般的TV正则化算法推广到磁共振扩散张量场,需要一个理想的坐标空间。本文在欧几里德坐标空间,把Cholesky分解引入到张量场,以保证张量的正定对称性。并采用耦合技术以更好的保护张量场的边缘特征。得到了较好的恢复效果,但是部分张量的各向异性降低了。由于欧几里德空间的局限性,黎曼空间成为张量场正则化一个更好的选择,本文采用对数欧几里德黎曼空间,在很好的恢复了张量场走向的同时,使恢复后的张量场更具有各向异性。
With the development of magnetic resonance imaging, Diffusion tensor magnetic resonance imaging, attracts wide attention for its non-invasive and multi-parameters. Diffusion weighted magnetic resonance imaging, introducing a large number of noise inevitably, leads to noisy tensor field with the messy and irregular direction. This results tracking by the lack of a smooth fiber structure, and may even get the wrong results. Regularization for tensor field in order to restore the real tensor field direction becomes a necessary step of the following fiber tracking work. Based on the variation partial differential equation principle, in this paper, we did a systematic and thorough study of the regularization method for tensor field, and its numerical realization was discussed.
Image restoration problem is a typical ill-posed problem. Regularization methods are known as the solution of ill-posed problem. TV regularization method, not only has successful restoration in gray-scale image, in the multi-valued image regularization also achieved good results.
Magnetic resonance diffusion tensor field is 3×3 field of the positive definite symmetric matrix. Compared with the normal vector, tensor has its unique prope-rty (symmetric positive-definite). Therefore, during the course of processing, it is necessary to ensure its positive-definite symmetry, to avoid going out of the tensor space. In order to generation the usual TV algorithm to diffusion tensor field, a consistent operational framework is needed. In order to ensure the positive-definite symmetry of tensor, in this paper, the Cholesky decomposition is introduced into tensor field on the Euclidean coordinate space, the experimental results is better, but the anisotropy of part tensors reduced. To remedy the limitation of the Euclidean space, Riemann space becomes a better choice for tensor field regularization. In this paper, Log-Euclidean Riemann space obtains a better effect of regularization, tensor is more anisotropic.
引文
1张建伟.基于变分方法的心脏核磁共振图像分割研究.南京理工大学博士论文. 2006:1~4
2汪红志,聂生东,张学龙.筹建核磁共振成像技术实验室的探索与思考.实验技术与管理. 2007, 24(5):1~2
3史浩,赵斌.心脏MRI新技术.医学影像学杂志. 2004, 14(3):1~2
4 Taylor DG, Bushell MC. The spatial mapping of translational diffusion coefficie-nts by the NMR imaging technique. Phys Med Biol. 1985, 16(2):3~8
5 Le Bihan, Breton E. Lallem. MR imaging of intravoxel incoherent motions appl-ication to diffusion and perfusion in neurologic Radiology. 1986, 16(2):12~15
6郭雪梅.扩散张量磁共振成像基本原理及应用.实用放射学杂志. 2003, 19(9):1~3
7耿辉,白玫,彭明辰.磁共振弥散张量成像的原理及分析.仪器原理与实用. 2004, 19(1):1~2
8曾洪武.磁共振扩散加权与弥散张量成像原理分析及比较.中国医学影像技术. 2005, 21(12):1~3
9章炜炜.扩散张量磁共振成像技术在分析阿尔茨海默病中的应用.清华大学硕士论文. 2005:8~19
10李俊胜,刘宗田.基于异性扩散-中值滤波的超声医学图像去噪方法.计算机应用与软件. 2009, 26(1):1~2
11 Martin Welk, Joachim Weickert and Florian Becker. Median and related local filters for tensor-valued images. Signal Processing. 2007, 12(6):4~16
12 A.N. Tikhonov, V.Y. Arsenin. Solutions of ill-posed problems. Winston and Sons Washington, D.C. 1977, 28(1):10~12
13 L. Rudin, S. Osher, E. Fatemi. Nonliear total variation based noise removal algo-rithms. Physica D. 1992, 30(6):2~4
14 C.A. Castano-Moraga, C. Lenglet, R. Deriche and J. Ruiz Alzola. A riemannian approach to anisotropic filtering of tensor fields. Preprint submitted to Elsevier Science. 2004, 5(2):2~5
15 M.A. Rodriguez Florido, C.F. Westin and J. Ruiz Alzola. DT-MRI reguliraziti-on using anisotropic tensor field filtering. IEEE. 2004, 26(7):2~4
16白衡,王世杰,罗立民. DT-MRI张量值图像的一种加权方向距离滤波方法.东南大学学报. 2006, 36(1):2~4
17张相芬,田蔚风,陈武凡.基于Gaussian-MRF的DTI图像的正则化.辽宁工程技术大学学报. 2007, 26(1):1~4
18张相芬,田蔚风,叶宏. DTI图像恢复的向量复扩散模型.计算机工程与设计. 2009, 30(3):1~4
19樊晓香,胡茂林,孙龙.一种基于偏微分方程的图像平滑技术.微机发展. 2005, 15(9):2~3
20石澄贤.几何图像模型及其在医学图像处理中的应用研究.南京理工大学博士论文. 2005:10~32
21肖亮.基于变分和多重分形的图像建模理论、算法与应用.南京理工大学博士论文. 2003:12~22
22 S.D. Babacan, R. Molina and A.K. Katsaggelos. Variational bayesian blind deco-nvolution using a total variation prior. IEEE Transactions on Image Processing. 2009, 21(7):5~9
23 M. Bertero, P. Boccacci. Introduction to inverse problems in imaging. Bristol an-d Philadelphia: IOP, institute of Physics Publishing, 1998:12~22
24 S.D. Babacan, R. Molina, and A.K. Katsaggelos. Total variation blind deconvo-lution using a variational approach to parameter, image, and blur estimation. Eu-ropean Signal Processing. 2007, 13(4):6~19
25 Victor Isakov. Inverse problems for partial differential equations. Springer Ver-lag. 2004, 6(3):4~6
26 Engl H.W. On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems. Approx. Theory. 1987, 49(5):15~21
27 G. frerer. Ana-posteriori parameter choice for ordinary and iterated Tikhonov re-gularization leading to optimal convergence rates. Math Comp. 1987, 49(6):5~12
28 Hou Zongyi and Li Henong. The general method for solving ill-po sed problems. Nonlinear Analysis. 1993, 21(6):19~20
29 Morozov A. Methods for Solving Incorrectly Posed Problems. Springer, New Y-ork. 1984, 21(6):6~12
30 Schock. Parameter choice by discrepancy principle for the approximate solution of ill-posed problems. Integral Equation and Operator Theory. 1984, 7(5):8~18
31林长好.不适定问题解的稳定性.数学的实践与认识. 1996, 26(3):2~6
32张红英,彭启宗.变分图像复原中PDE的推导及其实现.计算机工程与科学. 2006, 28(6):2~3
33程华,王美清.基于偏微分方程的图像正则化恢复算法.福州大学学报. 2006, 34(2):3~5
34 S.D. Babacan, R. Molina, and A.K. Katsaggelos. Total variation image restorati-on and parameter estimation using variational distribution approximation. Intern-ational Conference on Image Processing. 2007, 15(7):6~12
35 T.F .Chan, S. Osher and J. Shen. The Digital TV Filter and Nonlinear Denoising. IEEE Trans on Image Processing. 2001, 10(2):10~12
36 M. Moakher. A differential geometry approach to the geometric mean of symm-etric positive-definite matrices. SIAM Jour. on Mat. Anal. and Appl. 2005, 26(3):7~14
37 L. Rudin, S.J. Osher, and E. Fatemi. Total Variation Based Image Restoration with Free Local Constrains. Proc. 1st IEEE Int. Conf. Image Processing. 1994, 12(2):13~15
38 Pierpaoli C, Basser P.J. Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med. 1996, 36(6):3~8
39 L. Grady . Multilabel random walker image segmentation using prior models. In Proceedings of CVPR05. 2005, 15(6):12~22
40 Westin C.F., Maier S.E. and Mamata H. Processing and visualization for diffusi-on tensor MRI. Medical Image Analysis. 2002, 6(2):4~9
41 Kindlmann G. Superquadric tensor glyphs. In Proceedings of IEEE TVCG/EG Symposium on Visualization. 2004, 14(7):12~20
42 C. Poupon, C.A. Clark, V. Frouin, J. Regis, I. Bloch, D. Le Bihan and J.F. Mangin. Regularization of diffusion-based direction maps for the tracking of bra-in white matter fascicles. 2000, 12(2):18~25
43 C.F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz and R. Kikinis. Processing and visualization of diffusion tensor MRI. Media. 2002, 6(2):9~18
44 Z. Wang, B. Vemuri, Y. Chen and T.H. Mareci. A constrained variational princi-ple for simultaneous smoothing and estimation of the diffusion tensors from co-mplex DWI data. IEEE TMI. 2004, 23(8):13~19
45 C. Chefd, D. Tschumperl and O. Faugeras. Regularizing flows for constrained matrix-valued images. Math. Im. Vis. 2004, 20(6):14~16
46 O. Coulon, D. Alexander, and S. Arridge. Diffusion tensor magnetic resonanceimage regularization. Medical Image Analysis. 2004, 8(1):4~7
47 C. Lenglet, M. Rousson, R. Deriche and O. Faugeras. Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA. 2004, 12(2):10~12
48 Drebin R.A., Carpenter L. and Hanrahan P. Volume rendering. Computer Graph-ics. 1988, 22(4):2~7
49 Bihan D.L., Mangin J.F. Diffusion Tensor Imaging: Concepts and Application. Journal of MRI. 2001, 13(4):5~11
50 Shlomo Sternberg. Letures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall Inc. 1964, 12(6):12~14
51 Basse P.J., Mattiello J. MR Diffusion Tensor Spectroscopy and Imaging. Bioph-ysical Journal. 1994, 66(1):2~7
2汪红志,聂生东,张学龙.筹建核磁共振成像技术实验室的探索与思考.实验技术与管理. 2007, 24(5):1~2
3史浩,赵斌.心脏MRI新技术.医学影像学杂志. 2004, 14(3):1~2
4 Taylor DG, Bushell MC. The spatial mapping of translational diffusion coefficie-nts by the NMR imaging technique. Phys Med Biol. 1985, 16(2):3~8
5 Le Bihan, Breton E. Lallem. MR imaging of intravoxel incoherent motions appl-ication to diffusion and perfusion in neurologic Radiology. 1986, 16(2):12~15
6郭雪梅.扩散张量磁共振成像基本原理及应用.实用放射学杂志. 2003, 19(9):1~3
7耿辉,白玫,彭明辰.磁共振弥散张量成像的原理及分析.仪器原理与实用. 2004, 19(1):1~2
8曾洪武.磁共振扩散加权与弥散张量成像原理分析及比较.中国医学影像技术. 2005, 21(12):1~3
9章炜炜.扩散张量磁共振成像技术在分析阿尔茨海默病中的应用.清华大学硕士论文. 2005:8~19
10李俊胜,刘宗田.基于异性扩散-中值滤波的超声医学图像去噪方法.计算机应用与软件. 2009, 26(1):1~2
11 Martin Welk, Joachim Weickert and Florian Becker. Median and related local filters for tensor-valued images. Signal Processing. 2007, 12(6):4~16
12 A.N. Tikhonov, V.Y. Arsenin. Solutions of ill-posed problems. Winston and Sons Washington, D.C. 1977, 28(1):10~12
13 L. Rudin, S. Osher, E. Fatemi. Nonliear total variation based noise removal algo-rithms. Physica D. 1992, 30(6):2~4
14 C.A. Castano-Moraga, C. Lenglet, R. Deriche and J. Ruiz Alzola. A riemannian approach to anisotropic filtering of tensor fields. Preprint submitted to Elsevier Science. 2004, 5(2):2~5
15 M.A. Rodriguez Florido, C.F. Westin and J. Ruiz Alzola. DT-MRI reguliraziti-on using anisotropic tensor field filtering. IEEE. 2004, 26(7):2~4
16白衡,王世杰,罗立民. DT-MRI张量值图像的一种加权方向距离滤波方法.东南大学学报. 2006, 36(1):2~4
17张相芬,田蔚风,陈武凡.基于Gaussian-MRF的DTI图像的正则化.辽宁工程技术大学学报. 2007, 26(1):1~4
18张相芬,田蔚风,叶宏. DTI图像恢复的向量复扩散模型.计算机工程与设计. 2009, 30(3):1~4
19樊晓香,胡茂林,孙龙.一种基于偏微分方程的图像平滑技术.微机发展. 2005, 15(9):2~3
20石澄贤.几何图像模型及其在医学图像处理中的应用研究.南京理工大学博士论文. 2005:10~32
21肖亮.基于变分和多重分形的图像建模理论、算法与应用.南京理工大学博士论文. 2003:12~22
22 S.D. Babacan, R. Molina and A.K. Katsaggelos. Variational bayesian blind deco-nvolution using a total variation prior. IEEE Transactions on Image Processing. 2009, 21(7):5~9
23 M. Bertero, P. Boccacci. Introduction to inverse problems in imaging. Bristol an-d Philadelphia: IOP, institute of Physics Publishing, 1998:12~22
24 S.D. Babacan, R. Molina, and A.K. Katsaggelos. Total variation blind deconvo-lution using a variational approach to parameter, image, and blur estimation. Eu-ropean Signal Processing. 2007, 13(4):6~19
25 Victor Isakov. Inverse problems for partial differential equations. Springer Ver-lag. 2004, 6(3):4~6
26 Engl H.W. On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems. Approx. Theory. 1987, 49(5):15~21
27 G. frerer. Ana-posteriori parameter choice for ordinary and iterated Tikhonov re-gularization leading to optimal convergence rates. Math Comp. 1987, 49(6):5~12
28 Hou Zongyi and Li Henong. The general method for solving ill-po sed problems. Nonlinear Analysis. 1993, 21(6):19~20
29 Morozov A. Methods for Solving Incorrectly Posed Problems. Springer, New Y-ork. 1984, 21(6):6~12
30 Schock. Parameter choice by discrepancy principle for the approximate solution of ill-posed problems. Integral Equation and Operator Theory. 1984, 7(5):8~18
31林长好.不适定问题解的稳定性.数学的实践与认识. 1996, 26(3):2~6
32张红英,彭启宗.变分图像复原中PDE的推导及其实现.计算机工程与科学. 2006, 28(6):2~3
33程华,王美清.基于偏微分方程的图像正则化恢复算法.福州大学学报. 2006, 34(2):3~5
34 S.D. Babacan, R. Molina, and A.K. Katsaggelos. Total variation image restorati-on and parameter estimation using variational distribution approximation. Intern-ational Conference on Image Processing. 2007, 15(7):6~12
35 T.F .Chan, S. Osher and J. Shen. The Digital TV Filter and Nonlinear Denoising. IEEE Trans on Image Processing. 2001, 10(2):10~12
36 M. Moakher. A differential geometry approach to the geometric mean of symm-etric positive-definite matrices. SIAM Jour. on Mat. Anal. and Appl. 2005, 26(3):7~14
37 L. Rudin, S.J. Osher, and E. Fatemi. Total Variation Based Image Restoration with Free Local Constrains. Proc. 1st IEEE Int. Conf. Image Processing. 1994, 12(2):13~15
38 Pierpaoli C, Basser P.J. Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med. 1996, 36(6):3~8
39 L. Grady . Multilabel random walker image segmentation using prior models. In Proceedings of CVPR05. 2005, 15(6):12~22
40 Westin C.F., Maier S.E. and Mamata H. Processing and visualization for diffusi-on tensor MRI. Medical Image Analysis. 2002, 6(2):4~9
41 Kindlmann G. Superquadric tensor glyphs. In Proceedings of IEEE TVCG/EG Symposium on Visualization. 2004, 14(7):12~20
42 C. Poupon, C.A. Clark, V. Frouin, J. Regis, I. Bloch, D. Le Bihan and J.F. Mangin. Regularization of diffusion-based direction maps for the tracking of bra-in white matter fascicles. 2000, 12(2):18~25
43 C.F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz and R. Kikinis. Processing and visualization of diffusion tensor MRI. Media. 2002, 6(2):9~18
44 Z. Wang, B. Vemuri, Y. Chen and T.H. Mareci. A constrained variational princi-ple for simultaneous smoothing and estimation of the diffusion tensors from co-mplex DWI data. IEEE TMI. 2004, 23(8):13~19
45 C. Chefd, D. Tschumperl and O. Faugeras. Regularizing flows for constrained matrix-valued images. Math. Im. Vis. 2004, 20(6):14~16
46 O. Coulon, D. Alexander, and S. Arridge. Diffusion tensor magnetic resonanceimage regularization. Medical Image Analysis. 2004, 8(1):4~7
47 C. Lenglet, M. Rousson, R. Deriche and O. Faugeras. Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA. 2004, 12(2):10~12
48 Drebin R.A., Carpenter L. and Hanrahan P. Volume rendering. Computer Graph-ics. 1988, 22(4):2~7
49 Bihan D.L., Mangin J.F. Diffusion Tensor Imaging: Concepts and Application. Journal of MRI. 2001, 13(4):5~11
50 Shlomo Sternberg. Letures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall Inc. 1964, 12(6):12~14
51 Basse P.J., Mattiello J. MR Diffusion Tensor Spectroscopy and Imaging. Bioph-ysical Journal. 1994, 66(1):2~7