插值法在CT图像重建中的应用
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摘要
CT(Computed Tomography X射线计算机断层摄影术)自问世以来得到了越来越广泛的应用。CT的广泛应用反过来又推动了对它的研究,使它得到进一步的发展。在过去30年的发展过程中,它基本上经历了五次大的变革。CT的这些发展变化主要体现在两个方面,一是提高扫描速度,二是改善图像的质量。
Radon于1917年提出了著名的Radon变换,然而,直到计算机出现以后才实现了傅里叶变换及卷积的快速计算,这一成果才引起人们足够的注意。Radon变换理论现在已经成为CT重建技术的重要基础,这一技术不仅可以应用于医学成像,也同样适用于远程成像等许多图象处理学科。迄今为止,人们已经开发出基于Radon变换的多种比较成熟的重建算法。
本文首先介绍了反傅里叶变换重建,然后研究了二维平行束滤波反投影和扇束滤波反投影重建,最后介绍了局部凸组合插值重建方法。通过对CT图像的投影进行傅里叶变换,将变换所得到的数据在极坐标网格中按照角度排列在Radon变换域中,进行反傅里叶变换的时候,首先需要对极坐标下的Radon变换域进行插值。由于经过傅里叶变换之后的Radon变换域中的采样是均匀的,在靠近中心的区域出现过采样,而在远离中心的地方出现欠采样。因此本文提出了一种新的方法进行插值处理,即局部凸组合插值法,并且采用了内插和外插相结合的方法进行插值。插值的权重是由一距离的倒数的函数决定的。通过分别在投影空间和经过傅里叶变换的Radon变换域中进行插值,然后比较插值效果的方法。得出结论,在投影空间进行数据的内插和外插的效果要好于在傅里叶变换之后的Radon变换域中进行内插和外插,其原因主要是因为投影空间所得到的数据要比傅里叶变换空间中所得到的数据更均匀连续,图像仿真的结果也证明了这一点。
最后,通过Matlab程序验证了各种插值方法的可行性和正确性。
Since the birth of the first CT (Computed Tomography) scanner in 1972, CT has been widely used in the world. The wide uses of CT also generate great impetus for CT research. Several big changes have taken placed during the past 30 years. The changes mainly involve in two aspects-speed and image quality.
The Radon transform was introduced by J. Radon in 1917. Little computation attention was given to it until the advent of computers enabled the fast evaluation of Fourier transform and their convolutions. The Radon transform is now a mainstay of computerized tomography in medical imaging as well as many other remote-imaging sciences. So far, many reliable reconstruction methods based on Radon transform have been developed.
Firstly we introduce the reverse discrete Fourier transform reconstruction, and then we study the two dimensions filtered back projected, at last we propose the method of local convex combination interpolation. Radon domain can be filled by the Fourier transforms for projection images in a polar gridding format (radial lines for parallel projections, radon arcs for fan-beam projections). The Radon-based tomographic reconstruction requires regridding a polar radon domain into a rectilinear lattice before inverse Fourier transform. Since the radon domain is irregularly sampled by Fourier-transformed projections, i.e, oversampled around the central regions and undersampled at the peripheral regions, the polar-to-Cartesian coordinate grid conversion involves rebinning for oversampled central region, interpolation for undersampled peripheral region, and extrapolation for extending the peripheral boundary. In this paper, we propose a general data interpolation or extrapolation scheme to deal with the radon domain regridding, which is a local convex combination with weights determined by a function of inverse distances. For filling the unavailable entries at peripheral regions, we propose to calculate the corresponding entries in the projection domain, rather than in the radon domain, by interpolations and extrapolations. The interpolation for peripheral region allows us investigate the angular sampling for computed tomography scanning. The extrapolation leads to super-resolution tomographic reconstruction. We find that data interpolation in projection domain may produce better results than in radon domain. This finding may be justified by the fact that the data distribution is more continuous in projection domain than in Fourier domain.
Finally the feasibility and correctness of the interpolation are validated by using Matlab programs.
Radon于1917年提出了著名的Radon变换,然而,直到计算机出现以后才实现了傅里叶变换及卷积的快速计算,这一成果才引起人们足够的注意。Radon变换理论现在已经成为CT重建技术的重要基础,这一技术不仅可以应用于医学成像,也同样适用于远程成像等许多图象处理学科。迄今为止,人们已经开发出基于Radon变换的多种比较成熟的重建算法。
本文首先介绍了反傅里叶变换重建,然后研究了二维平行束滤波反投影和扇束滤波反投影重建,最后介绍了局部凸组合插值重建方法。通过对CT图像的投影进行傅里叶变换,将变换所得到的数据在极坐标网格中按照角度排列在Radon变换域中,进行反傅里叶变换的时候,首先需要对极坐标下的Radon变换域进行插值。由于经过傅里叶变换之后的Radon变换域中的采样是均匀的,在靠近中心的区域出现过采样,而在远离中心的地方出现欠采样。因此本文提出了一种新的方法进行插值处理,即局部凸组合插值法,并且采用了内插和外插相结合的方法进行插值。插值的权重是由一距离的倒数的函数决定的。通过分别在投影空间和经过傅里叶变换的Radon变换域中进行插值,然后比较插值效果的方法。得出结论,在投影空间进行数据的内插和外插的效果要好于在傅里叶变换之后的Radon变换域中进行内插和外插,其原因主要是因为投影空间所得到的数据要比傅里叶变换空间中所得到的数据更均匀连续,图像仿真的结果也证明了这一点。
最后,通过Matlab程序验证了各种插值方法的可行性和正确性。
Since the birth of the first CT (Computed Tomography) scanner in 1972, CT has been widely used in the world. The wide uses of CT also generate great impetus for CT research. Several big changes have taken placed during the past 30 years. The changes mainly involve in two aspects-speed and image quality.
The Radon transform was introduced by J. Radon in 1917. Little computation attention was given to it until the advent of computers enabled the fast evaluation of Fourier transform and their convolutions. The Radon transform is now a mainstay of computerized tomography in medical imaging as well as many other remote-imaging sciences. So far, many reliable reconstruction methods based on Radon transform have been developed.
Firstly we introduce the reverse discrete Fourier transform reconstruction, and then we study the two dimensions filtered back projected, at last we propose the method of local convex combination interpolation. Radon domain can be filled by the Fourier transforms for projection images in a polar gridding format (radial lines for parallel projections, radon arcs for fan-beam projections). The Radon-based tomographic reconstruction requires regridding a polar radon domain into a rectilinear lattice before inverse Fourier transform. Since the radon domain is irregularly sampled by Fourier-transformed projections, i.e, oversampled around the central regions and undersampled at the peripheral regions, the polar-to-Cartesian coordinate grid conversion involves rebinning for oversampled central region, interpolation for undersampled peripheral region, and extrapolation for extending the peripheral boundary. In this paper, we propose a general data interpolation or extrapolation scheme to deal with the radon domain regridding, which is a local convex combination with weights determined by a function of inverse distances. For filling the unavailable entries at peripheral regions, we propose to calculate the corresponding entries in the projection domain, rather than in the radon domain, by interpolations and extrapolations. The interpolation for peripheral region allows us investigate the angular sampling for computed tomography scanning. The extrapolation leads to super-resolution tomographic reconstruction. We find that data interpolation in projection domain may produce better results than in radon domain. This finding may be justified by the fact that the data distribution is more continuous in projection domain than in Fourier domain.
Finally the feasibility and correctness of the interpolation are validated by using Matlab programs.
引文
1.刘政凯.数字图像恢复与重建[M],合肥：中国科学技术出版社,1989,236-241.
2.霍修坤.锥束CT直接三维成像算法研究[D],合肥：安徽大学,2005.
3.王家文等.MATLAB6.5图形图像处理[M],北京：国防工业出版社,2004,210-221.
4.庄天戈.CT原理和算法[M],第二版,上海：上海交通大学出版社,1992,1-99.
5. Gabor T Herman. Image Reconstruction From Projections--The Fundamental of Computed Tomography [M], Academic Press,1980.
6. R L Siddon. Fast calculation of the exact radiological path for a three-dimensional CT array [J], Medical Physics, March 1985, Vol.12:252-255.
7.黄爱民等.数字图像处理与分析基础[M],北京：中国水利水电出版社,2005,108-112.
8.[美]罗纳德N布雷斯韦尔.傅里叶变换及其应用[M],第三版,西安：西安交通大学出版社,2005,173-180.
9. Zikuan Chen, Mohammad A Karim, Majeed M Hayat. Elimination of higher order aliasings by multiple interlaced sampling [J], Optical Engineering,1999,38(5):879-885.
10. Paul A Rattey, Allen G Lindgren. Sampling the 2-D Radon Transform [J], IEEE Transactions on Acoustics, Speech And Signal Processing,1981,29(5):994-1002.
11.邸燕,常宏宇.Radon变换在断层成像中的应用[J],数学的实践与认识,2004,34(12)：87-90.
12.陈天华.数字图像处理[M],北京：清华大学出版社,2007,101-109.
13.孙即祥.图像压缩与投影重建[M],第一版,北京：科学出版社,2005,171-187.
14.刘丹.计算机图像处理的数学和算法基础[M],北京：国防工业出版社,2005,146-152.
15.张铁,李建东,李长军等.CT成像的改进的Fourier算法及其实验[J],CT理论与应用研究,2000,9(4)：7-11.
16.张铁,阎家斌.求解Radon变换改进Fourier算法的误差分析[J],CT理论与应用研究,2000,9(1)：12-16.
17.吴梦秋,程正兴.反投影重建算法的改进算法[J],西安工业学院学报,2004,24(1)：76-81.
18. Guy M Besson. CT Image Reconstruction from Fan-Parallel Projection Data [J], IEEE, 1999,1644-1648.
19.王一多.快速滤波反投影算法研究[D],长春：吉林大学硕士学位论文,2002.
20. Brady, M. L. A Fast Discrete Approximation Algorithm for the Radon Transform [J], SIAM Journal of Computing,1998,27(1):107-119.
21. Nilsson. S. Application of Fast Backprojection Techniques for Some Inverse Problems of Integral Geometry [D], Department of Mathmatics, Linkoping University,1997.
22. Jiang Hsieh. Computed Tomography [M], Bellingham, Washington USA:SPIE-The International Society for Optical Engineering,2003,64-98.
23. Proksa, R, T. Kohler, M. Grass, and J. Timmer. The n-PI-Method for Helical Cone-Beam CT [J], IEEE Transactions on Medical Imaging,2000,19(9):848-863.
24.刘晓平.扇束卷积反投影法的程序优化[J],CT理论与应用研究,1996,5(1)：35-37.
25.周光湖.计算机断层摄影原理及应用[M],成都：成都电讯工程学院出版社,1986,1-211.
26.江根苗,连兵,刘晋军,王怀志.CT图像重建的算法优化和代码优化[J],CT理论与应用研究,2000,第9卷增刊.
27.张朋,张兆田.几种CT图像重建算法的研究和比较[J],CT理论与应用研究,2001,10(4)：4-8.
28.孙晓安,陈淑珍,吴志斌,柴亚萍.图象重建中的最优化方法[J],中国图像图形学报,1999,4(2)：105-109.
29.过传卫,胡福乔.扇束等距CT滤波反投影重建算法的研究[J],微计算机信息,2007,23(7-1)：292-294.
30. Zikuan Chen, Ruola Ning. Filling the Radon domain in computed tomography by local convex combination [J], APPLIED OPTICS,2003,42(35):7043-7050.
31. A C Kak, M Slaney. Principles of Computerized Tomographic Imaging [M], New York, IEEE Press,1988.
32. I. Svalbe, D. Spek. Reconstruction of tomographic images using analog projections and the digital Radon transform [J], Linear Algebr,2001, Appl.339,125-145.
33. I. Amidror. Scattered data interpolation methods for electronic imaging systems:a survey [J], Electron. Imaging 11,2002,157-176.
34. S. E. Parker. Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation [J], Compute. Phys,2002,178,520-532.
35. D. Rajan and S. Chaudhuri. Generalized interpolation and its application in super-resolution imaging [J], Image Vision Compute.19,2001,957-969.
36. W K Carey, D B Chuang, S S Hemami. Regularity-preserving image interpolation [J], IEEE Trans, Image Proc.1999,8,1293-1297.
37. J. A. Parker, R. V. Kenyon, and D. E. Troxel. Comparison of interpolating methods for image resampling [J], IEEE Trans. Med. Imaging 2,1983,31-39.
38.伯晓晨,李涛,刘路.Matlab工具箱应用指南[M],北京：电子工业出版社,2000,20-166.
39.薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M],北京：清华大学出版社,2006,260-297.
40.冈萨雷斯著,阮秋琦译.数字图像处理(MATLAB版)[M],北京：电子工业出版社,2005.
41.朱晕.利用MATLAB进行图像重建的算法研究[D],兰州：兰州大学硕士学位论文,2003.
2.霍修坤.锥束CT直接三维成像算法研究[D],合肥：安徽大学,2005.
3.王家文等.MATLAB6.5图形图像处理[M],北京：国防工业出版社,2004,210-221.
4.庄天戈.CT原理和算法[M],第二版,上海：上海交通大学出版社,1992,1-99.
5. Gabor T Herman. Image Reconstruction From Projections--The Fundamental of Computed Tomography [M], Academic Press,1980.
6. R L Siddon. Fast calculation of the exact radiological path for a three-dimensional CT array [J], Medical Physics, March 1985, Vol.12:252-255.
7.黄爱民等.数字图像处理与分析基础[M],北京：中国水利水电出版社,2005,108-112.
8.[美]罗纳德N布雷斯韦尔.傅里叶变换及其应用[M],第三版,西安：西安交通大学出版社,2005,173-180.
9. Zikuan Chen, Mohammad A Karim, Majeed M Hayat. Elimination of higher order aliasings by multiple interlaced sampling [J], Optical Engineering,1999,38(5):879-885.
10. Paul A Rattey, Allen G Lindgren. Sampling the 2-D Radon Transform [J], IEEE Transactions on Acoustics, Speech And Signal Processing,1981,29(5):994-1002.
11.邸燕,常宏宇.Radon变换在断层成像中的应用[J],数学的实践与认识,2004,34(12)：87-90.
12.陈天华.数字图像处理[M],北京：清华大学出版社,2007,101-109.
13.孙即祥.图像压缩与投影重建[M],第一版,北京：科学出版社,2005,171-187.
14.刘丹.计算机图像处理的数学和算法基础[M],北京：国防工业出版社,2005,146-152.
15.张铁,李建东,李长军等.CT成像的改进的Fourier算法及其实验[J],CT理论与应用研究,2000,9(4)：7-11.
16.张铁,阎家斌.求解Radon变换改进Fourier算法的误差分析[J],CT理论与应用研究,2000,9(1)：12-16.
17.吴梦秋,程正兴.反投影重建算法的改进算法[J],西安工业学院学报,2004,24(1)：76-81.
18. Guy M Besson. CT Image Reconstruction from Fan-Parallel Projection Data [J], IEEE, 1999,1644-1648.
19.王一多.快速滤波反投影算法研究[D],长春：吉林大学硕士学位论文,2002.
20. Brady, M. L. A Fast Discrete Approximation Algorithm for the Radon Transform [J], SIAM Journal of Computing,1998,27(1):107-119.
21. Nilsson. S. Application of Fast Backprojection Techniques for Some Inverse Problems of Integral Geometry [D], Department of Mathmatics, Linkoping University,1997.
22. Jiang Hsieh. Computed Tomography [M], Bellingham, Washington USA:SPIE-The International Society for Optical Engineering,2003,64-98.
23. Proksa, R, T. Kohler, M. Grass, and J. Timmer. The n-PI-Method for Helical Cone-Beam CT [J], IEEE Transactions on Medical Imaging,2000,19(9):848-863.
24.刘晓平.扇束卷积反投影法的程序优化[J],CT理论与应用研究,1996,5(1)：35-37.
25.周光湖.计算机断层摄影原理及应用[M],成都：成都电讯工程学院出版社,1986,1-211.
26.江根苗,连兵,刘晋军,王怀志.CT图像重建的算法优化和代码优化[J],CT理论与应用研究,2000,第9卷增刊.
27.张朋,张兆田.几种CT图像重建算法的研究和比较[J],CT理论与应用研究,2001,10(4)：4-8.
28.孙晓安,陈淑珍,吴志斌,柴亚萍.图象重建中的最优化方法[J],中国图像图形学报,1999,4(2)：105-109.
29.过传卫,胡福乔.扇束等距CT滤波反投影重建算法的研究[J],微计算机信息,2007,23(7-1)：292-294.
30. Zikuan Chen, Ruola Ning. Filling the Radon domain in computed tomography by local convex combination [J], APPLIED OPTICS,2003,42(35):7043-7050.
31. A C Kak, M Slaney. Principles of Computerized Tomographic Imaging [M], New York, IEEE Press,1988.
32. I. Svalbe, D. Spek. Reconstruction of tomographic images using analog projections and the digital Radon transform [J], Linear Algebr,2001, Appl.339,125-145.
33. I. Amidror. Scattered data interpolation methods for electronic imaging systems:a survey [J], Electron. Imaging 11,2002,157-176.
34. S. E. Parker. Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation [J], Compute. Phys,2002,178,520-532.
35. D. Rajan and S. Chaudhuri. Generalized interpolation and its application in super-resolution imaging [J], Image Vision Compute.19,2001,957-969.
36. W K Carey, D B Chuang, S S Hemami. Regularity-preserving image interpolation [J], IEEE Trans, Image Proc.1999,8,1293-1297.
37. J. A. Parker, R. V. Kenyon, and D. E. Troxel. Comparison of interpolating methods for image resampling [J], IEEE Trans. Med. Imaging 2,1983,31-39.
38.伯晓晨,李涛,刘路.Matlab工具箱应用指南[M],北京：电子工业出版社,2000,20-166.
39.薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M],北京：清华大学出版社,2006,260-297.
40.冈萨雷斯著,阮秋琦译.数字图像处理(MATLAB版)[M],北京：电子工业出版社,2005.
41.朱晕.利用MATLAB进行图像重建的算法研究[D],兰州：兰州大学硕士学位论文,2003.