一种改进的Grangeat方法和重建图像质量分析
摘要
随着工业CT在无损检测领域的广泛应用,如何能增进重建图像的质量和提高重建速度受到越来越多的关注。现有的反投影重建方法例如FDK方法虽然重建速度较快,但由于其锥角不能太大,所以重建的物体大小受到了一定的限制。之所以有这样的限制是因为单圆环的扫描轨迹不能满足由Smith提出的精确重建条件。本文引入了一种基于Grangeat理论的改进重建方法。由于这种方法在对Radon域阴影区进行插值后,采用滤波反投影方法进行重建,所以相对于Grangeat方法,运算复杂度较小,而相对于FDK方法,在大锥角的情况下重建图像质量会有所提高。
在重建过程中,重建图像会受到与投影系统相关的因素的影响,导致图像质量下降。图像模糊问题是其中一个比较严重的问题,为了克服这个问题,很多工作者已经作了很有意义的工作,例如采用点扩散函数(Point-spread function)或者调制传递函数(ModulationTransfer Function)进行反卷积。在本文中采用调制传递函数对重建图像质量进行分析,其中调制传递函数可以通过对点扩散函数进行傅立叶变换得到。文中还通过计算机模拟先后列举了对于调制传递函数产生影响的因素。采用了数学模拟模型,分析调制传递函数相对于位置的变化趋势。除此之外,射线源焦点大小、成像时的放大倍数、探测器精度和重建图像的栅格大小也是影响调制传递函数的重要因素。通过对影响重建图像质量因素的分析,本文的最终目的是建立起一个包含多种因素的调制传递函数模型。最后采用FDK和Grangeat两种方法对物体进行重建,通过试验结果对比,对两种重建方法在重建质量及受上述因素影响的程度进行了分析。
With the extensive application of industrial computed tomography in the field of non-destructive testing, how to improve the quality of the reconstructed image is receiving more and more concern. It is well known that in the existing cone-beam filtered backprojection reconstruction algorithms, such as FDK algorithm, the cone-angle is controlled within a narrow range. Thus the size of the tested work piece is limited. The reason of this limitation is that single circle orbit can not satisfy the sufficient and necessary conditions proposed by B.Smith. An improved cone-beam filtered back-projection reconstruction algorithm is introduced in this paper, which is based on the Grangeat's algorithm. The aim of our algorithm is to improve the quality of the reconstructed image on large cone-angle situation. After filling the shadow zone by interpolation, filter backprojection algorithm is used to reconstruct the image. Comparing with Grangeat algorithm, the computational complexity of this algorithm can be reduced by using FFT. And comparing with FDK algorithm, the quality of the reconstructed image on big cone-angle situation can be improved.
The qualities of initial reconstructed images also suffer from reduced resolution due to all kinds of factors which are relative to the cone-beam CT imaging system. The blurring is one of the most difficult problems caused by imaging system. To suppress the blurring in the reconstructed image, some research has been conducted on the deconvolution using a point spread function (PSF) or Modulation Transfer Function (MTF), MTF is used here instead of PSF. The MTF can be obtained by application of the two-dimensional (2-D) Fourier transform to the point-spread function (PSF).In this paper, the impact of various factors on MTF is subsequently illustrated with computer simulations. We use numerical phantom to calculate the MTF, and analyze its variation with position. The size of X-ray focus and magnification are also two important factors which influence MTF, so their effects on MTF are simulated. In addition, the grids of reconstruction and the sizes of pixel on the detector are also influence the MTF. Finally, we use 3D Feldkamp and Radon transform algorithm to reconstruct the images respectively. By comparing the results obtained from the two representative reconstruction algorithms, the quality of the reconstruction result can be evaluated by MTF.
在重建过程中,重建图像会受到与投影系统相关的因素的影响,导致图像质量下降。图像模糊问题是其中一个比较严重的问题,为了克服这个问题,很多工作者已经作了很有意义的工作,例如采用点扩散函数(Point-spread function)或者调制传递函数(ModulationTransfer Function)进行反卷积。在本文中采用调制传递函数对重建图像质量进行分析,其中调制传递函数可以通过对点扩散函数进行傅立叶变换得到。文中还通过计算机模拟先后列举了对于调制传递函数产生影响的因素。采用了数学模拟模型,分析调制传递函数相对于位置的变化趋势。除此之外,射线源焦点大小、成像时的放大倍数、探测器精度和重建图像的栅格大小也是影响调制传递函数的重要因素。通过对影响重建图像质量因素的分析,本文的最终目的是建立起一个包含多种因素的调制传递函数模型。最后采用FDK和Grangeat两种方法对物体进行重建,通过试验结果对比,对两种重建方法在重建质量及受上述因素影响的程度进行了分析。
With the extensive application of industrial computed tomography in the field of non-destructive testing, how to improve the quality of the reconstructed image is receiving more and more concern. It is well known that in the existing cone-beam filtered backprojection reconstruction algorithms, such as FDK algorithm, the cone-angle is controlled within a narrow range. Thus the size of the tested work piece is limited. The reason of this limitation is that single circle orbit can not satisfy the sufficient and necessary conditions proposed by B.Smith. An improved cone-beam filtered back-projection reconstruction algorithm is introduced in this paper, which is based on the Grangeat's algorithm. The aim of our algorithm is to improve the quality of the reconstructed image on large cone-angle situation. After filling the shadow zone by interpolation, filter backprojection algorithm is used to reconstruct the image. Comparing with Grangeat algorithm, the computational complexity of this algorithm can be reduced by using FFT. And comparing with FDK algorithm, the quality of the reconstructed image on big cone-angle situation can be improved.
The qualities of initial reconstructed images also suffer from reduced resolution due to all kinds of factors which are relative to the cone-beam CT imaging system. The blurring is one of the most difficult problems caused by imaging system. To suppress the blurring in the reconstructed image, some research has been conducted on the deconvolution using a point spread function (PSF) or Modulation Transfer Function (MTF), MTF is used here instead of PSF. The MTF can be obtained by application of the two-dimensional (2-D) Fourier transform to the point-spread function (PSF).In this paper, the impact of various factors on MTF is subsequently illustrated with computer simulations. We use numerical phantom to calculate the MTF, and analyze its variation with position. The size of X-ray focus and magnification are also two important factors which influence MTF, so their effects on MTF are simulated. In addition, the grids of reconstruction and the sizes of pixel on the detector are also influence the MTF. Finally, we use 3D Feldkamp and Radon transform algorithm to reconstruct the images respectively. By comparing the results obtained from the two representative reconstruction algorithms, the quality of the reconstruction result can be evaluated by MTF.
引文
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[2]H.K.Tuy.An inversion formula for cone-beam reconstruction.SIAM J.Appl.Math,1983,43:546-552
[3]G.L.Zeng,R.Clack,G.T.Gullberg.Implementation of Tuy's cone-beam inversion formula Phys.Med.Biol.,1994,39:493-507
[4]Bruce D.Smith.Image Reconstruction from Cone-Beam Projections:Necessary and Sufficient Conditions and Reconstruction Methods.IEEE Transaction on Medical Imaging,1985,MI-4(1):14-25
[5]B.D.Smith,J.Chen.Implementation,investigation,and improvement of a novel cone-beam reconstruction method.IEEE Trans.Med.Imag.,1992,11:360-266
[6]G.L.Zeng,G.T.Gullberg.A cone-beam tomography algorithm for orthogonal circle-and-line orbit.Phys.Med.Biol.,1992,37:563-577
[7]P.Grangeat.Mathematical framework of cone-beam 3-D reconstruction via the first derivative of the Radon transform.In Lecture Notes in Mathamatics NEW York:Springer-Vering,1991:66-97
[8]Hiroyuki Kudo,Tsuneo Saito.Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits.IEEE Transaction on Medical Imaging,1994,13(1):196-211
[9]P.M.Joseph.An improved algorithm for reprojecting rays through pixel images.IEEE Trans.Med.Imaging,1983,MI-1(3):192-196
[10]C.Axellon.Direct Fourier methods in 3-D-reconstruction from cone-beam data.Licentiate thesis,Linkoping Univ.,1994.
[11]C.Axelsson and P.Danielsson.Direct Fourier methods for 3-D tomography reconstruction.Presented at the IEEE 1992 Med.Imaging Conf.,1992:27-31
[12]S.Schaller,T.Flohr,and P.Steffen.An efficient Fourier method for 3-9 Radon inversion in exact cone-beam CT reconstruction.IEEE Trans.Med.Imaging.,1998,17:244-250
[13]H.Turbell.Three-dimensional image reconstruction in circular and computed tomography.Ph.D.dissertation,Dept.Elect.EngLinkoping Univ,Linkoping,Sweden,1999.
[14]Samit.Basu,Toram.Bresler.O(N3logN)Backprojection Algorithm for the 3-D Radon Transforms.IEEE Transactions on Medical Imaging,2002,21(2):76-88
[15]G.Wang and P.C.Cheng.Feldkamp-type cone-beam reconstruction:Rebisited.Proc.8~(th)Int.Conf.3-D Image Proc.Microscopy,1995,Zoological Studies 34:159-161
[16]Seung Wook Lee,Gyuseong Cho,Ge Wang.Artifacts associated with implementation of the Grangeat formula.Med.Phys.2002,29(12):2871-2880
[17]J.Drikssonand P.-E.Danielsson.Helical scan 3D reconstruction using the linogram method.in Proc.1995 Int.Meeting Fully Three-Dimensional Image Reconstruction Radiology Nuclear Medicine,1995:287-290
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[19]王本,王革.锥束CT图像重建算法.CT理论与应用研究,2001,10(2):1-8
[20]孙宏宇,王召巴.三维CT在锥角增大时对重建图像质量的影响.测试技术学报,2002,16(4):299-301
[21]王召巴.任意角度入射的三维CT投影方法.计量学报,2002,23(1)
[22]王召巴.旋转中心偏移对CT重建图像质量的影响.兵工学报,2001,22(3)
[23]王贤刚,郭志平,彭亚辉等.用几何参数表实现快速三维重建CT图像.清华大学学报(自然科学版),2002,42(5):655-658
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[25]曾理,徐问之,陈廷槐.多尺度小波分析用于工业CT投影数据的滤波.无损检测,1998,20(12):331-345
[26]刘杰,李政,康克军.小波多辨率CT成像及处理算法.光电工程,2002,29(2):48-51
[27]孙少华,高文华,张丽等.小波重建中的小波选取.核电子学与探测技术,2003,23(3):252-255
[28]尤江生,包尚联.完全三维图像重建的一个解析公式.北京大学学报(自然科学版),1996,32(5):594-601
[29]黎明,王厚枢.三维CT图像自动分割法.南京航空航天大学学报,1996,28(3):413-418
[30]吕东辉,王朔中,庄天戈.倒置结构锥形束计算机层析的精确成像与Grangeat算法实现.光学学报,2002,22(10):1187-1194
[31]丁克勤,李海超.基于工业CT的压力管道在线测厚仪研制.CT理论与应用研究,2003,12(2):1-5
[32]倪培君,李旭东,张维国等.工业X射线CT的应用.CT理论与应用研究,1997,6(3):36-42
[33]孙怡,侯颖,胡家升等.体积CT的投影数据的模拟方法.CT理论应用与研究.2005,14(1):1-6
[34]G.T.Herman.由投影重建图像,北京:科学出版社,1985
[35]庄天戈.CT原理与算法,上海:上海交通大学出版社,1992
[36]庄天戈,计算机在生物医学中的应用,北京:科学出版社,2000
[37]Henrik Turbell.Cone-Beam Reconstruction Using Filtered Backprojection.Institute of Technology,Linkopings Universitet,Sweden,2001.
[38]Egbert Buhr,Susanne Gunther-Kohfahl,Ulrich Neitzel.Simple method for modulation transfer function determination of digital imaging detectors from edge images.Medical Imaging 2003:Physics of Medical Imaging,Proceeding of SPIE 2003,5030:877-884
[39]P.B.Greer,T.van Doom.Evaluation of an algorithm for the assessment of the MTF using an edge method.Med.Phys.,2000,27:2048-2059
[54]Satyapal Rathee, Zoly J. Koles and Thomas R. Overton. Image Restoration in Computed Tomography: Restoration of Experimental CT Images. IEEE Transactions on Medical Imaging, 1992, 11(4) : 546-553
[55]Satyapal Rathee, Zoly J. Koles and Thomas R. Overton. Image Restoration in Computed Tomography: The Spatially Invariant Point Spread Function. IEEE Transactions on Medical Imaging, 1992, 11(4) :530-538
[56]Shyh-Shiaw Kuo and Richard J. Mammone. Resolution Enhancement of Tomographic Images Using the Row Action Projection Method. IEEE Transactions on Medical Imaging, 1991, 10(4) : 593-601
[57]Tamas Daboczi, and Tamds B. Bako. Inverse Filtering of Optical Images. IEEE Transactions on Instrumentation and Measurement, 2001, 50(4) : 991-994
[58]Chih-Chun Feng, and Chong-Yung Chi. Performance of Cumulant Based Inverse Filters for Blind Deconvolution. IEEE Transactions on Signal Processing, 1999, 47(7) :1922-1935
[59]Ling Guan and Rabab K. Wardm. Restoration of Randomly Blurred Images by the Wiener Filter. IEEE Transactions on Acoustics. Speech and Signal Processing. 1989, 37(4) : 589-592
[60]Xiao-Hong Yan and Richard M. Leahy. Derivation and Analysis of a Filtered Backprojection Algorithm for Cone Beam Projection Data. IEEE Transactions on Medical Imaging, 1991, 10(3) : 462-472
[6l]G T Gullbergt, G L Zengt, F L Datzz et al. Review of convergent beam tomography in single photon emission computed tomography. Phys. Med. Biol, 1992, 37(3):507-534.
[62]Hiroyuki Kudo, and Tsuneo Saito. Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits. IEEE Transactions on Medical Imaging, 1994, 13(1) :196-211
[63]Ramesh R. Galigekere, Karl Wiesent, and David W. Holdsworth. Cone-Beam Reprojection Using Projection-Matrices. IEEE Transactions on medical Imaging, 2003, 22(10) :1202-1214
[64]Naoyoshi, Nameda. Development of a Test Pattern for Measurement of Color Visual MTF Properties. IEEE Proceeding of the 18th International Conference on Advanced Information Networking and Application. 2004
[65]Zikuan Chen, Ruola Ning, David Conover et al. Quantitative assessment of cone-beam CT system by 3D point spread function. Medical Imaging 2004:Physics of Medical Imaging, Proceedings of SPIE 2004, 5368 : 574-585
[66]Nicolas Villain, Yves Goussard, Jerome Idier et al. Three-Dimensional Edge -Preserving Image Enhancement for Computed Tomography. IEEE Transactions on Medical Imaging, 2003, 22(10) : 1275-1287
[67] 投影数据采集实验,上海交通大学生命科学技术学院,生物医学图像处理,http://life. sjtu. edu. cn/biomed/Experiment/Experiment_Book5. asp
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[2]H.K.Tuy.An inversion formula for cone-beam reconstruction.SIAM J.Appl.Math,1983,43:546-552
[3]G.L.Zeng,R.Clack,G.T.Gullberg.Implementation of Tuy's cone-beam inversion formula Phys.Med.Biol.,1994,39:493-507
[4]Bruce D.Smith.Image Reconstruction from Cone-Beam Projections:Necessary and Sufficient Conditions and Reconstruction Methods.IEEE Transaction on Medical Imaging,1985,MI-4(1):14-25
[5]B.D.Smith,J.Chen.Implementation,investigation,and improvement of a novel cone-beam reconstruction method.IEEE Trans.Med.Imag.,1992,11:360-266
[6]G.L.Zeng,G.T.Gullberg.A cone-beam tomography algorithm for orthogonal circle-and-line orbit.Phys.Med.Biol.,1992,37:563-577
[7]P.Grangeat.Mathematical framework of cone-beam 3-D reconstruction via the first derivative of the Radon transform.In Lecture Notes in Mathamatics NEW York:Springer-Vering,1991:66-97
[8]Hiroyuki Kudo,Tsuneo Saito.Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits.IEEE Transaction on Medical Imaging,1994,13(1):196-211
[9]P.M.Joseph.An improved algorithm for reprojecting rays through pixel images.IEEE Trans.Med.Imaging,1983,MI-1(3):192-196
[10]C.Axellon.Direct Fourier methods in 3-D-reconstruction from cone-beam data.Licentiate thesis,Linkoping Univ.,1994.
[11]C.Axelsson and P.Danielsson.Direct Fourier methods for 3-D tomography reconstruction.Presented at the IEEE 1992 Med.Imaging Conf.,1992:27-31
[12]S.Schaller,T.Flohr,and P.Steffen.An efficient Fourier method for 3-9 Radon inversion in exact cone-beam CT reconstruction.IEEE Trans.Med.Imaging.,1998,17:244-250
[13]H.Turbell.Three-dimensional image reconstruction in circular and computed tomography.Ph.D.dissertation,Dept.Elect.EngLinkoping Univ,Linkoping,Sweden,1999.
[14]Samit.Basu,Toram.Bresler.O(N3logN)Backprojection Algorithm for the 3-D Radon Transforms.IEEE Transactions on Medical Imaging,2002,21(2):76-88
[15]G.Wang and P.C.Cheng.Feldkamp-type cone-beam reconstruction:Rebisited.Proc.8~(th)Int.Conf.3-D Image Proc.Microscopy,1995,Zoological Studies 34:159-161
[16]Seung Wook Lee,Gyuseong Cho,Ge Wang.Artifacts associated with implementation of the Grangeat formula.Med.Phys.2002,29(12):2871-2880
[17]J.Drikssonand P.-E.Danielsson.Helical scan 3D reconstruction using the linogram method.in Proc.1995 Int.Meeting Fully Three-Dimensional Image Reconstruction Radiology Nuclear Medicine,1995:287-290
[18]M.Defrise and R.Clack.A cone-beam reconstruction algorithm using shift-variant filtering cone-beam backprojection.IEEE Trans.Med.Imag,1994,13:186-195
[19]王本,王革.锥束CT图像重建算法.CT理论与应用研究,2001,10(2):1-8
[20]孙宏宇,王召巴.三维CT在锥角增大时对重建图像质量的影响.测试技术学报,2002,16(4):299-301
[21]王召巴.任意角度入射的三维CT投影方法.计量学报,2002,23(1)
[22]王召巴.旋转中心偏移对CT重建图像质量的影响.兵工学报,2001,22(3)
[23]王贤刚,郭志平,彭亚辉等.用几何参数表实现快速三维重建CT图像.清华大学学报(自然科学版),2002,42(5):655-658
[24]张朝宗,郭志平,董宇峰等.用几何参数表方法实现快速重建CT图像.清华大学学报(自然科学版),1998,38(7):332-335
[25]曾理,徐问之,陈廷槐.多尺度小波分析用于工业CT投影数据的滤波.无损检测,1998,20(12):331-345
[26]刘杰,李政,康克军.小波多辨率CT成像及处理算法.光电工程,2002,29(2):48-51
[27]孙少华,高文华,张丽等.小波重建中的小波选取.核电子学与探测技术,2003,23(3):252-255
[28]尤江生,包尚联.完全三维图像重建的一个解析公式.北京大学学报(自然科学版),1996,32(5):594-601
[29]黎明,王厚枢.三维CT图像自动分割法.南京航空航天大学学报,1996,28(3):413-418
[30]吕东辉,王朔中,庄天戈.倒置结构锥形束计算机层析的精确成像与Grangeat算法实现.光学学报,2002,22(10):1187-1194
[31]丁克勤,李海超.基于工业CT的压力管道在线测厚仪研制.CT理论与应用研究,2003,12(2):1-5
[32]倪培君,李旭东,张维国等.工业X射线CT的应用.CT理论与应用研究,1997,6(3):36-42
[33]孙怡,侯颖,胡家升等.体积CT的投影数据的模拟方法.CT理论应用与研究.2005,14(1):1-6
[34]G.T.Herman.由投影重建图像,北京:科学出版社,1985
[35]庄天戈.CT原理与算法,上海:上海交通大学出版社,1992
[36]庄天戈,计算机在生物医学中的应用,北京:科学出版社,2000
[37]Henrik Turbell.Cone-Beam Reconstruction Using Filtered Backprojection.Institute of Technology,Linkopings Universitet,Sweden,2001.
[38]Egbert Buhr,Susanne Gunther-Kohfahl,Ulrich Neitzel.Simple method for modulation transfer function determination of digital imaging detectors from edge images.Medical Imaging 2003:Physics of Medical Imaging,Proceeding of SPIE 2003,5030:877-884
[39]P.B.Greer,T.van Doom.Evaluation of an algorithm for the assessment of the MTF using an edge method.Med.Phys.,2000,27:2048-2059
[54]Satyapal Rathee, Zoly J. Koles and Thomas R. Overton. Image Restoration in Computed Tomography: Restoration of Experimental CT Images. IEEE Transactions on Medical Imaging, 1992, 11(4) : 546-553
[55]Satyapal Rathee, Zoly J. Koles and Thomas R. Overton. Image Restoration in Computed Tomography: The Spatially Invariant Point Spread Function. IEEE Transactions on Medical Imaging, 1992, 11(4) :530-538
[56]Shyh-Shiaw Kuo and Richard J. Mammone. Resolution Enhancement of Tomographic Images Using the Row Action Projection Method. IEEE Transactions on Medical Imaging, 1991, 10(4) : 593-601
[57]Tamas Daboczi, and Tamds B. Bako. Inverse Filtering of Optical Images. IEEE Transactions on Instrumentation and Measurement, 2001, 50(4) : 991-994
[58]Chih-Chun Feng, and Chong-Yung Chi. Performance of Cumulant Based Inverse Filters for Blind Deconvolution. IEEE Transactions on Signal Processing, 1999, 47(7) :1922-1935
[59]Ling Guan and Rabab K. Wardm. Restoration of Randomly Blurred Images by the Wiener Filter. IEEE Transactions on Acoustics. Speech and Signal Processing. 1989, 37(4) : 589-592
[60]Xiao-Hong Yan and Richard M. Leahy. Derivation and Analysis of a Filtered Backprojection Algorithm for Cone Beam Projection Data. IEEE Transactions on Medical Imaging, 1991, 10(3) : 462-472
[6l]G T Gullbergt, G L Zengt, F L Datzz et al. Review of convergent beam tomography in single photon emission computed tomography. Phys. Med. Biol, 1992, 37(3):507-534.
[62]Hiroyuki Kudo, and Tsuneo Saito. Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits. IEEE Transactions on Medical Imaging, 1994, 13(1) :196-211
[63]Ramesh R. Galigekere, Karl Wiesent, and David W. Holdsworth. Cone-Beam Reprojection Using Projection-Matrices. IEEE Transactions on medical Imaging, 2003, 22(10) :1202-1214
[64]Naoyoshi, Nameda. Development of a Test Pattern for Measurement of Color Visual MTF Properties. IEEE Proceeding of the 18th International Conference on Advanced Information Networking and Application. 2004
[65]Zikuan Chen, Ruola Ning, David Conover et al. Quantitative assessment of cone-beam CT system by 3D point spread function. Medical Imaging 2004:Physics of Medical Imaging, Proceedings of SPIE 2004, 5368 : 574-585
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