CT和定点式SAR中的直接傅立叶重建算法研究
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摘要
计算机层析成像技术(CT)和综合孔径雷达(SAR)是图像重建技术应用的两个重要领域。卷积反投影(CBP)与直接傅立叶重建(DF)同是图像重建中变换法的典型算法。一般说来，DF算法原理简单，重建速度快，但缺乏准确、高效的插值算法，致使重建质量不如CBP法。因而，进一步开展DF重建算法在CT和SAR图像重建领域中的应用研究具有重要的理论价值和现实意义。
论文从系统模型的建立入手，根据DF算法与CBP算法的内在联系，推算出使用雅可比加权的二维周期sinc核插值器的DF算法就等价于CBP算法，然后基于这一结论分别在CT和SAR系统模型中进行了仿真实现。文中讨论的各种形式的sinc核插值DF重建算法，都可概括为以下两步：
(ⅰ)使用雅可比加权的二维周期sinc核插值器(或简化形式的sinc核插值器)，对非笛卡尔形的傅立叶数据进行插值；
(ⅱ)二维傅立叶反变换。
由于SAR系统中近似笛卡尔形的采样数据使其插值过程简易化，因而DF算法在SAR中更具优势。对小角度采样的SAR而言，简单的低阶插值即可获得很好的DF重建。
要说明的是，雅可比加权的sinc核插值并不是最优插值器。也即，使用理想插值器的DF算法可望获得优于CBP法的重建效果。总之，寻求准确高效的DF层析重建算法就等同于寻求DF中准确高效、近似理想的插值方法。再者，用以FFT优化的DSP芯片的推陈出新，使得DF算法具有广阔的发展前景。
Computerized tomography (CT) and synthetic aperture radar (SAR) are two of the most important fields in image reconstruction. Both convolution Backprojection (CBP) method and direct-Fourier (DF) Reconstruction method are the representative algorithms of transform methods in image reconstruction. Generally speaking, DF algorithm has the simple theory and the fast reconstruction, but the lack of good efficient interpolation makes the quality of reconstruction is inferior to that of CBP method. Thus, the further investigation of DF algorithm has the important theoretic value and practical meaning in the image reconstruction field of CT and SAR.
In this paper, we begin with system modeling. Then, according to the internal relations of DF and CBP, we show that the CBP algorithm is equivalent to DF reconstruction by using a Jacobian-weighted 2D sine-kernel interpolator. At last, this conclusion is applied to the simulations of CT and S AR. All reconstruction algorithms discussed in this thesis can be considered as a method of implementing a two-step reconstruction procedure: (i) Interpolation of non-Cartesian Fourier data, using a 2D Jacobian-weighted
sinc-kernal (or simplified version of the sine-kernel) interpolator; (ii) 2D inverse FFT.
For the S AR nearly Cartesian shape of the sampling grid makes interpolation more facilitation, DF method used in SAR has more advantages than those in CT. When the data collection angle is very small, the DF method performs very well with interpolators of low complexity.
It is point out that this interpolator using a Jacobian-weighted 2D sine-kernel is not optimal, i.e. DF algorithms utilizing optimal interpolators may surpass CBP in image quality. In conclusion, searching for accurate and efficient tomographic reconstruction algorithms is equivalent to that for accurate and efficient methods for approximating the optimal interpolator in DF reconstruction. Furthermore, many new DSP chips are optimized for FFT operations, which make DF algorithms of great promise.
论文从系统模型的建立入手，根据DF算法与CBP算法的内在联系，推算出使用雅可比加权的二维周期sinc核插值器的DF算法就等价于CBP算法，然后基于这一结论分别在CT和SAR系统模型中进行了仿真实现。文中讨论的各种形式的sinc核插值DF重建算法，都可概括为以下两步：
(ⅰ)使用雅可比加权的二维周期sinc核插值器(或简化形式的sinc核插值器)，对非笛卡尔形的傅立叶数据进行插值；
(ⅱ)二维傅立叶反变换。
由于SAR系统中近似笛卡尔形的采样数据使其插值过程简易化，因而DF算法在SAR中更具优势。对小角度采样的SAR而言，简单的低阶插值即可获得很好的DF重建。
要说明的是，雅可比加权的sinc核插值并不是最优插值器。也即，使用理想插值器的DF算法可望获得优于CBP法的重建效果。总之，寻求准确高效的DF层析重建算法就等同于寻求DF中准确高效、近似理想的插值方法。再者，用以FFT优化的DSP芯片的推陈出新，使得DF算法具有广阔的发展前景。
Computerized tomography (CT) and synthetic aperture radar (SAR) are two of the most important fields in image reconstruction. Both convolution Backprojection (CBP) method and direct-Fourier (DF) Reconstruction method are the representative algorithms of transform methods in image reconstruction. Generally speaking, DF algorithm has the simple theory and the fast reconstruction, but the lack of good efficient interpolation makes the quality of reconstruction is inferior to that of CBP method. Thus, the further investigation of DF algorithm has the important theoretic value and practical meaning in the image reconstruction field of CT and SAR.
In this paper, we begin with system modeling. Then, according to the internal relations of DF and CBP, we show that the CBP algorithm is equivalent to DF reconstruction by using a Jacobian-weighted 2D sine-kernel interpolator. At last, this conclusion is applied to the simulations of CT and S AR. All reconstruction algorithms discussed in this thesis can be considered as a method of implementing a two-step reconstruction procedure: (i) Interpolation of non-Cartesian Fourier data, using a 2D Jacobian-weighted
sinc-kernal (or simplified version of the sine-kernel) interpolator; (ii) 2D inverse FFT.
For the S AR nearly Cartesian shape of the sampling grid makes interpolation more facilitation, DF method used in SAR has more advantages than those in CT. When the data collection angle is very small, the DF method performs very well with interpolators of low complexity.
It is point out that this interpolator using a Jacobian-weighted 2D sine-kernel is not optimal, i.e. DF algorithms utilizing optimal interpolators may surpass CBP in image quality. In conclusion, searching for accurate and efficient tomographic reconstruction algorithms is equivalent to that for accurate and efficient methods for approximating the optimal interpolator in DF reconstruction. Furthermore, many new DSP chips are optimized for FFT operations, which make DF algorithms of great promise.
引文
[1] 陈立成编著，《层析成像的数学方法与应用》，西安交通大学出版社，1994．
[2] 洪明乾，生物医学成像百年史，自然杂志，1995，17(4)，pp．208-210．
[3] Harish P. Hiriyannariah, X-ray computed tomography for medical imaging, IEEE signal processing magazine, March 1997, pp: 42-59.
[4] Takahiro Kozuka, Kazuo Minaguchi, Ryuji Yamaguchi, et al, Three dimensional imaging of tracheobronchial system using spiral CT, Computer Methods and Programs in Biomedicine 1998, 57, pp: 133-138.
[5] 丁志俊，地球物理勘查与层析成像技术，物探化探译丛，1995(2)，pp．20-33．
[6] 温俊海，当前图像重建研究中的几个热点，生物医学工程学杂志，1999，16(1)，pp．104-108．
[7] 张澄波编著，《综合孔径雷达——原理、系统分析与应用》，科学出版社，1989．
[8] P. G.. Lale, The Examination of Internal Tissues Using Gamma-Ray Scatter with Possible Extention to Megavoltage Radiograph, Phys. Med. Bio., 1959, 4:159.
[9] 吕维雪编著，《医学图像处理》，高等教育出版社，1989．9．
[10] 庄天戈编著，《计算机在生物医学中的应用》，科学出版社 2000．
[11] G.. N. Hounsfield, Computed Medical Imaging, Journal of Computer Assisted Tomography, Oct. 1980, 4(5), pp. 655～674.
[12] D. L. Snyder, An Overview of Reconstructive Tomography and Limitations Imposed by a Finite number of Projections, Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Terpogossian Eds. Baltimore MD: Univ. Park Press, 1977.
[13] 何振亚，信号复原与重建，东南大学出版社，1992．
[14] Aziz Chihoub, Fourier-based reconstruction for CT: a parallel processing perspective, IEEE ENGINEERING IN MEDICINE AND BIOLOGY, November/December 2002, pp.99-108.
[15] 郎锐编写，《数字图像处理学——Visual C++实现》，北京希望电子出版社 2003．
[16] G T．Herman，《由投影重建图像——CT的理论基础》，科学出版社，1985．
[17] C. W. Sherwin, etal., Some Early Developments in Synthetic Aperture Radar Systems, IRE, April 1962, Vol. MIL-6, NO.2, pp. 111-115.
[18] W. M. Brown, L. J. Porcello, An introduction to synthetic aperture radar, IEEE Spectrum, Sept. 1969, pp:52-62.
[19] S·A·霍凡尼斯恩著，倪汉昌译，《合成阵与成像雷达导论》，宇航出版社，1986．8．
[20] 魏钟铨等著，《合成孔径雷达卫星》，科学出版社，2001．
[21] 袁孝康，合成孔径雷达的发展现状与未来，上海航天，2002．5，pp．42-47．
[22] 陶望平编著，《雷达》，科学出版社，1978．2．
[23] 洪文、毛士艺等，一种改进的机载聚束式合成孔径雷达层析算法，《电子学报》，Mar．1998，VOL．26，No．3，pp, 27-35．
[24] David C. Munson, JR., Member, IEEE, James Dennis O'Brien, student member, IEEE, and W. Kenneth Jenkins, Senior Member, IEEE, A Tomographic Formulation of Spotlight-mode Synthetic Aperture Radar, Proceedings of the IEEE, August 1983, VOL. 71, No. 8, pp. 917-925.
[25] 袁孝康，论合成孔径雷达的模糊性，空间电子技术，2001(1)，pp．78-90．
[26] 袁孝康，雷达图像的分辨特性，系统工程与电子技术，1999，21(3)，pp．26-30．
[27] Hong Wen, Mao Shiyi, Fourier reconstruction in spotlight mode SAR imaging, Radar, 1996. Proceedings., CIE International Conference of, Oct. 1996, Vol.8, No.10, pp.338-341.
[28] Hyeokho Choi and David C. Munson, Jr, DIRECT-FOURIER RECONSTRUCTION IN TOMOGRAPHY AND SYNTHETIC APERTURE RADAR, 1997, pp.6-11.
[29] HARISH P.HIRIYANNAIAH, X-ray Computed Tomography for Medical Imaging, IEEE SIGNAL PROCESSING MAGAZINE, pp. 42-59. MARCH 1997.
[30] SUMUEL MATEJ and IVAN BAJLA, A High-Speed Reconstruction from
Projections Using Direct Fourier Method with Optimized Parameters—An Experimental Analysis, IEEE TRANSACTIONS ON MEDICAL IMAGING, DECEMBER 1990, 9(4), pp.421-429.
[31]. R. M. Mersereau, Direct Fourier transform techniques in 3-D image reconstruction, Comput. Biol. Med., 1976, vol. 6, pp. 247-258.
[32] H. Stark, J. W. Woods, I. Paul and R. Hingorani, An investigation of computerized tomography by direct Fourier inversion and optimum interpolation, IEEE Trans. Biomed. Eng., vol. BME-28, pp. 496-505, 1981.
[33] N. Niki, R. T. Mizutani, Y. Takahashi and T. Inouye, A high-speed computerized tomography image reconstruction using direct two-dimensional Fourier transform method, Syst. Comput. Controls, vol. 14, no. 3, pp.56-65, 1983.
[34] K．E．阿特金森著，匡蛟勋，王国荣译，《数值分析引论》，上海科学技术出版社 1986．7．
[35] Mita D. Desai, W. Kenneth Jenkins, Convolution backprojection image reconstruction for spotlight mode synthetic aperture radar, IEEE Trans. Image Processing, Oct. 1992, vol. 1, No. 4, pp. 505-517.
[36] J. L. Walker, Range-Doppler imaging of rotating objects, IEEE Trans. on Aerospace and Electronic Systems, January 1980, vol. AES-16, no. 1, pp. 23-52.
[37] D. C. Munson, Jr. and J. L. C. Sanz, Image reconstruction from frequency-offset Fourier data, Proceedings of the IEEE, June 1984, vol. 72, no. 6, pp. 661-669.
[38] C. V. Jakowatz, Jr., D. E. Wahl, R H. Eichel, D. C. Ghiglis and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach. Norwell, MA: Kluwer Academic Publishers, 1996.
[39] J. I. Jackson, C. H. Meyer, D. G. Nishimura and A. Macovski, Selection of a convolution function for Fourier inversion using gridding, IEEE Trans. on Medical Imaging, September 1991, vol. MI-10, no. 3, pp. 473-478.
[40] H. Choi and D. C. Munson, Jr., Analysis and design of minimax-optimal interpolators, proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, Detroit, MI., 1995, May 8-12, pp. 885-888.
[2] 洪明乾，生物医学成像百年史，自然杂志，1995，17(4)，pp．208-210．
[3] Harish P. Hiriyannariah, X-ray computed tomography for medical imaging, IEEE signal processing magazine, March 1997, pp: 42-59.
[4] Takahiro Kozuka, Kazuo Minaguchi, Ryuji Yamaguchi, et al, Three dimensional imaging of tracheobronchial system using spiral CT, Computer Methods and Programs in Biomedicine 1998, 57, pp: 133-138.
[5] 丁志俊，地球物理勘查与层析成像技术，物探化探译丛，1995(2)，pp．20-33．
[6] 温俊海，当前图像重建研究中的几个热点，生物医学工程学杂志，1999，16(1)，pp．104-108．
[7] 张澄波编著，《综合孔径雷达——原理、系统分析与应用》，科学出版社，1989．
[8] P. G.. Lale, The Examination of Internal Tissues Using Gamma-Ray Scatter with Possible Extention to Megavoltage Radiograph, Phys. Med. Bio., 1959, 4:159.
[9] 吕维雪编著，《医学图像处理》，高等教育出版社，1989．9．
[10] 庄天戈编著，《计算机在生物医学中的应用》，科学出版社 2000．
[11] G.. N. Hounsfield, Computed Medical Imaging, Journal of Computer Assisted Tomography, Oct. 1980, 4(5), pp. 655～674.
[12] D. L. Snyder, An Overview of Reconstructive Tomography and Limitations Imposed by a Finite number of Projections, Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Terpogossian Eds. Baltimore MD: Univ. Park Press, 1977.
[13] 何振亚，信号复原与重建，东南大学出版社，1992．
[14] Aziz Chihoub, Fourier-based reconstruction for CT: a parallel processing perspective, IEEE ENGINEERING IN MEDICINE AND BIOLOGY, November/December 2002, pp.99-108.
[15] 郎锐编写，《数字图像处理学——Visual C++实现》，北京希望电子出版社 2003．
[16] G T．Herman，《由投影重建图像——CT的理论基础》，科学出版社，1985．
[17] C. W. Sherwin, etal., Some Early Developments in Synthetic Aperture Radar Systems, IRE, April 1962, Vol. MIL-6, NO.2, pp. 111-115.
[18] W. M. Brown, L. J. Porcello, An introduction to synthetic aperture radar, IEEE Spectrum, Sept. 1969, pp:52-62.
[19] S·A·霍凡尼斯恩著，倪汉昌译，《合成阵与成像雷达导论》，宇航出版社，1986．8．
[20] 魏钟铨等著，《合成孔径雷达卫星》，科学出版社，2001．
[21] 袁孝康，合成孔径雷达的发展现状与未来，上海航天，2002．5，pp．42-47．
[22] 陶望平编著，《雷达》，科学出版社，1978．2．
[23] 洪文、毛士艺等，一种改进的机载聚束式合成孔径雷达层析算法，《电子学报》，Mar．1998，VOL．26，No．3，pp, 27-35．
[24] David C. Munson, JR., Member, IEEE, James Dennis O'Brien, student member, IEEE, and W. Kenneth Jenkins, Senior Member, IEEE, A Tomographic Formulation of Spotlight-mode Synthetic Aperture Radar, Proceedings of the IEEE, August 1983, VOL. 71, No. 8, pp. 917-925.
[25] 袁孝康，论合成孔径雷达的模糊性，空间电子技术，2001(1)，pp．78-90．
[26] 袁孝康，雷达图像的分辨特性，系统工程与电子技术，1999，21(3)，pp．26-30．
[27] Hong Wen, Mao Shiyi, Fourier reconstruction in spotlight mode SAR imaging, Radar, 1996. Proceedings., CIE International Conference of, Oct. 1996, Vol.8, No.10, pp.338-341.
[28] Hyeokho Choi and David C. Munson, Jr, DIRECT-FOURIER RECONSTRUCTION IN TOMOGRAPHY AND SYNTHETIC APERTURE RADAR, 1997, pp.6-11.
[29] HARISH P.HIRIYANNAIAH, X-ray Computed Tomography for Medical Imaging, IEEE SIGNAL PROCESSING MAGAZINE, pp. 42-59. MARCH 1997.
[30] SUMUEL MATEJ and IVAN BAJLA, A High-Speed Reconstruction from
Projections Using Direct Fourier Method with Optimized Parameters—An Experimental Analysis, IEEE TRANSACTIONS ON MEDICAL IMAGING, DECEMBER 1990, 9(4), pp.421-429.
[31]. R. M. Mersereau, Direct Fourier transform techniques in 3-D image reconstruction, Comput. Biol. Med., 1976, vol. 6, pp. 247-258.
[32] H. Stark, J. W. Woods, I. Paul and R. Hingorani, An investigation of computerized tomography by direct Fourier inversion and optimum interpolation, IEEE Trans. Biomed. Eng., vol. BME-28, pp. 496-505, 1981.
[33] N. Niki, R. T. Mizutani, Y. Takahashi and T. Inouye, A high-speed computerized tomography image reconstruction using direct two-dimensional Fourier transform method, Syst. Comput. Controls, vol. 14, no. 3, pp.56-65, 1983.
[34] K．E．阿特金森著，匡蛟勋，王国荣译，《数值分析引论》，上海科学技术出版社 1986．7．
[35] Mita D. Desai, W. Kenneth Jenkins, Convolution backprojection image reconstruction for spotlight mode synthetic aperture radar, IEEE Trans. Image Processing, Oct. 1992, vol. 1, No. 4, pp. 505-517.
[36] J. L. Walker, Range-Doppler imaging of rotating objects, IEEE Trans. on Aerospace and Electronic Systems, January 1980, vol. AES-16, no. 1, pp. 23-52.
[37] D. C. Munson, Jr. and J. L. C. Sanz, Image reconstruction from frequency-offset Fourier data, Proceedings of the IEEE, June 1984, vol. 72, no. 6, pp. 661-669.
[38] C. V. Jakowatz, Jr., D. E. Wahl, R H. Eichel, D. C. Ghiglis and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach. Norwell, MA: Kluwer Academic Publishers, 1996.
[39] J. I. Jackson, C. H. Meyer, D. G. Nishimura and A. Macovski, Selection of a convolution function for Fourier inversion using gridding, IEEE Trans. on Medical Imaging, September 1991, vol. MI-10, no. 3, pp. 473-478.
[40] H. Choi and D. C. Munson, Jr., Analysis and design of minimax-optimal interpolators, proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, Detroit, MI., 1995, May 8-12, pp. 885-888.