基于非均匀傅里叶变换的超声层析成像研究
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摘要
超声波成像技术作为现代医学四大影像技术之一,有着不可替代的作用。X光成像可以准确获得人体组织的信息,是一种很好的检测手段,但其对软组织的成像不理想,且因其有电离辐射特性,对于人体的影响较大,所以它的应用有局限性。超声成像仪器因其价格便宜,无电离辐射,可以制造成便携式设备等特点,在医院的使用频率很高。常用的B超能应用于常规性的体检或胎儿的检查等情况。但是B超的成像较模糊,对人体情况的判断需要依靠医生的经验,个人的主观性会影响判断的结果,因此超声波用于人体的定量的成像成为一个重要的研究方向。层析成像(Computed Tomography)是通过从许多不同方向照射物体,根据透射或反射的数据来重建断面图像的成像技术。
本文讨论的是考虑散射的情况下采用超声波这种安全的能源对人体组织进行定量成像的超声衍射层析成像问题。所完成的主要工作和创新点如下:
基于傅里叶散射投影定理,频域插值(Gridding)算法是在频域对变换后的散射场数据进行插值,得到均匀网格上的值,从而进行二维傅里叶反变换得到介质成像平面的分布。
非均匀傅里叶变换(Non-uniform Fourier Transform)包括前向NUFT和后向NUFT。其中前向NUFT的快速算法可以采用对过采样的FFT变换的频域值进行插值得到。min-max方法对其插值参数进行优化,得到基于min-max的快速非均匀傅里叶变换的算法。本文讨论的是后向NUFT,即二维情况下从非均匀样点到均匀样点的前向NUFT的反变换问题。因为计算的规模很大,不能采用直接求解逆矩阵的方法实现,所以采用最小二乘法的思想,采用迭代算法(CG)逐步接近最优化解。采用插值方法的成像结果作为迭代的初始值,可以减少迭代次数,减小运算量。经试验分析,采用本文的方法,成像效果得到提高。
对仿真数据的获取方法进行了讨论分析,在弱散射情况下,基于傅里叶散射投影定理,可以对Shepp-Logan模型的频域任一点的值进行解析求解,基于此可以得到不引入任何误差的用于仿真的数据。对多频率投影的情况,分析其频域分布特点,在多频率入射的情况下,可以减少成像的时间,在医学图像的提取中是一个好的发展方向。在成像过程中会出现不完整投影的情况,分析其不完整程度对成像效果的影响,和数据冗余以减少投影或采样点,使信息的利用率变高。对透射和反射模式下散射场所提供的信息量进行分析,在不增大采样时间的情况下可同时获取反射场和透射场的数据,将其进行结合,可以提高成像质量。
As one of modern medical imaging techniques, ultrasound imaging technique has its irreplaceable role. X-ray imaging technique can accurately obtain information on human tissue, but it is not ideal for soft tissues. X-ray imaging has a huge impact on the human body due to associated radiation hazards, so there are limits in application. Ultrasound imaging equipment is used highly frequency in the hospital for its cheap, non-ionizing radiation, and it could be made into portable devices. B-mode ultrasonic can be applied to common physical examination or routine fetal examination and so on, but the quality of imaging is fuzzy, to judge the situation on the human body need to rely on the experience of doctors which affect the individual's subjective judgments of the results, so the quantitative ultrasound imaging for human become an important area. Computed Tomography is an imaging technology which illuminating an object from many different directions and reconstructing the cross-section image of the object according to transmission data or reflection data.
We discussed the ultrasound diffraction tomography problem that a security energy-ultrasonic image the human tissue quantitatively in the case of scatter in this paper. The main work and contribution of this dissertation can be summarized as follows:
According to the Fourier diffraction projection theorem, the frequency domain interpolation (Gridding) algorithm is interpolating the scattered field data that transformed into the frequency domain to get the value on the uniform grid. We get the imaging of the media plane through inverse 2-D Fourier transform.
Non-uniform Fourier transform include forward and backward NUFT. We can get the fast forward NUFT through interpolating the over-sampling FFT transform of the data in the frequency domain. Adopt to the min-max method to optimize the parameters in the interpolation, We get the non-uniform fast Fourier transform algorithm based on min-max. We discussed the backward NUFT samples from the non-uniform to uniform samples that the inverse transform of the forward NUFT problem in the two-dimensional. Because the calculation is large, we can?t solve the inverse matrix using direct method. We introduce the least square method idea using the iterative algorithm (CG) closing to the optimal resolve gradually. Using the imaging results in the in the interpolation method as the initial value of iteration, we can reduce the number of iterations and reduce the computation. Through test analysis , imaging effect is improved using this method.
Access to the simulation data are discussed .In the case of weak scattering, we can get the values of Shepp-Logan model in the frequency domain at any point using the analytic solution based on the Fourier projection theorem. We can get the simulated data without introducing any error based on this. The multi-frequency ultrasonic projections can reduce the imaging time. The standpoint is a good development of medical images. The less integrity projection occured in the imaging process, we analyze the effect of the adequate extent and data redundancy on the imaging to reduce the projector or sampling points, the utilization of information becomes higher. Analysing the information in the transmission and reflection mode scattering, we can get the transmission and reflection data without increasing the sampling time. We can improve the image quality with combining the data.
本文讨论的是考虑散射的情况下采用超声波这种安全的能源对人体组织进行定量成像的超声衍射层析成像问题。所完成的主要工作和创新点如下:
基于傅里叶散射投影定理,频域插值(Gridding)算法是在频域对变换后的散射场数据进行插值,得到均匀网格上的值,从而进行二维傅里叶反变换得到介质成像平面的分布。
非均匀傅里叶变换(Non-uniform Fourier Transform)包括前向NUFT和后向NUFT。其中前向NUFT的快速算法可以采用对过采样的FFT变换的频域值进行插值得到。min-max方法对其插值参数进行优化,得到基于min-max的快速非均匀傅里叶变换的算法。本文讨论的是后向NUFT,即二维情况下从非均匀样点到均匀样点的前向NUFT的反变换问题。因为计算的规模很大,不能采用直接求解逆矩阵的方法实现,所以采用最小二乘法的思想,采用迭代算法(CG)逐步接近最优化解。采用插值方法的成像结果作为迭代的初始值,可以减少迭代次数,减小运算量。经试验分析,采用本文的方法,成像效果得到提高。
对仿真数据的获取方法进行了讨论分析,在弱散射情况下,基于傅里叶散射投影定理,可以对Shepp-Logan模型的频域任一点的值进行解析求解,基于此可以得到不引入任何误差的用于仿真的数据。对多频率投影的情况,分析其频域分布特点,在多频率入射的情况下,可以减少成像的时间,在医学图像的提取中是一个好的发展方向。在成像过程中会出现不完整投影的情况,分析其不完整程度对成像效果的影响,和数据冗余以减少投影或采样点,使信息的利用率变高。对透射和反射模式下散射场所提供的信息量进行分析,在不增大采样时间的情况下可同时获取反射场和透射场的数据,将其进行结合,可以提高成像质量。
As one of modern medical imaging techniques, ultrasound imaging technique has its irreplaceable role. X-ray imaging technique can accurately obtain information on human tissue, but it is not ideal for soft tissues. X-ray imaging has a huge impact on the human body due to associated radiation hazards, so there are limits in application. Ultrasound imaging equipment is used highly frequency in the hospital for its cheap, non-ionizing radiation, and it could be made into portable devices. B-mode ultrasonic can be applied to common physical examination or routine fetal examination and so on, but the quality of imaging is fuzzy, to judge the situation on the human body need to rely on the experience of doctors which affect the individual's subjective judgments of the results, so the quantitative ultrasound imaging for human become an important area. Computed Tomography is an imaging technology which illuminating an object from many different directions and reconstructing the cross-section image of the object according to transmission data or reflection data.
We discussed the ultrasound diffraction tomography problem that a security energy-ultrasonic image the human tissue quantitatively in the case of scatter in this paper. The main work and contribution of this dissertation can be summarized as follows:
According to the Fourier diffraction projection theorem, the frequency domain interpolation (Gridding) algorithm is interpolating the scattered field data that transformed into the frequency domain to get the value on the uniform grid. We get the imaging of the media plane through inverse 2-D Fourier transform.
Non-uniform Fourier transform include forward and backward NUFT. We can get the fast forward NUFT through interpolating the over-sampling FFT transform of the data in the frequency domain. Adopt to the min-max method to optimize the parameters in the interpolation, We get the non-uniform fast Fourier transform algorithm based on min-max. We discussed the backward NUFT samples from the non-uniform to uniform samples that the inverse transform of the forward NUFT problem in the two-dimensional. Because the calculation is large, we can?t solve the inverse matrix using direct method. We introduce the least square method idea using the iterative algorithm (CG) closing to the optimal resolve gradually. Using the imaging results in the in the interpolation method as the initial value of iteration, we can reduce the number of iterations and reduce the computation. Through test analysis , imaging effect is improved using this method.
Access to the simulation data are discussed .In the case of weak scattering, we can get the values of Shepp-Logan model in the frequency domain at any point using the analytic solution based on the Fourier projection theorem. We can get the simulated data without introducing any error based on this. The multi-frequency ultrasonic projections can reduce the imaging time. The standpoint is a good development of medical images. The less integrity projection occured in the imaging process, we analyze the effect of the adequate extent and data redundancy on the imaging to reduce the projector or sampling points, the utilization of information becomes higher. Analysing the information in the transmission and reflection mode scattering, we can get the transmission and reflection data without increasing the sampling time. We can improve the image quality with combining the data.
引文
[Norton1979]Norton S J,Linzer M. Ultrasonic reflectivity tomography: Reconstruction with circular transducer arrays[J]. Ultasonic Imaging.1979,1(2):154-184.
[Kak 1979]Kak, A C. Computerized Tomography with X-Ray Emission and Ultrasound Source[J]. Proceedings of IEEE.1979,67(9):1245-1272.
[Devaney1980] Devaney A J. A new approach to emission and transmission CT[C]. 1980 Ultrasonics Symposium.1980:979-983
[Greenleaf1981] Greenleaf J F, Bahn R C. Clinical Imaging with Transmissive Ultrasonic Computerized Tomography[J].IEEE Transactions on Biomedical Engineering.1981, BME28(2):177-185.
[Norton1981] Norton S J, Stephen J. Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Plane, Cylindrical, and Spherical Apertures[J]. IEEE Engineering in Medicine and Biology Society.1981 BME-28(2):202-220.
[anon1982]-.A filtered backpropagation algorithm for diffraction images[J].Ultrasonic Imaging.1982,4:336-250
[Tracy1983]Tracy M. L. and Steven.A.Johnson. Inverse Scattering Solutions by A Sinc Basis, Multiple Source, Moment Method—Part II: Numerical Evaluations, Ultrasonic Imaging 5, 1983, p376--392.
[Pan1983]Pan S.X. and A.C.Kak,―A Computational Study of Reconstruction Algorithms for Diffraction Tomography: Interpolation Versus Filtered Backpropagation‖, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-31, No.5, October 1983, pp.1262-1275.
[Chew1990]W. C.Chew and Y.M. Wang, Reconstruction of Two-Dimensional Permittivity Distribution Using the Distorted Born Iterative Method, IEEE Transactions on Medical Imaging. 1990,9(2):218-225.
[Schomberg1995]Hermann Schomberg ,Jan Timmer, The Gridding Method for Image Reconstruction by Fourier Transformation, IEEE Transactions on Medical Imaging. 1995,14(3):596-607.
[Haddadin1998]Haddadin Osama S. Imaging Strongly Scattering Media Using a Multiple Frequency Distorted Born Iterative Method[J].IEEE transactions on ultrasonics, ferroelectrics, and frequency control.1998,45(6):1485-1496.
[Liu1998]Liu,Q.H , Nguyen N. Anaccurate algorithm for non-uniform fast Fourier transforms(NUFFT‘s)[J].Microwave and Guided Wave Letters,IEEE.1998, 8:18-20.
[Liu1998] Liu QingHuo, Tang Xueyuan.Iterative algorithm for nonuniform inverse fast FourierTransform(NU-IFFT)[J].Electronics Letters.1998,34(20):1913-1914.
[Kak 1999]Kak A.C,Slaney Malcolm. Principles of Computerized Tomographic Imaging[M]. The Institute of Electronics Engineers,Inc.New York,IEEE Press.1999.
[Tsihrintzis1999] Tsihrintzis G, Devaney A,Heyman E. Estimation of object location from wideband scattering data[J], IEEE Transactions on Image Processing. 1999,8(7):996-1001.
[Calvetti1999]Calvetti D, Reichel L.Iterative solution methods for large linear discrete ill-problem, A pplied and Computational Control[J].Signals and Circuits.1999,1:313-367.
[Anstasio2000]Anastasio M A, Pan Xiaochuan. A New Reconstruction Approach for Reflection Mode Diffraction Tomagraphy[J]. IEEE Transactions on Image Processing. 2000,9(7):1262-1271.
[Xu2000]Xu XueMin, Liu QingHuo.The conjugate-gradient nonuniform fast Fourier transform (CG-NUFFT) methods for one- and two-dimensional media[C]. Antennas and Propagation Society International Symposium, IEEE.2000,4:2332-2335.
[Fessler2001]Fessler J.A;Sutton B.P.A min-max Approach to the Multidimensional Non-uniform FFT: Application to Tomographic Image Reconstruction[J]. International Conference on Image Processing.2001,1:706-709
[Bronstein2002]Bronstein Michael M,Bronstein Alexander M.Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT[J].IEEE Transactions on Medical Imaging. 2002, 21(11):1395-1401.
[Fessler2002] Fessler Jeffrey A. Iterative Tomography Image Reconstruction Using Nonuniform Fast Fourier Transform, NUFFT-Reprojection[M]. Created 2001-12-21.Version April 16,2002.
[Fessler2003]Fessler Jeffrey A, Sutton Bradly P. Nonuniform Fast Fourier Transforms Using Min-Max Interpolation[J]. IEEE Transactions on Signal Processing.2003,51(2):560-574.
[Pan2002]Pan Xiaochuan,Anastasio Mark A. On a limited-View Reconstruction Problem Diffraction Tomography[J].IEEE Transactions on Medical Imaging.2002,21(4):413-416.
[Anastasio2002]Anastasio Mark A,Zou Yu,Pan Xiaochuan.Reflectivity Tomography using Temporally Truncated Data[J]. Proceedings of the Second Joint .2002, 2:921-922.
[Matej2004]Samuel Matej, Fessler Jerrey A. Iterative Tomographic Image Reconstruction Using Fourier-Based Forward and Back-Projection[J].IEEE Transactions on Medical Imaging.2004,23(4):401-412.
[程建春2004]程建春.数学物理方程及其近似方法[M].北京,科学出版社,2004.
[Zhang2004] Zhang Xiaodong , Broschat Shira L. A Numerical Study of Conjugate Gradient Directions for an Ultrasound Inverse ProbIem[J]. Joumal of Computational Acoustics.2004,l 2(4):587—604.
[刘超2005]刘超.超声逆散射成像问题中的正则化方法研究[J].浙江大学学报.2005,6:195-199.
[Pan2005] Pan Xiaochuan,Yu L,Kao C M.Image-resolution enhancement in computed tomography[J]. IEEE Transactions on Medical Imaging.2005,24:246-253.
[Shieh2006]Shieh Hsin M,bryne Charles L. Iterative image Reconstruction Using Prior Kownledge[J]. Optical Society of America.2006,23(6):1292-1300.
[王彦飞2007]王彦飞.反演问题的计算方法及其应用[M].北京,高等教育出版社.2007.
[Fessler2007]Fessler Jerry A.On NUFFT-based gridding for non-Cartesian MRI[J].Journal of Megnetic Resonance.2007,188(2):191-195.
[LaRoque2008]LaRoque S,Sidky E,Pan Xiaochuan.Accurate image reconstruction from few-veiw and limit-angle data in diffraction tomography[J].Optical Society of America. 2008,25:1772-1782.
[Mathews2009]Mathews Jacob.Optimized non-uniform fast Fourier transform(NUFFT) for iterative tomographic reconstruction[C]. ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing– Proceedings. 2009: 673-676.
[陶进绪2009]陶进绪,张东文.频域法超声散射成像[J].信号处理.2009,25(2):169-173.
[王朔中2010]王朔中,方针.声衍射层析成像进展[J].声学技术.2010,29(2):117-122.
[Kak 1979]Kak, A C. Computerized Tomography with X-Ray Emission and Ultrasound Source[J]. Proceedings of IEEE.1979,67(9):1245-1272.
[Devaney1980] Devaney A J. A new approach to emission and transmission CT[C]. 1980 Ultrasonics Symposium.1980:979-983
[Greenleaf1981] Greenleaf J F, Bahn R C. Clinical Imaging with Transmissive Ultrasonic Computerized Tomography[J].IEEE Transactions on Biomedical Engineering.1981, BME28(2):177-185.
[Norton1981] Norton S J, Stephen J. Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Plane, Cylindrical, and Spherical Apertures[J]. IEEE Engineering in Medicine and Biology Society.1981 BME-28(2):202-220.
[anon1982]-.A filtered backpropagation algorithm for diffraction images[J].Ultrasonic Imaging.1982,4:336-250
[Tracy1983]Tracy M. L. and Steven.A.Johnson. Inverse Scattering Solutions by A Sinc Basis, Multiple Source, Moment Method—Part II: Numerical Evaluations, Ultrasonic Imaging 5, 1983, p376--392.
[Pan1983]Pan S.X. and A.C.Kak,―A Computational Study of Reconstruction Algorithms for Diffraction Tomography: Interpolation Versus Filtered Backpropagation‖, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-31, No.5, October 1983, pp.1262-1275.
[Chew1990]W. C.Chew and Y.M. Wang, Reconstruction of Two-Dimensional Permittivity Distribution Using the Distorted Born Iterative Method, IEEE Transactions on Medical Imaging. 1990,9(2):218-225.
[Schomberg1995]Hermann Schomberg ,Jan Timmer, The Gridding Method for Image Reconstruction by Fourier Transformation, IEEE Transactions on Medical Imaging. 1995,14(3):596-607.
[Haddadin1998]Haddadin Osama S. Imaging Strongly Scattering Media Using a Multiple Frequency Distorted Born Iterative Method[J].IEEE transactions on ultrasonics, ferroelectrics, and frequency control.1998,45(6):1485-1496.
[Liu1998]Liu,Q.H , Nguyen N. Anaccurate algorithm for non-uniform fast Fourier transforms(NUFFT‘s)[J].Microwave and Guided Wave Letters,IEEE.1998, 8:18-20.
[Liu1998] Liu QingHuo, Tang Xueyuan.Iterative algorithm for nonuniform inverse fast FourierTransform(NU-IFFT)[J].Electronics Letters.1998,34(20):1913-1914.
[Kak 1999]Kak A.C,Slaney Malcolm. Principles of Computerized Tomographic Imaging[M]. The Institute of Electronics Engineers,Inc.New York,IEEE Press.1999.
[Tsihrintzis1999] Tsihrintzis G, Devaney A,Heyman E. Estimation of object location from wideband scattering data[J], IEEE Transactions on Image Processing. 1999,8(7):996-1001.
[Calvetti1999]Calvetti D, Reichel L.Iterative solution methods for large linear discrete ill-problem, A pplied and Computational Control[J].Signals and Circuits.1999,1:313-367.
[Anstasio2000]Anastasio M A, Pan Xiaochuan. A New Reconstruction Approach for Reflection Mode Diffraction Tomagraphy[J]. IEEE Transactions on Image Processing. 2000,9(7):1262-1271.
[Xu2000]Xu XueMin, Liu QingHuo.The conjugate-gradient nonuniform fast Fourier transform (CG-NUFFT) methods for one- and two-dimensional media[C]. Antennas and Propagation Society International Symposium, IEEE.2000,4:2332-2335.
[Fessler2001]Fessler J.A;Sutton B.P.A min-max Approach to the Multidimensional Non-uniform FFT: Application to Tomographic Image Reconstruction[J]. International Conference on Image Processing.2001,1:706-709
[Bronstein2002]Bronstein Michael M,Bronstein Alexander M.Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT[J].IEEE Transactions on Medical Imaging. 2002, 21(11):1395-1401.
[Fessler2002] Fessler Jeffrey A. Iterative Tomography Image Reconstruction Using Nonuniform Fast Fourier Transform, NUFFT-Reprojection[M]. Created 2001-12-21.Version April 16,2002.
[Fessler2003]Fessler Jeffrey A, Sutton Bradly P. Nonuniform Fast Fourier Transforms Using Min-Max Interpolation[J]. IEEE Transactions on Signal Processing.2003,51(2):560-574.
[Pan2002]Pan Xiaochuan,Anastasio Mark A. On a limited-View Reconstruction Problem Diffraction Tomography[J].IEEE Transactions on Medical Imaging.2002,21(4):413-416.
[Anastasio2002]Anastasio Mark A,Zou Yu,Pan Xiaochuan.Reflectivity Tomography using Temporally Truncated Data[J]. Proceedings of the Second Joint .2002, 2:921-922.
[Matej2004]Samuel Matej, Fessler Jerrey A. Iterative Tomographic Image Reconstruction Using Fourier-Based Forward and Back-Projection[J].IEEE Transactions on Medical Imaging.2004,23(4):401-412.
[程建春2004]程建春.数学物理方程及其近似方法[M].北京,科学出版社,2004.
[Zhang2004] Zhang Xiaodong , Broschat Shira L. A Numerical Study of Conjugate Gradient Directions for an Ultrasound Inverse ProbIem[J]. Joumal of Computational Acoustics.2004,l 2(4):587—604.
[刘超2005]刘超.超声逆散射成像问题中的正则化方法研究[J].浙江大学学报.2005,6:195-199.
[Pan2005] Pan Xiaochuan,Yu L,Kao C M.Image-resolution enhancement in computed tomography[J]. IEEE Transactions on Medical Imaging.2005,24:246-253.
[Shieh2006]Shieh Hsin M,bryne Charles L. Iterative image Reconstruction Using Prior Kownledge[J]. Optical Society of America.2006,23(6):1292-1300.
[王彦飞2007]王彦飞.反演问题的计算方法及其应用[M].北京,高等教育出版社.2007.
[Fessler2007]Fessler Jerry A.On NUFFT-based gridding for non-Cartesian MRI[J].Journal of Megnetic Resonance.2007,188(2):191-195.
[LaRoque2008]LaRoque S,Sidky E,Pan Xiaochuan.Accurate image reconstruction from few-veiw and limit-angle data in diffraction tomography[J].Optical Society of America. 2008,25:1772-1782.
[Mathews2009]Mathews Jacob.Optimized non-uniform fast Fourier transform(NUFFT) for iterative tomographic reconstruction[C]. ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing– Proceedings. 2009: 673-676.
[陶进绪2009]陶进绪,张东文.频域法超声散射成像[J].信号处理.2009,25(2):169-173.
[王朔中2010]王朔中,方针.声衍射层析成像进展[J].声学技术.2010,29(2):117-122.