生物力学模型导引的心肌运动与材料参数对偶估计
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摘要
缺血性心脏病是对人体健康危害最大的心脏疾病之一,由于发病初期无明显临床症状,病人常常因忽视病情而延误治疗,因此缺血性心肌病的早期诊断对人类健康状态的改善有重要意义。理论研究和大量临床试验表明,左心室的运动变形状态和心肌材料属性是反映心肌功能的两个重要指标,它们的异常也是诊断缺血性心肌病的重要依据。弹性力学中,应变和弹性模量分别是描述物体变形程度和反映物体变形难易程度的两个力学量,通过对左心室心肌这两个量的定量研究,医生可以以此为缺血性心脏病建立一个量化的诊断标准。
目前对心脏的应变成像和弹性成像分为基于物理的方法和基于模型的方法两种。基于物理的应变成像和弹性成像由于物理成像方法本身的局限性,一般都存在缺点。如超声成像只能对一维方向成像,光学相干断层弹性成像只能在体外进行成像等。传统基于模型的方法将运动和参数分析作为两个独立过程,忽视了它们的紧密联系,并且分析工作是用确定的模型为前提,分析过程将模型参数确定为某个先验值不变,因此也不可避免具有缺陷。
为鲁棒地实现心肌应变成像和材料成像,本文基于对偶滤波器思想,在生物力学约束模型和状态空间框架下,用有限元方法进行心肌域的表达,并将运动状态向量和材料参数向量分别置于两组状态空间方程中,用相应的滤波器进行交替迭代估计,从而实现心肌运动状态和材料参数的同时重建。针对参数估计器中的参数测量方程,本文推导了其不同形式,并应用不同的滤波器进行优化求解,分别对应双KF、KF-EKF、KF-UKF对偶滤波器算法。
在心脏运动估计的被动模型中,一般构造图像力作为驱动模型运动的力,或是用罚函数法计算反作用力,但这违背了心脏运动主动力的特性或是利用的信息的不充分,这类方法所构造出来的外力往往不准确,影响运动分析的准确性。本文利用流体力学的Navier-Stokes方程,通过流体速度反演出心肌壁压力,并将压力信息加入到运动估计框架中,试图提高运动估计的准确性和鲁棒性
Ischemic heart disease(IHD) severely endangers health of mankind.In the early stages, patients suffer from IHD are not conscious of the sickness because of no obvious symptoms. So, it's much meaningful for early diagnosis of IHD. According to a large number of researches and experimental, abnormalities of motion and material of left ventricle are main indicators for IHD. Strain and elastic modulus are two parameters that can reflect the motion state and material property of myocardium. By quantitative analysis of these two parameters, we can have better diagnosis for IHD.
At present there are two kinds of method for strain imaging and elastic imaging. One is based on hardware of imaging; the other is based on model technique. Because of the inherent drawback of each method of heart imaging technique, there are relevant defects for hardware based method. The majority of traditional model techniques deal with cardiac motion analysis and material parameters estimation separately without consideration of their underlying close connection. So there are limitations for traditional model based method.
In this paper, we present a new method of estimating deformation and material properties based on a biomechanical guided dual filter framework, which grows out our earlier joint estimation. This approach leads to fast convergence and avoid cross covariance between parameter and state, which are main drawbacks of joint estimation method. In our current implementation, at each time step, we rely on techniques from Kaman filter to first generate estimates of heart kinematics with suboptimal material parameter estimates, and then recover the elasticity property given these kinematic state estimates based on an extended Kalman filter(EKF), unscented Kalman filter(UKF) or traditional Kalman filter(KF) techniques according to different measure equations in parameter filter. These coupled iterative steps are repeated as necessary until convergence.
In addition, we proposed a new approach of computation of the force that drives the moving heart. We use the difference method to solve the equations of Navies-Stokes from fluid mechanics by giving velocity of all point of myocardium. We try to insert this estimation force into our optimization framework, in order to improve the robust and accuracy of estimation results.
目前对心脏的应变成像和弹性成像分为基于物理的方法和基于模型的方法两种。基于物理的应变成像和弹性成像由于物理成像方法本身的局限性,一般都存在缺点。如超声成像只能对一维方向成像,光学相干断层弹性成像只能在体外进行成像等。传统基于模型的方法将运动和参数分析作为两个独立过程,忽视了它们的紧密联系,并且分析工作是用确定的模型为前提,分析过程将模型参数确定为某个先验值不变,因此也不可避免具有缺陷。
为鲁棒地实现心肌应变成像和材料成像,本文基于对偶滤波器思想,在生物力学约束模型和状态空间框架下,用有限元方法进行心肌域的表达,并将运动状态向量和材料参数向量分别置于两组状态空间方程中,用相应的滤波器进行交替迭代估计,从而实现心肌运动状态和材料参数的同时重建。针对参数估计器中的参数测量方程,本文推导了其不同形式,并应用不同的滤波器进行优化求解,分别对应双KF、KF-EKF、KF-UKF对偶滤波器算法。
在心脏运动估计的被动模型中,一般构造图像力作为驱动模型运动的力,或是用罚函数法计算反作用力,但这违背了心脏运动主动力的特性或是利用的信息的不充分,这类方法所构造出来的外力往往不准确,影响运动分析的准确性。本文利用流体力学的Navier-Stokes方程,通过流体速度反演出心肌壁压力,并将压力信息加入到运动估计框架中,试图提高运动估计的准确性和鲁棒性
Ischemic heart disease(IHD) severely endangers health of mankind.In the early stages, patients suffer from IHD are not conscious of the sickness because of no obvious symptoms. So, it's much meaningful for early diagnosis of IHD. According to a large number of researches and experimental, abnormalities of motion and material of left ventricle are main indicators for IHD. Strain and elastic modulus are two parameters that can reflect the motion state and material property of myocardium. By quantitative analysis of these two parameters, we can have better diagnosis for IHD.
At present there are two kinds of method for strain imaging and elastic imaging. One is based on hardware of imaging; the other is based on model technique. Because of the inherent drawback of each method of heart imaging technique, there are relevant defects for hardware based method. The majority of traditional model techniques deal with cardiac motion analysis and material parameters estimation separately without consideration of their underlying close connection. So there are limitations for traditional model based method.
In this paper, we present a new method of estimating deformation and material properties based on a biomechanical guided dual filter framework, which grows out our earlier joint estimation. This approach leads to fast convergence and avoid cross covariance between parameter and state, which are main drawbacks of joint estimation method. In our current implementation, at each time step, we rely on techniques from Kaman filter to first generate estimates of heart kinematics with suboptimal material parameter estimates, and then recover the elasticity property given these kinematic state estimates based on an extended Kalman filter(EKF), unscented Kalman filter(UKF) or traditional Kalman filter(KF) techniques according to different measure equations in parameter filter. These coupled iterative steps are repeated as necessary until convergence.
In addition, we proposed a new approach of computation of the force that drives the moving heart. We use the difference method to solve the equations of Navies-Stokes from fluid mechanics by giving velocity of all point of myocardium. We try to insert this estimation force into our optimization framework, in order to improve the robust and accuracy of estimation results.
引文
[1]DE COUTO G, OUZOUNIAN M, LIU P P. Early detection of myocardial dysfunction and heart failure[J]. Nature Reviews Cardiology,2010,7(6):334-344.
[2]WEYMAN A E. The year in echocardiography[J]. Journal of the American College of Cardiology,2004,43(1):140.
[3]SENGUPTA P P, KORINEK J, BELOHLAVEK M, et al. Left Ventricular Structure and Function::Basic Science for Cardiac Imaging[J]. Journal of the American College of Cardiology,2006,48(10):1988-2001.
[4]LIPTON M J, BOGAERT J, BOXT L M, et al. Imaging of ischemic heart disease[J]. European radiology,2002,12(5):1061-1080.
[5]MARKL M, SCHNEIDER B, HENNIG J. Fast phase contrast cardiac magnetic resonance imaging:improved assessment and analysis of left ventricular wall motion[J]. Journal of Magnetic Resonance Imaging,2002,15(6):642-653.
[6]MIRSKY I, PARMLEY W W. Assessment of passive elastic stiffness for isolated heart muscle and the intact heart[J]. Circulation research,1973,33(2):233.
[71 EDVARDSEN T, GERBER B L, GAROT J, et al. Quantitative assessment of intrinsic regional myocardial deformation by Doppler strain rate echocardiography in humans:validation against three-dimensional tagged magnetic resonance imaging[J]. Circulation,2002,106(1):50.
[8]吴家龙.弹性力学[M].上海：同济大学出版社,1993.
[9]D'HOOGE J, KONOFAGOU E, JAMAL F, et al. Two-dimensional ultrasonic strain rate measurement of the human heart in vivo[J]. IEEE Trans Ultrason Ferroelectr Freq Control,2002,49(2):281-286.
[10]罗建文,白净.超声弹性成像的研究进展[J].中国医疗器械信息2005,11(005)：23-31.
[11]ROBERT B, SINKUS R, GENNISSON J L, et al. Application of DENSE-MR-elastography to the human heart[J]. Magnetic Resonance in Medicine,2009,62(5):1155-1163.
[12]MANDUCA A, OLIPHANT T E, DRESNER M A, et al. Magnetic resonance elastography:non-invasive mapping of tissue elasticity[J]. Med Image Anal, 2001,5(4):237-254.
[13]ELGETI T, RUMP J, HAMHABER U, et al. Cardiac magnetic resonance elastography: initial results[J]. Investigative radiology,2008,43(11):762.
[14]KIRKPATRICK S J. Optical elastography,2001 [C].
[15]FRANGI A F, NIESSEN W J, VIERGEVER M A. Three-dimensional modeling for functional analysis of cardiac images, a review[J]. Medical Imaging, IEEE Transactions on,2001,20(1):2-5.
[16]PARK J, METAXAS D, AXEL L. Analysis of left ventricular wall motion based on volumetric deformable models and MRI-SPAMM[J]. Med Image Anal, 1996,1(1):53-71.
[17]BILLET F, SERMESANT M, DELINGETTE H, et al. Cardiac motion recovery and boundary conditions estimation by coupling an electromechanical model and cine-mri data,2009[C].
[18]ABD-ELMONIEM K Z, STUBER M, PRINCE J L. Direct three-dimensional myocardial strain tensor quantification and tracking using zHARP[J]. Medical image analysis, 2008,12(6):778-786.
[19]VAN LINT J, HOOGENDOORN S P, HEGYI A. Dual EKF state and parameter estimation in multi-class first-order traffic flow models,2008[C].
[20]HU Z, METAXAS D, AXEL L. In vivo strain and stress estimation of the heart left and right ventricles from MRI images[J]. Med Image Anal,2003,7(4):435-444.
[21]LIU H, SHI P. State-space analysis of cardiac motion with biomechanical constraints[J]. Image Processing, IEEE Transactions on,2007,16(4):901-917.
[22]SHI P, SINUSAS A J, CONSTABLE R T, et al. Volumetric deformation analysis using mechanics-based data fusion:Applications in cardiac motion recovery[J]. International Journal of Computer Vision,1999,35(1):87-107.
[23]LIU H, SHI P. Meshfree representation and computation:Applications to cardiac motion analysis,2003 [C].
[24]RAGHAVAN K R, YAGLE A E. Forward and inverse problems in elasticity imaging of soft tissues[J]. Nuclear Science, IEEE Transactions on,1994,41(4):1639-1648.
[25]SHI P, SINUSAS A J, CONSTABLE R T, et al. Point-tracked quantitative analysis of left ventricular surface motion from 3-D image sequences[J]. Medical Imaging, IEEE Transactions on,2000,19(1):36-50.
[26]BENAYOUN S, AYACHE N. Dense non-rigid motion estimation in sequences of medical images using differential constraints[J]. International journal of computer vision,1998,26(1):25-40.
[27]LEDESMA-CARBAYO M J, KYBIC J, DESCO M, et al. Spatio-temporal nonrigid registration for ultrasound cardiac motion estimation[J]. IEEE Trans Med Imaging, 2005,24(9):1113-1126.
[28]LIU H, HU H, WONG K, et al. Cardiac Motion Analysis Using Nonlinear Biomechanical Constraints,2005[C].
[29]PAPADEMETRIS X, SINUSAS A J, DIONE D P, et al. Estimation of 3D left ventricular deformation from echocardiography[J]. Medical Image Analysis,2001,5(1):17-28.
[30]PAPADEMETRIS X, SINUSAS A J, DIONE D P, et al. Estimation of 3-D left ventricular deformation from medical images using biomechanical models[J]. IEEE Trans Med Imaging,2002,21(7):786-800.
[31]SHI P, LIU H. Stochastic finite element framework for simultaneous estimation of cardiac kinematic functions and material parameters [J]. Medical Image Analysis, 2003,7(4):445-464.
[32]LIU H, SHI P. Robust identification of object elasticity [J]. Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis, 2004:423-435.
[33]LIU Y, WEN H, GORMAN R C, et al. Reconstruction of myocardial tissue motion and strain fields from displacement-encoded MR imaging[J]. American Journal of Physiology-Heart and Circulatory Physiology,2009,297(3):H1151.
[34]FRITZSCHE P J, STARK D D. MRI of the body[M]. Raven,1993.
[35]HASHEMI R H, BRADLEY W G, LISANTI C J. MRI:The Basics[M]. Lippincott Williams\& Wilkins,2010.
[36]DENNEY JR T S. Estimation and detection of myocardial tags in MR image without user-defined myocardial contours[J]. Medical Imaging, IEEE Transactions on, 1999,18(4):330-344.
[37]MASOOD S, YANG G Z, PENNELL D J, et al. Investigating intrinsic myocardial mechanics:the role of MR tagging, velocity phase mapping, and diffusion imaging[J]. J Magn Reson Imaging,2000,12(6):873-883.
[38]WANG X, CHEN T, ZHANG S, et al. Lv motion and strain computation from tmri based on meshless deformable models[J]. Medical Image Computing and Computer-Assisted Intervention--MICCAI 2008,2008:636-644.
[39]KERWIN W S, PRINCE J L. Tracking MR tag surfaces using a spatiotemporal filter and interpolator [J]. International journal of imaging systems and technology, 1999,10(2):128-142.
[40]OSMAN N F, MCVEIGH E R, PRINCE J L. Imaging heart motion using harmonic phase MRI[J]. Medical Imaging, IEEE Transactions on,2000,19(3):186-202.
[41]LARSON A C, WHITE R D, LAUB G, et al. Self-gated cardiac cine MRI[J]. Magnetic resonance in medicine,2004,51(1):93-102.
[42]LIU H, CHEN Y, CHEN W. Neighborhood aided implicit active contours[J].2006.
[43]HAN X, XU C, PRINCE J L. A topology preserving level set method for geometric deformable models[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003:755-768.
[44]王勖成.有限单元法[M].北京：清华大学出版社,2003.
[45]庄玲.基于三维图像序列的左心室分割与运动联合估计[D].浙江大学,2008.
[46]SONG S M, LEAHY R M, BOYD D P, et al. Determining cardiac velocity fields and intraventricular pressure distribution from a sequence of ultrafast CT cardiac images[J]. Medical Imaging, IEEE Transactions on,1994,13(2):386-397.
[47]EBBERS T, WIGSTR O M L, BOLGER A F, et al. Noninvasive measurement of time-varying three-dimensional relative pressure fields within the human heart [J]. Journal of biomechanical engineering,2002,124:288.
[48]常春馨.现代控制理论基础[M].机械工业出版社,1988.
[49]线性系统理论[M].清华大学出版社,2002.
[50]MOIREAU P, CHAPELLE D, TALLEC P L. Joint state and parameter estimation for distributed mechanical systems[J]. Computer methods in applied mechanics and engineering,2008,197(6-8):659-677.
[51]ZHANG H, LI B, YOUNG A A, et al. Recovery of myocardial kinematic function without the time history of external loads[J]. EURASIP Journal on Advances in Signal Processing,2010,2010:7.
[52]GOVE J H, HOLLINGER D Y. Application of a dual unscented Kalman filter for simultaneous state and parameter estimation in problems of surface-atmosphere exchange[J]. J. Geophys. Res,2006,111:1-21.
[53]WAN E A, VAN DER MERWE R, NELSON A T. Dual estimation and the unscented transformation[J]. Advances in Neural Information Processing Systems, 2000,12:666-672.
[54]MORADKHANI H, SOROOSHIAN S, GUPTA H V, et al. Dual state-parameter estimation of hydrological models using ensemble Kalman filter[J]. Advances in Water Resources,2005,28(2):135-147.
[2]WEYMAN A E. The year in echocardiography[J]. Journal of the American College of Cardiology,2004,43(1):140.
[3]SENGUPTA P P, KORINEK J, BELOHLAVEK M, et al. Left Ventricular Structure and Function::Basic Science for Cardiac Imaging[J]. Journal of the American College of Cardiology,2006,48(10):1988-2001.
[4]LIPTON M J, BOGAERT J, BOXT L M, et al. Imaging of ischemic heart disease[J]. European radiology,2002,12(5):1061-1080.
[5]MARKL M, SCHNEIDER B, HENNIG J. Fast phase contrast cardiac magnetic resonance imaging:improved assessment and analysis of left ventricular wall motion[J]. Journal of Magnetic Resonance Imaging,2002,15(6):642-653.
[6]MIRSKY I, PARMLEY W W. Assessment of passive elastic stiffness for isolated heart muscle and the intact heart[J]. Circulation research,1973,33(2):233.
[71 EDVARDSEN T, GERBER B L, GAROT J, et al. Quantitative assessment of intrinsic regional myocardial deformation by Doppler strain rate echocardiography in humans:validation against three-dimensional tagged magnetic resonance imaging[J]. Circulation,2002,106(1):50.
[8]吴家龙.弹性力学[M].上海：同济大学出版社,1993.
[9]D'HOOGE J, KONOFAGOU E, JAMAL F, et al. Two-dimensional ultrasonic strain rate measurement of the human heart in vivo[J]. IEEE Trans Ultrason Ferroelectr Freq Control,2002,49(2):281-286.
[10]罗建文,白净.超声弹性成像的研究进展[J].中国医疗器械信息2005,11(005)：23-31.
[11]ROBERT B, SINKUS R, GENNISSON J L, et al. Application of DENSE-MR-elastography to the human heart[J]. Magnetic Resonance in Medicine,2009,62(5):1155-1163.
[12]MANDUCA A, OLIPHANT T E, DRESNER M A, et al. Magnetic resonance elastography:non-invasive mapping of tissue elasticity[J]. Med Image Anal, 2001,5(4):237-254.
[13]ELGETI T, RUMP J, HAMHABER U, et al. Cardiac magnetic resonance elastography: initial results[J]. Investigative radiology,2008,43(11):762.
[14]KIRKPATRICK S J. Optical elastography,2001 [C].
[15]FRANGI A F, NIESSEN W J, VIERGEVER M A. Three-dimensional modeling for functional analysis of cardiac images, a review[J]. Medical Imaging, IEEE Transactions on,2001,20(1):2-5.
[16]PARK J, METAXAS D, AXEL L. Analysis of left ventricular wall motion based on volumetric deformable models and MRI-SPAMM[J]. Med Image Anal, 1996,1(1):53-71.
[17]BILLET F, SERMESANT M, DELINGETTE H, et al. Cardiac motion recovery and boundary conditions estimation by coupling an electromechanical model and cine-mri data,2009[C].
[18]ABD-ELMONIEM K Z, STUBER M, PRINCE J L. Direct three-dimensional myocardial strain tensor quantification and tracking using zHARP[J]. Medical image analysis, 2008,12(6):778-786.
[19]VAN LINT J, HOOGENDOORN S P, HEGYI A. Dual EKF state and parameter estimation in multi-class first-order traffic flow models,2008[C].
[20]HU Z, METAXAS D, AXEL L. In vivo strain and stress estimation of the heart left and right ventricles from MRI images[J]. Med Image Anal,2003,7(4):435-444.
[21]LIU H, SHI P. State-space analysis of cardiac motion with biomechanical constraints[J]. Image Processing, IEEE Transactions on,2007,16(4):901-917.
[22]SHI P, SINUSAS A J, CONSTABLE R T, et al. Volumetric deformation analysis using mechanics-based data fusion:Applications in cardiac motion recovery[J]. International Journal of Computer Vision,1999,35(1):87-107.
[23]LIU H, SHI P. Meshfree representation and computation:Applications to cardiac motion analysis,2003 [C].
[24]RAGHAVAN K R, YAGLE A E. Forward and inverse problems in elasticity imaging of soft tissues[J]. Nuclear Science, IEEE Transactions on,1994,41(4):1639-1648.
[25]SHI P, SINUSAS A J, CONSTABLE R T, et al. Point-tracked quantitative analysis of left ventricular surface motion from 3-D image sequences[J]. Medical Imaging, IEEE Transactions on,2000,19(1):36-50.
[26]BENAYOUN S, AYACHE N. Dense non-rigid motion estimation in sequences of medical images using differential constraints[J]. International journal of computer vision,1998,26(1):25-40.
[27]LEDESMA-CARBAYO M J, KYBIC J, DESCO M, et al. Spatio-temporal nonrigid registration for ultrasound cardiac motion estimation[J]. IEEE Trans Med Imaging, 2005,24(9):1113-1126.
[28]LIU H, HU H, WONG K, et al. Cardiac Motion Analysis Using Nonlinear Biomechanical Constraints,2005[C].
[29]PAPADEMETRIS X, SINUSAS A J, DIONE D P, et al. Estimation of 3D left ventricular deformation from echocardiography[J]. Medical Image Analysis,2001,5(1):17-28.
[30]PAPADEMETRIS X, SINUSAS A J, DIONE D P, et al. Estimation of 3-D left ventricular deformation from medical images using biomechanical models[J]. IEEE Trans Med Imaging,2002,21(7):786-800.
[31]SHI P, LIU H. Stochastic finite element framework for simultaneous estimation of cardiac kinematic functions and material parameters [J]. Medical Image Analysis, 2003,7(4):445-464.
[32]LIU H, SHI P. Robust identification of object elasticity [J]. Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis, 2004:423-435.
[33]LIU Y, WEN H, GORMAN R C, et al. Reconstruction of myocardial tissue motion and strain fields from displacement-encoded MR imaging[J]. American Journal of Physiology-Heart and Circulatory Physiology,2009,297(3):H1151.
[34]FRITZSCHE P J, STARK D D. MRI of the body[M]. Raven,1993.
[35]HASHEMI R H, BRADLEY W G, LISANTI C J. MRI:The Basics[M]. Lippincott Williams\& Wilkins,2010.
[36]DENNEY JR T S. Estimation and detection of myocardial tags in MR image without user-defined myocardial contours[J]. Medical Imaging, IEEE Transactions on, 1999,18(4):330-344.
[37]MASOOD S, YANG G Z, PENNELL D J, et al. Investigating intrinsic myocardial mechanics:the role of MR tagging, velocity phase mapping, and diffusion imaging[J]. J Magn Reson Imaging,2000,12(6):873-883.
[38]WANG X, CHEN T, ZHANG S, et al. Lv motion and strain computation from tmri based on meshless deformable models[J]. Medical Image Computing and Computer-Assisted Intervention--MICCAI 2008,2008:636-644.
[39]KERWIN W S, PRINCE J L. Tracking MR tag surfaces using a spatiotemporal filter and interpolator [J]. International journal of imaging systems and technology, 1999,10(2):128-142.
[40]OSMAN N F, MCVEIGH E R, PRINCE J L. Imaging heart motion using harmonic phase MRI[J]. Medical Imaging, IEEE Transactions on,2000,19(3):186-202.
[41]LARSON A C, WHITE R D, LAUB G, et al. Self-gated cardiac cine MRI[J]. Magnetic resonance in medicine,2004,51(1):93-102.
[42]LIU H, CHEN Y, CHEN W. Neighborhood aided implicit active contours[J].2006.
[43]HAN X, XU C, PRINCE J L. A topology preserving level set method for geometric deformable models[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003:755-768.
[44]王勖成.有限单元法[M].北京：清华大学出版社,2003.
[45]庄玲.基于三维图像序列的左心室分割与运动联合估计[D].浙江大学,2008.
[46]SONG S M, LEAHY R M, BOYD D P, et al. Determining cardiac velocity fields and intraventricular pressure distribution from a sequence of ultrafast CT cardiac images[J]. Medical Imaging, IEEE Transactions on,1994,13(2):386-397.
[47]EBBERS T, WIGSTR O M L, BOLGER A F, et al. Noninvasive measurement of time-varying three-dimensional relative pressure fields within the human heart [J]. Journal of biomechanical engineering,2002,124:288.
[48]常春馨.现代控制理论基础[M].机械工业出版社,1988.
[49]线性系统理论[M].清华大学出版社,2002.
[50]MOIREAU P, CHAPELLE D, TALLEC P L. Joint state and parameter estimation for distributed mechanical systems[J]. Computer methods in applied mechanics and engineering,2008,197(6-8):659-677.
[51]ZHANG H, LI B, YOUNG A A, et al. Recovery of myocardial kinematic function without the time history of external loads[J]. EURASIP Journal on Advances in Signal Processing,2010,2010:7.
[52]GOVE J H, HOLLINGER D Y. Application of a dual unscented Kalman filter for simultaneous state and parameter estimation in problems of surface-atmosphere exchange[J]. J. Geophys. Res,2006,111:1-21.
[53]WAN E A, VAN DER MERWE R, NELSON A T. Dual estimation and the unscented transformation[J]. Advances in Neural Information Processing Systems, 2000,12:666-672.
[54]MORADKHANI H, SOROOSHIAN S, GUPTA H V, et al. Dual state-parameter estimation of hydrological models using ensemble Kalman filter[J]. Advances in Water Resources,2005,28(2):135-147.