多尺度系统建模、估计与融合方法研究
|
摘要
多尺度系统理论的提出和发展，为更全面、更精确地描述复杂大系统提供了
新的思想和方法。本文将多尺度分析方法与动态系统的卡尔曼滤波、线性系统理
论相结合，提出了一套体系较为完整的多尺度系统建模、估计与融合方法。在此
基础上得到的实时算法可以给出多个层次（尺度）观测数据的最优综合，或在任
一层次（尺度）上利用全局信息对局部信号进行最优估计。论文的主要贡献如下：
1．以小波逆变换为基础在无限网格上建立了多尺度模型，网格被严格定义为二
叉树形式，用移动算子来表示尺度之间的递推关系。这样建立的多尺度模型
已经不只局限在小波变换的范畴，而是具有更广泛的意义，它可以描述信号
在某一尺度上的一般随机过程特征；
2．将时间离散随机过程的马尔可夫性推广到多尺度随机系统，提出了多尺度树
上的马尔可夫性这一新概念。研究了多尺度模型的内部实现和外部实现方法，
给出了选取内部矩阵的基本原则，并着重分析了为马尔可夫自由场建立内部
模型的方法；
3．将Kalman滤波和Rauch-Tung-Striebel平滑算法推广到多尺度状态空间，给出
了多尺度估计与融合算法。这个算法具有很好的并行计算特性，扩展到2-D
数据时计算复杂度增加不多；
4．提出了一种多尺度动态递归估计算法，将尺度动态方程与时间动态方程相结
合，建立了尺度框架下的动态系统。与传统的动态估计方法比较，在解决大
型系统的估计问题时，计算量大大减少；
5．与时间动态系统的状态空间分析相对照，定义了多尺度状态空间的能达性、
能控性、能观性和可重构性，并给出了各自的相应条件。分析了最优多尺度
估计算法的误差协方差特性，研究了误差的动态方程和黎卡提方程的稳定性
和渐近稳定性。
6．提出了一种与前面形式不同的动态系统多尺度估计和融合新算法，它无需建
立多尺度模型，在最小估计误差方差意义下具有最优性。随后，进一步对算
法的实时性进行了改进。特别是给出了在某些尺度观测数据残缺的情况下，
构造等效观测方程和等效观测值的方法，使该算法更具实用价值。
The multiscale system theory provides a new idea and method to describe the complex and
large-scale systems more completely and accurately. In this dissertation, a systemic multiscale
modeling, estimation and data fusion theory is developed on the basis of multiscale analysis method,
Kalman filtering for dynamic systems and linear system theory. The real-time algorithms derived
from this theory can obtain the optimal global fusion of multiresolution (multiscale) data, or optimal
estimation at a certain scale. The main contributions are as follows:
I. Based on wavelet inverse translation, the multiscale models are built on infinite lattice. The
infinite lattice is defined as dyadic tree strictly and a scale-to-scale relationship is specified by
the shift operator. In this case the multiscale model is more abstract than wavelet transform. It
can represent the features of the signal at a certain scale.
2. Markov property of time discrete stochastic systems is generalized to multisale stochastic
systems and a new concept桵arkov property of multisale tree is proposed. The internal
realization and external models are given. The principle of choosing internal matrix is
developed at the same time. It is analyzed how to build internal realization models for Markov
random field.
3. By generalizing Kalman filtering and Rauch-Tung-Striebe smoothing algorithm, a multiscale
estimation and data fusion algorithm is presented. This algorithm is highly parallelizable and
computationally efficient. When the same idea is extended to 2-D data, the computation burden
will not increase significantly.
4. A multiscale dynamic recursive estimation algorithm is proposed and the dynamic systems on
multiscale frame are built. While modeling and estimating for large-scale system, this method
can reduce computations greatly compared with conventional optimal estimation methods.
5. The reachability, controllability, observability and reconstructibilty for multiscale models are
defined as compared to their counterparts for ordinary state-space models. The conditions are
given under which the system is reachable, controllable and observable. The properties of error
covariance for optimal multiscale estimation algorithm, and the stability and asymptotic
behavior of the error dynamics and Riccati equation are analyzed.
6. A new multiscale estimation and data fusion algorithm is given. The advantages are that there is
no need to build multiscale model and the algorithm is optimal in the sense of linear least
square estimation. A proposal to reduce delay time is put forward. It is discussed how to obtain
equivalent measurement equation and measurements in the case of no measurement data at
some scale.
新的思想和方法。本文将多尺度分析方法与动态系统的卡尔曼滤波、线性系统理
论相结合，提出了一套体系较为完整的多尺度系统建模、估计与融合方法。在此
基础上得到的实时算法可以给出多个层次（尺度）观测数据的最优综合，或在任
一层次（尺度）上利用全局信息对局部信号进行最优估计。论文的主要贡献如下：
1．以小波逆变换为基础在无限网格上建立了多尺度模型，网格被严格定义为二
叉树形式，用移动算子来表示尺度之间的递推关系。这样建立的多尺度模型
已经不只局限在小波变换的范畴，而是具有更广泛的意义，它可以描述信号
在某一尺度上的一般随机过程特征；
2．将时间离散随机过程的马尔可夫性推广到多尺度随机系统，提出了多尺度树
上的马尔可夫性这一新概念。研究了多尺度模型的内部实现和外部实现方法，
给出了选取内部矩阵的基本原则，并着重分析了为马尔可夫自由场建立内部
模型的方法；
3．将Kalman滤波和Rauch-Tung-Striebel平滑算法推广到多尺度状态空间，给出
了多尺度估计与融合算法。这个算法具有很好的并行计算特性，扩展到2-D
数据时计算复杂度增加不多；
4．提出了一种多尺度动态递归估计算法，将尺度动态方程与时间动态方程相结
合，建立了尺度框架下的动态系统。与传统的动态估计方法比较，在解决大
型系统的估计问题时，计算量大大减少；
5．与时间动态系统的状态空间分析相对照，定义了多尺度状态空间的能达性、
能控性、能观性和可重构性，并给出了各自的相应条件。分析了最优多尺度
估计算法的误差协方差特性，研究了误差的动态方程和黎卡提方程的稳定性
和渐近稳定性。
6．提出了一种与前面形式不同的动态系统多尺度估计和融合新算法，它无需建
立多尺度模型，在最小估计误差方差意义下具有最优性。随后，进一步对算
法的实时性进行了改进。特别是给出了在某些尺度观测数据残缺的情况下，
构造等效观测方程和等效观测值的方法，使该算法更具实用价值。
The multiscale system theory provides a new idea and method to describe the complex and
large-scale systems more completely and accurately. In this dissertation, a systemic multiscale
modeling, estimation and data fusion theory is developed on the basis of multiscale analysis method,
Kalman filtering for dynamic systems and linear system theory. The real-time algorithms derived
from this theory can obtain the optimal global fusion of multiresolution (multiscale) data, or optimal
estimation at a certain scale. The main contributions are as follows:
I. Based on wavelet inverse translation, the multiscale models are built on infinite lattice. The
infinite lattice is defined as dyadic tree strictly and a scale-to-scale relationship is specified by
the shift operator. In this case the multiscale model is more abstract than wavelet transform. It
can represent the features of the signal at a certain scale.
2. Markov property of time discrete stochastic systems is generalized to multisale stochastic
systems and a new concept桵arkov property of multisale tree is proposed. The internal
realization and external models are given. The principle of choosing internal matrix is
developed at the same time. It is analyzed how to build internal realization models for Markov
random field.
3. By generalizing Kalman filtering and Rauch-Tung-Striebe smoothing algorithm, a multiscale
estimation and data fusion algorithm is presented. This algorithm is highly parallelizable and
computationally efficient. When the same idea is extended to 2-D data, the computation burden
will not increase significantly.
4. A multiscale dynamic recursive estimation algorithm is proposed and the dynamic systems on
multiscale frame are built. While modeling and estimating for large-scale system, this method
can reduce computations greatly compared with conventional optimal estimation methods.
5. The reachability, controllability, observability and reconstructibilty for multiscale models are
defined as compared to their counterparts for ordinary state-space models. The conditions are
given under which the system is reachable, controllable and observable. The properties of error
covariance for optimal multiscale estimation algorithm, and the stability and asymptotic
behavior of the error dynamics and Riccati equation are analyzed.
6. A new multiscale estimation and data fusion algorithm is given. The advantages are that there is
no need to build multiscale model and the algorithm is optimal in the sense of linear least
square estimation. A proposal to reduce delay time is put forward. It is discussed how to obtain
equivalent measurement equation and measurements in the case of no measurement data at
some scale.
引文
[1] Hamid Krim, Walter Willinger, Anatoli Juditski, David Tse. Introduction to the Special Issue on Multiscale Statistical Signal Analysis and Its Application, IEEE trans. on Information Theory, 1999, 45(3):825-827
[2] M. Basseville, A. Benveniste, K. Chou, S. Golden, R. Nikoukhah, A. S. Willsky. Modeling and estimation of multiresolution stochastic process, IEEE Trans. on Information Theory, 1992,38(2): 766-784.
[3] K. Chou, A. S. Willsky, A. Benveniste. Multiscale recursive estimation, data fusion and regularization, IEEE Trans. on Automatic Control, 1994,39(3): 464-478.
[4] Benveniste, R. Nikoukhah, A. S. Willsky. Multiscale system theory, in Proc. 29~(th) IEEE Conference on Decision and Control, Honolulu, Hawaii, Dec. 1990:2484-2487.
[5] M. Basseville, A. Benveniste, A. S. Willsky. Multiscale autoregressive processes, Part Ⅰ:Schur-Levinson parametrizations, IEEE Trans.on Signal Processing, 1992, 40(8):1915-1934.
[6] M. Basseville, A. Benveniste, A. S. Willsky. Multiscale autoregressive processes, Part Ⅱ:lattice structure for whitening and modeling, IEEE Trans.on Signal Processing, 1992, 40(8):1935-1953.
[7] K. Daoudi, A. B. Frakt, A. S. Willsky. Multiscale autoregressive models and wavelets, IEEE trans, on Information Theory, 1999, 45(3): 828-845.
[8] K.E. Timmermann, R. D. Nowak. Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging, IEEE trans, on Information Theory, 1999, 45(3):846-862.
[9] G.W. Womell, A.V. Oppenheim. Wavelet-based representations for a class of self-similar with application to fractal modulation, IEEE trans. on Information Theory, 1992, 38(2): 785-800.
[10] K. Chou, A. S. Willsky. Kalman filtering and Riccati equations for multiscale processes, in Proc. 29~(th) IEEE Conference on Decision and Control, Honolulu, Hawaii, Dec. 1990:841-846.
[11] K. Chou, S. A. Golden, A. S. Willsky. Multiresolution stochastic models, data fusion and wavelet transform, Signal Processing, 1993,34(3): 257-282.
[12] K. Chou, A. S. Willsky, R. Nikoukhah. Multiscale systems, Kalman filters and Riccati equations, IEEE Trans. on Automatic Control, 1994, 39(3): 479-492.
[13] M. Bello, A. S. Willsky, B. Levy, D. Castanon. Smoothing error dynamics and their use in the solution of smoothing and mapping problems, IEEE Trans. on Information Theory, 1986, 32(4): 483-495.
[14] M. Bello, A. S. Willsky, B. Levy. Construction and application of discrete-time smoothing error models, Int. J. Contr., 1989, 50(1): 203-223.
[15] M.R. Luettgen, A. S. Willsky. Multiscale smoothing error models, IEEE Trans. on Automat. Contr., 1995, 40(1): 173-175.
[16] M. R. Luettgen. Image processing with multiscale stochastic models. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, 1993.
[17] M. R. Luettgen, A. S. Willsky. Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination, IEEE Trans. on Inage Processing, 1995, 4(2): 194-207.
[18] P.W. Feiguth, A. S. Willsky. Fractal estimation using model on multiscale trees, IEEE Trans. on Signal Processing, 1996, 44(5): 1297-1300.
[19] P. Fieguth, W. Karl, A. S. Willsky, C. Wunsch. Multiresolution optimal interpolation and tatistical analysis of TOPEX/POSEIDON satellite altimetry, IEEE Trans. on Geoscience and Remote Sensing, 1995, 33(2): 280-292.
[20] P. Fieguth, D. Menemenlis, T. Ho, A. S. Willsky, C. Wunsch. Mapping Mediterranean altimeter data with a multiresolution optimal interpolation algorithm, J. of Atmospheric and Oceanic Technology, 1998, 15(4): 535-546.
[21] M.R. Luettgen, W. C. Karl, A. S. Willsky. Multiscale representation of Markov random fields, IEEE Trans. on Signal Processing, 1993, 41(12): 3377-3395.
[22] D. Menemenlis, P. Fieguth, C. Wunsch, A. S. Willsky. Adaptation of a fast optimal interpolation algorithm to the mapping of oceanographic data, J. of Geophysical Research, 1997, 102(C5): 10573-10584.
[23] W. Irving, P. Fieguth, A. S. Willsky. An overlapping tree approach to multiscale stochastic modeling and estimation, IEEE Trans. on Image Processing, 1997, 6(11): 1517-1729.
[24] T.T. Ho. Multiscale modeling and estimation of large-scale dynamic systems. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, 1998.
[25] M. Daniel, A. S. Willsky. A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing, Proc. IEEE, 1997, 85(1): 164-183.
[26] P. Moulin, J. A. O'Sullivan, D. L. Snyder. A method of sieves for multiresolution spectrum estimation and radar imaging, IEEE Trans. on Information Theory, 1992,38(2): 801-813.
[27] A. Barb. A level-crossing based scaling-dimensionality transform applied to stationary Gaussian processes, IEEE Trans. on Information Theory, 1992,38(2): 814-823.
[28] M. Luettgen, W. Karl, A. S. Willsky. Efficient multiscale regulation with application to the computation of optical flow, IEEE Trans. on Image Processing, 1994, 3(1): 41-64.
[29] P. Fieguth, W. Karl, A. S. Willsky. Efficient multiresolution counterparts to variational methods in surface reconstruction, Computer Vision and Image Understanding, 1998, 70(2): 157-176.
[30] M. Schneider. Multiscale methods for the segmentation of images, [Master's thesis], Massachusetts Institute of Technology, Cambridge, MA, 1996.
[31] A.O. Hero, R. Piramuthu, J. A. Fessler, S. R. Titus. Minimax emission computed tomography using high-resolution anatomical side information and B-spline models, IEEE trans, on Information Theory, 1999, 45(3): 920-938.
[32] D. Leporini, J. C. Pesquet. High-order wavelet packets and cumulant field analysis, IEEE trans, on Information Theory, 1999, 45(3): 863-877.
[33] B. Pesquet-Popescu. Wavelet packet decompositions for the analysis of 2-D field with stationary fractional increments, IEEE trans, on Information Theory, 1999, 45(3): 1033-1038.
[34] L. Hong. Multiresolution filtering using wavelet transform, IEEE trans. on Aerospace and Electronic System, 1993, 29(4): 1244-1251.
[35] L. Hong. Multiresolution distributed filtering, IEEE trans. On Automatic Control, 1994, 39(4): 853-856.
[36] L. Hong. Multiresolution target tracking using the wavelet transform, in Proc. 32~(nd) IEEE Conference on Decision and Control, San Antonio, Texas, Dec. 1993: 924-929.
[37] L. Hong. Multiresolutional multiple-model target tracking, IEEE trans, on Aerospace and Electronic System, 1994, 30(2): 518-524.
[38] L. Hong, Werthmann, J. R. Werthmann, G. S. Bierman, R. A. Wood. Real-time multiresolution target tracking, in Signal and Data Processing of Small Targets, Orlando, FL, Apr. 1993: 233-244.
[39] L. Hong. Two-level JPDA-NN and NN-JPDA tracking algorithms, in Proc. Of the Amerian Control Conference, Maryland, June, 1994: 1057-1061.
[40] L. Hong, C. Wang, Z. Ding. Multiresolutional decomposition and modeling with an application to joint probabilistic data association, Mathl. Comput. Modeling, 1997, 25(12): 19-32.
[41] S. Cong, L. Hong, R. A. Wood. Spatial domain multiresolutional measurement and model decomposition, in Signal and Data Processing of Small Targets, San Diego, CA, July, 1997: 11-21.
[42] L. Hong. Multiplatform multisensor fusion with adaptive-rate data communication, IEEE trans. on Aerospace and Electronic System, 1997, 33(1): 274-281.
[43] 孙红岩，毛士艺．多传感器目标识别的数据融合，电子学报，1995，23(10)：187-193．
[44] 孙红岩，毛士艺，林品兴．多传感器数据分层融合性质，电子学报，1996，24(16)：55-61．
[45] 孙红岩，毛士艺．推广的多传感器数据的分层融合算法，北京航空航天大学学报，1996，24(16)：55-61．
[46] 毛士艺等．多传感器数据融合，中国电子学会第五界全国信号处理学术论文集，武汉、1994．
[47] 邵远等．多传感器信息融合浅析，电子学报，1994、22(5)：
[48] 郑容，文成林，施晨呜，张洪才．多分辨多模型目标跟踪，电子学报，1998，26(12)：115-118．
[49] 潘泉，王培德，张洪才，周宏仁．一种有效的IMM自适应跟踪算法，西北工业大学学报，1993，11(1)：24-29．
[50] 潘泉，张洪才，向阳朝，王培德．组合式快速JPDA算法，航空学报，1994，15(5)：558-563．
[51] 潘泉，戴冠中，张洪才．交互式多模型滤波器及其并行实现研究，控制理论及应用、1997、14(4)：544-550．
[52] 文成林．多尺度估计理论及应用，西安：西北工业大学博士论文，1999．
[53] 水鹏朗．广义内插小波和递归内插小波理论及应用的研究，西安：西安电子科技大学博士论文，1998．
[54] 杨福生．小波变换的工程分析与应用，科学山版社，2000．
[55] 张磊．子波域滤波理论与算法，西安：西北工业大学硕士论文，1998．
[56] Benveniste, R. Nikoukhah, A. S. Willsky. Multiscale system theory, IEEE trans, on circuit and systems-Ⅰ, 1994, 41(1): 2-15.
[57] 王自果，田铮．随机过程，西北工业大学出版社，1990．
[58] H. Derin, P. Kelly. Discrete-index Markov-type random processes, Proceedings of IEEE, 1989, 77(10):1485--1510.
[59] T.P. Speed, H. T. Kiiveri. Gaussian Markov distributions over finite graphs, The Annals of Statistics, 1986, 14(1):138--150.
[60] M. R. Luettgen, W. C. Karl, A. S. Willsky, R. R. Tenney. Multiscale representations of Markov random Fields, IEEE International Conference on Acoustics, Speech, and Signal Processing, Piscataway, NJ, 1993, 41-44.
[61] W.W. Irving. Theory for multiscale stochastic realization and identification. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, September 1995.
[62] H. Akaike. Markovian representation of stochastic processes by canonical variables, SIAM Journal of Control、1975, 13(1).
[63] Ingrid Daubechies. Orthonormal bases of compactly supported wavelets, Communications of Pure and Applied Mathematics, 41:909-996, November 1988.
[64] Gilbert Strang, Truong Nguyeun. Wavelets and Filter Banks, MIT 18.325 Class Notes, 1995.
[65] K.C. Chou. A stochastic modeling approach to multiscale signal processing, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, May 1991.
[66] Patrick Flandrin. On the spectrum of fractional Brownian motions, IEEE Transactions on Information Theory, 1989, 35(1): 197-199.
[67] Patrick Flandrin. Wavelet analysis and synthesis of fractional Brownian motion, IEEE Transactions on Information Theory, 1992, 38(2): 910-917.
[68] Tewfik, M. Kim. Correlation structure of the discrete Wavelet coefficients of fractional Brownian motion, IEEE Transactions on Information Theory, 1992, 38(2): 904-909.
[69] M.S. Keshner. 1/f Noise, Proceedings of IEEE, 70:212-218, 1982.
[70] G. W. Wornell. A Karhunen-Love-like expansion for 1/f processes via Wavelets, IEEE Transactions on Information Theory, 1990, 36(4): 859-861.
[71] G. C. Verghese, T. Kailath. A further note on backwards Markovian models, IEEE Transactions on Information Theory, 1979, vol.25: 121--124.
[72] W.W. Irving. A Canonical correlation approach to multiscale stochastic realization, IEEE Transactions on Automatic Control, 1998.
[73] M Daniel. Multiresolution statistical modeling with application to modeling groundwater flow. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, February 1997.
[74] A. Frakt, A. Willsky. Efficient multiscale stochastic realization, Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Seattle, Washington, May 1998.
[75] A. B. Frakt. A realization theory for multiscale auto-regressive stochastic models, Ph.D. Thesis Proposal, MIT, Cambridge, MA, September 1996.
[76] A.B. Frakt. Multiscale hypothesis testing with application to anomaly characterization from tomograhic projections, Master thesis, Massachusetts Institute of Technology, Cambridge, MA, May 1996.
[77] A.B. Frakt, W. C. Karl, A. S. Willsky. A multiscale hypothesis testing approach to anomaly detection and localization from noisy tomograhic data, IEEE Transactions on Image Processing, 7(6): 825--837, June 1998.
[78] C. Fosgate, H. Krim, W. Irving, A. Willsky. Multiscale segmentation and anomaly enhancement of SAR imagery. IEEE Transactions on Image Processing, 6(1): 7--20, January 1997.
[79] H.T. Banks, K. Kunisch. Estimation techniques for distributed parameter systems. Birkhauser, Boston, 1989.
[80] S. Omatu, J.H. Seifeld, T. Soeda, Y. Sawaragi. Estimation of Nitrogen Dioxide concentration in the vicinity of a roadway by optimal filtering theory. Automatica, 24(1): 19--29, January 1988.
[81] A. da Silva, J. Guo. Documentation of the physical-space statistical analysis system (PSAS) Part Ⅰ: The conjugate gradient solver version PSAS-1.00. DAO Office Notes 96-02, Data Assimilation Office, Goddard Laboratory for Atmospheres, NASA, February 1996.
[82] W. L. Briggs. A multigrid tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.
[83] T.M. Chin. Dynamic estimation in computational vision. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, October 1991.
[84] T. M. Chin, W. C. Karl, A. S. Willsky. Sequential filtering for Multi-frame visual reconstruction, Signal Processing, 28:311-- 333, August 1992.
[85] T.M. Chin, W. C. Karl, A. S. Willsky. Probabilistic and sequential computation of optical flow using temporal coherence, IEEE Transactions on Image Processing, 3(6): 773--788, November 1994.
[86] T.M. Chin, A. J. Mariano. Wavelet-based compression of co-variances in Kalman filtering of geophysical flows, In Proceedings of SPIE, vol.2242L: 842--850, April 1994.
[87] S. Aihara. On adaptive boundary control for stochastic parabolic systems, IEEE Transactions on Automatic Control, 42(3):350--363, March 1997.
[88] M. Athans. Toward a practical theory for distributed parameter systems, IEEE Transactions on Automatic Control, pages 245--247, April 1970.
[89] T.M. Chin, W. C. Karl, A. S. Willsky. A distributed and iterative method for square root filtering in space-time estimation. Automatic, 31(1):67--82, 1995.
[90] J.M. Coleman. Gaussian space-time models: Markov field properties. PhD thesis, University of California at Davis, David, CA, August 1995.
[91] M. Daniel, A. Willsky. Modeling and estimation of fractional Brownian motion using multiresolution stochastic processes, Fractals in Engineering, Springer, 1997:124--137.
[92] P. Fieguth. Application of multiscale estimation to large scale multidimensional time varying problems, Ph.D. Thesis Proposal, MIT, Cambridge, MA, November 1993.
[93] P. Fieguth, M. R. Allen, M. J. Murray. Hierarchical methods for global-scale estimation problems, Proceedings of Canadian Conference on Electrical Engineering, Waterloo, Canada, 1998:161--164
[94] P. Gaspar, C. Wunsch. Estimates from altimeter data of barotropic Rossby waves in the Northwestern Atlantic Ocean, Journal of Physical Oceanography, 19(12): 1821--1844, 1989.
[95] Alan George. Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1981.
[96] E. Hildreth. Computations underlying the measurement of visual motion. Artificial Intelligence, 23:309--354, 1984.
[97] T. Ho, P. Fieguth, A. Willsky. Multiresolution stochastic models for the efficient solution of large-scale space-time estimation problems. Proceedings of International Conference on Acoustics, Speech, and Signal Processing, vol 6, Atlanta, Georgia, May 1996: 3098—4001.
[98] T. Ho, P. Fieguth, A. Willsky. Computationally efficient multiscale estimation of large-Scale dynamic systems. Proceedings of International Conference on Image Processing, Chicago, Illinois, 1998.
[99] T. Ho, A. Frakt, A. Willsky. Multiscale realization and estimation for space and space-time problems. Proceedings of IEEE International Symposium on Information Theory, Cambridge, Massachusetts, 1998.
[100] T. Ho. Large-scale multiscale estimation of dynamic systems. Ph.D. Thesis Proposal, MIT, Cambridge, MA, November 1996.
[101] T Kailath. The divergence and Bhattacharrya distance measures in signal selection". IEEE Transactions on Communication Technology, 15(1), February 1967.
[102] H. Lev-Ari, S. R. Parker, T. Kailath. Multidimensional maximum-entropy covariance extension, IEEE Transactions on Information Theory, 35(3):497--508, May 1989.
[103] H. E. Rauch, F. Tung, C. T. Striebel. Maximum likelihood estimates of linear dynamic Systems. AIAA Journal, 3(8): 1445--1450, 1965.
[104] J. C. Strikwerda. Finite difference schemes and partial differential equations. Chapman and Hall, New York, 1989.
[105] P.-Y. Le Traon, P. Gaspar, F. Bouyssel, H. Makhmara. Using TOPEX/ POSEIDON data to enhance ERS-1 data. Journal of Atmospheric and Oceanic Technology, 12: 161--170, 1995.
[106] P.-Y. Le Traon, J. Stum, J. Dorandeau, P. Gaspar. Global statistical analysis of TOPEX and POSEIDON data. Journal of Geophysical Research, 99(C12): 25619--25631, 1994.
[107] C. Wunsch, D. Stammer. The global frequency-wavenumber spectrum of oceanic variability estimated from TOPEX/POSEIDON altimetric measurements, Journal of Geophysical Research, 100(C12): 24895--24910, 1995.
[108] G. Simone, F. C. Morabito, A. Farina. Radar Imagine Fusion by multiscale Kalman filtering, the 3~(rd) International Conference of information fusion, Paris France, July 10-13~(th) 2000, WED3-2.
[2] M. Basseville, A. Benveniste, K. Chou, S. Golden, R. Nikoukhah, A. S. Willsky. Modeling and estimation of multiresolution stochastic process, IEEE Trans. on Information Theory, 1992,38(2): 766-784.
[3] K. Chou, A. S. Willsky, A. Benveniste. Multiscale recursive estimation, data fusion and regularization, IEEE Trans. on Automatic Control, 1994,39(3): 464-478.
[4] Benveniste, R. Nikoukhah, A. S. Willsky. Multiscale system theory, in Proc. 29~(th) IEEE Conference on Decision and Control, Honolulu, Hawaii, Dec. 1990:2484-2487.
[5] M. Basseville, A. Benveniste, A. S. Willsky. Multiscale autoregressive processes, Part Ⅰ:Schur-Levinson parametrizations, IEEE Trans.on Signal Processing, 1992, 40(8):1915-1934.
[6] M. Basseville, A. Benveniste, A. S. Willsky. Multiscale autoregressive processes, Part Ⅱ:lattice structure for whitening and modeling, IEEE Trans.on Signal Processing, 1992, 40(8):1935-1953.
[7] K. Daoudi, A. B. Frakt, A. S. Willsky. Multiscale autoregressive models and wavelets, IEEE trans, on Information Theory, 1999, 45(3): 828-845.
[8] K.E. Timmermann, R. D. Nowak. Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging, IEEE trans, on Information Theory, 1999, 45(3):846-862.
[9] G.W. Womell, A.V. Oppenheim. Wavelet-based representations for a class of self-similar with application to fractal modulation, IEEE trans. on Information Theory, 1992, 38(2): 785-800.
[10] K. Chou, A. S. Willsky. Kalman filtering and Riccati equations for multiscale processes, in Proc. 29~(th) IEEE Conference on Decision and Control, Honolulu, Hawaii, Dec. 1990:841-846.
[11] K. Chou, S. A. Golden, A. S. Willsky. Multiresolution stochastic models, data fusion and wavelet transform, Signal Processing, 1993,34(3): 257-282.
[12] K. Chou, A. S. Willsky, R. Nikoukhah. Multiscale systems, Kalman filters and Riccati equations, IEEE Trans. on Automatic Control, 1994, 39(3): 479-492.
[13] M. Bello, A. S. Willsky, B. Levy, D. Castanon. Smoothing error dynamics and their use in the solution of smoothing and mapping problems, IEEE Trans. on Information Theory, 1986, 32(4): 483-495.
[14] M. Bello, A. S. Willsky, B. Levy. Construction and application of discrete-time smoothing error models, Int. J. Contr., 1989, 50(1): 203-223.
[15] M.R. Luettgen, A. S. Willsky. Multiscale smoothing error models, IEEE Trans. on Automat. Contr., 1995, 40(1): 173-175.
[16] M. R. Luettgen. Image processing with multiscale stochastic models. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, 1993.
[17] M. R. Luettgen, A. S. Willsky. Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination, IEEE Trans. on Inage Processing, 1995, 4(2): 194-207.
[18] P.W. Feiguth, A. S. Willsky. Fractal estimation using model on multiscale trees, IEEE Trans. on Signal Processing, 1996, 44(5): 1297-1300.
[19] P. Fieguth, W. Karl, A. S. Willsky, C. Wunsch. Multiresolution optimal interpolation and tatistical analysis of TOPEX/POSEIDON satellite altimetry, IEEE Trans. on Geoscience and Remote Sensing, 1995, 33(2): 280-292.
[20] P. Fieguth, D. Menemenlis, T. Ho, A. S. Willsky, C. Wunsch. Mapping Mediterranean altimeter data with a multiresolution optimal interpolation algorithm, J. of Atmospheric and Oceanic Technology, 1998, 15(4): 535-546.
[21] M.R. Luettgen, W. C. Karl, A. S. Willsky. Multiscale representation of Markov random fields, IEEE Trans. on Signal Processing, 1993, 41(12): 3377-3395.
[22] D. Menemenlis, P. Fieguth, C. Wunsch, A. S. Willsky. Adaptation of a fast optimal interpolation algorithm to the mapping of oceanographic data, J. of Geophysical Research, 1997, 102(C5): 10573-10584.
[23] W. Irving, P. Fieguth, A. S. Willsky. An overlapping tree approach to multiscale stochastic modeling and estimation, IEEE Trans. on Image Processing, 1997, 6(11): 1517-1729.
[24] T.T. Ho. Multiscale modeling and estimation of large-scale dynamic systems. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, 1998.
[25] M. Daniel, A. S. Willsky. A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing, Proc. IEEE, 1997, 85(1): 164-183.
[26] P. Moulin, J. A. O'Sullivan, D. L. Snyder. A method of sieves for multiresolution spectrum estimation and radar imaging, IEEE Trans. on Information Theory, 1992,38(2): 801-813.
[27] A. Barb. A level-crossing based scaling-dimensionality transform applied to stationary Gaussian processes, IEEE Trans. on Information Theory, 1992,38(2): 814-823.
[28] M. Luettgen, W. Karl, A. S. Willsky. Efficient multiscale regulation with application to the computation of optical flow, IEEE Trans. on Image Processing, 1994, 3(1): 41-64.
[29] P. Fieguth, W. Karl, A. S. Willsky. Efficient multiresolution counterparts to variational methods in surface reconstruction, Computer Vision and Image Understanding, 1998, 70(2): 157-176.
[30] M. Schneider. Multiscale methods for the segmentation of images, [Master's thesis], Massachusetts Institute of Technology, Cambridge, MA, 1996.
[31] A.O. Hero, R. Piramuthu, J. A. Fessler, S. R. Titus. Minimax emission computed tomography using high-resolution anatomical side information and B-spline models, IEEE trans, on Information Theory, 1999, 45(3): 920-938.
[32] D. Leporini, J. C. Pesquet. High-order wavelet packets and cumulant field analysis, IEEE trans, on Information Theory, 1999, 45(3): 863-877.
[33] B. Pesquet-Popescu. Wavelet packet decompositions for the analysis of 2-D field with stationary fractional increments, IEEE trans, on Information Theory, 1999, 45(3): 1033-1038.
[34] L. Hong. Multiresolution filtering using wavelet transform, IEEE trans. on Aerospace and Electronic System, 1993, 29(4): 1244-1251.
[35] L. Hong. Multiresolution distributed filtering, IEEE trans. On Automatic Control, 1994, 39(4): 853-856.
[36] L. Hong. Multiresolution target tracking using the wavelet transform, in Proc. 32~(nd) IEEE Conference on Decision and Control, San Antonio, Texas, Dec. 1993: 924-929.
[37] L. Hong. Multiresolutional multiple-model target tracking, IEEE trans, on Aerospace and Electronic System, 1994, 30(2): 518-524.
[38] L. Hong, Werthmann, J. R. Werthmann, G. S. Bierman, R. A. Wood. Real-time multiresolution target tracking, in Signal and Data Processing of Small Targets, Orlando, FL, Apr. 1993: 233-244.
[39] L. Hong. Two-level JPDA-NN and NN-JPDA tracking algorithms, in Proc. Of the Amerian Control Conference, Maryland, June, 1994: 1057-1061.
[40] L. Hong, C. Wang, Z. Ding. Multiresolutional decomposition and modeling with an application to joint probabilistic data association, Mathl. Comput. Modeling, 1997, 25(12): 19-32.
[41] S. Cong, L. Hong, R. A. Wood. Spatial domain multiresolutional measurement and model decomposition, in Signal and Data Processing of Small Targets, San Diego, CA, July, 1997: 11-21.
[42] L. Hong. Multiplatform multisensor fusion with adaptive-rate data communication, IEEE trans. on Aerospace and Electronic System, 1997, 33(1): 274-281.
[43] 孙红岩，毛士艺．多传感器目标识别的数据融合，电子学报，1995，23(10)：187-193．
[44] 孙红岩，毛士艺，林品兴．多传感器数据分层融合性质，电子学报，1996，24(16)：55-61．
[45] 孙红岩，毛士艺．推广的多传感器数据的分层融合算法，北京航空航天大学学报，1996，24(16)：55-61．
[46] 毛士艺等．多传感器数据融合，中国电子学会第五界全国信号处理学术论文集，武汉、1994．
[47] 邵远等．多传感器信息融合浅析，电子学报，1994、22(5)：
[48] 郑容，文成林，施晨呜，张洪才．多分辨多模型目标跟踪，电子学报，1998，26(12)：115-118．
[49] 潘泉，王培德，张洪才，周宏仁．一种有效的IMM自适应跟踪算法，西北工业大学学报，1993，11(1)：24-29．
[50] 潘泉，张洪才，向阳朝，王培德．组合式快速JPDA算法，航空学报，1994，15(5)：558-563．
[51] 潘泉，戴冠中，张洪才．交互式多模型滤波器及其并行实现研究，控制理论及应用、1997、14(4)：544-550．
[52] 文成林．多尺度估计理论及应用，西安：西北工业大学博士论文，1999．
[53] 水鹏朗．广义内插小波和递归内插小波理论及应用的研究，西安：西安电子科技大学博士论文，1998．
[54] 杨福生．小波变换的工程分析与应用，科学山版社，2000．
[55] 张磊．子波域滤波理论与算法，西安：西北工业大学硕士论文，1998．
[56] Benveniste, R. Nikoukhah, A. S. Willsky. Multiscale system theory, IEEE trans, on circuit and systems-Ⅰ, 1994, 41(1): 2-15.
[57] 王自果，田铮．随机过程，西北工业大学出版社，1990．
[58] H. Derin, P. Kelly. Discrete-index Markov-type random processes, Proceedings of IEEE, 1989, 77(10):1485--1510.
[59] T.P. Speed, H. T. Kiiveri. Gaussian Markov distributions over finite graphs, The Annals of Statistics, 1986, 14(1):138--150.
[60] M. R. Luettgen, W. C. Karl, A. S. Willsky, R. R. Tenney. Multiscale representations of Markov random Fields, IEEE International Conference on Acoustics, Speech, and Signal Processing, Piscataway, NJ, 1993, 41-44.
[61] W.W. Irving. Theory for multiscale stochastic realization and identification. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, September 1995.
[62] H. Akaike. Markovian representation of stochastic processes by canonical variables, SIAM Journal of Control、1975, 13(1).
[63] Ingrid Daubechies. Orthonormal bases of compactly supported wavelets, Communications of Pure and Applied Mathematics, 41:909-996, November 1988.
[64] Gilbert Strang, Truong Nguyeun. Wavelets and Filter Banks, MIT 18.325 Class Notes, 1995.
[65] K.C. Chou. A stochastic modeling approach to multiscale signal processing, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, May 1991.
[66] Patrick Flandrin. On the spectrum of fractional Brownian motions, IEEE Transactions on Information Theory, 1989, 35(1): 197-199.
[67] Patrick Flandrin. Wavelet analysis and synthesis of fractional Brownian motion, IEEE Transactions on Information Theory, 1992, 38(2): 910-917.
[68] Tewfik, M. Kim. Correlation structure of the discrete Wavelet coefficients of fractional Brownian motion, IEEE Transactions on Information Theory, 1992, 38(2): 904-909.
[69] M.S. Keshner. 1/f Noise, Proceedings of IEEE, 70:212-218, 1982.
[70] G. W. Wornell. A Karhunen-Love-like expansion for 1/f processes via Wavelets, IEEE Transactions on Information Theory, 1990, 36(4): 859-861.
[71] G. C. Verghese, T. Kailath. A further note on backwards Markovian models, IEEE Transactions on Information Theory, 1979, vol.25: 121--124.
[72] W.W. Irving. A Canonical correlation approach to multiscale stochastic realization, IEEE Transactions on Automatic Control, 1998.
[73] M Daniel. Multiresolution statistical modeling with application to modeling groundwater flow. [PhD thesis], Massachusetts Institute of Technology, Cambridge, MA, February 1997.
[74] A. Frakt, A. Willsky. Efficient multiscale stochastic realization, Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Seattle, Washington, May 1998.
[75] A. B. Frakt. A realization theory for multiscale auto-regressive stochastic models, Ph.D. Thesis Proposal, MIT, Cambridge, MA, September 1996.
[76] A.B. Frakt. Multiscale hypothesis testing with application to anomaly characterization from tomograhic projections, Master thesis, Massachusetts Institute of Technology, Cambridge, MA, May 1996.
[77] A.B. Frakt, W. C. Karl, A. S. Willsky. A multiscale hypothesis testing approach to anomaly detection and localization from noisy tomograhic data, IEEE Transactions on Image Processing, 7(6): 825--837, June 1998.
[78] C. Fosgate, H. Krim, W. Irving, A. Willsky. Multiscale segmentation and anomaly enhancement of SAR imagery. IEEE Transactions on Image Processing, 6(1): 7--20, January 1997.
[79] H.T. Banks, K. Kunisch. Estimation techniques for distributed parameter systems. Birkhauser, Boston, 1989.
[80] S. Omatu, J.H. Seifeld, T. Soeda, Y. Sawaragi. Estimation of Nitrogen Dioxide concentration in the vicinity of a roadway by optimal filtering theory. Automatica, 24(1): 19--29, January 1988.
[81] A. da Silva, J. Guo. Documentation of the physical-space statistical analysis system (PSAS) Part Ⅰ: The conjugate gradient solver version PSAS-1.00. DAO Office Notes 96-02, Data Assimilation Office, Goddard Laboratory for Atmospheres, NASA, February 1996.
[82] W. L. Briggs. A multigrid tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.
[83] T.M. Chin. Dynamic estimation in computational vision. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, October 1991.
[84] T. M. Chin, W. C. Karl, A. S. Willsky. Sequential filtering for Multi-frame visual reconstruction, Signal Processing, 28:311-- 333, August 1992.
[85] T.M. Chin, W. C. Karl, A. S. Willsky. Probabilistic and sequential computation of optical flow using temporal coherence, IEEE Transactions on Image Processing, 3(6): 773--788, November 1994.
[86] T.M. Chin, A. J. Mariano. Wavelet-based compression of co-variances in Kalman filtering of geophysical flows, In Proceedings of SPIE, vol.2242L: 842--850, April 1994.
[87] S. Aihara. On adaptive boundary control for stochastic parabolic systems, IEEE Transactions on Automatic Control, 42(3):350--363, March 1997.
[88] M. Athans. Toward a practical theory for distributed parameter systems, IEEE Transactions on Automatic Control, pages 245--247, April 1970.
[89] T.M. Chin, W. C. Karl, A. S. Willsky. A distributed and iterative method for square root filtering in space-time estimation. Automatic, 31(1):67--82, 1995.
[90] J.M. Coleman. Gaussian space-time models: Markov field properties. PhD thesis, University of California at Davis, David, CA, August 1995.
[91] M. Daniel, A. Willsky. Modeling and estimation of fractional Brownian motion using multiresolution stochastic processes, Fractals in Engineering, Springer, 1997:124--137.
[92] P. Fieguth. Application of multiscale estimation to large scale multidimensional time varying problems, Ph.D. Thesis Proposal, MIT, Cambridge, MA, November 1993.
[93] P. Fieguth, M. R. Allen, M. J. Murray. Hierarchical methods for global-scale estimation problems, Proceedings of Canadian Conference on Electrical Engineering, Waterloo, Canada, 1998:161--164
[94] P. Gaspar, C. Wunsch. Estimates from altimeter data of barotropic Rossby waves in the Northwestern Atlantic Ocean, Journal of Physical Oceanography, 19(12): 1821--1844, 1989.
[95] Alan George. Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1981.
[96] E. Hildreth. Computations underlying the measurement of visual motion. Artificial Intelligence, 23:309--354, 1984.
[97] T. Ho, P. Fieguth, A. Willsky. Multiresolution stochastic models for the efficient solution of large-scale space-time estimation problems. Proceedings of International Conference on Acoustics, Speech, and Signal Processing, vol 6, Atlanta, Georgia, May 1996: 3098—4001.
[98] T. Ho, P. Fieguth, A. Willsky. Computationally efficient multiscale estimation of large-Scale dynamic systems. Proceedings of International Conference on Image Processing, Chicago, Illinois, 1998.
[99] T. Ho, A. Frakt, A. Willsky. Multiscale realization and estimation for space and space-time problems. Proceedings of IEEE International Symposium on Information Theory, Cambridge, Massachusetts, 1998.
[100] T. Ho. Large-scale multiscale estimation of dynamic systems. Ph.D. Thesis Proposal, MIT, Cambridge, MA, November 1996.
[101] T Kailath. The divergence and Bhattacharrya distance measures in signal selection". IEEE Transactions on Communication Technology, 15(1), February 1967.
[102] H. Lev-Ari, S. R. Parker, T. Kailath. Multidimensional maximum-entropy covariance extension, IEEE Transactions on Information Theory, 35(3):497--508, May 1989.
[103] H. E. Rauch, F. Tung, C. T. Striebel. Maximum likelihood estimates of linear dynamic Systems. AIAA Journal, 3(8): 1445--1450, 1965.
[104] J. C. Strikwerda. Finite difference schemes and partial differential equations. Chapman and Hall, New York, 1989.
[105] P.-Y. Le Traon, P. Gaspar, F. Bouyssel, H. Makhmara. Using TOPEX/ POSEIDON data to enhance ERS-1 data. Journal of Atmospheric and Oceanic Technology, 12: 161--170, 1995.
[106] P.-Y. Le Traon, J. Stum, J. Dorandeau, P. Gaspar. Global statistical analysis of TOPEX and POSEIDON data. Journal of Geophysical Research, 99(C12): 25619--25631, 1994.
[107] C. Wunsch, D. Stammer. The global frequency-wavenumber spectrum of oceanic variability estimated from TOPEX/POSEIDON altimetric measurements, Journal of Geophysical Research, 100(C12): 24895--24910, 1995.
[108] G. Simone, F. C. Morabito, A. Farina. Radar Imagine Fusion by multiscale Kalman filtering, the 3~(rd) International Conference of information fusion, Paris France, July 10-13~(th) 2000, WED3-2.