基于几何代数的时空场数据特征分析与运动表达
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摘要
近年来,随着全球观测系统的逐步健全与完善,传感器网络以及遥感等技术的广泛应用,时空场数据在连续地理现象的表达方面发挥越来越明显的作用。传统时空场分析多涉及DEM、遥感影像等标量场数据,对矢量场数据关注较少。此外,多数时空场分析方法仅能处理规则阵列数据,对海量不规则场数据处理和分析能力相对不足。本文引入Clifford代数构建不同维度、不同类型时空场数据的统一表达与运算框架,进而实现维度融合视角的时空场数据结构与运动特征的解析和提取算法。并通过典型案例分析对所相关算法的准确性、有效性及地学解释进行探讨。
基于Clifford代数的多维统一分析框架实现了时空场数据基本特征及其之间关系的统一表达；实现对不同维度时空场数据及上述特征参数的融合表达；实现对标量场、矢量场以及多重向量场特征参数在算法结构的统一；从多维融合的视角对时空场数据演化特征参数统一表达。
自适应模板匹配算法通过Rotor算子对欧氏变换的统一表达实现对原始数据演化构型的自适应模板匹配和分类,统一了传统卷积运算模板的表达形式。对基于卫星测高数据的海面地形坡度和海面波动特征分析结果表明该方法可有效表达地学时空场数据结构特征；特征模板匹配算法基于Clifford FFT,实现地学分析导向的时空场特征提取。以全球海面卫星测高数据为案例数据,构建ENSO暖舌的特征模板,提取的特征参数较好地解析出ENSO对全球海面影响的空间格局特征。
基于时空代数的时空分割可实现观察者视点相关的时空分割与透视,从而可以更好的再现原始时空过程的真实特性。针对给定地理现象的演化,构建一系列不同参数的Lorenz变换,从不同视角获取时空场观测数据的内在结构特征,从而可对地理现象的运动过程特征及演化参数进行有效解析。基于卫星测高数据模拟了赤道太平洋海区海面变化的时空演化过程,所提取的运动特征参数与MEI指数具有较好的对应性。
In recent years, sensor networks and remote sensing technology are widely used, and the global observing system is gradually improved and perfected. Temporal-spatial field data play a more and more important role in the continuous expression of geographical phenomena. The traditional analysis of temporal-spatial field involves only the scalar field data such as digital elevation model (DEM), remote sensing image and so on, but pays less attention to the vector field data. In addition, most temporal-spatial field analyses apply only to regularly arranged data, but lack the processing and analysis capabilities of irregular and massive field data. This paper introduces the Geometric Algebra which can construct a unified framework for expression and computing of temporal-spatial field data of different dimensions and different types. And then, the structure and movement characteristics extracting algorithms of temporal-spatial field data are implemented in the perspective of integration of dimension. At last, the accuracy, validity and geographical interpretation of mentioned algorithms are discussed through the typical case studies.
(1) This paper expresses the basic characterization parameters of temporal-spatial field data and their relationship uniformly based on Clifford algebra. (2)The different dimension temporal-spatial field data and their characterization parameters are expressed in a unified form. (3)Algorithms of consistent structure are constructed to achieving the unified extraction of the characterization parameters in scalar field, vector field and multivector field. (4)Finally, the temporal-spatial evolution parameters of field data are express in the multidimensional-fused perspective.
The adaptive template matching algorithm unify the traditional forms of template convolution by the classification and template matching of the original data's structure and evolution features through uniformly expressing of Euclidean transformation based on Rotor operator. The analysis results of the slope of sea surface topography and surface fluctuation characteristics based on altimetry data show that the method can effectively extract the geological structure of temporal-spatial field data. Feature template matching algorithm based on Clifford FFT (Fast Fourier Transform) realizes the extraction of Geo-oriented temporal-spatial field feature. Take the global sea surface altimetry data for example, this paper construct a features template of ENSO warm tongue and apply it to template matching algorithm. It Implement with the result of some significant characterization parameters which reflect the impact of ENSO (El Nino and Southern Oscillation) on the spatial pattern of variability of global sea surface.
Through the segmentation of time and space based on space-time algebra, the segmentation of observer-view-dependent perspective can be achieved. For the evolution of a given geographical phenomena, this paper construct a series of Lorenz transformation to get the internal structure characters of temporal-spatial field data from different perspectives. With the algorithm, movement characteristics and evolution parameters of geographical phenomena can be effectively extracted. Apply the algorithm to altimeter data, the temporal-spatial evolution process of sea-level change in equatorial Pacific Ocean is simulated, what's more, the extracted movement parameters and the MEI (Multivariate ENSO Index) have good correspondence.
基于Clifford代数的多维统一分析框架实现了时空场数据基本特征及其之间关系的统一表达；实现对不同维度时空场数据及上述特征参数的融合表达；实现对标量场、矢量场以及多重向量场特征参数在算法结构的统一；从多维融合的视角对时空场数据演化特征参数统一表达。
自适应模板匹配算法通过Rotor算子对欧氏变换的统一表达实现对原始数据演化构型的自适应模板匹配和分类,统一了传统卷积运算模板的表达形式。对基于卫星测高数据的海面地形坡度和海面波动特征分析结果表明该方法可有效表达地学时空场数据结构特征；特征模板匹配算法基于Clifford FFT,实现地学分析导向的时空场特征提取。以全球海面卫星测高数据为案例数据,构建ENSO暖舌的特征模板,提取的特征参数较好地解析出ENSO对全球海面影响的空间格局特征。
基于时空代数的时空分割可实现观察者视点相关的时空分割与透视,从而可以更好的再现原始时空过程的真实特性。针对给定地理现象的演化,构建一系列不同参数的Lorenz变换,从不同视角获取时空场观测数据的内在结构特征,从而可对地理现象的运动过程特征及演化参数进行有效解析。基于卫星测高数据模拟了赤道太平洋海区海面变化的时空演化过程,所提取的运动特征参数与MEI指数具有较好的对应性。
In recent years, sensor networks and remote sensing technology are widely used, and the global observing system is gradually improved and perfected. Temporal-spatial field data play a more and more important role in the continuous expression of geographical phenomena. The traditional analysis of temporal-spatial field involves only the scalar field data such as digital elevation model (DEM), remote sensing image and so on, but pays less attention to the vector field data. In addition, most temporal-spatial field analyses apply only to regularly arranged data, but lack the processing and analysis capabilities of irregular and massive field data. This paper introduces the Geometric Algebra which can construct a unified framework for expression and computing of temporal-spatial field data of different dimensions and different types. And then, the structure and movement characteristics extracting algorithms of temporal-spatial field data are implemented in the perspective of integration of dimension. At last, the accuracy, validity and geographical interpretation of mentioned algorithms are discussed through the typical case studies.
(1) This paper expresses the basic characterization parameters of temporal-spatial field data and their relationship uniformly based on Clifford algebra. (2)The different dimension temporal-spatial field data and their characterization parameters are expressed in a unified form. (3)Algorithms of consistent structure are constructed to achieving the unified extraction of the characterization parameters in scalar field, vector field and multivector field. (4)Finally, the temporal-spatial evolution parameters of field data are express in the multidimensional-fused perspective.
The adaptive template matching algorithm unify the traditional forms of template convolution by the classification and template matching of the original data's structure and evolution features through uniformly expressing of Euclidean transformation based on Rotor operator. The analysis results of the slope of sea surface topography and surface fluctuation characteristics based on altimetry data show that the method can effectively extract the geological structure of temporal-spatial field data. Feature template matching algorithm based on Clifford FFT (Fast Fourier Transform) realizes the extraction of Geo-oriented temporal-spatial field feature. Take the global sea surface altimetry data for example, this paper construct a features template of ENSO warm tongue and apply it to template matching algorithm. It Implement with the result of some significant characterization parameters which reflect the impact of ENSO (El Nino and Southern Oscillation) on the spatial pattern of variability of global sea surface.
Through the segmentation of time and space based on space-time algebra, the segmentation of observer-view-dependent perspective can be achieved. For the evolution of a given geographical phenomena, this paper construct a series of Lorenz transformation to get the internal structure characters of temporal-spatial field data from different perspectives. With the algorithm, movement characteristics and evolution parameters of geographical phenomena can be effectively extracted. Apply the algorithm to altimeter data, the temporal-spatial evolution process of sea-level change in equatorial Pacific Ocean is simulated, what's more, the extracted movement parameters and the MEI (Multivariate ENSO Index) have good correspondence.
引文
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[4]李延芳,顾耀林.矢量场数据演示的快速Clifford傅立叶变换.计算机工程与设计,2007,(21)：5177-5189
[5]李岩山.基于Clifford代数的数字图像水印技术.电子学报.2008,36(5)：852-855
[6]吴亚波,邓雪梅,赵国明等Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用.物理学报,2005,11：4994-4998
[7]顾耀林,李延芳.基于模板匹配的流场涡旋识别.计算机应用,2007,27(9)：2246-2248
[8]谢维信,曹文明,蒙山.基于Clifford代数的混合型传感器网络覆盖理论分析.中国科学(E),2007,37(8)：1018-1031
[9]袁林旺,俞肇元,谢志仁等.1965～2005年西北太平洋边缘海区海面变化记录的ENSO信号及其时空分异特征.中国科学D辑：地球科学,2009,39(5)：664-676
[10]Acar E, Yener B.Unsupervised Multiway Data Analysis:A Literature Survey. EEE Transactions on Knowledge and Data Engineering,2009,21(1):6-20
[11]Acar E, Camtepe S A, Yener B. Collective sampling and analysis of high order tensors for chatroom communications. In Proc. of IEEE Internaional Conference on Intelligence and Security Informatics.2006:213-224
[12]Andersen C M, Bro R. Practical aspects of parafac modelling of fluorescence excitation-emission data. J. of Chemometrics,2003,17(4):200-215
[13]Bayro-Corrochano E, Falcon L E. Geometric algebra of points, lines, planes and spheres for computer vision and robotics. Robotica,2005.23(6):755-770
[14]Besprosvany J. Representations and interactions from field equations on an extended spin space. Paris, France:Institute of Physics Publishing,2003
[15]Blackmore T, Norton G H. Bounds on the state complexity of geometric Goppa codes. Serrento:Institute of Electrical and Electronics Engineers Inc,2000
[16]Brackx F, Schepper N D, Sommen F. The two-dimensional Clifford-Fourier transform. Journal of Mathematical Imaging and Vision,2006,26(1-2):5-18
[17]Brett W B, Michael W B, Murray B. Discussion tracking in Enron email using PARAFAC In Text Mining 2007, Workshop at the SIAM International Conference on Data Mining,2007
[18]Bro R, Harshman R. Sidiropoulos N D. Modeling multi-way data with linearly dependent loadings. Technical Report,2005
[19]Buchholz S, Sommer G. On Clifford neurons and Clifford multi-layer perceptrons. Neural Networks,2008,21(7):925-935
[20]Cameron J, Lasenby J. Oriented conformal geometric algebra. Advances in Applied Clifford Algebras,2008,18(3-4):523-538
[21]Cannon A J. Nonlinear Principal Predictor Analysis:Application to the Lorenz System. J. Climate,2006,19,579-589
[22]Carroll J E. Complex signals can be real. European Journal of Physics,2007.28(6): 1151-1160
[23]Chew P A, Bader T G, Kolda, et al. Cross-Language Information Retrieval Using PARAFAC2. Proc.13th ACM SIGKDD Int'l Conf. Knowledge Discovery and Data Mining (KDD'07),2007:143-152
[24]Chys P. Application of geometric algebra for the description of polymer conformations. Journal of Chemical Physics,2008.128(10):104-107
[25]Ding C Ye J P. Two-Dimensional Singular Value Decomposition for 2D Maps and Images. Proc. SIAM Int'l Conf. Data Mining (SDM'05),2005:32-43
[26]Doran C, Lasenby A. Geometric Algebra for Physicists. Cambridge University. Press, Cambridge,2003:228-264
[27]Dorst L, Fontijne D, Mann S. Geometric algebra for computer science. The Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann, Elsevier,2008
[28]Ebling J, Scheuermann G. Template Matching on Vector Fields using Clifford Algebra. IKM 2006,2006
[29]Etzel K R, McCarthy J M. Interpolation of spatial displacements using the Clifford Algebra of E4. Journal of Mechanical Design, Transactions of the ASME,1999,121(1): 39-44
[30]Gebken C, Perwass C, Sommer G. Parameter estimation from uncertain data in Geometric Algebra. Advances in Applied Clifford Algebras,18(3-4):647-664
[31]Gekhtman M, Shapiro M. On the properties of the exchange graph of a cluster algebra. Mathematical Research Letters,2008,15(2-3):321-330
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