【作者】 程永强；

【导师】 黎湘；

【作者基本信息】 国防科学技术大学 ， 信息与通信工程， 2012， 博士

【摘要】 信息几何是在黎曼流形上采用现代微分几何方法来研究统计学问题的基础性、前沿性学科,被誉为是继香农开辟现代信息理论之后的又一新的理论变革,在信息科学与系统理论研究领域展现出了巨大的发展潜力。本文以信息几何理论为基础,探索其在信号处理、尤其是现代雷达信号处理技术中的应用,以全新的视角对雷达信号处理的基础性、科学性问题展开全面、系统的研究,提出了一系列新概念、新思想和新方法。特别地,对雷达系统的信息分辨、信号检测、参数估计、传感器的信息获取、信息累积等科学问题进行了深入研究,提出了一套新的分析方法,同时也为问题的解决提供了崭新的思路和手段。第一章为绪论,精炼了信息几何的科学内涵,综述了信息几何理论的发展历史与应用的研究现状,归纳和总结了其中所体现的信息几何的基本思想和基本方法。第二章深入浅出地介绍信息几何的基本原理和数学基础,以直观而形象的方式阐述复杂而抽象的理论与方法,为后续各章的研究奠定基础。第三章围绕雷达分辨力这一主题,对分辨力的定义、意义、度量及其对目标检测和跟踪性能的影响进行了全面而系统的研究,使分辨理论成为一个体系。首次提出雷达系统“信息分辨力”概念,将雷达波形特征、测量模型和信噪比相结合,统一描述为测量的似然函数,在由似然函数构成的统计流形中度量系统的分辨能力,拓展了Woodward于上世纪50年代提出的基于信号模糊函数的雷达分辨理论,为描述雷达系统的实际分辨能力提供了度量,同时也为衡量检测与跟踪系统的联合分辨提供了依据。由信息分辨力概念得出了一系列重要的结论,丰富了对分辨力的传统认识和理解。基于检测分辨单元,研究了雷达测量→检测分辨→目标跟踪测量数据提取的关键问题。提出了基于误判概率的双源分辨单元定义,分析了信噪比对分辨力的影响、分辨力对检测和跟踪性能的影响,并依据所提分辨单元,研究了测量数据的提取策略,为提升雷达系统的联合检测与跟踪性能及波形优化设计提供了理论支持。第四章研究了信号检测的信息几何方法。建立了确定性信号和随机信号检测的信息几何解释。在该框架下,信号有无的两种假设和检测器都可以视为统计流形上的几何对象,而检测问题则转变为对两种假设所对应的概率分布的辨别问题,将检测问题转变为纯粹的几何问题来研究。建立了广义似然比检测(GLRT)的信息几何解释,提出了弯曲指数分布族GLRT的两幅几何图景,并从几何角度探讨了有限样本条件下GLRT可能带来的检测信息损失。提出了微弱信号局部最大势检验(LMP)方法的信息几何解释,建立了LMP检测与统计流形上优化问题之间的联系,提出了扩展的局部最大势检验统计量(ELMP),将经典的仅适用于单参数、单边检验的LMP算法,扩展到多变量、“双边”检验的形式,大大扩展了LMP检验的适用范围。从理论上揭示了检测问题与识别问题的等价关系,将各种检测问题统一到一个几何框架中予以分析和解决,将检测问题的研究提升到了一个新的高度。第五章研究了参数估计的信息几何方法。阐述了信息几何关于参数估计信息损失的基本理论,并以实例揭示了由于观测模型的非线性给参数估计带来的固有信息损失,通过计算给出了相比CRLB更确切的估计方差下界。针对弯曲指数分布族的参数估计问题,提出了基于统计流形上自然梯度的迭代最大似然估计(NG-MLE)算法。将非线性估计问题与统计流形上的确定性优化问题相统一,为算法的收敛性提供了明确的几何解释。将NG-MLE算法应用于相位干涉定位系统的目标定位参数估计问题,并建立了定位误差传播的统计模型,提出了一种有效控制定位误差传播的方法,为分析和解决大规模分布式传感器网络逐次定位误差传播问题提供了可选的途径。第六章研究了目标跟踪传感器网络的信息几何。面向目标跟踪应用,探究信息几何与目标跟踪传感器网络性能之间的联系,集中研究传感器网络的信息获取能力、信息累积效应以及传感器测量模型的几何表示等基本问题,并对上述成果在目标分辨、状态估计、传感器布站、目标跟踪航迹优化等问题中的应用展开具体研究。针对目标跟踪中的数据关联、目标分辨问题,采用统计流形上的Fisher信息距离、Kullback-Leibler分离度和能量函数来度量传感器测量对邻近目标的区分能力。采用Levi-Civita仿射联络和黎曼曲率张量、Ricci曲率等几何量来表征统计流形的结构,建立了Ricci曲率张量场与传感器信息获取能力以及信息变化率之间的联系。研究了统计流形在欧氏空间中的仿射嵌入表示方法,直观表现传感器测量模型的流形结构,为传感器网络的优化配置提供潜在的方法和应用。研究了目标与传感器之间相对运动形成的信息累积效应,并基于信息累积量最大准则,解决被动目标跟踪传感器的最优机动问题。提出的一整套分析方法为传感器系统的分析与目标跟踪中相关问题的研究提供了理论指导。第七章研究了基于信息散度的相位测量模糊鉴别问题。将模糊的相位测量映射到目标状态空间,建立了映射的查找表,通过查找表来求解目标位置估计所涉及的丢番图方程组。针对动目标跟踪中的相位测量模糊问题,提出采用数据关联和滤波技术去除模糊的方法,并给出了运动目标跟踪模糊鉴别的例子。针对微动目标定位中的相位测量模糊问题,分析了微动目标相位测量模糊的可分辨性,推导了目标微动条件下的相位测量概率分布函数,而后基于样本分布与理论分布的KLD距离,提出了微动目标模糊鉴别和定位算法,并给出了算法的流形解释。所提方法与基于流形和流形上距离度量的分类和识别方法具有异曲同工之处,为识别问题的解决提供了借鉴。研究成果为基于连续波相位测量的微动目标定位与跟踪提供了技术途径。第八章总结全文,并提出了信息几何在信号处理领域中的若干开放性问题。本文研究成果不仅丰富了统计信号处理和雷达信号处理的基础理论,同时也建立了信息几何与更广泛的统计学问题之间的联系,为信息几何在信号处理领域的应用提供了有益的借鉴。更多还原

【Abstract】 Information Geometry is the fundamental and cutting-edge discipline which studiesstatistical problems on Riemannian manifolds of probability distributions using the methodsof Differential Geometry. It is identified as the second generation of modern InformationTheory pioneered by Shannon and exhibits great potential for development in the field ofinformation science and systems theory. The underlying thesis is to explore the applications ofinformation geometry to signal processing, especially to radar signal processing, and studiesthe fundamental and scientific problems in radar signal processing from a bran-new viewpoint.In particular, the scientific problems such as information resolution of radar systems, signaldetection, parameter estimation, the information gathering capacity and accumulativeinformation of sensor networks are explored in depth, with the development of a new set ofanalysis methods as well as a new set of methods to deal with the existing problems.The first chapter refines the scientific content of information geometry and elaborates thehistory and applications of information geometry as well as its basic ideas and basic methods.The second chapter introduces the principles and mathematical foundations of infor-mation geometry and provides a basis for the following chapters.The third chapter explores the definition, significance and measure of radar resolution aswell as its influence on the performance of target detection and tracking. A new concept calledinformation resolution for a sensor measurement system, which is defined in the frameworkof information geometry, is proposed. In particular, the information resolution of radarsystems is generalized from the work on existing radar resolution pioneered by Woodwardand defined on statistical manifolds where the intrinsic geometrical structure of waveform,measurement and noise models of the underlying sensing devices are convenientlycharacterized in terms of the Fisher information metric. Information resolution provides ametric to measure the practical resolution capacity of radar systems as well as the resolutionof joint detection-tracking systems. A set of important conclusions of information resolutionenriches the conventional understanding of radar resolution. Based on resolution cells fortarget detection, the key problems of radar measurement, detection and resolution, as well asthe measurement extraction schemes are studied. A definition of detection-based radarresolution, called differential resolution, is developed to describe the system’s capacity todistinguish two closely spaced targets. The SNR effects on resolution are analyzed and themeasurement-extraction scheme based on the differential resolution cell is discussed andsimulations show that an enhanced tracking performance can be obtained by such adevelopment.The fourth chapter studies the information geometric methods of signal detection. Aconcise geometric interpretation of deterministic and random signal detection in the theory ofinformation geometry is established. In such a framework, both hypotheses and detector canbe treated as geometrical objects on the statistical manifold of a parameterized family ofprobability distributions. Both the detector and detection performance are elucidated geometrically in terms of the Kullback-Leibler divergence. Then, the generalized likelihoodratio test (GLRT) for composite hypothesis testing problems is considered from a geometricviewpoint. Two pictures of the GLRT for curved exponential families are presented, based onwhich the performance deterioration when performing the GLRT under finite number ofsamples is discussed. Further more, a concrete geometric interpretation of the locally mostpowerful (LMP) test for weak signal detection is presented. In particular, the LMP test isidentified as the norm of natural whitened gradient on the statistical manifold, which indicatesthat the LMP test pursues the steepest learning directions from the null hypothesis to theempirical distribution of the observed data on the manifold. Due to this nicety, an immediateextension of the LMP test, called the ELMP test which removes the scalar and the one-sidedrestrictions in the LMP test, is proposed. The above analyses reveal equivalence between thedetection problems and the discrimination problems and provide a unified geometricframework for the analysis of detection problems, which extends the existing analyses to anew level.The fifth chapter investigates information geometric methods of parameter estimation.Firstly, the basic theory of information geometry on information loss of parameter estimationis summarized and examples of the inherent information loss caused by the nonlinearity ofmeasurement model are presented, while a tighter lower bound of the error variance withrespect to the well-known Cramér-Rao Lower Bound (CRLB) is calculated. Secondly, anatural gradient-based maximum likelihood estimator (NG-MLE) on statistical manifolds isproposed to deal with the nonlinear parameter estimation problem of curved exponentialfamilies. We demonstrate that the nonlinear estimation problem can be simply viewed as adeterministic optimization problem with respect to the structure of a statistical manifold. Inthis way, information geometry offers an elegant geometric interpretation for the definitionand convergence of the estimator. The theory is interpreted via the analysis of a distributedmote network localization problem where the Radio Interferometric Positioning System(RIPS) measurements are used for free mote location estimation. The analysis resultspresented demonstrate the advanced computational philosophy of the proposed methodology.Moreover, a noisy measurement model that takes the location uncertainties of anchor nodesinto account in the node localization process has been derived, which effectively deals withthe problem of progressive localization when the localization error is nonlinearly propagatedover the sensor network.The sixth chapter studies the information geometry of target tracking sensor networks.The connections between information geometry and performance of sensor networks fortarget tracking are explored to pursue a better understanding of placement, planning andscheduling issues. Firstly, the integrated Fisher information distance (IFID) between the statesof two targets is analyzed by solving the geodesic equations and is adopted as a measure oftarget resolvability by the sensor. The differences between the IFID and the well knownKullback-Leibler divergence (KLD) are highlighted. We also explain how the energyfunctional, which is the “integrated, differential” KLD, relates to the other distance measures.Secondly, the structures of statistical manifolds are elucidated by computing the canonical Levi-Civita affine connection as well as Riemannian and scalar curvatures. We show therelationship between the Ricci curvature tensor field and the amount of information that canbe obtained by the network sensors. Thirdly, an analytical presentation of statistical manifoldsas an immersion in the Euclidean space for distributions of exponential type is given. Thesignificance and potential to address system definition and planning issues using informationgeometry, such as the sensing capability to distinguish closely spaced targets, calculation ofthe amount of information collected by sensors and the problem of optimal scheduling ofnetwork sensor and resources, etc., are demonstrated. Finally, the cumulative effect ofinformation when there is relative motion between the target and sensor is discussed. Theaccumulative information is used as a criterion for the sensor trajectory scheduling problem inbearings-only tracking.The seventh chapter explores the target tracking and localization problem in the presenceof phase measurement ambiguities. The main focus of this chapter is to deal with theambiguities caused by phase measurements and to elucidate how to identify and remove theseambiguities in tracking and localization context. Specifically, we combine the aboveapproaches in terms of creating mappings between target location and phase measurementspaces so that the nonlinear and indeterminate Diophantine problem reduces to the acquisitionof a finite set of possible target locations over a region of interest. Then firstly, we show thatwhen the target motion is significant between data sampling intervals the location ambiguitycan be resolved over time via known target-in-cluster tracking techniques. Secondly, when thetarget is undergoing micromotions which results in the same collection of candidate locationsfrom phase measurements over time, the location ambiguity can be resolved using a novelphase distribution discrimination method. In this method a probability density function of theambiguous phase-only measurement is derived that takes both sensor noise and target motiondistributions into account based on directional statistics. Optimal locations are inferred fromsuch distributions. The inference algorithm is interpreted from the viewpoint of manifoldlearning, which provides a reference for solving identification problems based on manifold.The eighth chapter makes a summary of the thesis, while several open problems ofinformation geometry in applications of signal processing are proposed.In conclusion, the studies and results in this paper not only enrich the basic theory ofstatistical signal processing and radar signal processing, but also establish comprehensiveconnections between statistics and information geometry, and provides an exemplification ofadvantages of the geometrical perspective on studying statistical problems.更多还原