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发布时间: 2020-01-16 |
遥感图像处理 |
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张荫华, 杨萌 |
收稿日期: 2019-04-22; 修回日期: 2019-06-14; 预印本日期: 2019-06-21
基金项目: 国家自然科学基金项目（61501152）
第一作者简介:
张荫华, 1994年生, 男, 硕士研究生, 主要研究方向为信息几何及其应用。E-mail:zyh986@163.com.
中图法分类号: TP391.4
文献标识码: A
文章编号: 1006-8961(2020)01-0206-08
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摘要
目的 舰船目标检测是合成孔径雷达（SAR）图像在海事监测领域中的一项重要应用。由于海面微波散射的复杂性，SAR图像中海杂波分布具有非均匀性、非平稳性等特点，传统的基于恒虚警率（CFAR）的SAR图像舰船检测算法难以适应复杂多变的海杂波环境，无法实现实时有效的智能检测任务。鉴于此，本文提出了基于信息几何的SAR图像船舰目标检测方法，旨在分析统计流形及其在参数空间中的几何结构，探讨信息几何在SAR图像目标检测应用中的切入点，从新的角度提升该应用领域的理论与技术水平。方法 首先，运用威布尔分布族对SAR图像中的海杂波进行统计建模，利用最大似然方法估计SAR图像局部邻域像素的分布参数，并将不同参数下的统计分布作为威布尔流形上的不同点；其次，融合高斯分布的费歇耳度量来构造威布尔流形空间中概率分布之间的测度，实现目标与背景区域的差异性表征；最后，利用最大类间方差法，实现SAR图像舰船目标检测。结果 实验和分析表明，相比于传统的基于恒虚警率的检测算法，信息几何方法可以有效地区分舰船目标和海杂波背景，降低虚警率，实现舰船目标显著性表示与检测。结论 由于舰船目标的复杂后向散射特性，如何有效地表征这一差异，是统计类检测算法的关键所在。本文依据信息几何理论，将概率分布族的参数空间视为微分流形，在参数流形上构造合适的黎曼度量，对SAR图像中各像素局部邻域进行测度表征，可以显著性表示目标与背景杂波之间的统计差异，实现舰船目标检测。
关键词
SAR图像; 目标检测; 信息几何; 统计流形; 黎曼度量
Zhang Yinhua, Yang Meng Abstract
Objective Among synthetic aperture radar (SAR) image applications, automatic ship detection in SAR images is an active research field and plays a crucial role in various related military and civil applications, such as ocean traffic surveillance, protection against illegal fisheries, and ship rescuing. Many algorithms have been developed for ship detection in SAR images. Among them, constant false alarm rate (CFAR) algorithms, which have minimal operational complexity and a regular structure, are the most commonly used for ship detection in SAR imagery. CFAR-based methods are simple and effective, and the corresponding adaptive threshold preserves a constant false alarm probability. However, due to the non-homogeneity of sea clutter in the intensity domain, which is caused by the complexity of microwave scattering on the ocean surface, traditional CFAR-based detection methods cannot easily adapt to the variability and complexity of the sea clutter environment, and they cannot realize a robust detection of targets within sea clutter. Another approach to detect targets in a sea clutter background is to extract the features of targets in SAR images. The detection strategy relies on the feature description and analysis of targets in high-resolution SAR images. However, each feature representation for targets has its strengths and weaknesses and should be evaluated according to practical application scenarios. Additionally, the resolution of most SAR images is often not sufficiently high to extract effective detailed target information. In view of these situations, a high-performance ship detector based on information geometry is proposed in this study. Method Information geometry originated from the study of the intrinsic properties of manifolds of probability distributions. This theory is a combination of mathematical statistic models and geometrical methods. The development of geometrical theory and numerical techniques has extended the applicability of information geometry to the field of signal/image analysis. The purpose of this research is to obtain an improved understanding and analysis of the statistical manifold and its geometric structure in parameter space. This work explores the application of information geometry theory in ship detection from SAR images and analyzes detection problems from a new perspective. The manifold model is a good representation of the structural information of the pixel distribution controlled by a set of parameter. On this basis, an effective ship detection approach in SAR images is developed in this study. First, the Weibull distribution is used to model clutter, and the maximum likelihood estimation method is adopted to estimate the distribution parameters of the local neighborhood pixels of the SAR image. Second, the statistical distribution under different parameters is regarded as the difference point in the Weibull manifold. Third, a novel Riemannian metric is constructed to realize distance measurement between probability distributions in manifold space. Finally, the targets are extracted using an automatic threshold selection method. Result According to the theory of modern geometry, two points that are similar in Euclidean space may be far apart in non-Euclidean space. A significant statistical difference exists between ship targets and sea clutter because of the complex backscattering feature of ships. The proposed method based on information geometry utilizes this feature and geometrical methods to implement non-Euclidean metrics between classes (ship targets and background clutter to achieve saliency representation and detection of targets). Detection experiments are conducted on real SAR imagery. The results of the conventional Weibull-based CFAR detector is also provided for comparison to validate the effectiveness of the proposed method in real data. Conventional CFAR detection methods fail to yield satisfactory results due to low signal-to-clutter ratio and varying local clutter. Compared with conventional CFAR approaches, the proposed method can enhance targets and measure the local dissimilarity between a target and its neighborhood by using the information geometrical structure. Experimental results also show that the proposed method based on information geometry is effective in discriminating between ships and sea clutter and has good performance in ship detection in SAR images. Conclusion Information geometry began as the application of differential geometry to statistical theory. It has been applied to study the geometrical structure of a manifold of probability distributions. Information geometry has developed and continues to develop with the types of geometric statement used and in its application areas. In reality, no geometric statement is true or false by nature. Sometimes, it is merely a question of choice. Given the discovery of the geometric meaning of Fisher information, which contributes to the development of information geometry in a concise and intuitive manner, the geometric structure of a set of positive densities in a given statistical manifold space has elicited the interest of many researchers. Moreover, the Riemannian metric is not unique. Many important families of probability distributions possess a series of metric structures. Each metric corresponds to a different geometric structure. For these reasons, extensive research has focused on identifying new geometrical structures of parametric statistical models. It provides statistical science with a highly efficient method for constructing abstract models that maximize the use of space in signal/image processing. The aim of this study is to show the benefits of statistical manifolds suitable for ship detection in SAR imagery and based on information geometry theory. The principal tool in this work is the metric construction by means of building new metrics from old ones. Theoretical analysis and experimental results show that information geometry provides detection problems with a new perspective to view the structure of the investigated statistical manifold.
Key words
SAR image; target detection; information geometry; statistical manifold; Riemannian metric
0 引言
合成孔径雷达(SAR)具有全天时、全天候、高分辨、宽测绘带等特点，是海洋信息获取与监测中不可或缺的重要手段(Wang和Chen，2017)。随着SAR成像技术的不断进步，获取高分辨、海量数据的能力得到大幅度提升，基于SAR图像的舰船目标检测问题受到了相关科研、工程技术工作者的广泛重视，成为海洋遥感技术领域的重要研究内容之一(李健伟等，2018)。
目前的SAR图像舰船目标检测算法主要包括基于自适应阈值的检测方法和基于目标特征表示的检测方法(Wang等，2019)。基于自适应阈值的检测方法，本质上是针对幅度图像，依据舰船目标与海杂波电磁后向散射特性的幅度信息差异来构建检测阈值的二分类方法(Wang等，2017)。在此类方法中，具有代表性的当属以简单、快速、有效而著称的恒虚警率(CFAR)检测算法和其相应的改进算法(Gao和Shi，2017)。然而，由于SAR系统的相干成像特点以及海面状况的复杂性，导致SAR图像中杂波分布具有较强的时变性、非平稳性、非高斯性等特征，使得CFAR检测算法具有较高的虚警率。并且，CFAR检测算法在很大程度上依赖于对杂波模型、虚警率、目标窗口、保护窗口和背景窗口的准确估计，使得该类算法的检测性能在实际应用中存在较大的不确定性(胡炎等，2019)。基于目标特征表示的检测算法是一类更为广义的检测方法，其借助于机器学习、模式识别等理论，通过对SAR图像目标与背景杂波的特征描述与提取，通过模式分类技术，实现目标检测(Li等，2018)。该类方法通常需要挖掘SAR图像中诸如散射特征、极化特征、几何特征、统计特征、纹理特征、时—频特征、融合特征等特征信息，比较适合于具有高分辨、特征丰富的目标检测问题(Schwegmann等，2017)。然而，SAR图像舰船目标检测中的特征表示方法都具有其自身的优点和不足，需要依据实际应用场景加以评估。此外，大多数星载SAR图像的分辨率往往不足以提取足够详细的舰船目标特征信息，使得算法的适用性受到限制。
信息几何是将现代几何理论方法应用于信息学领域而发展起来的一套理论体系(Malagò等，2018)，在非线性问题分析处理中具有独特的优势(Li和Montúfar，2018)。依据信息几何理论，针对研究问题本身具有的非线性特点，可通过构建微分流形的方式，定义具有线性结构的切丛(Wong，2018)，从而能像欧氏空间一样在切丛上定义内积，将非线性问题转化为局部线性问题求解(Naudts和Zhang，2018)。
本文旨在研究指数分布族的参数流形及其度量，分析参数空间的几何结构，探索微分几何理论应用于SAR图像舰船目标检测问题的切入点(Amari，2016)。本文算法从指数分布族出发，构建参数流形，引入参数化的费歇耳信息度量(Arwini和Dodson，2008)，构造切向量，对待检测目标区域进行显著性表示，实现SAR图像舰船目标检测。
本文方法的主要创新点为：1)依据切丛理论探索信息几何应用于SAR图像处理领域的切入点; 2)融合高斯统计流形与威布尔统计流形实现目标特征的显著性表示。
1 统计流形及其几何结构
1.1 费歇耳信息度量
依据信息几何理论，在不同的不变性原则下，统计流性可以赋予不同的黎曼度量，即不同的几何结构。由于费歇耳信息矩阵所具有的统计与几何特性，使其成为信息几何理论与应用研究中的切入点，常常用于构建统计流形上的黎曼度量(Cao等，2008)。
不失一般性，考虑指数分布族的密度函数
| $ \int_{ - \infty }^{ + \infty } {{p_\theta }} (x){\rm{d}}x = 1,\theta \in \mathit{\boldsymbol{ \boldsymbol{\varTheta} }} $ | (1) |
式中，
记
| $ \int_{ - \infty }^{ + \infty } {{\partial _i}} {\partial _j}{p_\theta }(x){\rm{d}}x = 0,\theta \in \mathit{\boldsymbol{ \boldsymbol{\varTheta} }} $ | (2) |
于是，费歇耳信息度量
| $ {g_{ij}} = \int_{ - \infty }^{ + \infty } {{p_\theta }} (x){\partial _i}{\partial _j}{p_\theta }(x){\rm{d}}x $ | (3) |
令
| $ \mathit{\boldsymbol{u}} = \left\{ {{u^i}{\partial _i}} \right\},\mathit{\boldsymbol{v}} = \left\{ {{v^i}{\partial _i}} \right\} $ | (4) |
则有
| $ g\left( {\mathit{\boldsymbol{u}},\mathit{\boldsymbol{v}}} \right) = {g_{ij}}{u^i}{v^j} $ | (5) |
1.2 高斯统计流形
高斯分布具有极其广泛的实际背景，雷达工程中很多随机变量的概率分布都可以近似地用正态分布来描述(黎湘等，2016)，其在信息几何中也占有相当重要的位置，是统计几何分析的核心研究内容之一(赵兴刚等，2017)。
考虑高斯分布族的密度函数
| $ p\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sigma \sqrt {2{\rm{ \mathsf{ π} }}} }}\exp \left( { - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right) $ | (6) |
式中，
对式(6)取自然对数得
| $ \ln p\left( {x;\mu ,\sigma } \right) = \frac{{\theta _1^2}}{{4{\theta _2}}} - \frac{1}{2}\ln \left( { - \frac{{\rm{ \mathsf{ π} }}}{{{\theta _2}}}} \right) + {\theta _1}x + {\theta _2}{x^2} $ | (7) |
式中，
| $ \varphi \left( {{\theta _1},{\theta _2}} \right) = \frac{1}{2}\ln \left( { - \frac{{\rm{ \mathsf{ π} }}}{{{\theta _2}}}} \right) - \frac{{\theta _1^2}}{{4{\theta _2}}} $ | (8) |
于是，式(6)可以写为
| $ \ln p\left( {x;\mu ,\sigma } \right) = {\theta _1}x + {\theta _2}{x^2} - \varphi \left( {{\theta _1},{\theta _2}} \right) $ | (9) |
对式(9)两端关于(
| $ {\partial _i}{\partial _j}\ln p\left( {x;\mu ,\sigma } \right) = - {\partial _i}{\partial _j}\varphi \left( {{\theta _1},{\theta _2}} \right) $ | (10) |
于是
| $ \int_{ - \infty }^{ + \infty } p \left( {x;\mu ,\sigma } \right){\partial _i}{\partial _j}\ln p\left( {x;\mu ,\sigma } \right){\rm{d}}x = - {\partial _i}{\partial _j}\varphi \left( {{\theta _1},{\theta _2}} \right) $ | (11) |
则费歇耳信息度量(孙华飞等，2018)为
| $ {g_{ij}} = - {\partial _i}{\partial _j}\varphi \left( {{\theta _1},{\theta _2}} \right) $ | (12) |
或
| $ \left[ {{g_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} {{\sigma ^\tau }}&{2\mu {\sigma ^\tau }}\\ {2\mu {\sigma ^\tau }}&{4{\mu ^3}{\sigma ^\tau } + 2\mu {\sigma ^{2\tau }}} \end{array}} \right] $ | (13) |
1.3 威布尔统计流形
威布尔分布是工程领域中广泛应用的一类经验统计模型，尤其适用于海杂波的拖尾分布形式。并且，威布尔分布可利用概率值较易估计相应的分布参数(Gao，2019)，因此，本文采用威布尔分布对SAR图像杂波数据进行统计建模。
威布尔分布族的密度函数
| $ p\left( {x;\lambda ,\kappa } \right) = \frac{\kappa }{\lambda }{\left( {\frac{x}{\lambda }} \right)^{\kappa - 1}}\exp \left( { - {{\left( {\frac{x}{\lambda }} \right)}^\kappa }} \right),\;\;\;x \ge 0 $ | (14) |
式中，
| $ \ln p\left( {x;\lambda ,\kappa } \right) = \ln \kappa - \kappa \ln \lambda + \left( {\kappa - 1} \right)\ln x - {\left( {\frac{x}{\lambda }} \right)^\kappa } $ | (15) |
令
| $ {g_{ij}} = - E\left( {\frac{{{\partial ^2}\ln p\left( {x;\lambda ,\kappa } \right)}}{{\partial {\theta _i}\partial {\theta _j}}}} \right) $ | (16) |
式中，
| $ \left[ {{g_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{{\lambda ^2}}}{{{\kappa ^2}}}}&{\frac{{\gamma - 1}}{\kappa }}\\ {\frac{{\gamma - 1}}{\kappa }}&{\frac{{{{\left( {\gamma - 1} \right)}^2} + {{\rm{ \mathsf{ π} }}^2}/6}}{{{\lambda ^2}}}} \end{array}} \right] $ | (17) |
式中，
2 SAR图像目标检测算法
威布尔分布是常用的海杂波幅度统计分布模型。在分布模型参数估计应用上，常用的两种重要的估计方法为矩估计方法和极大似然估计方法。总体来讲，矩估计法较为易行，但其性质不如极大似然估计好，因此，本文采用极大似然估计法进行小样本下的分布模型参数估计。依据极大似然估计法，威布尔分布模型中的尺度参数
| $ \lambda = \frac{m}{{\left( {1/\kappa } \right)\sum\limits_{i = 1}^m {x_i^\lambda } \ln {x_i} - \sum\limits_{i = 1}^m {\ln } {x_i}}} $ | (18) |
和
| $ \kappa = {\left[ {\left( {\frac{1}{m}} \right)\sum\limits_{i = 1}^m {\ln } x_i^\lambda } \right]^{1/\lambda }} $ | (19) |
联立得到。其中，样本
| $ \mu ' = \lambda \mathit{\Gamma }\left[ {\left( {\kappa + 1} \right)/\kappa } \right] $ | (20) |
| $ \sigma ' = \sqrt {{\lambda ^2}\mathit{\Gamma }\left[ {\left( {\kappa + 2} \right)/\kappa } \right] - {\mu ^2}} $ | (21) |
式中，
依据式(13)构建费歇耳信息度量
| $ \left[ {{g_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\sigma '}^\tau }}&{2\mu '{{\sigma '}^\tau }}\\ {2\mu '{{\sigma '}^\tau }}&{4{{\mu '}^3}{{\sigma '}^\tau } + 2\mu '{{\sigma '}^{2\tau }}} \end{array}} \right] $ | (22) |
令
| $ \mathit{\boldsymbol{v}} = \left\{ {{v^i}{\partial _i}} \right\} $ | (23) |
式中
| $ {v^1} = \frac{\lambda }{{\sqrt {{\lambda ^2} + {\kappa ^2}} }},{v^2} = \frac{\kappa }{{\sqrt {{\lambda ^2} + {\kappa ^2}} }} $ | (24) |
则
| $ g\left( {\mathit{\boldsymbol{v}},\mathit{\boldsymbol{v}}} \right) = {g_{ij}}{v^i}{v^j} $ | (25) |
即
| $ {\left| \mathit{\boldsymbol{v}} \right|^2} = \mathit{\boldsymbol{v}}\left[ {{g_{ij}}} \right]{\mathit{\boldsymbol{v}}^{\rm{T}}} $ | (26) |
式中，向量
本文基于信息几何的SAR图像舰船目标检测算法步骤为：
1) 选取大小为
2) 对获取SAR图像
3) 由所估计的尺度参数
4) 将各像素所对应的
5) 针对
3 实验结果与分析
3.1 实验结果
为了验证本文检测方法的有效性，实验基于MATLAB仿真平台进行仿真，所采用SAR图像大小为153×151像素，如图 1所示，其中，图 1(a)和图 1(b)各含有一个待检测目标。本文采用威布尔分布对海杂波进行统计建模。
在实验过程中，设置滑动窗口大小
不失一般性，运用最大类间方差法分别对如图 2和图 3所示的目标显著性图像进行二分类，得到最终检测结果，如图 4所示。
在实验中，本文算法运用威布尔分布对图像各像素局部邻域数据进行统计建模，借助舰船目标与其背景电磁散射的统计差异性，在参数流形上进行几何结构分析，将SAR图像各像素邻域数据映射成切丛中的切向量长度，提高目标与背景之间的对比度，实现SAR图像舰船目标检测。实验结果表明，本文算法提供了将信息几何理论应用于SAR图像检测的一个好的切入点。
3.2 实验分析
由于SAR系统的相干成像特点以及复杂的海面状况，导致SAR图像中海杂波分布具有较强的时变性、非平稳性、非高斯性等特征，严重影响了基于统计模型的传统SAR图像目标检测算法性能。
如图 5和图 6所示，SAR图像中存在大量与目标比较相似的杂波尖峰，使得海杂波呈现出非高斯特性，具有拖尾分布特征。图 7和图 8分别给出了图 1(a)(b)所示图像数据的概率密度分布图以及对其进行威布尔参数估计的概率分布曲线。如图 7和图 8所示, 由威布尔参数估计所得到的概率分布曲线在一定程度上能够拟合原SAR图像数据的概率密度分布图。
为了进一步证明本文算法的有效性，运用基于威布尔分布模型的CFAR检测算法(Gao，2019)对图 1进行检测，结果如图 9所示。在实验过程中，相应的参数设置为：目标窗口大小为25×25像素，恒虚警率为10－7。与本文检测方法相比，传统的基于CFAR检测算法对海杂波中与目标比较相似的杂波尖峰比较敏感，检查结果往往具有较高的虚警率。
4 结论
本文针对SAR图像舰船目标检测中杂波数据的非均匀性和非平稳性特点，研究信息几何理论应用于SAR图像目标检测领域的切入点，以及基于统计流形几何结构特征融合的图像感兴趣目标显著性表示方法。首先对高斯分布族的费歇耳信息度量引入了尺度调置参数，优化统计流形上各点之间的距离测度，并运用威布尔分布族对SAR图像海杂波进行建模。结合高斯统计流形上的费歇耳信息度量形式，在威布尔统计流形上构建黎曼度量，以归一化的尺度参数和形状参数构造切向量，实现对SAR图像目标的显著性表示。理论分析和实验结果表明，信息几何检测方法有潜力成为超越现有基于传统统计学原理的检测方法，获得更加深刻、更加本质的结论。
然而，由于信息几何理论与应用基础研究需要复杂而抽象的数学知识，并且具有多学科交融性特点，使得相关研究工作者既要具有较强的数理基础，又需要对所研究问题有很深的理解和把握，才能构建合适、有效的切入点。本文只是从局部黎曼度量的角度引入信息几何的处理方法，对统计流形的几何结构并未做更为深入的分析。事实上，信息几何蕴含着丰富的几何结构(如联络、测地线、曲率等)，在诸多应用领域仍处于起步阶段，在这一领域所取得的进步将会是具有开拓意义的研究成果。
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