基于几何代数的太赫兹时域光谱信号分析及物质识别方法研究
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摘要
太赫兹技术是目前科技领域的研究前沿，太赫兹时域光谱（THz-TDS）技术是太赫兹科学技术研究领域中的重要探测技术之一。太赫兹时域光谱(THz-TDS)信号蕴含了表征物质特征指纹的丰富信息，可以给出物质独一无二的指纹光谱，因此可被用于物质识别。如何对THz-TDS信号进行有效表示和处理分析，如何充分利用信号内含的信息有效地进行物质识别等，是THz-TDS技术研究中必须解决的重要课题。
近年来的研究发现很多物质在太赫兹波段没有单独的特征峰，需要采用整个测量波段的THz-TDS光谱作为物质的指纹特征来进行分析处理。整个测量波段的THz-TDS信号可被视为多维矢量空间中的一个多维矢量。为了提取并有效利用THz-TDS信号中蕴含的巨大信息量，迫切需要系统深入地研究和发展THz-TDS信号处理和分析的新理论新方法。为此，本文针对THz-TDS信号特点，将几何代数引入了THz-TDS信号分析和处理的研究中，提出了基于几何代数的THz-TDS信号物质识别的新方法。
本文的研究成果可以概括为如下五部分：
1.基于THz-TDS透射模式系统的物理机理，构建了THz-TDS信号的几何代数框架，提出了THz-TDS信号的几何代数分析方法。该方法充分利用了THz光谱的幅度和相位信息，将整个有效频域内的THz-TDS信号视为高维实矢量空间中的矢量，研究了THz-TDS信号的几何分布特性及代数关系，揭示了物质的光学常数和THz-TDS信号矢量间存在的对应关系。
2.基于THz-TDS信号的几何代数分析研究结果，定义了太赫兹时域光谱信号的外积相似性函数和欧氏距离，推导了这两种相似性度量的具体计算公式，讨论了计算复杂度，提出基于距离(外积相似性函数)的分类方法来实现信号分类。实验验证了这两种度量的有效性和可行性。本文还研究了THz-TDS信号及其梯度特征矢量的联合熵和互信息相似性度量，并与基于几何积的相似性度量(外积相似性函数和欧氏距离)进行了对比实验。实验验证了本文提出的外积相似性函数的优越性。
3.对THz-TDS信号的射影分解方法及其性质进行了研究，还研究了THz-TDS信号矢量在主分量分析变换(PCA)和射影分解变换(PS)下的性质，提出了基于PCA的THz-TDS信号射影分解的维数简约方法，并应用于信号分类和识别。该方法可以有效地实现THz-TDS信号的维数简约和去噪，进行THz-TDS信号的显著特征提取，有助于对THz-TDS信号的分类和辨识。THz-TDS信号的分类实验验证了该方法的有效性，仿真信号的辨识实验表明该方法优于传统的PCA方法，具有较好的抗噪能力。
4.对信号的共形分解变换进行了研究，提出了太赫兹时域光谱信号的共形分解方法，并构建了特征子空间的辨识准则，进而提出了基于共形分解的THz-TDS信号物质辨识方法。该方法对待辨识样品的THz-TDS信号矢量进行基于已知物质的面片的共形分解，将矢量线性映射到各物质对应的二维子空间，在子空间中，采用辨识标准来识别样品的物质。物质辨识实验显示了该方法具有很高的辨识成功率，不仅验证了该方法的可行性，而且与空间图样成份分析(CSPA)方法相比具有优越性。
5.将信号稀疏表示理论和THz-TDS信号几何代数分析的研究结果相结合，对THz-TDS信号矢量进行基于冗余字典的稀疏表示，提出了基于稀疏表示的THz-TDS信号分类方法。基于THz-TDS信号矢量的几何分布和代数结构特性，该方法采用了冗余字典原子的优化选取方法；通过引入噪声因素，修正了信号分类模型；在确定类别标识的问题上，分别提出了以最大系数和最小残差作为分类依据的方法。实验结果验证了该方法在等间隔采样的THz-TDS信号分类问题中的可行性和有效性。
本文的研究结果表明几何代数分析方法在THz-TDS信号的分析和处理研究中具有极大的潜力，将有助于基于THz-TDS信号的物质识别技术的发展和应用。
Terahertz technology is the frontier of the current science and technology studyfields, the terahertz time-domain spectroscopy (THz-TDS) is an important explorationtechnology in the research field of the terahertz science and technology. THz-TDSsignals contain abundant information, and present a unique fingerprint of the material,which can be used for the material identification. How to represent, process and analyzeTHz-TDS signals effectively, and how to utilize the information of signals effectively inthe substance identification, those are the important problems needed to be solved in thestudy of THz-TDS technology.
Recent studies show that there are often no obvious peaks in THz-TDS signals, andit is needed to adopt the entire THz-TDS spectra as the substance fingerprint foranalysis and processing. One THz-TDS signal of the entire spectral band can beregarded as a vector in the multi-dimensional vector space. In order to extract andutilize the tremendous information of THz-TDS signals effectively, there is an urgentdemand to develop new signal processing and analysis methods systematically anddeeply. Therefore, the geometric algebra (GA) is brought into the THz-TDS signalanalysis and processing to the characters of THz-TDS signals in this paper, newsubstance identification methods of THz-TDS signals are developed based on GA.
Specifically, major contributions of this dissertation are as follows:
1. Based on the physical mechanism of the transmission model THz-TDS system,the geometric algebra framework for the THz-TDS signals analysis is constructed, andthe signal analysis method is developed. In the method, both the magnitude and phaseinformation of THz-TDS signals are utilized, and one THz-TDS signal of the entirespectral band is regarded as a vector in the high-dimensional real vector space.Geometrical distribution characteristics and algebraical relations of the THz-TDS signalvectors are studied, which reveal the corresponding relationships between the opticalconstants of the substance and THz-TDS signal vectors.
2. Based on the results of the geometric algebra analysis of THz-TDS signals, anouter product similarity function and the Euclidean distance of the THz-TDS signals aredefined respectively using the geometric product. Detailed computation formulas ofthese two similarity measures are derived, with their computational complexitiesdiscussed. The substance classification method based on the THz-TDS signals based on those metrics is proposed. Experiments on the substance classification verify thefeasibility and validity of the similarity function. Other similarity metrics of THz-TDSsignals and their gradient feature vectors using the joint entropy and mutual informationare studied and compared with the metrics defined by the geometric procduct.Comparative experiment shows the advantages of the outer product similarity function.
3. The projective split of the THz-TDS signal and its properties are studied. Byanalyzing properties of THz-TDS signals’ metrics under the principle componentanalysis (PCA) and PS, a new dimensionality reduction method of THz-TDS signalsbased on PCA is proposed and applied to signal classification and recognition. Thismethod can effectively realize the dimensionality reduction and the signal denoising,facilitate the feature extraction, contribute to the accurate classification andidentification. Experiments on the classification of THz signals not only verify theeffectiveness of our method, but also show its potential ability in the visualization ofTHz signals and images. The simulated experiments on the signals identification alsoshow that our method is better than the traditional PCA and has better anti-noise ability.
4. The conformal split and its properties is studied, the conformal split ofTHz-TDS signal is presented with properties studied. Accordingly, two criteria forsubstance identification are proposed, and a novel substance identification method viathe conformal split is presented. In the method, using the conformal split with respect to2-blades of the “known” substances, one THz-TDS signal is related to vectors insubspaces of substances. The substance can be identified using criteria in the subspaces.The presented methods are applied in experiments of the substance identification, andalso compared with the component spatial pattern analysis (CSPA) method.Experiments show its feasibility and greater accuracy.
5. From the signal sparse representation and the results of the THz-TDS analysis,THz-TDS signal vectors can be sparsely represented in an overcomplete dictionary, anda signal classification method via the sparse representation is presented. In the method,an optimal construction of the overcomplete dictionary is developed based on thegeometrical distribution and the algebraic structure properties of signal vectors, thesignal classification model is modified to account for possibly noise, and the test sampleis classified using either the maximized coefficient or the minimized residual.Feasibility and effectiveness of the method is confirmed by experiments presented.
Study results in this paper show that the geometric algebra analysis method haspotential ability in the THz-TDS signal analysis and processing, and it will help thedevelopment and application of the substance identification technology based on THz-TDS signals.
近年来的研究发现很多物质在太赫兹波段没有单独的特征峰，需要采用整个测量波段的THz-TDS光谱作为物质的指纹特征来进行分析处理。整个测量波段的THz-TDS信号可被视为多维矢量空间中的一个多维矢量。为了提取并有效利用THz-TDS信号中蕴含的巨大信息量，迫切需要系统深入地研究和发展THz-TDS信号处理和分析的新理论新方法。为此，本文针对THz-TDS信号特点，将几何代数引入了THz-TDS信号分析和处理的研究中，提出了基于几何代数的THz-TDS信号物质识别的新方法。
本文的研究成果可以概括为如下五部分：
1.基于THz-TDS透射模式系统的物理机理，构建了THz-TDS信号的几何代数框架，提出了THz-TDS信号的几何代数分析方法。该方法充分利用了THz光谱的幅度和相位信息，将整个有效频域内的THz-TDS信号视为高维实矢量空间中的矢量，研究了THz-TDS信号的几何分布特性及代数关系，揭示了物质的光学常数和THz-TDS信号矢量间存在的对应关系。
2.基于THz-TDS信号的几何代数分析研究结果，定义了太赫兹时域光谱信号的外积相似性函数和欧氏距离，推导了这两种相似性度量的具体计算公式，讨论了计算复杂度，提出基于距离(外积相似性函数)的分类方法来实现信号分类。实验验证了这两种度量的有效性和可行性。本文还研究了THz-TDS信号及其梯度特征矢量的联合熵和互信息相似性度量，并与基于几何积的相似性度量(外积相似性函数和欧氏距离)进行了对比实验。实验验证了本文提出的外积相似性函数的优越性。
3.对THz-TDS信号的射影分解方法及其性质进行了研究，还研究了THz-TDS信号矢量在主分量分析变换(PCA)和射影分解变换(PS)下的性质，提出了基于PCA的THz-TDS信号射影分解的维数简约方法，并应用于信号分类和识别。该方法可以有效地实现THz-TDS信号的维数简约和去噪，进行THz-TDS信号的显著特征提取，有助于对THz-TDS信号的分类和辨识。THz-TDS信号的分类实验验证了该方法的有效性，仿真信号的辨识实验表明该方法优于传统的PCA方法，具有较好的抗噪能力。
4.对信号的共形分解变换进行了研究，提出了太赫兹时域光谱信号的共形分解方法，并构建了特征子空间的辨识准则，进而提出了基于共形分解的THz-TDS信号物质辨识方法。该方法对待辨识样品的THz-TDS信号矢量进行基于已知物质的面片的共形分解，将矢量线性映射到各物质对应的二维子空间，在子空间中，采用辨识标准来识别样品的物质。物质辨识实验显示了该方法具有很高的辨识成功率，不仅验证了该方法的可行性，而且与空间图样成份分析(CSPA)方法相比具有优越性。
5.将信号稀疏表示理论和THz-TDS信号几何代数分析的研究结果相结合，对THz-TDS信号矢量进行基于冗余字典的稀疏表示，提出了基于稀疏表示的THz-TDS信号分类方法。基于THz-TDS信号矢量的几何分布和代数结构特性，该方法采用了冗余字典原子的优化选取方法；通过引入噪声因素，修正了信号分类模型；在确定类别标识的问题上，分别提出了以最大系数和最小残差作为分类依据的方法。实验结果验证了该方法在等间隔采样的THz-TDS信号分类问题中的可行性和有效性。
本文的研究结果表明几何代数分析方法在THz-TDS信号的分析和处理研究中具有极大的潜力，将有助于基于THz-TDS信号的物质识别技术的发展和应用。
Terahertz technology is the frontier of the current science and technology studyfields, the terahertz time-domain spectroscopy (THz-TDS) is an important explorationtechnology in the research field of the terahertz science and technology. THz-TDSsignals contain abundant information, and present a unique fingerprint of the material,which can be used for the material identification. How to represent, process and analyzeTHz-TDS signals effectively, and how to utilize the information of signals effectively inthe substance identification, those are the important problems needed to be solved in thestudy of THz-TDS technology.
Recent studies show that there are often no obvious peaks in THz-TDS signals, andit is needed to adopt the entire THz-TDS spectra as the substance fingerprint foranalysis and processing. One THz-TDS signal of the entire spectral band can beregarded as a vector in the multi-dimensional vector space. In order to extract andutilize the tremendous information of THz-TDS signals effectively, there is an urgentdemand to develop new signal processing and analysis methods systematically anddeeply. Therefore, the geometric algebra (GA) is brought into the THz-TDS signalanalysis and processing to the characters of THz-TDS signals in this paper, newsubstance identification methods of THz-TDS signals are developed based on GA.
Specifically, major contributions of this dissertation are as follows:
1. Based on the physical mechanism of the transmission model THz-TDS system,the geometric algebra framework for the THz-TDS signals analysis is constructed, andthe signal analysis method is developed. In the method, both the magnitude and phaseinformation of THz-TDS signals are utilized, and one THz-TDS signal of the entirespectral band is regarded as a vector in the high-dimensional real vector space.Geometrical distribution characteristics and algebraical relations of the THz-TDS signalvectors are studied, which reveal the corresponding relationships between the opticalconstants of the substance and THz-TDS signal vectors.
2. Based on the results of the geometric algebra analysis of THz-TDS signals, anouter product similarity function and the Euclidean distance of the THz-TDS signals aredefined respectively using the geometric product. Detailed computation formulas ofthese two similarity measures are derived, with their computational complexitiesdiscussed. The substance classification method based on the THz-TDS signals based on those metrics is proposed. Experiments on the substance classification verify thefeasibility and validity of the similarity function. Other similarity metrics of THz-TDSsignals and their gradient feature vectors using the joint entropy and mutual informationare studied and compared with the metrics defined by the geometric procduct.Comparative experiment shows the advantages of the outer product similarity function.
3. The projective split of the THz-TDS signal and its properties are studied. Byanalyzing properties of THz-TDS signals’ metrics under the principle componentanalysis (PCA) and PS, a new dimensionality reduction method of THz-TDS signalsbased on PCA is proposed and applied to signal classification and recognition. Thismethod can effectively realize the dimensionality reduction and the signal denoising,facilitate the feature extraction, contribute to the accurate classification andidentification. Experiments on the classification of THz signals not only verify theeffectiveness of our method, but also show its potential ability in the visualization ofTHz signals and images. The simulated experiments on the signals identification alsoshow that our method is better than the traditional PCA and has better anti-noise ability.
4. The conformal split and its properties is studied, the conformal split ofTHz-TDS signal is presented with properties studied. Accordingly, two criteria forsubstance identification are proposed, and a novel substance identification method viathe conformal split is presented. In the method, using the conformal split with respect to2-blades of the “known” substances, one THz-TDS signal is related to vectors insubspaces of substances. The substance can be identified using criteria in the subspaces.The presented methods are applied in experiments of the substance identification, andalso compared with the component spatial pattern analysis (CSPA) method.Experiments show its feasibility and greater accuracy.
5. From the signal sparse representation and the results of the THz-TDS analysis,THz-TDS signal vectors can be sparsely represented in an overcomplete dictionary, anda signal classification method via the sparse representation is presented. In the method,an optimal construction of the overcomplete dictionary is developed based on thegeometrical distribution and the algebraic structure properties of signal vectors, thesignal classification model is modified to account for possibly noise, and the test sampleis classified using either the maximized coefficient or the minimized residual.Feasibility and effectiveness of the method is confirmed by experiments presented.
Study results in this paper show that the geometric algebra analysis method haspotential ability in the THz-TDS signal analysis and processing, and it will help thedevelopment and application of the substance identification technology based on THz-TDS signals.
引文
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