Covariance and Gramian matrices in control and systems theory and pattern recognition are studied in the context of reduction of dimensionality and hence complexity of large-scale systems. This is achieved by the removal of redundant or 'almost' redundant information contained in the covariance and Grarrdan matrices. The Karhunen-Loeve expansion (principal component analysis) and its extensions and the singular value decomposition of matrices provide the framework for the work presented in the thesis. The results given for linear dynamical systems are based on controllability and observability Gramians and some new developments in singular perturbational analysis are also presented.

This degree project deals with Wavelet transform and Karhunen-Loeve transform. Through the mathematic description to understand and simulation to investigate the denoise ability of WT and the de-correlation ability of KLT. Mainly prove that the new algorithm which is the joint of these two algorithms is feasible.

Abstract In this dissertation, person identification systems based on Karhunen-Loeve transform (KLT) are investigated. Both speaker and face recognition are considered in our design. Among many aspects of the system design issues, three important problems: how to improve the correct classification rate, how to reduce the computational cost and how to increase the robustness property of the system, are addressed in this thesis. Improvement of the correct classification rate and reduction of the computational cost for the person identification system can be accomplished by appropriate feature design methodology. KLT and hard-limited KLT (HLKLT) are proposed here to extract class related features. Theoretically, KLT is the optimal transform in minimum mean square error and maximal energy packing sense. The transformed data is totally uncorrelated and it contains most of the classification information in the first few coordinates. Therefore, satisfactory correct classification rate can be achieved by using only the first few KLT derived eigenfeatures. In the above data transformation process, the transformed data is calculated from the inner products of the original samples and the selected eigenvectors. The computation is of course floating point arithmetic. If this linear transformation process can be further reduced to integer arithmetic, the time used for both person feature training and person classification will be greatly reduced. The hard-limiting process (HLKLT) here is used to extract the zero-crossing information in the eigenvectors, which is hypothesized to contain important information that can be used for classification. This kind of feature tremendously simplifies the linear transformation process since the computation is merely integer arithmetic. In this thesis, it is demonstrated that the hard-limited KL transform has much simpler structure than that of the KL transform and it possess approximately the same excellent performances for both speaker identification system and face recognition system. Moreover, a hybrid KLT/GMM speaker identification system is proposed in this thesis to improve classification rate and to save computational time. The increase of the correct rate comes from the fact that two different sets of speech features, one from the KLT features, the other from the MFCC features of the Gaussian mixture speaker model (GMM), are applied in the hybrid system. Furthermore, this hybrid system performs classification in a sequential manner. In the first stage, the relatively faster KLT features are used as the initial candidate selection tool to discard those speakers with larger separability. Then in the second stage, the GMM is utilized as the final speaker recognition means to make the ultimate decision. Therefore, only a small portion of the speakers needed to be discriminated in the time-consuming GMM stage. Our results show that the combination is beneficial to both classification accuracy and computational cost. The above hybrid KLT/GMM design is also applied to a robust speaker identification system. Under both additive white Gaussian noise (AWGN) and car noise environments, it is demonstrated that accuracy improvement and computational saving compared to the conventional GMM model can be achieved. Genetic algorithm (GA) is proposed in this thesis to improve the speaker identification performance of the vector quantizer (VQ) by avoiding typical local minima incurred in the LBG process. The results indicates that this scheme is useful for our application on recognition and practice.

The Karhunen-Loeve Transform (KLT) is a popular transform used in multiple image processing scenarios. Sometimes, the application of the KLT is not carried out as a single transform over an entire image Rather, the image is divided into smaller spatial regions (segments), each of which is transformed by a smaller dimensional KLT. Such a situation may penalize the transform efficiency. An improvement for the segmented KLT, aiming at mitigating discontinuities arising on the edge of adjacent regions, is proposed in this paper. In the case of moderately varying image regions, discontinuities occur as the consequence of disregarded similarity between transform domains, as the order and sign of eigenvectors in the transform matrices are mismatched. In the proposed method, the KLT is adjusted to guarantee the best achievable similarity via the optimal assignment and sign correspondence for eigenvectors. Experimental results indicate that the proposed transform improves the similarity between transform domains, and reduces RMSE on the edge of adjacent regions. In consequence, images processed by the adjusted KLT present better cohesion and continuity between independently transformed regions.

The objective of this thesis is to increase the correct recognition rate of the two-word Mandarin phrases. The reason for inaccuracy is due to the ambiguities of the syllables and the intonations. For the syllable ambiguity, a balanced speech training dataset is designed and the weights of the state observation probabilities on vowels and consonants are adjusted. For the tone ambiguity, both the pitch contour and the spectrum evolution property derived from the Karhunen-Loéve transform are applied. The experimental results indicate that an 85% correct rate can be achieved, that is a 6% increase in the performance for the system without the above improvements.

The design of a face recognition system ( FRS ) can been separated into two major modules ¡V face detection and face recognition. In the face detection part, we combine image pre-processing techniques with maximum-likelihood estimation to detect the nearest frontal face in a single image. Under limited restrictions, our detection method overcomes some of the challenging tasks, such as variability in scale, location, orientation, facial expression, occlusion ( glasses ), and lighting change. In the face recognition part, we use both Karhunen-Loeve transform and linear discrimant analysis ( LDA ) to perform feature extraction. In this feature extraction process, the features are calculated from the inner products of the original samples and the selected eigenvectors. In general, as the size of the face database is increased, the recognition time will be proportionally increased. To solve this problem, hard-limited Karhunen-Loeve transform ( HLKLT ) is applied to reduce the computation time in our FRS.

[pt] Nesta tese, as vibrações não-lineares e a estabilidade de uma casca cilíndrica contendo um fluido são estudadas com base em modelos de dimensão reduzida, isto é, modelos com um número reduzido de graus de liberdade. A partir dos funcionais de energia potencial e cinética de uma casca cilíndrica, deduzem-se suas equações de movimento. O campo de deformações da casca cilíndrica segue a teoria não- linear de Donnell para cascas abatidas. O fluido é considerado interno à casca irrotacional, não-viscoso e incompressível, sendo descrito a partir de um potencial de velocidade que leva em consideração a interação entre o fluido e a estrutura. Para resolver o sistema de equações de equilíbrio da casca, desenvolve-se um procedimento analítico que permite obter os campos de deslocamento axial e circunferencial em função dos deslocamentos laterais, além de atender as condições de contorno do problema. Desta forma, reduz-se o sistema de equações de equilíbrio a uma única equação diferencial parcial que é resolvida com o método de Galerkin. A determinação dos deslocamentos laterais é feita a partir de técnicas de perturbação que ordena os modos não-lineares de acordo com sua importância na solução da casca cilíndrica. Comprova-se essa ordenação através do método de Karhunen-Loève que fornece, também, uma expansão ótima para os deslocamentos laterais. Além dessas técnicas, apresenta-se uma redução polinomial que relacionam as amplitudes dos modos não-lineares com a amplitude do modo linear, criando uma expansão modal com 1 GDL. Apresentam-se respostas no tempo, fronteiras de instabilidade e diagramas de bifurcação para uma casca cilíndrica submetida a dois tipos de carregamentos harmônicos, pressão lateral e carga axial. A seguir, são propostos alguns critérios para a análise da a integridade do sistema dinâmico tanto para um sistema com 1 GDL quanto para um sistema multidimensional através da evolução e erosão das bacias de atração. Por fim, estuda-se o comportamento de cascas cilíndricas parcialmente cheias, mostrando a influência da altura do fluido nas fronteiras de instabilidade e curvas de ressonância da casca cilíndrica. / [en] The nonlinear vibrations and stability of a fluid-filled cylindrical shell is investigated using reduced order models. First, the nonlinear equations of motion of the cylindrical shell are deduced based on the expressions for the potential and kinetic energy, which are obtained using Donnell shallow shell theory. The internal fluid is considered to be irrotational, non- viscous and incompressible. It is described by a velocity potential that takes into account the fluid-shell interaction. A procedure is proposed to obtain analytically the axial and circumferential displacements of the shell, satisfying the in-plane equations of motion and the associated boundary conditions. So, the problem is reduced to one partial differential equation of motion which is solved by the Galerkin method. The transversal displacement field is obtained by perturbation techniques. This enables one to identify the relevance of each term in the nonlinear expansion of the vibration modes. Then, the Karhunen-Loève method is employed to investigate de relative importance of each mode obtained by the perturbation analysis on the nonlinear response and to deduce optimal interpolation function to be used in the Galerkin procedure. A SDOF model is also obtained by relating the modal amplitudes of the nonlinear modes to the vibration amplitude of the linear mode. Time responses, instability boundaries and ifurcation diagrams are obtained for cylindrical shells subjected to harmonic lateral and axial loads. Different procedures for the analysis of the shell integrity are proposed based on the evolution and erosion of the basins of attraction in state-space. Finally, the influence of the fluid height on the stability boundaries and resonance curves is studied.

[pt] Neste trabalho a Análise de Vibrações é tratada no contexto da formulação fraca. Um sistema contínuo é formulado abstratamente em um espaço de Hilbert e uma base de projeção é escolhida para a dinâmica. Um esquema de convergência para a aproximação é garantido à medida em que se aumenta o número de funções da base usada para representar a resposta do problema. Esta é a idéia por traz de métodos como o Método dos Elementos Finitos e o Método dos Modos Supostos, que derivam do Método de Galerkin. Esta estratégia é diferente do que comumente é ensinado nos cursos de vibrações, onde um sistema massa-mola é analisado, e sistemas discretos formados por massas, molas e amortecedores são discutidos. Nestes casos não se sabe qual é o erro cometido na análise numérica. A Análise de Vibrações é muito usada na manutenção preditiva de máquinas rotativas. Alguns fenômenos observados nesses equipamentos motivaram o desenvolvimento de um modelo numérico que pudesse reproduzir tais fenômenos para melhor entendê-los. Um sistema rotor-mancal é modelado e sua resposta dinâmica comparada qualitativamente com a resposta dinâmica captada através de acelerômetros fixados nos mancais de um exaustor da Companhia Siderúrgica de Tubarão (CST). Durante o trabalho diversos programas foram desenvolvidos através da plataforma MATLAB. / [en] Vibration Analysis is treated in the context of weak formulation. A continuous system is formulated in the Hilbert space and one base is selected to project the dynamics. An approximation scheme is guaranteed by increasing the number of functions in the base used to represent the response. This is the idea behind methods like the Finite Element Method and Assumed Modes Method, which derive from Galerkin Method. This strategy is different from what is commonly taught in vibration courses, where a mass-spring system is analyzed and discrete systems composed by masses, springs and dashpots are discussed. In those cases the error of the numerical analysis is not known. Vibration Analysis is very used in predictive maintenance of rotating machines. Some phenomenons observed in those machines motivated the development of a numerical model that could reproduce such phenomenons to better understand them. A rotor-bearing system is modelled and its dynamic response is qualitative compared to the dynamic response captured by accelerometers fixed on the bearings of a blower of the steel company Companhia Siderúrgica de Tubarão (CST). During this work several programs were developed using MATLAB software.

L'analyse en composantes principales et l'analyse des correspondances sont généralisées pour étudier des ensembles de trajectoires de processus numériques ou qualitatifs. On étudie également certains problèmes de régression avec une infinité non dénombrable de variables: régression sur un processus, régression d'un processus sur un autre.

A speech recognition system for Chinese names based on Karhunen Loeve transform (KLT), MFCC, hidden Markov model (HMM) and Viterbi algorithm is proposed in this thesis. KLT is the optimal transform in minimum mean square error and maximal energy packing sense to reduce data. HMM is a stochastic approach which characterizes many of the variability in speech signal by recording the state transitions. For the speaker-dependent case, the correct identification rate can be achieved 93.97% within 3 seconds in the laboratory environment.