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.

Universiẗat, Diss., 1999--Kassel. / Lizenzpflichtig.

The Karhunen-Loève transform, which optimally extracts coherent information from multichannel data, has been applied to several problems in reflection seismic data processing. The transform is derived by a least-squares construction of an orthogonal set of principal components and eigenvectors, with corresponding eigenvalues. Data are reconstructed as a linear combination of the principal components. The mathematical properties of the Karhunen-Loève transform which render it applicable to problems in seismic data processing are reviewed, and a number of new algorithms developed. Most algorithms are tested on synthetic and real data examples, and 'production-line' industrially viable versions of some of the programs have been developed. A new signal-to-noise ratio enhancement technique, based on reconstruction of stacked seismic sections, has proved to be successful on real data. Reconstruction of less coherent information to emphasize anomalous features in stacked seismic data ("misfit" reconstruction) shows some promise. Diffraction hyperbolae isolated by misfit reconstruction are used to estimate residual migration velocities with some success. And, the ability of the transform to segregate coherent information is used successfully as the basis of a new multiple suppression technique. An anomaly identification scheme, based on cluster analysis of the eigenvectors of the transform, works well on the synthetic data used, and gives promising results when applied to real data. A new velocity analysis method, utilizing a ratio of the eigenvalues, works well for good data at early travel times, and offers a potential for high resolution velocity inversion studies. Use of the eigenvalues in evaluation of a constant phase approximation to dispersion for synthetic data provides promising results, leading to quantification of dispersion in terms of relative phase shifts. As part of this development, an analysis of the effect of dispersion on Vibroseis© data acquisition, which represents an original investigation, is presented. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

The objective of this study is to characterize the mechanism of drag reduction by comparing the dynamical eigenfunctions of a turbulent pipe flow against those of two known cases of drag reduced flows. The first is forced drag reduction by spanwise wall oscillation, and the second is natural drag reduction found in relaminarizing flow. The dynamics are examined through a Karhunen-Lo`eve (KL) expansion of the direct numerical simulation flow field results. The direct numerical simulation (DNS) is performed using NEK5000, a spectral element Navier-Stokes solver, the first exponentially convergent investigation of DNS of turbulence in a pipe. The base flow is performed at a Reynolds number of Re = 150, resulting in a KL dimension of D_KL = 2130. As in turbulent channel flow, propagating modes are found, characterized with constant phase speed, and contribute of 80.58% of the total fluctuating energy. Based upon wavenumber characteristics and coherent vorticity visualization, four subclasses of propagating modes and two subclasses of non-propagating modes are discovered, qualitatively similar to the horseshoe (hairpin) vortex structure reported in literature. The drag reduced case is performed at the same Reynolds number with a spanwise velocity A+ = 20, a period of T+ = 50, and is driven by a constant pressure gradient. This results in a increase of flow rate by 27 %, and the KL dimension is reduced to D_KL = 102, a 96% reduction. The propagating modes, in particular the wall modes, are pushed away from the wall, resulting in a 34% increase in their advection speed, and a shift away from the wall of the root-mean-square and Reynolds stress peaks. The relaminarizing case observes the chugging motion of the mean flow rate when the Reynolds number is barely turbulent, at Re = 95. This chugging motion is the relaminarization of the flow, resulting in an increased flow rate, and then before complete relaminarization, the flow regains its turbulent state. This occurs because the lift modes, which are responsible for the majority of the energy in the inertial range of the energy spectra, decrease by two or three orders of magnitude. The chugging ends when the wall modes restart the turbulent cascade, and the lift modes are repopulated with energy. A model for the energy path is developed, with energy going from the pressure gradient to the shear modes, then to the roll modes, then to the wall modes, and then finally to the lift modes. It is concluded that drag reduction in a flow can be achieved by disrupting any leg of this model, thus disrupting the self-sustaining mechanism of turbulence. The spanwise wall oscillation shortened the life span of the wall modes, thus limiting their ability to pass energy to the lift modes. Likewise, the low Reynolds number did not provide enough energy to sustain the lift modes, and so relaminarization began. The contribution of this work is twofold. Firstly, the structure of turbulent pipe flow is examined and visualized for the first time using the Karhunen-Lo`eve method. The second, and perhaps greatest contribution of this work, is that the mechanism of drag reduction has been characterized as the link between the wall modes and the lift modes. This will allow future work on developing real methods of drag reduction, and eventually porting it to high Reynolds number flows, like that of an oil pipeline at Re= 40, 000. To achieve this, certain questions remain to be answered, such as what is the most efficient method of disrupting the wall-lift mechanism? Is there a single structure that can be identified and manipulated that gives a similar eect? Once answered, this will allow for a new generation of pipelines to be developed, and considering the implications in petroleum industry alone, will result in a significant contribution to the economy of the world. / Ph. D.

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.

La probabilité de défaillance d’une structure sous séisme est représentée à l’aide de « courbes de fragilité ». Pour les estimer, nous proposons d’enrichir une base de données pré-existante à l’aide du modèle de l’article de F. Poirion et I. Zentner, Stochastic model construction of natural hazards given experimental data, qui se fonde sur la représentation de Karhunen-Loève. Les signaux générés sont triés par classes d’indicateur de nocivité sismique à l’aide d’un algorithme de partitionnement de données. Malgré la ressemblance certaine que présentent plusieurs signaux simulés, et les conséquences que nous tirons de ce problème, ils sont soumis à la structure. Les signaux de réponses résultants sont eux aussi enrichis, en prenant en compte certaines incertitudes afin de construire un intervalle autour de la courbe. La méthode fonctionne pour tout indicateur de nocivité sismique, et permet de s’affranchir de plusieurs hypothèses simplificatrices courantes. Les caractéristiques du scénario sismique sont conservées lors de l’enrichissement, et le processus modélisant le mouvement du sol garde toute sa généralité. Notre démarche est validée d’abord sur un cas simple, puis sur un cas industriel. / The failure probability of a structure under earthquake is represented with « fragility curves ». To estimate them, we propose to enrich a pre-existing data basis using the model of the article Stochastic model construction of natural hazards given experimental data, written by F. Poirion et I. Zentner, which is based on the Karhunen-Loeve expansion. The generated signals are sorted by seismic indicator classes using a data partitioning algorithm. Despite the resemblance between some simulated signals, and the consequences we draw from this problem, the structure is submitted to them. The resulting response signals are also enriched, taking into account uncertainties to construct an interval around the curve. The method works for any seismic indicator, and overcomes several common simplifying assumptions. The characteristics of the seismic scenario are preserved during the enrichment, and the process modeling the ground motion retains its generality. Our approach is first validated on a simple case, then on an industrial case.

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.

Tradicionalmente, a técnica de síntese aditiva com funções base de Fourier é a que tem apresentado melhores resultados no que diz respeito à qualidade dos sons gerados. Entretanto, a carga computacional imposta por esta técnica é extremamente alta, dificultando assim a sua implementação em tempo real. Substituindo as funções de Fourier por outras mais complexas, derivadas das técnicas estatísticas de Karhunen-Loève, consegue-se uma redução na quantidade de operações necessárias. Neste trabalho foi estudado e implementado em microcomputador um modelo de síntese aditiva baseada em análise utilizando a transformada de Karhunen-Loève. / Traditionally, the Fourier based aditive synthesis is the method that have achieved the best results concerning to the quality of the generated sounds. However, the computacional load imposed by this technique is extremely high, difliculting its real time implementation. Substituting the Fourier functions by another set of more complex functions, derived from the Karhunen-Loève statistical techniques, one achieves a reduction on the amount of the necessary operations so as to viabilize its real time implementation.

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.