Tesi sul tema "Quaternion analysis"

A Wearable Motion Analysis System (WMAS) was developed to evaluate gait, particularly parameters that are indicative of mild traumatic brain injury. The WMAS consisted on six Opal IMUs attached on the sternum, waist, left and right thigh and left and right shank. Algorithms were developed to calculate the knee flexion angle, stride length and cadence parameters during slow, normal and fast gait speeds. The WMAS was validated for repeatability using a robotic arm and accuracy using the Vicon motion capture system, the gold standard for gait analysis. The WMAS calculated the gait parameters to within a clinically acceptable range and is a powerful tool for gait analysis and potential concussion diagnosis outside of a laboratory setting.

Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt
The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable

En recherche d’information, les documents sont le plus souvent considérés comme des "sacs-de-mots". Ce modèle ne tient pas compte de la structure temporelle du document et est sensible aux bruits qui peuvent altérer la forme lexicale. Ces bruits peuvent être produits par différentes sources : forme peu contrôlée des messages des sites de micro-blogging, messages vocaux dont la transcription automatique contient des erreurs, variabilités lexicales et grammaticales dans les forums du Web. . . Le travail présenté dans cette thèse s’intéresse au problème de la représentation de documents issus de sources bruitées.La thèse comporte trois parties dans lesquelles différentes représentations des contenus sont proposées. La première partie compare une représentation classique utilisant la fréquence des mots à une représentation de haut-niveau s’appuyant sur un espace de thèmes. Cette abstraction du contenu permet de limiter l’altération de la forme de surface du document bruité en le représentant par un ensemble de caractéristiques de haut-niveau. Nos expériences confirment que cette projection dans un espace de thèmes permet d’améliorer les résultats obtenus sur diverses tâches de recherche d’information en comparaison d’une représentation plus classique utilisant la fréquence des mots.Le problème majeur d’une telle représentation est qu’elle est fondée sur un espace de thèmes dont les paramètres sont choisis empiriquement.La deuxième partie décrit une nouvelle représentation s’appuyant sur des espaces multiples et permettant de résoudre trois problèmes majeurs : la proximité des sujets traités dans le document, le choix difficile des paramètres du modèle de thèmes ainsi que la robustesse de la représentation. Partant de l’idée qu’une seule représentation des contenus ne peut pas capturer l’ensemble des informations utiles, nous proposons d’augmenter le nombre de vues sur un même document. Cette multiplication des vues permet de générer des observations "artificielles" qui contiennent des fragments de l’information utile. Une première expérience a validé cette approche multi-vues de la représentation de textes bruités. Elle a cependant l’inconvénient d’être très volumineuse,redondante, et de contenir une variabilité additionnelle liée à la diversité des vues. Dans un deuxième temps, nous proposons une méthode s’appuyant sur l’analyse factorielle pour fusionner les vues multiples et obtenir une nouvelle représentation robuste,de dimension réduite, ne contenant que la partie "utile" du document tout en réduisant les variabilités "parasites". Lors d’une tâche de catégorisation de conversations,ce processus de compression a confirmé qu’il permettait d’augmenter la robustesse de la représentation du document bruité.Cependant, lors de l’élaboration des espaces de thèmes, le document reste considéré comme un "sac-de-mots" alors que plusieurs études montrent que la position d’un terme au sein du document est importante. Une représentation tenant compte de cette structure temporelle du document est proposée dans la troisième partie. Cette représentation s’appuie sur les nombres hyper-complexes de dimension appelés quaternions. Nos expériences menées sur une tâche de catégorisation ont montré l’efficacité de cette méthode comparativement aux représentations classiques en "sacs-de-mots"
In the Information Retrieval field, documents are usually considered as a "bagof-words". This model does not take into account the temporal structure of thedocument and is sensitive to noises which can alter its lexical form. These noisescan be produced by different sources : uncontrolled form of documents in microbloggingplatforms, automatic transcription of speech documents which are errorprone,lexical and grammatical variabilities in Web forums. . . The work presented inthis thesis addresses issues related to document representations from noisy sources.The thesis consists of three parts in which different representations of content areavailable. The first one compares a classical representation based on a term-frequencyrepresentation to a higher level representation based on a topic space. The abstractionof the document content allows us to limit the alteration of the noisy document byrepresenting its content with a set of high-level features. Our experiments confirm thatmapping a noisy document into a topic space allows us to improve the results obtainedduring different information retrieval tasks compared to a classical approach based onterm frequency. The major problem with such a high-level representation is that it isbased on a space theme whose parameters are chosen empirically.The second part presents a novel representation based on multiple topic spaces thatallow us to solve three main problems : the closeness of the subjects discussed in thedocument, the tricky choice of the "right" values of the topic space parameters and therobustness of the topic-based representation. Based on the idea that a single representationof the contents cannot capture all the relevant information, we propose to increasethe number of views on a single document. This multiplication of views generates "artificial"observations that contain fragments of useful information. The first experimentvalidated the multi-view approach to represent noisy texts. However, it has the disadvantageof being very large and redundant and of containing additional variability associatedwith the diversity of views. In the second step, we propose a method based onfactor analysis to compact the different views and to obtain a new robust representationof low dimension which contains only the informative part of the document whilethe noisy variabilities are compensated. During a dialogue classification task, the compressionprocess confirmed that this compact representation allows us to improve therobustness of noisy document representation.Nonetheless, during the learning process of topic spaces, the document is consideredas a "bag-of-words" while many studies have showed that the word position in a7document is useful. A representation which takes into account the temporal structureof the document based on hyper-complex numbers is proposed in the third part. Thisrepresentation is based on the hyper-complex numbers of dimension four named quaternions.Our experiments on a classification task have showed the effectiveness of theproposed approach compared to a conventional "bag-of-words" representation

Les formes automorphes sont des objets centraux en théorie des nombres. En dépit de leur omniprésence, elles demeurent mystérieuses et leur comportement est loin d'être entièrement compris. Considérer ces formes automorphes au sein de familles a un effet régularisant, et ouvre la voie aux résultats en moyenne : voilà l'esprit des statistiques arithmétiques. La famille de toutes les représentations automorphes d'un groupe réductif donné, appelée famille universelle du groupe, est particulièrement importante. Dans le cas des formes intérieures de GL(2), autrement dit les groupes d'unités d'algèbres de quaternions, la formule des traces de Selberg est une puissante méthode d'approche. Il existe une notion de taille sur les formes automorphes, le conducteur analytique, permettant de tronquer la famille universelle en un ensemble fini pour lequel ces problèmes de statistiques arithmétiques ont un sens.Une loi de comptage pour la famille universelle tronquée est établie, avec un terme d'erreur gagnant par une puissance dans le cas totalement défini, et une constante à forte teneur géométrique. Cette loi de Weyl est généralisée en un résultat d'équirépartition par rapport à une mesure explicite, et mène à vérifier les conjectures de Sato-Tate dans ce cadre. Des statistiques sur les petits zéros des fonctions L associées sont établies, menant à dévoiler partiellement le type de symétrie des algèbres de quaternions.Plusieurs indices sont mentionnés laissant à croire que d'autres groupes sont abordables par les mêmes méthodes, et les lois de comptage conjecturales pour certains groupes unitaires et symplectiques de petits rangs sont énoncées
Automorphic forms are central objects in modern number theory. Despite their ubiquity, they remain mysterious and their behavior is far from understood. Embedding them in wider families has a smoothing effect, allowing results on average: these are the aims of arithmetic statistics. The whole family of automorphic representations of a given reductive group, referred to as its universal family, is of fundamental importance. In the case of inner forms of GL(2), that is to say groups of units of quaternion algebras, the Selberg trace formula is a powerful method to handle it. There is a way to define a suitable notion of size, the analytic conductor, allowing to truncate the universal family to a finite one amenable to arithmetic statistics methods. A counting law for the truncated universal family is established, with a power savings error term in the totally definite case and a geometrically meaningful constant. This Weyl's law is generalized to an equidistribution result with respect to an explicit measure, and leads to answer the Sato-Tate conjectures in this case. Statistics on low-lying zeros are provided, leading to uncover part of the type of symmetry of quaternion algebras.Strong evidence is provided that further ground groups should be amenable as well to the same methods and conjectural counting laws are given in the case of symplectic and unitary groups of low ranks

Les signaux bivariés apparaissent dans de nombreuses applications (optique, sismologie, océanographie, EEG, etc.) dès lors que l'analyse jointe de deux signaux réels est nécessaire. Les signaux bivariés simples ont une interprétation naturelle sous la forme d'une ellipse dont les propriétés (taille, forme, orientation) peuvent évoluer dans le temps. Cette propriété géométrique correspondant à la notion de polarisation en physique est fondamentale pour la compréhension et l'analyse des signaux bivariés. Les approches existantes n'apportent cependant pas de description directe des signaux bivariés ou des opérations de filtrage en termes de polarisation. Cette thèse répond à cette limitation par l'introduction d'une nouvelle approche générique pour l'analyse et le filtrage des signaux bivariés. Celle-ci repose sur deux ingrédients essentiels : (i) le plongement naturel des signaux bivariés -- vus comme signaux à valeurs complexes -- dans le corps des quaternions H et (ii) la définition d'une transformée de Fourier quaternionique associée pour une représentation spectrale interprétable de ces signaux. L'approche proposée permet de définir les outils de traitement de signal usuels tels que la notion de densité spectrale, de filtrage linéaire ou encore de spectrogramme ayant une interprétation directe en termes d'attributs de polarisation. Nous montrons la validité de l'approche grâce à des garanties mathématiques et une implémentation numériquement efficace des outils proposés. Diverses expériences numériques illustrent l'approche. En particulier, nous démontrons son potentiel pour la caractérisation de la polarisation des ondes gravitationnelles
Bivariate signals appear in a broad range of applications (optics, seismology, oceanography, EEG, etc.) where the joint analysis of two real-valued signals is required. Simple bivariate signals take the form of an ellipse, whose properties (size, shape, orientation) may evolve with time. This geometric feature of bivariate signals has a natural physical interpretation called polarization. This notion is fundamental to the analysis and understanding of bivariate signals. However, existing approaches do not provide straightforward descriptions of bivariate signals or filtering operations in terms of polarization or ellipse properties. To this purpose, this thesis introduces a new and generic approach for the analysis and filtering of bivariate signals. It essentially relies on two key ingredients: (i) the natural embedding of bivariate signals -- viewed as complex-valued signals -- into the set of quaternions H and (ii) the definition of a dedicated quaternion Fourier transform to enable a meaningful spectral representation of bivariate signals. The proposed approach features the definition of standard signal processing quantities such as spectral densities, linear time-invariant filters or spectrograms that are directly interpretable in terms of polarization attributes. More importantly, the framework does not sacrifice any mathematical guarantee and the newly introduced tools admit computationally fast implementations. Numerical experiments support throughout our theoretical developments. We also demonstrate the potential of the approach for the nonparametric characterization of the polarization of gravitational waves

Thesis (Master)--İzmir Institute of Technology, İzmir, 2006.
Keywords: Biokinematic analysis, quaternionlar, structural groups, mechanism theory,robot manipulators. Includes bibliographical references (leaves 78-85).

The geometric properties of the complex numbers and the quaternions, particularly their behavior under fractional linear transformations, make them very useful for modeling certain types of geometric objects. In this thesis, we will connect the characteristics of fractional linear transformations of both the complex numbers and quaternions to the problem of developing and modifying discrete surfaces for problems in computer graphics and engineering.

Master's thesis is dealing with desgin of a measuring unit incorporating inertial sensors, used for analysis in canoe sprint. Data from a three-axis accelerometer and a three-axis gyroscope were combined using an extended Kalman filter, yielding speed, roll, pitch and yaw of the boat and stroke rate. Calculated values were verified by a GPS. Furthermore, parameters describing dynamic behaviour of the system were identified, allowing an inclusion of dynamic quantities like force and power into the analysis.

The master’s thesis deals with a design of the computational method for the analysis of swimmers training with the use of an inertial measurement unit. The developed algorithm uses quaternion-based Unscented Kalman filter and merges accelerometer and gyroscope measurements. The proposed method enables analysis of velocity, acceleration and inclination of a swimmer. Verification of the method was based on an underwater video camera capturing and a tethered velocity meter.

Approved for public release; distribution unlimited
This thesis done in cooperation with the MOVES Institute
Mathematicians have used quaternions for about a hundred years. Today they are an important part of computer graphics and simulation systems. This thesis takes an analytical approach to quaternions by using them in the construction of a virtual human for sourceless Magnetic Accelerometer Rate Sensor (MARG) body tracking system. Virtual citizens will be a reflection of our personalities in cyberspace. Prophecies say they may take control in the virtual world and govern themselves too. One of the objectives of this thesis is to design a seamless and realistic humanoid from laser scan data clouds. This humanoid will be compatible with motion capture systems and networked virtual environments. Second objective of this thesis is to search for the answers related to the optimal real-time representation of an articulated virtual human, maintaining a high level of visual fidelity within networked cyberspace. While visual detail and fidelity have been and will continue to be a major ongoing interest within the computer graphics community, the idea of sourceless body tracking is still in its early stages. MARG body tracking is one of the successful approaches to body tracking systems. This thesis proposes a networked quaternion based real-time virtual human interface for the MARG body tracking system.

Doutoramento em Matemática e Aplicações (PDMA)
Compressed sensing is a new paradigm in signal processing which states that for certain matrices sparse representations can be obtained by a simple l1-minimization. In this thesis we explore this paradigm for higher-dimensional signal. In particular three cases are being studied: signals taking values in a bicomplex algebra, quaternionic signals, and complex signals which are representable by a nonlinear Fourier basis, a so-called Takenaka-Malmquist system.
Amostragem Compressiva é um novo paradigma em processamento de sinal, no qual se assegura, para determinadas matrizes, que as representações esparsas de sinais podem ser obtidas por intermédio de um simples procedimento de l1-minimização. Nesta tese, exploramos este paradigma para sinais em dimensões superiores. Estudaremos três casos particulares: sinais com valores na álgebra bi-complexa, sinais quaterniónicos e, finalmente, sinais complexos representáveis por uma base de Fourier não-linear, dito sistema de Takenaka-Malmquist.

Orientador: Waldyr Alves Rodrigues Junior
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica
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Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto
Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book
Mestrado
Ágebra
Mestre em Matemática

De nombreux algorithmes de traitement d'image numérique (compression, restauration, analyse) sont basés sur une représentation en ondelettes. Les outils mathématiques disponibles étant souvent pensés pour des signaux 1D à valeurs scalaires (comme le son), ils sont mal adaptés aux signaux 2D vectoriels comme les images couleur. Les méthodes les plus répandues utilisent ces outils indépendamment sur chaque ligne et chaque colonne (méthodes " séparables "), de chaque plan couleur (méthode " marginale ") de l'image. Ces techniques trop simples ne donnent pas accès aux informations visuelles élémentaires, aboutissant à des traitements qui risquent d'introduire des artefacts rectangulaires et de fausses couleurs. Notre axe de recherche se situe autour des représentations analytiques qui utilisent un modèle oscillatoire des signaux. Ces outils de traitement du signal sont connus pour être bien adaptés à la perception humaine (auditive et visuelle), et leur extension à des dimensions supérieures est un sujet encore très actif, qui révèle des propriétés intéressantes pour l'analyse de la géométrie locale. Dans cette thèse, nous faisons une revue des ondelettes analytiques existantes pour l'image en niveaux de gris (dites complexes, quaternioniques et monogènes), et nous proposons des expérimentations qui valident leur intérêt pratique. Nous définissons ensuite une extension vectorielle qui permet de manipuler facilement le contenu géométrique d'une image couleur, ce que nous validons à travers des expérimentations en codage et analyse d'image.

博士
輔仁大學
應用科學與工程研究所博士班
105
Quaternion Heisenberg groups can be referred as subRiemannian manifolds, of which, solving heat equations and characterizing vector fields being conservative are main topics to be explored in this work. Using the Hamiltonian formalism, an integral representation for the heat kernel of a subLaplacian is obtained. Lengths and formulas of geodesics are also calculated for some certain circumstances. On the other hand, a necessary and sufficient condition for a vector field being conservative is deduced. In which case, the corresponding potential function can be written out explicitly.

碩士
國立暨南國際大學
電機工程學系
100
With the development of science and technology, the technology provide users with more convenience and reliability. The biometric technology such as the face recognition, voice recognition, fingerprint recognition, palm print recognition and iris recognition are widely used. Pattern recognition is important in computer vision. One of the important method is that the translation, scaling, rotation remain unchanged in moments invariant. In 1961, Hu proposed seven moments invariant for Pattern recognition and have been used widely. In the thesis, the iris recognition system uses the quaternion with moments invariant and discuss with the invariant of replacing the kernel function. The experimental results reveal that the improved quaternion moments invariant can be applied to iris recognition system. Although the preliminary experimental results are not satisfactory. However, When the improved quaternion moments compared to traditional methods, the reduce the a pretreatment steps think that strengthening the image preprocessing module or classification identification module replacement can enhance the recognition rate.

The present dissertation develops an invariant framework for 3D gesture comparison studies. 3D gesture comparison without Lagrangian models is challenging not only because of the lack of prediction provided by physics, but also because of a dual geometry representation, spatial dimensionality and non-linearity associated to 3D-kinematics. In 3D spaces, it is difficult to compare curves without an alignment operator since it is likely that discrete curves are not synchronized and do not share a common point in space. One has to assume that each and every single trajectory in the space is unique. The common answer is to assert the similitude between two or more trajectories as estimating an average distance error from the aligned curves, provided that the alignment operator is found. In order to avoid the alignment problem, the method uses differential geometry for position and orientation curves. Differential geometry not only reduces the spatial dimensionality but also achieves view invariance. However, the nonlinear signatures may be unbounded or singular. Yet, it is shown that pattern recognition between intrinsic signatures using correlations is robust for position and orientation alike. A new mapping for orientation sequences is introduced in order to treat quaternion and Euclidean intrinsic signatures alike. The new mapping projects a 4D-hyper-sphere for orientations onto a 3D-Euclidean volume. The projection uses the quaternion invariant distance to map rotation sequences into 3D-Euclidean curves. However, quaternion spaces are sectional discrete spaces. The significance is that continuous rotation functions can be only approximated for small angles. Rotation sequences with large angle variations can only be interpolated in discrete sections. The current dissertation introduces two multi-scale approaches that improve numerical stability and bound the signal energy content of the intrinsic signatures. The first is a multilevel least squares curve fitting method similar to Haar wavelet. The second is a geodesic distance anisotropic kernel filter. The methodology testing is carried out on 3D-gestures for obstetrics training. The study quantitatively assess the process of skill acquisition and transfer of manipulating obstetric forceps gestures. The results show that the multi-scale correlations with intrinsic signatures track and evaluate gesture differences between experts and trainees.

碩士
中華大學
應用數學學系(所)
96
This thesis mainly studies Denavit-Hartenberg coordinate and varies the method and Quaternion coordinate and varies the method, come calculator device people insert claw extreme point coordinate and posture relation of space of component separately, analyze its two kinds of methods' pluses and minuses, and apply two kinds of methods mechanically to probe into the tow-armed robot that we research and develop.

Research Doctorate - Doctor of Philosophy (PhD)
Fourier analysis has long been studied as a method to analyse real-valued or complex-valued signals. The Clifford-Fourier transform recently developed by Brackx, De Schepper, and Sommen has led to the development of Fourier analytic methods for hypercomplex or Clifford-valued signals. In the quaternionic case, Brackx et al. have found the kernel of the Quaternionic Fourier transform which allows for much easier calculation, and we focus much of our attention in this thesis on the quaternionic case. We define the continuous wavelet transform of quaternion-valued signals on the plane and prove a Calderón reproducing formula. We also define the monogenic signal, a generalization of the analytic signal of a function on the real line. We provide a characterization of translation-invariant operators and submodules of the quaternionic L₂ module. We develop several fundamental analogues of classical orthogonal wavelet theory pioneered by Cohen, Daubechies, Mallat, and Meyer to quaternion-valued functions on the plane. We include design conditions required to produce wavelets which have compact support and desired regularity. We also develop the basic theory needed for constructing a biorthogonal wavelet basis and construct an example. For a general Clifford algebra, we develop a condition on f so that f*g satisfies a convolution theorem. We also develop a Clifford-Fourier characterization of the Clifford-valued Hardy spaces on ℝ^d.

D.Ing.
Aerial Vehicle (UAV)-based observation system, by using the principles of strapdown inertial measurement and navigation systems. Effort is concentrated around the mathematical implementation thereof and analysis and proof of the concept in a computer simulation environment. Although the principles of the strapdown system approach to camera-to-target vector estimation are universal to any type of airborne platform that can carry the observation payload, the application thereof is specifically tailored for a UAV system. More specifically, the operational scenario and UAV parameters of a typical close-range UAV system that is used for artillery observation, is used in the derivation of the models and equations. The secondary objective of this research is to derive a realizable mathematical implementation for this strapdown system based camera-to-target vector estimation methodology, by performing a systematic tradeoff between the use of Euler angles and quaternions for describing the camera-to-target vector, and by incorporating the principles of Kalman filtering. This dissertation fully describes the approach that was followed in the derivation of the strapdown system equations for the camera-to-target vector estimation. The mathematical models and principles used are universal for any airborne targeting application with a real-time video down-link. The results as presented in this dissertation, prove that the methodology provides satisfactory results in both a pure digital computer simulation environment, as well as in a digital computer simulation that is hybridized with experimentally determined sensor outputs. It has led to a realizable and workable implementation that could form the basis of practical implementation thereof in operational targeting systems. It further proves that the slant range between a camera and a stationary target on the ground, can be estimated effectively without the use of a laser rangefinder.

Research Doctorate - Doctor of Philosophy (PhD)
Fourier Analysis is a primary technique in the analysis of images, yet it has several limitations when it comes to the higher dimensional case of colour images. This thesis seeks to address some of these limitations through two main areas. First, we consider the recently developed Clifford-Fourier Transform of Brackx et al, which has the advantage over the classical Fourier Transform of combining the different channels of a colour image. We characterise the Hardy Spaces of this transform and show that functions in these Hardy spaces have monogenic extensions with bounded integral averages. We also characterise the Paley-Wiener spaces and show that functions in a Paley-Wiener space with radius R have monogenic extensions with integral averages that grow according to the radius R. Second, we consider the case of two dimensional compactly supported wavelets with orthonormal shifts and develop projection algorithms to find compactly supported, continuous wavelets with orthonormal shifts and dilations and 2 vanishing moments which are not tensor products of one dimensional wavelets. We also apply these techniques in one dimension and discover an example of an anti-symmetric, compactly supported, continuous wavelet with orthonormal shifts and dilations and 2 vanishing moments.

The occasional ambiguity of traditional mathematical notations, and the increasing complexity of proofs, has led to the situation that the use of a proof-verification system is desirable if not, in some situations, unavoidable. Moreover, by merging calculation techniques and proofs, theorem provers also allow a deep interconnection of the three basic mathematical activities, that is, defining, calculating and reasoning. In this thesis we explore such activities from different points of view, dealing with complex and hypercomplex analysis and computability theory using the HOL Light theorem prover. More precisely, the work is divided into four parts, each independent from the others. In the first we report on a formal development of quaternions and their algebraic structure, and we discuss automatic and certified procedures to perform calculations on them. The second part is dedicated to investigate the formalization of possible applications of our framework about quaternions. They are interesting theories on their own and, at the same time, a test for our work. In particular, we formalize basics definitions and theorems about two of the most recent and stimulating theories based on quaternions, that is, ''Slice regular quaternionic functions'' and ''Pythagorean-Hodograph curves''. Slice regular functions extend, in a suitable way, the notion of complex holomorphic function to the quaternionic case whereas, PH-curves are a class of polynomial functions with many theoretical properties and several significant computational advantages in many fields like computer-aided design (CAD), digital motion control, path planning, robotics applications and animation. The main points of the work presented in the first two parts has been published in proceedings of the 8th International Conference on Interactive Theorem Proving in Brasilia 2017. In part three and four we consider computability as a theory on its own. In particular, we focus on two radically different models of computation (but equally important), namely ''Turing Machines'' and ''quantum computing''. We give the basic definitions and we develop two certified systems to simulate computations in such models. Moreover, by implementing the concepts of Turing machines and quantum circuits in HOL Light, we explore these different approaches formalizing some simple different problems they can solve.

abstract: Continuous monitoring of sensor data from smart phones to identify human activities and gestures, puts a heavy load on the smart phone's power consumption. In this research study, the non-Euclidean geometry of the rich sensor data obtained from the user's smart phone is utilized to perform compressive analysis and efficient classification of human activities by employing machine learning techniques. We are interested in the generalization of classical tools for signal approximation to newer spaces, such as rotation data, which is best studied in a non-Euclidean setting, and its application to activity analysis. Attributing to the non-linear nature of the rotation data space, which involve a heavy overload on the smart phone's processor and memory as opposed to feature extraction on the Euclidean space, indexing and compaction of the acquired sensor data is performed prior to feature extraction, to reduce CPU overhead and thereby increase the lifetime of the battery with a little loss in recognition accuracy of the activities. The sensor data represented as unit quaternions, is a more intrinsic representation of the orientation of smart phone compared to Euler angles (which suffers from Gimbal lock problem) or the computationally intensive rotation matrices. Classification algorithms are employed to classify these manifold sequences in the non-Euclidean space. By performing customized indexing (using K-means algorithm) of the evolved manifold sequences before feature extraction, considerable energy savings is achieved in terms of smart phone's battery life.
Dissertation/Thesis
M.S. Electrical Engineering 2014