This thesis constitutes a comprehensive study of the fundamental properties of an analytic function theory over quaternions, which is known as slice regularity, as introduced in Gentili et al. (Gentili et al., 2022). In 2006, an approach to analysis over quaternions was introduced by G. Gentili and D.C. Struppa. This approach gave rise to a new notion of regularity for quaternion-valued functions called slice regularity. Functions satisfying this regularity were termed slice-regularity functions. This new theory heavily relies on complex analysis and has found applications in geometry. The primary objective of this project is to present this new theory, encompassing polynomials and power series in slice regular functions with quaternionic coefficients. Various properties, such as slice regular products, extend to polynomials, and convergent power series in the form of slice regular functions are defined.Consequently, this thesis offers an overview of the latest developments in regularity theory and the fundamentals of quaternion-valued slice regularity functions. I assume the reader isfamiliar with the basics of complex analysis or has read my bachelor’s thesis (Fathian Pourkondori, 2022) in advance. Given that my master’s thesis builds upon the foundations laidin my bachelor’s thesis, I have refrained from rewriting certain chapters on quaternions and Fueter’s regularity theory. These chapters are prerequisites for understanding slice regularity, and I provide the link to the previous thesis (Fathian Pourkondori, 2022) for reference.

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This dissertation reports the development of a technique for automated face recognition, using color images. One of the more powerful techniques for recognition of faces in monochromatic images has been extended to color by the use of hypercomplex numbers called quaternions. Two software implementations have been written of the new method and the analogous method for use on monochromatic images. Test results show that the new method is superior in accuracy to the analogous monochrome method. Although color images are generally collected, the great majority of published research efforts and of commercially available systems use only the intensity features. This surprising fact provided motivation to the three thesis statements proposed in this dissertation. The first is that the use of color information can increase face recognition accuracy. Face images contain many features, some of which are only easily distinguishable using color while others would seem more robust to illumination variation when color is considered. The second thesis statement is that the currently popular technique of graph-based face analysis and matching of features extracted from application of a family of Gabor filters can be extended to use with color. A particular method of defining a filter appropriate for color images is used; the usual complex Gabor filter is adapted to the domain of quaternions.. Four alternative approaches to the extension of complex Gabor filters to quaternions are defined and discussed; the most promising is selected and used as the basis for subsequent implementation and experimentation. The third thesis statement is that statistical analysis can identify portions of the face image that are highly relevant — i.e., locations that are especially well suited for use in face recognition systems. Conventionally, the Gabor-based graph method extracts features at locations that are equally spaced, or perhaps selected manually on a non-uniform graph. We have defined a relevance image, in which the intensity values are computed from the intensity variance across a number of images from different individuals and the mutual information between the pixel distributions of sets of images from different individuals and the same individual. A complete software implementation of the new face recognition method has been developed. Feature vectors called jets are extracted by application of the novel quaternion Gabor filter, and matched against models of other faces. In order to test the validity of the thesis statements, a parallel software implementation of the conventional monochromatic Gabor graph method has been developed and side-by-side testing has been conducted. Testing results show accuracy increases of 3% to 17% in the new color-based method over the conventional monochromatic method. These testing results demonstrate that color information can indeed provide a significant increase in accuracy, that the extension of Gabor filters to color through the use of quaternions does give a viable feature set, and that the face landmarks chosen via statistical methods do have high relevance for face discrimination. / Ph. D.

This thesis deals with the usefulness of the application of quaternions in representing robot kinematics. It begins by showing the relationship of quaternions to the more commonly-known complex numbers and how it can represent rotations in three-dimensions. Then, the dual quaternions are introduced to represent both the three-dimensional rotation and translation. It will then be used to derive the forward and inverse kinematics, particularly, for the Universal Robot UR3 which is a 6-DOF robotic arm. Lastly, an actual application of dual quaternions in robot programming will be demonstrated

The accurate determination of spacecraft attitude has always been a critical issue in many applications. The presence of imperfect sensors introduces errors in the system and affects the outcome of the mission. One of the most significant sensors is the rate gyroscope. Particularly, the rate gyros are known to degrade with time, introducing random noise and bias. This calls for estimation algorithms which process the measured data in order to reduce the effects of the disturbances to a minimum. This research presents an approach which takes full advantage on the nonlinear dynamics and possibly non-Gaussian disturbances. It is based on recent work involving particle filters, where the probability density functions are approximated by a relatively large number of parameters. It is shown that accurate attitude estimation can be obtained with a manageable number of particles.

"Ultrasound has become a widely used form of medical imaging because it is low-cost, safe, and portable. However, it is heavily dependent on the skill of the operator to capture quality images and properly detect abnormalities. Training is a key component of ultrasound, but the limited availability of training courses and programs presents a significant obstacle to the wider use of ultrasound systems. The goal of this work was to design and implement an interactive training system to help train and evaluate sonographers. This Virtual Training System for Diagnostic Ultrasound is an inexpensive, software-based training system in which the trainee scans a generic scan surface with a sham transducer containing position and orientation sensors. The observed ultrasound image is generated from a pre-stored 3D image volume and is controlled interactively by the user€™s movements of the sham transducer. The patient in the virtual environment represented by the 3D image data may depict normal anatomy, exhibit a specific trauma, or present a given physical condition. The training system provides a realistic scanning experience by providing an interactive real-time display with adjustable image parameters similar to those of an actual diagnostic ultrasound system. This system has been designed to limit the amount of hardware needed to allow for low-cost and portability for the user. The system is able to utilize a PC to run the software. To represent the patient to be scanned, a specific scan surface has been produced that allows for an optical sensor to track the position of the sham transducer. The orientation of the sham transducer is tracked by using an inexpensive inertial measurement unit that relies on the use of quaternions to be integrated into the system. The lack of a physical manikin is overcome by using a visual implementation of a virtual patient in the software along with a virtual transducer that reflects the movements of the user on the scan surface. Pre-processing is performed on the selected 3D image volume to provide coordinate transformation parameters that yield a least-mean square fit from the scan surface to the scanning region of the virtual patient. This thesis presents a prototype training system accomplishing the main goals of being low-cost, portable, and accurate. The ultrasound training system can provide cost-effective and convenient training of physicians and sonographers. This system has the potential to become a powerful tool for training sonographers in recognizing a wide variety of medical conditions."

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves - ). Also available for download via the World Wide Web; free to University of Oregon users.

Capes / A descoberta dos quatérnios pelo matemático britânico William Rowan Hamilton (1805-1865) permitiu uma nova abordagem na resolução de equações algébricas, fornecendo uma estrutura algébrica mais geral onde buscar soluções. Generalizando o caso clássico (sobre os complexos) apresentamos neste trabalho um tratamento da equação algébrica geral com coeficientes quatérnios. Verificamos que o número de raízes pode ser maior que o grau, e muitas vezes, pode mesmo ser infinito. Damos ênfase ao caso da equação quadrática, obtendo fórmulas para as raízes. Também nos detemos na obtenção de uma raiz enésima quatérnia de um quatérnio e de um número real. / The discovery of quartenions by the mathematician Willian Rowan Hamilton (1805-1865) allowed a new approach regards solving algebraic equations, providing a broad algebraic structure to seek solutions. As a generalization of the classical case (about the complexes), here we present a treatment of the general algebraic equation with quaternions coefficients. We found that the number of roots can be greater than the degree and often can be infinite. We give emphasis to the case of the quadratic equation, obtaining its solution formulas. We also dealt with obtaining a quaternary root of a quaternion and a real number.

Capes / Este trabalho foi desenvolvido a partir da constatação das dificuldades e falta de motivação dos alunos do ensino médio no aprendizado de números complexos. O desenvolvimento consistiu em realizar uma linha do tempo no estudo dos números complexos desde Cardano até Sir Hamilton, buscando contribuir para sua melhor compreensão, associando as propriedades algébricas com a interpretação geométrica visando melhorar o entendimento do uso dos números complexos na resolução de problemas. Ainda, a história da introdução da unidade imaginária i e a representação algébrica em duas dimensões (2D) a+bi, estendendo esta representação aos quatérnios (4D) a+bi+cj+dk e suas outras formas menos usuais, como a forma matricial, com vetores, incluindo o procedimento utilizado na rotação, apresentando sua importância como motivação no ensino de geometria, na física e na computação gráfica. / This work was developed after observing the difficulties and unmotivated of the high school students to learning complex numbers. The development consisted in create a timeline in the study of complex numbers since Cardano at Sir Hamilton, expecting to contribute to understanding of this subject, associating algebraic properties and geometric interpretation, seeing to improve the understanding of the use of complex numbers to solve problems. Also, the history of imaginary unit i introduction and representation two-dimensional (2D) complex numbers a+bi, extending this for four-dimensions(4D) quaternions numbers a+bi+cj+dk, and its less usual forms like matrix form, vector form, including the procedure used in the rotation, showing your importance as motivation in the teaching of geometry, in physics and graphic computation.

Capes / Este trabalho foi desenvolvido a partir da constatação das dificuldades e falta de motivação dos alunos do ensino médio no aprendizado de números complexos. O desenvolvimento consistiu em realizar uma linha do tempo no estudo dos números complexos desde Cardano até Sir Hamilton, buscando contribuir para sua melhor compreensão, associando as propriedades algébricas com a interpretação geométrica visando melhorar o entendimento do uso dos números complexos na resolução de problemas. Ainda, a história da introdução da unidade imaginária i e a representação algébrica em duas dimensões (2D) a+bi, estendendo esta representação aos quatérnios (4D) a+bi+cj+dk e suas outras formas menos usuais, como a forma matricial, com vetores, incluindo o procedimento utilizado na rotação, apresentando sua importância como motivação no ensino de geometria, na física e na computação gráfica. / This work was developed after observing the difficulties and unmotivated of the high school students to learning complex numbers. The development consisted in create a timeline in the study of complex numbers since Cardano at Sir Hamilton, expecting to contribute to understanding of this subject, associating algebraic properties and geometric interpretation, seeing to improve the understanding of the use of complex numbers to solve problems. Also, the history of imaginary unit i introduction and representation two-dimensional (2D) complex numbers a+bi, extending this for four-dimensions(4D) quaternions numbers a+bi+cj+dk, and its less usual forms like matrix form, vector form, including the procedure used in the rotation, showing your importance as motivation in the teaching of geometry, in physics and graphic computation.