Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Optimization of the holographic process for imaging and lithography

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 272-297).Since their invention in 1948 by Dennis Gabor, holograms have demonstrated to be important components of a variety of optical systems and their implementation in new fields and methods is expected to continue growing. Their ability to encode 3D optical fields on a 2D plane opened the possibility of novel applications for imaging and lithography. In the traditional form, holograms are produced by the interference of a reference and object waves recording the phase and amplitude of the complex field. The holographic process has been extended to include different recording materials and methods. The increasing demand for holographic-based systems is followed by a need for efficient optimization tools designed for maximizing the performance of the optical system. In this thesis, a variety of multi-domain optimization tools designed to improve the performance of holographic optical systems are proposed. These tools are designed to be robust, computationally efficient and sufficiently general to be applied when designing various holographic systems. All the major forms of holographic elements are studied: computer generated holograms, thin and thick conventional holograms, numerically simulated holograms and digital holograms. Novel holographic optical systems for imaging and lithography are proposed. In the case of lithography, a high-resolution system based on Fresnel domain computer generated holograms (CGHs) is presented. The holograms are numerically designed using a reduced complexity hybrid optimization algorithm (HOA) based on genetic algorithms (GAs) and the modified error reduction (MER) method. The algorithm is efficiently implemented on a graphic processing unit. Simulations as well as experimental results for CGHs fabricated using electron-beam lithography are presented. A method for extending the system's depth of focus is proposed. The HOA is extended for the design and optimization of multispectral CGHs applied for high efficiency solar concentration and spectral splitting. A second lithographic system based on optically recorded total internal reflection (TIR) holograms is studied. A comparative analysis between scalar and (cont.) vector diffraction theories for the modeling and simulation of the system is performed.A complete numerical model of the system is conducted including the photoresist response and first order models for shrinkage of the holographic emulsion. A novel block-stitching algorithm is introduced for the calculation of large diffraction patterns that allows overcoming current computational limitations of memory and processing time. The numerical model is implemented for optimizing the system's performance as well as redesigning the mask to account for potential fabrication errors. The simulation results are compared to experimentally measured data. In the case of imaging, a segmented aperture thin imager based on holographically corrected gradient index lenses (GRIN) is proposed. The compound system is constrained to a maximum thickness of 5mm and utilizes an optically recorded hologram for correcting high-order optical aberrations of the GRIN lens array. The imager is analyzed using system and information theories. A multi-domain optimization approach is implemented based on GAs for maximizing the system's channel capacity and hence improving the information extraction or encoding process. A decoding or reconstruction strategy is implemented using the superresolution algorithm. Experimental results for the optimization of the hologram's recording process and the tomographic measurement of the system's space-variant point spread function are presented. A second imaging system for the measurement of complex fluid flows by tracking micron sized particles using digital holography is studied. A stochastic theoretical model based on a stability metric similar to the channel capacity for a Gaussian channel is presented and used to optimize the system. The theoretical model is first derived for the extreme case of point source particles using Rayleigh scattering and scalar diffraction theory formulations. The model is then extended to account for particles of variable sizes using Mie theory for the scattering of homogeneous dielectric spherical particles. The influence and statistics of the particle density dependent cross-talk noise are studied. Simulation and experimental results for finding the optimum particle density based on the stability metric are presented. For all the studied systems, a sensitivity analysis is performed to predict and assist in the correction of potential fabrication or calibration errors.by José Antonio Domínguez-Caballero.Ph.D

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral kodierte multispektrale Lichtfelder untersucht, wie sie von einer Lichtfeldkamera mit einem spektral kodierten Mikrolinsenarray aufgenommen werden. Für die Rekonstruktion der kodierten Lichtfelder werden zwei Methoden entwickelt, eine basierend auf den Prinzipien des Compressed Sensing sowie eine Deep Learning Methode. Anhand neuartiger synthetischer und realer Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail evaluiert

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In this work, spatio-spectrally coded multispectral light fields, as taken by a light field camera with a spectrally coded microlens array, are investigated. For the reconstruction of the coded light fields, two methods, one based on the principles of compressed sensing and one deep learning approach, are developed. Using novel synthetic as well as a real-world datasets, the proposed reconstruction approaches are evaluated in detail

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral codierte multispektrale Lichtfelder, wie sie von einer Lichtfeldkamera mit einem spektral codierten Mikrolinsenarray aufgenommen werden, untersucht. Für die Rekonstruktion der codierten Lichtfelder werden zwei Methoden entwickelt und im Detail ausgewertet. Zunächst wird eine vollständige Rekonstruktion des spektralen Lichtfelds entwickelt, die auf den Prinzipien des Compressed Sensing basiert. Um die spektralen Lichtfelder spärlich darzustellen, werden 5D-DCT-Basen sowie ein Ansatz zum Lernen eines Dictionary untersucht. Der konventionelle vektorisierte Dictionary-Lernansatz wird auf eine tensorielle Notation verallgemeinert, um das Lichtfeld-Dictionary tensoriell zu faktorisieren. Aufgrund der reduzierten Anzahl von zu lernenden Parametern ermöglicht dieser Ansatz größere effektive Atomgrößen. Zweitens wird eine auf Deep Learning basierende Rekonstruktion der spektralen Zentralansicht und der zugehörigen Disparitätskarte aus dem codierten Lichtfeld entwickelt. Dabei wird die gewünschte Information direkt aus den codierten Messungen geschätzt. Es werden verschiedene Strategien des entsprechenden Multi-Task-Trainings verglichen. Um die Qualität der Rekonstruktion weiter zu verbessern, wird eine neuartige Methode zur Einbeziehung von Hilfslossfunktionen auf der Grundlage ihrer jeweiligen normalisierten Gradientenähnlichkeit entwickelt und gezeigt, dass sie bisherige adaptive Methoden übertrifft. Um die verschiedenen Rekonstruktionsansätze zu trainieren und zu bewerten, werden zwei Datensätze erstellt. Zunächst wird ein großer synthetischer spektraler Lichtfelddatensatz mit verfügbarer Disparität Ground Truth unter Verwendung eines Raytracers erstellt. Dieser Datensatz, der etwa 100k spektrale Lichtfelder mit dazugehöriger Disparität enthält, wird in einen Trainings-, Validierungs- und Testdatensatz aufgeteilt. Um die Qualität weiter zu bewerten, werden sieben handgefertigte Szenen, so genannte Datensatz-Challenges, erstellt. Schließlich wird ein realer spektraler Lichtfelddatensatz mit einer speziell angefertigten spektralen Lichtfeldreferenzkamera aufgenommen. Die radiometrische und geometrische Kalibrierung der Kamera wird im Detail besprochen. Anhand der neuen Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail bewertet. Es werden verschiedene Codierungsmasken untersucht -- zufällige, reguläre, sowie Ende-zu-Ende optimierte Codierungsmasken, die mit einer neuartigen differenzierbaren fraktalen Generierung erzeugt werden. Darüber hinaus werden weitere Untersuchungen durchgeführt, zum Beispiel bezüglich der Abhängigkeit von Rauschen, der Winkelauflösung oder Tiefe. Insgesamt sind die Ergebnisse überzeugend und zeigen eine hohe Rekonstruktionsqualität. Die Deep-Learning-basierte Rekonstruktion, insbesondere wenn sie mit adaptiven Multitasking- und Hilfslossstrategien trainiert wird, übertrifft die Compressed-Sensing-basierte Rekonstruktion mit anschließender Disparitätsschätzung nach dem Stand der Technik

Robustness, scalability and interpretability of equivariant neural networks across different low-dimensional geometries

In this thesis we develop neural networks that exploit the symmetries of four different low-dimensional geometries, namely 1D grids, 2D grids, 3D continuous spaces and graphs, through the consideration of translational, rotational, cylindrical and permutation symmetries. We apply these models to applications across a range of scientific disciplines demonstrating the predictive ability, robustness, scalability, and interpretability. We develop a neural network that exploits the translational symmetries on 1D grids to predict age and species of mosquitoes from high-dimensional mid-infrared spectra. We show that the model can learn to predict mosquito age and species with a higher accuracy than models that do not utilise any inductive bias. We also demonstrate that the model is sensitive to regions within the input spectra that are in agreement with regions identified by a domain expert. We present a transfer learning approach to overcome the challenge of working with small, real-world, wild collected data sets and demonstrate the benefit of the approach on a real-world application. We demonstrate the benefit of rotation equivariant neural networks on the task of segmenting deforestation regions from satellite images through exploiting the rotational symmetry present on 2D grids. We develop a novel physics-informed architecture, exploiting the cylindrical symmetries of the group SO+ (2, 1), which can invert the transmission effects of multi-mode optical fibres (MMFs). We develop a new connection between a physics understanding of MMFs and group equivariant neural networks. We show that this novel architecture requires fewer training samples to learn, better generalises to out-of-distribution data sets, scales to higher-resolution images, is more interpretable, and reduces the parameter count of the model. We demonstrate the capability of the model on real-world data and provide an adaption to the model to handle real-world deviations from theory. We also show that the model can scale to higher resolution images than was previously possible. We develop a novel architecture which provides a symmetry-preserving mapping between two different low-dimensional geometries and demonstrate its practical benefit for the application of 3D hand mesh generation from 2D images. This models exploits both the 2D rotational symmetries present in a 2D image and in a 3D hand mesh, and provides a mapping between the two data domains. We demonstrate that the model performs competitively on a range of benchmark data sets and justify the choice of inductive bias in the model. We develop an architecture which is equivariant to a novel choice of automorphism group through the use of a sub-graph selection policy. We demonstrate the benefit of the architecture, theoretically through proving the improved expressivity and improved scalability, and experimentally on a range of widely studied benchmark graph classification tasks. We present a method of comparison between models that had not been previously considered in this area of research, demonstrating recent SOTA methods are statistically indistinguishable