Continuous and Discrete Data-Processing on Non-Cartesian Lattices
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2024-06-21
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Abstract
This thesis focuses on the challenges and practical considerations involved in approximating natural phenomena on regular, yet non-square (non-Cartesian) grids. At a high level, the most simple illustrative example is the move away from square pixels, to say, hexagonal pixels, which have much nicer symmetry compared to square pixels. The focus of this work is geared towards pragmatic solutions, building theory when needed, but with careful consideration to the practical aspects of data processing found in many sub-domains of computer science. The key sub-domains we explore are primarily scientific visualization and machine learning; but the techniques within this thesis extend much further into the numerical sciences. Central to our exploration is the development of the lattice tensor. This data structure is de- signed to encapsulate the complexities inherent in handling non-Cartesian grids. The lattice tensor is simple enough in its formulation so as to be integrated within an (auto)-differentiable com- putational framework. This immediately opens machine learning to the world of non-Cartesian methods. In addition to introducing the lattice tensor, this thesis proposes and evaluates various practical methods for processing and interpolating data within this framework. These methods have been created with a focus on practicality. The culmination of this work is showcased in the final chapter, where we venture into the realm of machine learning. Here, we explore the potential applications and implications of lattice ten- sors in machine learning research, underscoring their utility and effectiveness. This exploration not only demonstrates the practical applicability of our proposed methods but also opens new av- enues for research in machine learning, offering fresh perspectives and tools for tackling complex computational problems. In essence, this thesis presents a comprehensive study that bridges the gap between theoretical concepts and practical applications in the approximation of natural phenomena on non-Cartesian grids. Through the introduction of the lattice tensor and associated methodologies, this work con- tributes significantly to the field, providing a robust foundation for future research and development in both computational science and machine learning.
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Horacsek, J. (2024). Continuous and discrete data-processing on non-Cartesian lattices (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.