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Kurmanbek, Bakytzhan

University of British Columbia

This thesis addresses the multivariate super-resolution problem commonly encountered in pixelized images. The study uses concepts from measure theory, linear algebra, and functional analysis. It builds on previous work conducted in the field of two-dimensional image analysis by Eftekhari, Bendory, and Tang [14]. Specifically, this research addresses an open problem posted by Schiebinger, Robeva, and Recht [21], with a particular emphasis on scenarios where the higher dimensional point-spread function can be decomposed. The first chapter of this thesis provides an introduction to the super-resolution image problem. It acquaints the reader with the topic by presenting an overview of previous works and research. Additionally, a mathematical problem is introduced that focuses on the case where the point-spread function of the imaging device can be decomposed component-wise. Chapter 2 presents a comprehensive overview of the subject matter, along with the foundational theoretical concepts that will be vital for understanding the subsequent chapters. Chapter 3 presents the findings of the study. The research demonstrates that accurate recovery of true images is possible under mild conditions of component-wise functions, even in the absence of noise. Furthermore, the study shows the construction of dual certificates using T-systems and T*-systems. Chapter 4 outlines future work on extending this research to include the general Gaussian case, defining the generalization of T-systems to higher dimensions, and contextualizing their relevance in this context. Overall, this research contributes to the field of image analysis by providing valuable insights into the complexities of the multivariate super-resolution problem.

Text

2023-04-19

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/84333

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/