Theses and Dissertations

Date of Award

7-1-2024

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Mrinal Kanti Roychowdhury

Second Advisor

Andras Balogh

Third Advisor

Hansapani Rodrigo

Abstract

Bucklew and Wise (1982) established that the quantization dimension of an absolutely continuous probability measure on a Euclidean space equals the Euclidean dimension of the space, and that the quantization coefficient is finite and positive. This thesis explores the variability of quantization coefficients for uniform distributions on various geometric structures, including line segments, circles, and regular polygons. We derive the conditional optimal sets of n-points and calculate the n-th conditional quantization errors for uniform distributions under different conditions. On line segments, we identify optimal point distributions as either equally spaced or geometrically spaced towards fixed end-points. For circles, we analyze both fixed points on the circumference and uniformly distributed points. For regular polygons, we use affine transformations to simplify the computation of optimal point distributions and errors by mapping the boundary to a linear interval. A key finding is that increasing the number of quantization points consistently reduces the quantization error across all structures. This underscores the efficiency gained with more quantization points. The geometric properties of each structure, such as segment length, circle radius, and polygon sides, significantly impact the quantization coefficients. These insights deepen the understanding of quantization processes for uniform distributions and offer practical guidance for optimizing quantization in digital communications, data compression, and signal processing. Further research could extend these findings to higher dimensions, non-uniform distributions, and dynamic quantization methods.

Comments

Copyright 2024 Evans Nyanney. https://proquest.com/docview/3115398733

Download

Included in

Mathematics Commons

Share

COinS