Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves
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Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves Gabriela Pena1 · Hansapani Rodrigo1 Josef Sifuentes1 · Erwin Suazo1
· Mrinal Kanti Roychowdhury1 ·
Received: 25 December 2019 / Accepted: 12 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form n = 6k for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. Keywords Uniform distribution · Optimal quantizers · Quantization error · Quantization dimension · Quantization coefficient
Communicated by Juan-Enrique Martinez Legaz.
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Hansapani Rodrigo [email protected] Gabriela Pena [email protected] Mrinal Kanti Roychowdhury [email protected] Josef Sifuentes [email protected] Erwin Suazo [email protected]
1
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201, W University Dr., Edinburg, TX 78539-2909, USA
123
Journal of Optimization Theory and Applications
Mathematics Subject Classification 60Exx · 94A34
1 Introduction Quantization is a process of approximation with broad application in engineering and technology [1–3]. For the mathematical treatment of quantization, one is referred to Graf–Luschgy’s book [4]. Recently, optimal quantization for uniform distributions on different regions has been investigated by several authors [5–8]. On the contrary, optimal quantization for uniform distributions on curves has not yet been much investigated. Such kind of problems has rigorous applications in many areas, including signal processing. In this note, we would like to list two such applications. The first application comes within the area of signal processing. When we drive long distances, quite often cellular signals get cut off. This happens because we are either far away from the tower, or there is no tower nearby to catch the signal. In optimal quantization, one of our goals is to find the exact locations of the towers so that while driving we can get the best signal to our cellular phones. The second application comes within the area of agriculture. The amount of agricultural wat
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