Authors: Huxley, Martin
Žunić, Joviša 
Title: The Number of Different Digital N-Discs
Journal: Journal of Mathematical Imaging and Vision
Volume: 56
Issue: 3
First page: 403
Last page: 408
Issue Date: 1-Nov-2016
Rank: M21a
ISSN: 0924-9907
DOI: 10.1007/s10851-016-0643-y
Abstract: 
Configurations of integer lattice points inside a closed curve are a permanent topic in different research areas: from number theory to digital image analysis. This paper deals with the digital N-discs—the sets consisting of N integer lattice points which fall inside a circle. A digital N-disc corresponds to a digital image of a real circular discs, which consists of N pixels. We show that, for a large positive integer N, the average number of digital N-discs is asymptotic to 2N when N runs through a set of intervals, whose length is upper bounded by N47 / 58. In addition, we show that 2N is the best possible asymptotic estimate (the error term is ignored), for the number of digital N-discs. Such an asymptotic estimate is reached for almost all digital N-discs, for N in an interval, of the length N47 / 58. This improves result of Huxley and Žunić (IEEE Trans Pattern Anal Mach Intell 29:159–161, 2007), where an O(N) upper bound, for the number of digital N-discs, has been proven.
Keywords: Digital disc | Digital geometry | Image processing
Publisher: Springer Link

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