Properties of Farey Sequence and their Applications to Digital Image Processing

TL;DR

This paper introduces an efficient integer-based algorithm using the Regula Falsi method and Farey tables to search for fractions in Farey sequences, with applications demonstrated in digital image processing.

Contribution

It presents a novel algorithm for searching fractions in Farey sequences using Regula Falsi and Farey tables, optimized for integer computations.

Findings

Algorithm is efficient and accurate in finding closest fractions.

Applications in image processing demonstrate practical utility.

Experimental results confirm the method's effectiveness.

Abstract

Farey sequence has been a topic of interest to the mathematicians since the very beginning of last century. With the emergence of various algorithms involving the digital plane in recent times, several interesting works related with the Farey sequence have come up. Our work is related with the problem of searching an arbitrary fraction in a Farey sequence and its relevance to image processing. Given an arbitrary fraction p/q (0 < p < q) and a Farey sequence Fn of order n, we propose a novel algorithm using the Regula Falsi method and the concept of Farey table to efficiently find the fraction of Fn closest to p/q. All computations are in the integer domain only, which is its added benefit. Some contemporary applications of image processing have also been shown where such concepts can be incorporated. Experimental results have been furnished to demonstrate its efficiency and elegance.