On Euclidean norm approximations

Abstract

Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches [1,2,3] were based on minimizing the maximum error. Recently, Seol and Cheun [4] proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.

Comments on “On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension”

“…Note the striking similarity between (5) and (6). Interestingly, a similar but less rigorous approach had been published earlier by Ohashi [9].…”

“…In a recent study [6], we examined various Euclidean norm approximations in detail and compared their average and maximum errors using numerical simulations. Here we show that two of those approximations, namely Barni et al's norm [1,7] and Seol and Cheun's norm [8], are viable alternatives to D M .…”

Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

“…Note also that these two norms are much more naturally linearizable norms than the Euclidean norm. The most recent works ( [7], [19]) on the subject showed that in the Euclidean plane, the best maximum relative error of an optimized combination of the L 1 and L ∞ norms with respect to the L 2 norm is approximately equal to 5.6%. In the same paper the best combination of an overestimating norm, the euclidean Chamfering weighted distance in 2-D, and an underestimated norm, the Inverse square root weighted t-cost distance in 2-D provide the best empirical MRE of 1.29%.…”

Automated Fabric Defect Detection Based on Multiple Gabor Filters and KPCA

“…Assuming d 2 ,r) is considered (in the calibration procedure we will show how to calculate out these average features). Afterwards, the Euclidean norm of local feature vectors is calculated [23] by the following formula, respectively: Obtained. It should be underlined that, differently from [24], norm-1 has been used.…”