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

Abstract

Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in Z n . In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in R n . We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.

Error analysis of octagonal distances defined by periodic neighborhood sequences for approximating Euclidean metrics in arbitrary dimension

“…We also consider the fact that scaling of WtCWD ( n ) ( u ; W , , a , b ), by an appropriate scale factor k may further reduce the MRE as it is observed in a few other works reported previously [1,4,6] . In the following, we provide the expression for optimum scale factor ( k opt ) at which the MRE becomes minimum, as well as the scale adjusted optimum MRE value.…”