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

Authors:
M. Emre Celebi;Hassan A. Kingravi;Fatih Celiker
Affiliations:
Department of Computer Science, Louisiana State University, Shreveport, LA, USA;School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA;Department of Mathematics, Wayne State University, Detroit, MI, USA
Venue:
Pattern Recognition Letters
Year:
2012

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Abstract

Mukherjee [Mukherjee, J., 2011. On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension. Pattern Recognition Lett. 32, 824-831] 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.