Comments on "On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension"
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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.
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