Comments on "On Approximating Euclidean Metrics by Weighted t-Cost   Distances in Arbitrary Dimension"
  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 $\mathbb{Z}^n$. 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 $\mathbb{R}^n$. We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors. 
M. Emre Celebi, Hassan A. Kingravi, Fatih Celiker
cs.NA cs.CV