Investigating the Relationship between the Weighted Figure of Merit and Rosin's Measure

Many studies have been conducted to solve the problem of approximating a digital boundary by piece straight-line segments for the further processing required in computer vision applications. The authors of these studies compared their schemes to determine the best one. The initial measure used to assess the goodness of fit of a polygonal approximation was the figure of merit. Later,it was noted that this measure was not an appropriate metric for a valid reason which is why Rosin-through mathematical analysis-introduced a measure called merit. However,this measure involves an optimal scheme of polygonal approximation,so it is time-consuming to compute it to assess the goodness of fit of an approximation. This led many researchers to use a weighted figure of merit as a substitute for Rosin's measure to compare sub optimal schemes. An attempt is made in this communication to investigate whether the two measures-weighted figure of merit and Rosin's measure-are related so that one can be used instead of the other, and toward this end, theoretical analysis, experimental investigation and statistical analysis are carried out. The mathematical formulas for the weighted figure of merit and Rosin's measure are analyzed, and through proof of theorems,it is found that the two measures are theoretically independent of each other. The graphical analysis of experiments carried out using a public dataset supports the results of the theoretical analysis. The statistical analysis via Pearson's correlation coefficient and non-linear correlation measure also revealed that the two measures are uncorrelated. This analysis leads one to conclude that if a suboptimal scheme is found to be better (worse) than some other suboptimal scheme,as indicated by Rosin's measure,then the same conclusion cannot be drawn using a weighted figure of merit,so one cannot use a weighted figure of merit instead of Rosin's measure.