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

Many studies had been conducted to solve the problem of approximating a digital boundary by piece straight-line segments for 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 a polygonal approximation was figure of merit. Later, it was pointed out that this measure was not an appropriate metric for a valid reason and this is why Rosin - through mathematical analysis - introduced a measure called merit. However, this measure involves optimal scheme of polygonal approximation and so it is time-consuming to compute it to assess the goodness of an approximation. This led many researchers to use weighted figure of merit as a substitute for Rosin's measure to compare among 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 towards this end theoretical analysis, experimental investigation and statistical analysis are carried out. The mathematical formula for weighted figure of merit and Rosin's measure are analyzed and through proof of theorems it is found that the two measures are independent of each other theoretically. The graphical analysis of experiments carried out using public dataset supports theoretical analysis. The statistical analysis using Pearson's correlation coefficient also establishes that the two measures are uncorrelated. This analysis leads one to conclude that if a sub-optimal scheme is found to be better (worse) than some other sub-optimal scheme as indicated by Rosin's measure then the same conclusion cannot be drawn using weighted figure of merit and so one cannot use weighted figure of merit instead of Rosin's measure.