Measuring Linearity of Ordered Point Sets

It is often practical to measure how linear a certain ordered set of points is. We are interested in linearity measures which are invariant to rotation, scaling, and translation. These linearity measures should also be calculated very quickly and be resistant to protrusions in the data set. No such measures exist in literature. We propose several such measures here: average sorted orientations, triangle sides ratio, and the product of a monotonicity measure and one of the existing measures for linearity of unordered point sets. The monotonicity measure is also a contribution here. All measures are tested on a set of 25 curves. Although they appear to be conceptually very different approaches, six monotonicity based measures appear mutually highly correlated (all correlations are over .93). Average sorted orientations and triangle side ratio appear as effectively different measures from them (correlations are about .8) and mutually relatively close (correlation .93). When compared to human measurements, the average sorted orientations and triangle side ratio methods prove themselves to be closest. We also apply our linearity measures to design new polygonal approximation algorithms for digital curves. We develop two polygonization algorithms: linear polygonization, and a binary search polygonization. Both methods search for the next break point with respect to a known starting point. The break point is decided by applying threshold tests based on a linearity measure.