Measuring Linearity of a Finite Set of Points

It is often useful to measure how linear a certain set of points is. Our goal is to design algorithms that give a linearity measurement in the interval [0, 1]. There is no explicit discussion on linearity in literature, although some existing shape measures may be adapted. 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. The measures of eccentricity and contour smoothness were adapted from literature, the other five being triangle heights, triangle perimeters, rotation correlation, average orientations, and ellipse axis ratio. The algorithms are tested on 30 sample curves and the results are compared against the linear classifications of these curves by human subjects. It is found that humans and computers typically easily identify sets of points that are clearly linear, and sets of points that are clearly not linear. They have trouble measuring sets of points which are in the gray area in between. Although they appear to be conceptually very different approaches, we prove, theoretically and experimentally, that eccentricity and rotation correlations yield exactly the same linearity measurements. They however provide results which are furthest from human measurements. The average orientations method provides the closest results to human perception, while the other algorithms proved themselves to be very competitive