Shape based circularity measures of planar point sets

We are interested in circularity measures which are invariant to rotation, scaling, and translation, are calculated very quickly and are resistant to protrusions in the data set. We propose several measures here, all of which are based on existing linearity measures that have been adapted to measure circularity. In order to make use of these linearity measures, we transfer the Cartesian coordinates of the input set into polar coordinates. The linearity of the polar coordinate set corresponds to the circularity of the original input set given a suitable center. We separately consider the circularity of ordered and unordered point sets. The circularity of unordered data is determined directly from the linearity measure, whereas the circularity of ordered data is derived by multiplying the unordered data circularity measure by a monotonicity factor. We discuss two ways of determining the center of the shape. The circularity measures are tested on a set of 25 curves. The proposed algorithms work on both open and closed curves, whereas all competing algorithms (except one) are linked with exclusively closed curves. The measures were compared with human measurements of circularity of the same set. The new methods are have been found to best correspond to human perceptions.