Measuring conic properties and shape orientations of two-dimensional point sets

Abstract

We propose new methods for computing a shape's orientation and several shape measures for elongation, linearity, circularity, ellipticity, hyperbolicity, and parabolicity of 2D point sets. Measures for both ordered and unordered data sets which are invariant to rotation, scaling, and translation interest us. These measures should also be calculated very quickly. Moment based and average pair wise direction based calculations of orientation are proposed here. We describe linearity measures for unordered data sets called eccentricity, triangle perimeters, triangle heights, triplet smoothness, rotation correlation, average orientations, and ellipse axis ratio. Linearity measures for sorted data sets include average sorted orientations, triangle sides ratio, and the product of a new monotonicity measure and one of the existing measures for linearity of unordered point sets. The monotonicity measure is the ratio of signed and non-signed sums of piecewise projections onto the orientation line. In order to measure circularity, 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. Our ellipse fit will determine the optimal location of the foci of the fitted ellipse along the orientation line (symmetrically with respect to the shape center) such that it minimizes the variance of sums of distances of points to the foci. In order to find ellipticity (hyperbolicity), we made use of the property that the sum (difference, respectively) of distances from each point on the ellipse to both foci is constant. We also propose an ellipticity measure based on the average ratio of distances of each point to the ellipse and to its center. The parabolicity measure is based on a similar idea of maintaining a constant sum of distances to the focus and a line parallel to the directrix line for each point. We discover that the definition of elongation highly correlates with the definition of linearity. All of the shape measures are tested on digital curves and compared with existing methods. All of the methods work in real time