Direct Ellipse Fitting and Measuring Based on Shape Boundaries

Measuring ellipticity is an important area of computer vision systems. Most existing ellipticity measures are area based and cannot be easily applied to point sets such as extracted edges from real world images. We are interested in ellipse fitting and ellipticity measures which rely exclusively on shape boundary points which are practical in computer vision. They should also be calculated very quickly, be invariant to rotation, scaling and translation. Direct ellipse fitting methods are guaranteed to specifically return an ellipse as the fit rather than any conic. We argue that the only existing direct ellipse fit method does not work properly and propose a new simple scheme. It 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. We next propose a novel way of measuring the accuracy of ellipse fits against the original point set. The evaluation of fits proceeds by our novel ellipticity measure which transforms the point data into polar representation where the radius is equal to the sum of distances from the point to both foci, and the polar angle is equal to the one the original point makes with the center relative to the x-axis. The linearity of the polar representation will correspond to the quality of the ellipse fit for the original data. We also propose an ellipticity measure based on the average ratio of distances to the ellipse and to its center. The choice of center for each shape impacts the overall ellipticity measure. We discuss two ways of determining the center of the shape. The measures are tested on a set of shapes. The proposed algorithms work well on both open and closed curves.