Measuring the Related Properties of Linearity and Elongation of Point Sets

The concept of elongation is generally well understood. However, there is no clear, precise, mathematical definition of elongation in any dictionary we could find. We propose that the definition of elongation should overlap with the definition of linearity since we will show that these two measures produce results that are highly correlated when applied to different types of 2D shapes. Our experiments consist of testing known methods of linearity and elongation on sets of closed shapes contours, shapes whose areas are filled, and shapes with open contours. We tested each algorithm on 25 different shapes in each category. It was found that the Average Orientations linearity measure from [10] best correlates to the elongation measures found in literature. It has a correlation value of above 0.9 with measures of elongation for open and closed curves. Also, we have discovered that the standard measure of elongation, applied to its intended area based shapes, gives almost identical results when it is applied to just the boundary pixels of the same area based shapes. They are over .98 correlated. This leads to a new linearity/elongation measure which is fast, applicable to both open and closed shapes, is given by a closed formula, and highly agrees with existing measures.