Fitting an ellipse to an arbitrary shape: implications for strain analysis

An ellipse can be fit to an arbitrary shape using a linear least squares approach applied to boundary data. Alternatively, this problem can also be solved by calculating the second moments of the entire region, a technique popular in image analysis applications. If the irregular shape can be approximated by a polygon then Greens theorem allows efficient calculation of the second moments. If the shape is pixelated then the second moments can be calculated by a simple summation process. By considering the behaviour of these fitting methods with increasing deformation it is shown that as an arbitrary shape passively deforms, the best-fit ellipse also behaves as if it were deforming passively. This implies that all techniques of strain analysis that were previously restricted to populations of elliptical objects may now be applied to populations of arbitrary shapes, provided the best-fit ellipse is calculated by one of the methods described here. Furthermore it implies that selective sampling based on shape or methods of weighting based upon shape are invalid and tend to bias the raw data.