On the precision of curve length estimation in the plane

Abstract

The estimator of planar curve length based on intersection counting with a square grid, called the Buffon- Steinhaus estimator, is simple, design unbiased and efficient. However, the prediction of its error variance from a single grid superimposition is a non trivial problem. A previously published predictor is checked here by means of repeated Monte Carlo superimpositions of a curve onto a square grid, with isotropic uniform randomness relative to each other. Nine curvilinear features (namely flattened DNA molecule projections) were considered, and complete data are shown for two of them. Automatization required image processing to transform the original tiffimage of each curve into a polygonal approximation consisting of between 180 and 416 straight line segments or 'links' for the different curves. The performance of the variance prediction formula proved to be satisfactory for practical use (at least for the curves studied).