Closed-form bias reduction for shape estimation with polygon models

We look at the task of estimating the parameters of a geometric constraint from noisy points in 2D. The classical approach of minimizing the Euclidean distance error between points and constraint generally yields biased estimates for nonlinear constraints and higher noise levels. To deal with this issue, the expected distribution of the distance error can be explicitly incorporated in the estimator. However, for piecewise linear constraints, e.g., polygons, only computationally demanding sampling-based approaches are available. We propose two major contributions in order to resolve this issue. First, we derive closed-form expressions for the probability density of the signed distance between noisy points and a polygon angle. Second, based on this result, we develop a bias reduction method for polygons, which can be calculated in closed-form as well. We demonstrate that the quality of our approach can compete with its sampling-based alternatives, but only demands a fraction of their computational cost.