Scale Selection for the Analysis of Point-Sampled Curves

An important task in the analysis and reconstruction of curvilinear structures from unorganized 3-D point samples is the estimation of tangent information at each data point. Its main challenges are in (1) the selection of an appropriate scale of analysis to accommodate noise, density variation and sparsity in the data, and in (2) the formulation of a model and associated objective function that correctly expresses their effects. We pose this problem as one of estimating the neighborhood size for which the principal eigenvector of the data scatter matrix is best aligned with the true tangent of the curve, in a probabilistic sense. We analyze the perturbation on the direction of the eigenvector due to finite samples and noise using the expected statistics of the scatter matrix estimators, and employ a simple iterative procedure to choose the optimal neighborhood size. Experiments on synthetic and real data validate the behavior predicted by the model, and show competitive performance and improved stability over leading polynomial-fitting alternatives that require a preset scale.