Assessing Approaches for Consecutive B-Spline Model Adaption Based on Point Clouds with Varying Structural Complexity

Abstract

The representation of point clouds by mathematical models yields benefits in terms of data and noise reduction whilst simultaneously serving as basis to derive parameter-based epoch comparison resulting in a space-continuous deformation analysis. B-spline curves resp. surfaces serve as well-suited models to approximate point clouds with superior complexity. Their most general form is a tensor-product (TP) b-spline, which is built upon a regular parameter grid. This representation is sufficient for point clouds depicting objects with a continuous and smooth surface. The approximation of point clouds that show a more heterogeneous course by TP b-spline surfaces has two possible drawbacks: Either the description is too coarse, as systematic local point cloud deviations are neglected or the representation is of superior complexity leading to a large number of parameters to be estimated and thus being computational expensive. The transition of TP b-splines to strategies of local refinement tackles these challenges by increasing the model complexity in detached regions whilst keeping parameter sparsity where sufficient. This paper suggests and assesses three approaches for the identification of regions that require local refinement. The first approach is founded on residual analysis. The second one is based on interepochal differences of control points and the third one uses information about curvature deviations between the approximating b-spline model and the data to identify local insufficiencies. The application of the approaches is tested on b-spline curves that approximate synthetic profile point clouds. Results show that all three approaches correctly identify regions, where model adaption is needed. However, among the approaches, only the residual- based refinement may lead to full alignment of the refined B-spline curve’s control points with the nominal ones.