Convergence analysis for B-spline geometric interpolation

In this paper, we propose a sufficient condition for the convergence of a geometric algorithm for interpolating a given polygon using non-uniform cubic B-splines. Geometric interpolation uses the given polygon as the initial shape of the control polygon of the B-spline and reduces the approximate error by iteratively updating the control points with the deviations from the corresponding interpolated vertices to their nearest footpoints on the current B-spline curve. The convergence condition is derived by employing a spectral radius estimation technique. The primary goal is to find for each control point a parametric interval within which the nearest footpoint should be confined such that the spectral radius of the error iteration matrix is smaller than 1. A convergent condition for the geometric interpolation of uniform B-splines can be derived as a special case of the new scheme.

Graphical abstractInterpolation generation of the Chinese character ‘Good’: (a) original character; (b) polygon approximation of the character contour; (c–e) (middle row): the interpolation results of variants 1, 2, and 3; bottom row: zoomed in images corresponding to the part of images marked by the dotted rectangles in the middle row.Figure optionsDownload full-size imageDownload high-quality image (185 K)Download as PowerPoint slideHighlights► We proposed a sufficient condition for non-uniform B-spline which guarantees the convergence of the geometric interpolation algorithm. ► We investigated the efficiency of all variant algorithms. ► We showed the spectral of the composite matrices. ► We compared the quality of the generated curves.