Fitting B-splines to scattered data new and old parameterization

Parametric representation for curves is important in computer-aided geometric design, medical imaging, computer vision, computer graphics, shape matching, and face/object recognition. They are far better alternatives to free form representation, which are plagued with unboundedness and stability problems. This paper deals with the problem of fitting approximating B-spline curves with high compression ratios (control to data point ratios) to scattered data, where the data might be noisy or locally deformed, and where the curve sample points might be non-uniformly sampled across the curve. The approximating B-spline is robust to noise and local deformation. B-splines are bounded, continuous, invariant to affine and perspective transformations, and have local shape controllability. A parameterization based on curvature is proposed as a meaningful choice for the topological parameter of the B-spline that leads to a superior capturing of the curve shape and of preserving corner points. We compare several parameterization methods both theoretically and experimentally, and show that the curvature based parameterization is superior for noise free data, noisy data, local deformation, and non-uniform sampling. The fitting method is fast, and non-iterative. The fitting errors are, by and large, sub-resolution, i.e., they are below or at the resolution of the scattered data (defined here as the average distance between curve points). We also show the extension of the method to 3D surface fitting.