A divide-and-conquer algorithm for curve fitting

Abstract

Curve fitting is still an open problem which draws attention from many applications, such as computer-aided design, computer-aided manufacturing and reverse engineering. Splines such as Bézier, B-Spline and NURBS curves are usually employed in engineering applications and are intensively used for fitting purposes. The optimization of their shapes and localization parameters, however, is a very complex task. The literature presents many methods which empirically set some important parameters, such as the number of control points. As guessing such a value is difficult, this paper presents a new method to choose it through a multi-curve fitting method, based on linear least square optimizations, using a divide-and-conquer algorithm and an error tolerance threshold. Four prime procedures compose the method: the conquer step fits curves over subset point clouds; the combine step glues curve segments together with some selective continuity; the divide step splits subsets which are not properly fitted yet; and the merge step blends curve segments together. Several curve setups were tested in well-known benchmarks, using four-division strategies: bisection, error balance, point with the greatest curvature and point with the smallest curvature. The developed method allows for fast computation even for larger point clouds, and it was able to properly reconstruct each tested shape, even with the addition of synthetic noise. We also demonstrate that it can be significantly faster than a single-curve fitting using the same number of control points. • A multiple curve fitting approach using iterative multi-control-point insertion. • A divide-and-conquer algorithm allows for fast curve fitting. • Point cloud division strategies have their own pros and cons in curve fitting. • Adequate error tolerance balances quality and simplicity in curve fitting.