Abstract
Algebraic curve fitting based on the algebraic distance is simple, but it has the disadvantage of inclining to a trivial solution. Researchers therefore introduce some constraints into the objective function in order to avoid the trivial solution. However, this often causes additional branches. Fitting based on geometric distance can avoid additional branches, but it does not offer sufficient fitting precision. In this paper we present a novel algebraic B-spline curve fitting method which combines both geometric distance and algebraic distance. The method first generates an initial curve by a distance field fitting that takes geometric distance as the objective function. Then local topology-preserving calibrations based on algebraic distance are performed so that each calibration does not produce any additional branches. In this way, we obtain an additional branch free fitting result whose precision is close to or even better than that produced by purely algebraic distance based methods. The adopted precision criterion is the geometric distance error rather than the algebraic one. In addition, we find a calibration fatigue phenomenon about calibrating strategy and propose a hybrid mode to solve it.
Similar content being viewed by others
References
Stamati, V., Fudos, I.: A feature based approach to re-engineering objects of freeform design by exploiting point cloud morphology. In: Proceedings of the 2007 ACM symposium on Solid and Physical Modeling, pp. 347–353 (2007)
Hudson, J.: Processing Large Point Cloud Data in Computer Graphics. The Ohio State University, Columbus (2003)
Mahmoudi, M., Sapiro, G.: Three-dimensional point cloud recognition via distributions of geometric distances. Graph. Models 71, 22–31 (2009)
Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, Fifth Edition, pp. 119–146. Morgan Kaufmann, San Francisco (2002)
Ahn, S.: Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space, pp. 1–16. Springer, Berlin, Heidelberg (2004)
Pratt, V.: Direct least-squares fitting of algebraic surfaces. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH), pp. 145–152 (1987)
Taubin, G.: Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 13, 1115–1138 (1991)
Jüttler, B., Chalmovianský, P., Shalaby, M., Wurm, E.: Approximate algebraic methods for curves and surfaces and their applications. In: Proceedings of the 21st Spring Conference on Computer Graphics, pp. 13–18 (2005)
Ahn, S., Rauh, W., Warnecke, H.-J.: Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola. Pattern Recogn. 34, 2283–2303 (2001)
Redding, N.: Implicit polynomials, orthogonal distance regression, and the closest point on a curve. IEEE Trans. Pattern Anal. Mach. Intell. 22, 191–199 (2000)
Elber, G., Kim, M.-S.: Geometric constraint solver using multivariate rational spline functions. In: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 1–10 (2001)
Aigner, M., Jüttler, B.: Robust computation of foot points on implicitly defined curves. In: Duhlen, M., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces: Robust Footpoint Computation. Nashboro Press (2005)
Krystek, M., Anton, M.: A weighted total least-squares algorithm for fitting a straight line. Meas. Sci. Technol. 18, 3438–3442 (2007)
Wang, W., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph. 25, 214–238 (2006)
Sederberg, T., Zheng, J., Klimaszewski, K., Dokken, T.: Approximate implicitization using monoid curves and surfaces. Graph. Models Image Process. 61, 177–198 (1999)
Jüttler, B., Felis, A.: Least-squares fitting of algebraic spline surfaces. Adv. Comput. Math. 17, 135–152 (2002)
Yang, Z., Deng, J., Chen, F.: Fitting unorganized point clouds with active implicit B-spline curves. Vis. Comput. 21, 831–839 (2005) (2005)
Li, Y., Feng, J., Jin, X.: Algebraic B-spline curve reconstruction based on signed distance field. J. Softw. 18, 2306–2317 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, M., Feng, J. & Zheng, J. An additional branch free algebraic B-spline curve fitting method. Vis Comput 26, 801–811 (2010). https://doi.org/10.1007/s00371-010-0476-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-010-0476-4