Abstract
Computing the geometric distance between a point and an implicitly defined curve is widely used in implicit curve fitting and geometric processing. Iterative numerical methods are often used to compute the distance, which may give quite accurate solutions but are usually time-consuming and non-robust. In this paper, a circle double-and-bisect algorithm is proposed to reliably evaluate the accurate geometric distance of a point to an implicit curve. The method’s prerequisite is merely the Lipschitz continuity of the implicit function. Moreover, by using gradient norm’s upper bound estimation and minor arcs evolution, the efficiency can be improved apparently. Experimental results show that the evaluated distance can describe the fitting result with more fidelity and that some fitting algorithms (for example, the genetic algorithm) re-implemented with the proposed geometric distance converge much faster. Due to these contributions, the proposed method can be applied in diverse applications such as error evaluation, curve fitting and random point cloud generator.











Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Ahn, S.J., Rauh, W., Warnecke, H.-J.: Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola. Pattern Recogn. 34(12), 2283–2303 (2001)
Ahn, S.J.: Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space. Springer, Berlin (2004)
Aigner, M., Jüttler, B.: Robust computation of foot points on implicitly defined curves. In: Dæhlen, M., Mørken, K., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces: Tromsø 2004, pp. 1–10. Nashboro Press, Brentwood (2005)
Bajaj, C.L., Bernardini, F., Xu, G.: Reconstructing surfaces and functions on surfaces from unorganized three-dimensional data. Algorithmica 1997(19), 243–261 (1997)
Bo, P., Ling, R., Wang, W.: A revisit to fitting parametric surfaces to point clouds. Comput. Graph. 36(5), 534–540 (2012)
Cheng, K.-S.D., Wang, W., Qin, H., Wong, K.-Y.K., Yang, H., Liu, Y.: Design and analysis of optimization methods for subdivision surface fitting. IEEE Trans. Vis. Comput. Graph. 13(5), 878–890 (2007)
De La Fraga, L.G., Silva, I.V., Cruz-Cortes, N.: Euclidean distance fit of ellipses with a genetic algorithm. In: Proceedings of Applications of Evolutionary Computing—EvoWorkshops 2007, pp. 359–366 (2007)
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, Ann Arbor, pp. 1–10 (2001)
Gotardo, P., Bellon, O., Boyer, K., Silva, L.: Range image segmentation into planar and quadric surfaces using an improved robust estimator and genetic algorithm. IEEE Trans. Syst. Man Cybern. Part B Cybern. 34(6), 2303–2316 (2004)
Hartmann, E.: On the curvature of curves and surfaces defined by normal forms. Comput. Aided Geom. Des. 16(5), 355–376 (1999)
Hu, M., Feng, J., Zheng, J.: An additional branch free algebraic B-spline curve fitting method. Vis. Comput. 26(6), 801–811 (2010)
Hunyadi, L., Vajk, I.: Fitting a model to noisy data using low-order implicit curves and surfaces. In: Proceedings of 2nd Eastern European Regional Conference on the Engineering of Computer Based Systems, pp. 106–114 (2011)
Jüttler, B., Felis, A.: Least-squares fitting of algebraic spline surfaces. Adv. Comput. Math. 17, 135–152 (2002)
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)
Kanatani, K., Sugaya, Y.: Unified computation of strict maximum likelihood for geometric fitting. J. Math. Imaging Vis. 38(1), 1–13 (2010)
Kanatani, K., Rangarajan, P.: Hyper least squares fitting of circles and ellipses. Comput. Stat. Data Anal. 55, 2197–2208 (2011)
Li, Y., Feng, J., Jin, X.: Algebraic B-spline curve reconstruction based on signed distance field. J. Softw. 18(9), 2306–2317 (2007)
Mullen, P., De Goes, F., Desbrun, M., Cohen-Steiner, D., Alliez, P.: Signing the unsigned: robust surface reconstruction from raw pointsets. Comput. Graph. Forum 29(5), 1733–1741 (2010)
Nishita, T., Sederberg, T.W., Kakimoto, M.: Ray tracing trimmed rational surface patches. Comput. Graph. (SIGGRAPH’90) 24(4), 337–345 (1990)
Redding, N.J.: Implicit polynomials, orthogonal distance regression, and the closest point on a curve. IEEE Trans. Pattern Anal. Mach. Intell. 22(2), 191–199 (2000)
Rouhani, M., Sappa, A.D.: Implicit polynomial representation through a fast fitting error estimation. IEEE Trans. Image Process. 21(4), 2089–2098 (2012)
Sampson, P.D.: Fitting conic sections to very scattered data: an iterative refinement of the Bookstein algorithm. Comput. Graph. Image Process. 18(1), 97–108 (1982)
Sappa, A.D., Rouhani, M.: Efficient distance estimation for fitting implicit quadric surfaces. In: Proceedings of International Conference on Image Processing (ICIP), pp. 3521–3524 (2009)
Schulz, T., Jüttler, B.: Envelope computation in the plane by approximate implicitization. Appl. Algebra Eng. Commun. Comput. 22(4), 265–288 (2011)
Sherbrooke, E.C., Patrikalakis, N.M.: Computation of the solutions of nonlinear polynomial systems. Comput. Aided Geom. Des. 10(5), 379–405 (1993)
Song, X., Jüttler, B.: Modeling and 3D object reconstruction by implicitly defined surfaces with sharp features. Comput. Graph. 33(3), 321–330 (2009)
Song, X., Jüttler, B., Poteaux, A.: Hierarchical spline approximation of the signed distance function. In: SMI’10 Proceedings of the 2010 Shape Modeling International Conference, pp. 241–245 (2010)
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(11), 1115–1138 (1991)
Upreti, K., Song, T., Tambat, A., Subbarayan, G.: Algebraic distance estimations for enriched isogeometric analysis. Comput. Methods Appl. Mech. Eng. 280, 28–56 (2014)
Upreti, K., Subbarayan, G.: Signed algebraic level sets on NURBS surfaces and implicit Boolean signed algebraic level sets on NURBS surfaces and implicit Boolean. Comput. Aided Des. 82, 112–126 (2017)
Wang, W., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph. 25(2), 214–238 (2006)
Zagorchev, L.G., Goshtasby, A.A.: A curvature-adaptive implicit surface reconstruction for irregularly spaced points. IEEE Trans. Vis. Comput. Graph. 18(9), 1460–1473 (2012)
Acknowledgements
This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY14F020032.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, M., Zhou, Y. & Li, X. Robust and accurate computation of geometric distance for Lipschitz continuous implicit curves. Vis Comput 33, 937–947 (2017). https://doi.org/10.1007/s00371-017-1370-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-017-1370-0