Abstract
Given parallel cross-sections data of a smooth object in 
, one of the ways of generating approximation to the object is by interpolating the signed-distance functions corresponding to the cross-sections. This well-known method is useful in many applications, and yet its approximation properties are not fully established. The known result is that away from cross-sections that are parallel to the boundary of the object, this method gives high approximation order. However, near such tangent cross-sections the approximation order is drastically reduced. This is due to the singular behaviour of the signed-distance function near tangent cross-sections. In this paper we suggest a way to restore the high approximation order everywhere. The new method involves a recent development in the approximation of functions with singularities. We present the application of this approach to our case, analyze its approximation properties, and discuss the numerical issues involved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bajaj, C.L., Coyle, E.J., Lin, K.N.: Arbitrary topology shape reconstruction from planar cross sections. Graph. Models Image Process. 58(6), 524–543 (1996)
Bermano, A., Vaxman, A., Gotsman, C.: Online reconstruction of 3D objects from arbitrary cross-sections. ACM Trans. Grap. (TOG) 30(5), 113 (2011)
Boissonnat, J.-D.: Shape reconstruction from planar cross sections. Comput. Vis. Graph. Image Process. 44(1), 1–29 (1988)
Boissonnat, J.-D., Memari, P.: Shape reconstruction from unorganized cross-sections. In: Symposium on Geometry Processing, pp. 89–98 (2007)
Cohen-Or, D., Solomovic, A., Levin, D.: Three-dimensional distance field metamorphosis. ACM Trans. Graph. (TOG) 17(2), 116–141 (1998)
Fuchs, H., Kedem, Z.M., Uselton, S.P.: Optimal surface reconstruction from planar contours. Commun. ACM 20(10), 693–702 (1977)
Herman, G.T., Zheng, J., Bucholtz, C.A.: Shape-based interpolation. IEEE Comput. Graph. Appl. 12(3), 69–79 (1992)
Kels, S., Dyn, N.: Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets. Comput. Graph. 35(3), 741–746 (2011)
Kels, S., Dyn, N.: Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric. Found. Comput. Math. 13(5), 835–865 (2013)
Levin, D.: Multidimensional reconstruction by set-valued approximations. IMA J. Numer. Anal. 6(2), 173–184 (1986)
Lipman, Y., Levin, D.: Approximating piecewise-smooth functions. IMA J. Numer. Anal. 30(4), 1159–1183 (2010)
Nurzynska, K.: 3D object reconstruction from parallel cross-sections. In: Bolc, L., Kulikowski, J.L., Wojciechowski, K. (eds.) ICCVG 2008. LNCS, vol. 5337, pp. 111–122. Springer, Heidelberg (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Kagan, Y., Levin, D. (2015). High Order Reconstruction from Cross-Sections. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-22804-4_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22803-7
Online ISBN: 978-3-319-22804-4
eBook Packages: Computer ScienceComputer Science (R0)