We present a multi-level partition of unity algebraic set surfaces (MPU-APSS) for surface reconstruction which can be represented by either a projection or in an implicit form. An algebraic point set surface (APSS) defines a smooth surface from a set of unorganized points using local moving least-squares (MLS) fitting of algebraic spheres. However, due to the local nature, APSS does not work well for geometry editing and modeling. Instead, our method builds an implicit approximation function for the scattered point set based on the partition of unity approach. By using an octree subdivision strategy, we first adaptively construct local algebraic spheres for the point set, and then apply weighting functions to blend together these local shape functions. Finally, we compute an error-controlled approximation of the signed distance function from the surface. In addition, we present an efficient projection operator which makes our representation suitable for point set filtering and dynamic point resampling. We demonstrate the effectiveness of our unified approach for both surface reconstruction and geometry modeling such as surface completion.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Levin D. Mesh-Independent Surface Interpolation. Geometric Modeling for Scientific Visualization, Springer, 2003, pp.37–49.
Alexa M, Behr J, Cohen-Or D, Fleishman S, Levin D, Silva C T. Computing and rendering point set surfaces. IEEE Transactions on Visualization and Computer Graphics, 2003, 9(1): 3–15.
Gross M, Pfister H. Point-Based Graphics. Morgan Kaufmann, 2007.
Amenta N, Kil Y J. Defining point-set surfaces. ACM Transactions on Graphics, 2004, 23(3): 264–270.
Alexa M, Adamson A. On normals and projection operators for surfaces defined by point sets. In Proc. Symposium on Pint-Based Graphics, Zurich, Switzerland, Jun. 2–4, 2004, pp.150–155.
Guennebaud G, Gross M. Algebraic point set surfaces. ACM Transactions on Graphics, 2007, 26(3): Article No.3.
Guennebaud G, Germann M, Gross M. Dynamic sampling and rendering of algebraic point set surfaces. Computer Graphics Forum, 2008, 27(2): 653–662.
Babuska I, Melenk J M. The partition of unity method. International Journal for Numerical Methods in Engineering, 1997, 40(4): 727–758.
Ohtake Y, Belyaev A, Alexa M, Turk G, Seidel H P. Multi-level partition of unity implicits. In Proc. International Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 2003), San Diego, USA, Jul. 27–31, 2005, pp.463–470.
Xiao C, Feng G, Chu Y, Du Z, Yang X. Multi-level partition of unity algebraic point set surfaces. International Conference on Multimedia Information Networking and Security, Wuhan, China, Nov. 18–20, 2009, pp.645–649.
Pauly M, Keiser R, Kobbelt L P, Gross M. Shape modeling with point-sampled geometry. ACM Transactions on Graphics, 2003, 22(3): 641–650.
Alexa M, Adamson A. Interpolatory point set surfaces-convexity and Hermite data. ACM Transactions on Graphics, 2009, 28(2): Article No. 20.
Fleishman S, Cohen-Or D, Silva C T. Robust moving least-squares fitting with sharp features. In Proc. International Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 2005), Los Angeles, USA, Jul. 31-Aug. 4, 2005, pp.544–552.
Pratt V. Direct least-squares fitting of algebraic surfaces. ACM SIGGRAPH Computer Graphics, 1987, 21(4): 145–152.
Hoppe H, De Rose T, Duchamp T et al. Piecewise smooth surface reconstruction. In Proc. SIGGRAPH 1994, Orlando, USA, Jul. 24–29, 1994, pp.295–302.
Amenta N, Bern M, Kamvysselis M. A new Voronoi-based surface reconstruction algorithm. In Proc. SIGGRAPH 1998, Orlando, USA, Jul. 19–24, 1998, pp.415–421.
Carr J C, Beatson R K, Cherrie J B, Mitchell T J, Fright W R, McCallum B C, Evans T R. Reconstruction and representation of 3D objects with radial basis functions. In Proc. SIG-GRAPH 2003, Los Angeles, USA, Aug. 12–17, 2001, pp.67–76.
Morse B S, Yoo T S, Rheingans P, Chen D T, Subramanian K R. Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. In Proc. SMI 2001 International Conference on Shape Modeling and Applications, Genova, Italy, May 7–11, 2001, pp.89–98.
Dinh H Q, Turk G, Slabaugh G. Reconstructing surfaces using anisotropic basis functions. In Proc. the Eighth IEEE International Conference on Computer Vision (ICCV 2001), Vancouver, Canada, Jul. 9–12, 2001, pp.606–613.
Kazhdan M, Bolitho M, Hoppe H. Poisson surface reconstruction. In Proc. the 4th Eurographics Symposium on Geometry Processing, Cagliari, Italy, Jun. 26–28, 2006, pp.61–70.
Xie H, McDonnell K T, Qin H. Surface reconstruction of noisy and defective data sets. In Proc. the Conference on Visualization ’2004, Vancouver, Canada, Jul. 7–14, 2004, pp.259–266.
Ohtake Y, Belyaev A, Alexa M. Sparse low-degree implicit surfaces with applications to high quality rendering, feature extraction, and smoothing. In Proc. the 3rd Eurographics Symposium on Geometry Processing, Vienna, Austria, Jul. 4–6, 2005, Article No. 149.
Turk G. Re-tiling polygonal surfaces. Computer Graphics Association for Computing Machinery, 1992, 26(2): 55–64.
Pauly M, Gross M, Kobbelt L P, Hochschule E T, Zurich S. Efficient simplification of point-sampled surfaces. In Proc. VIS 2002, Boston, USA, Oct. 27-Nov. 1, 2002, pp.163–170.
Guennebaud G, Barthe L, Paulin M. Dynamic surfel set refinement for high-quality rendering. Computers & Graphics, 2004, 28(6): 827–838.
Guennebaud G, Barthe L, Paulin M. Interpolatory refinement for real-time processing of point-based geometry. Computer Graphics Forum, 2005, 24(3): 657–666.
Xiao C, Zheng W, Peng Q, Forrest A R. Robust morphing of point-sampled geometry. Computer Animation and Virtual Worlds, 2004, 15(3/4): 201–210.
Xiao C, Zheng W, Miao Y, Zhao Y, Peng Q. A unified method for appearance and geometry completion of point set surfaces. The Visual Computer, 2007, 23(6): 433–443.
Xiao C, Miao Y, Liu S, Peng Q. A dynamic balanced flow for filtering point-sampled geometry. The Visual Computer, 2006, 22(3): 210–219.
Xiao C, Fu H, Tai C L. Hierarchical aggregation for efficient shape extraction. The Visual Computer, 2009, 25(3): 267–278.
Miao Y W, Feng J Q, Xiao C X, Peng Q S, Forrest A R. Differentials-based segmentation and parameterization for point-sampled surfaces. Journal of Computer Science and Technology, 2007, 22(5): 749–760.
Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W. Surface reconstruction from unorganised points. Computer Graphics, 1992, 26(2): 71–77.
Franke R, Nielson G. Smooth interpolation of large sets of scattered data. International Journal for Numerical Methods in Engineering, 1980, 15(11): 1691–1704.
Taubin G. Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1991, 13(11): 1115–1138.
This work was partly supported by the National Natural Science Foundation of China under Grant Nos. 60803081, 61070081, the National High Technology Research and Development 863 Program of China under Grant No. 2008AA121603, the Fundamental Research Funds for the Central Universities under Grant No. 6081005, and the National Research Foundation for the Doctoral Program of Higher Education of China under Grant No. 200804861038.
About this article
Cite this article
Xiao, CX. Multi-Level Partition of Unity Algebraic Point Set Surfaces. J. Comput. Sci. Technol. 26, 229–238 (2011). https://doi.org/10.1007/s11390-011-9429-2
- moving least squares
- surface reconstruction
- implicit modeling
- partition of unity approximation