This paper presents a new scheme of flaw searching in surface modeling based on Euler Characteristic. This scheme can be applied to surface construction or reconstruction in computer. It is referred to as Euler Accompanying Test (EAT) algorithm in this paper. Two propositions in algebraic topology are presented, which are the foundation of the EAT algorithm. As the modeling is the first step for rendering in the animation and visualization, or computer-aided design (CAD) in related applications, the flaws can bring some serious problems in the final image or product, such as an artificial sense in animation rendering or a mistaken product in industry. To verify the EAT progressive procedure, a three-dimensional (3D) stamp model is constructed. The modeling process is accompanied by the EAT procedure. The EAT scheme is verified as the flaws in the stamp model are found and modified.
Guo, X., Xiao, J. and Wang, Y. (2018) A Survey on Algorithms of Hole Filling in 3D Surface Reconstruction. Visual Computer, 34, 93-103. https://doi.org/10.1007/s00371-016-1316-y
Jun, Y. (2005) A Piecewise Hole Filling Algorithm in Reverse Engineering. Computer-Aided Design, 37, 263-270. https://doi.org/10.1016/j.cad.2004.06.012
Attene, M., Campen, M. and Kobbelt, L. (2013) Polygon Mesh Repairing: An Application Perspective. ACM Computing Surveys, 45, Article No. 15. https://doi.org/10.1145/2431211.2431214
Casciola, G., Lazzaro, D., Montefusco, L.B. and Morigi, S. (2005) Fast Surface Reconstruction and Hole Filling Using Positive Definite Radial Basis Functions. Numerical Algorithms, 39, 289-305. https://doi.org/10.1007/s11075-004-3643-8
Wu, X. and Chen, W. (2014) A Scattered Point Set Hole-Filling Method Based on Boundary Extension and Convergence. Proceedings of the 11th World Congress on Intelligent Control and Automation, Shenyang, 29 June-4 July 2014, 5329-5334. https://doi.org/10.1109/WCICA.2014.7053624
Doria, D. and Radke, R.J. (2012) Filling Large Holes in LiDAR Data by Inpainting Depth Gradients. Proceedings of 2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, Providence, 16-21 June 2012, 65-72. https://doi.org/10.1109/CVPRW.2012.6238916
Yang, N.-E., Kim, Y.-G. and Park, R.-H. (2012) Depth Hole Filling Using the Depth Distribution of Neighboring Regions of Depth Holes in the Kinect Sensor. Proceedings of 2012 IEEE International Conference on Signal Processing, Communication and Computing, Hong Kong, 31 July-3August 2012, 658-661. https://doi.org/10.1109/ICSPCC.2012.6335696
Quinsat, Y. and Iartigue, C. (2015) Filling Holes in Digitized Point Cloud Using a Morphing-Based Approach to Preserve Volume Characteristics. The International Journal of Advanced Manufacturing Technology, 81, 411-421. https://doi.org/10.1007/s00170-015-7185-0
Bendels, G.H., Schnabel, R. and Klein, R. (2006) Detecting Holes in Point Set Surface. Journal of WSCG, 14, 89-96.
Methirumangalath, S., Kannan, S.S., Parakkat, A.D. and Muthuganapathy, R. (2017) Hole Detection in a Planar Point Set: An Empty Disk Approach. Computers & Graphics, 66, 124-134. https://doi.org/10.1016/j.cag.2017.05.006
Cunningham, S. (2008) Computer Graphics: Programming in OpenGL for Visual Communication. Prentice Hall Inc., Upper Saddle River.
Botsch, M. and Sorkine, O. (2008) On Linear Variational Surface Deformation Methods. IEEE Transactions on Visualization and Computer Graphics, 14, 213-230. https://doi.org/10.1109/TVCG.2007.1054
Xu, Q., Wang, J. and An, X. (2014) A Pipeline for Surface Reconstruction of 3-Dimentional Point Cloud. Proceedings of 2014 International Conference on Audio, Language and Image Processing, Shanghai, 7-9 July 2014, 822-826. https://doi.org/10.1109/ICALIP.2014.7009909
Wiemann, T., Mitschke, I., Mock, A. and Hertzberg, J. (2018) Surface Reconstruction from Arbitrarily Large Point Clouds. Proceedings of 2018 Second IEEE International Conference on Robotic Computing, Laguna Hills, 31 January-2 February 2018, 278-281. https://doi.org/10.1109/IRC.2018.00059
Odesanya, O.S., Waggenspack, W.N. and Thompson, D.E. (1993) Construction of Biological Surface Models from Cross-Sections. IEEE Transactions on Biomedical Engineering, 40, 329-334. https://doi.org/10.1109/10.222325
Shi, J., Zhu, L., Li, Z., Yang, J. and Wang, X. (2017) A Design and Fabrication Method for a Heterogeneous Model of 3D Bio-Printing. IEEE Access, 5, 5347-5353. https://doi.org/10.1109/ACCESS.2017.2692248
Fuhrmann, M. and Goesele, M. (2014) Floating Scale Surface Reconstruction. ACM Transactions on Graphics, 33, Article No. 46. https://doi.org/10.1145/2601097.2601163
Lávička, M., Šír, A. and Vršek, J. (2016) Smooth Surface Interpolation Using Patches with Rational Offset. Computer Aided Geometric Design, 48, 75-85. https://doi.org/10.1016/j.cagd.2016.09.002
Shen, J., Kosinka, J., Sabin, M. and Dodgson, N. (2016) Converting a CAD Model into a Non-Uniform Subdivision Surface. Computer Aided Geometric Design, 48, 17-35. https://doi.org/10.1016/j.cagd.2016.07.003
Sederberg, T.W., Zheng, J., Bakenov, A. and Nasri, A. (2003) T-Splines and T-NURCCs. ACM Transactions on Graphics, 22, 477-484. https://doi.org/10.1145/882262.882295
Coêlho, J., Gattass, M. and Lopes, H. (2019) ARTMe: A New Array-Based Algorithm for Adaptive Refinement of Triangle Meshes. Engineering with Computers, 35, 1-20. https://doi.org/10.1007/s00366-018-0579-5
Bock, K. and Stiller, J. (2018) Optimizing Triangular High-Order Surface Meshes by Energy-Minimization. Engineering with Computers, 34, 659-670. https://doi.org/10.1007/s00366-017-0565-3
Chen, H. and Shen, J. (2018) Denoising of Point Cloud Data for Computer-Aided Design, Engineering and Manufacturing. Engineering with Computers, 34, 523-541. https://doi.org/10.1007/s00366-017-0556-4
Hoschek, J. and Lasser, D. (1993) Fundamentals of Computer Aided Geometric Design. A K Peters, Wellesely.
Baker, H. (2004) Computer Graphics with OpenGL. Third Edition, Pearson Prentice Hall, Upper Saddle River.
Wang, Y., Zheng, J. and Wang, H. (2019) Fast Mesh Simplification Method for Three-Dimensional Geometric Models with Feature-Preserving Efficiency. Scientific Programming, 2019, Article ID: 4926190. https://doi.org/10.1155/2019/4926190
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J. and Stuetzle, W. (1992) Surface Reconstruction from Unorganized Points. Proceedings of SIGGRAPH’92, Chicago, 26-31 July 1992, 71-78. https://doi.org/10.1145/133994.134011
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J. and Stuetzle, W. (1993) Mesh Optimization. Proceedings of SIGGRAPH’93, Anaheim, 2-6 August 1993, 19-26. https://doi.org/10.1145/166117.166119
Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., Mcdonald, J., Schweitzer, J. and Stuetzle, W. (1994) Piecewise Smooth Surface Reconstruction. Proceedings of SIGGRAPH’94, Orlando, 24-29 July 1994, 295-302. https://doi.org/10.1145/192161.192233
Turk, G. and Levoy, M. (1994) Zippered Polygon Meshes from Range Images. Proceedings of SIGGRAPH’94, Orlando, 24-29 July 1994, 311-318. https://doi.org/10.1145/192161.192241
Catmull, E. and Clark, J. (1978) Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes. Computer-Aided Design, 10, 350-355. https://doi.org/10.1016/0010-4485(78)90110-0
Doo, D.W.H. and Sabin, M.A. (1978) Behaviour of Recursive Division Surfaces near Extraordinary Points. Computer-Aided Design, 10, 356-360. https://doi.org/10.1016/0010-4485(78)90111-2
Bolz, J. and Schröder, P. (2002) Rapid Evaluation of Catmull-Clark Subdivision Surfaces. Proceedings of the Seventh International Conference on 3D Web Technology, Tempe, 24-28 February 2002, 11-17. https://doi.org/10.1145/504502.504505
Sederberg, T.W., Zheng, J., Sewell, D. and Sabin, M. (1998) Non-Uniform Recursive Subdivision Surfaces. Proceedings of ACM SIGGRAPH’98, Orlando, 19-24 July 1998, 387-394. https://doi.org/10.1145/280814.280942
Velho, L. and Zorin, D. (2001) 4-8 Subdivision. Computer Aided Geometric Design, 18, 397-427. https://doi.org/10.1016/S0167-8396(01)00039-5
Müller, K., Reusche, L. and Fellner, D. (2006) Extended Subdivision Surfaces: Building a Bridge between NURBS and Catmull-Clark Surface. ACM Transactions on Graphics, 25, 268-292. https://doi.org/10.1145/1138450.1138455
Cashman, T.J., Augsdörfer, U.H., Dodgson, N.A. and Sabin, M.A. (2009) NURBS with Extraordinary Points: High-Degree, Non-Uniform, Rational Subdivision Schemes. ACM Transactions on Graphics, 28, Article No. 46. https://doi.org/10.1145/1531326.1531352
Cashman, T.J. (2010) NURBS-Compatible Subdivision Surfaces. Doctoral Thesis, University of Cambridge, Cambridge. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.608572
Shen, J., Kosinka, J., Sabin, M.A. and Dodgson, N.A. (2014) Conversion of Trimmed NURBS Surfaces to Catmull-Clark Subdivision Surfaces. Computer Aided Geometric Design, 31, 486-498. https://doi.org/10.1016/j.cagd.2014.06.004
Campen, M. and Zorin, D. (2017) Similarity Maps and Field-Guided T-Splines: A Perfect Couple. ACM Transactions on Graphics, 36, Article No. 91. https://doi.org/10.1145/3072959.3073647
Massy, W.S. (1991) Algebraic Topology: An Introduction. Springer-Verlag, Berlin/Heidelberg/New York.