對離散點之三維表面重建問題是一個被電腦圖學廣泛討論的題目，並被廣泛應用於不同的領域上，如科學，工程和藝術方面等問題。對具有足夠採樣點的三維模型，許多演算法能夠正確重建出其原始表面。然而，在現實中的表面模型通常包含銳利區域，例如摺痕線，邊緣，圓錐尖點等。一般的三維取樣技術通常無法在這些銳利區域上取樣。事實上，幾乎沒有一個取樣點會落在這些銳利區域。因此，這些銳利區域的資訊將在取點過程中遺失。這篇論文中，我們提出一個新穎且有效的方法，不僅可以順利重建平滑區域表面，也能成功恢復存在於表面的的尖銳特徵。我們利用polar normal和cocone algorithm對物體的平滑地區附近的尖銳特徵來估計鄰近銳利區域的新點位置。對於不同的銳利區域特徵，我們進行了一系列實驗並成功地重建了物體表面。此外，我們也進行了對noise data的各項測試，實驗結果證明，我們的演算法成功地提升了各種含銳利區域模型之表面重建結果。

Surface reconstruction problem is a classical problem which finds wide applications in different disciplines, such as science, engineering and arts. For well-sampled point set from a smooth surface, there have been many algo-
rithms to correctly reconstruct a triangular mesh from the point set. However, surface models in reality usually contains some sharp features, such as crease lines, corners and apices where C-1 continuity is not guaranteed on
the surface. The acquisition process of point clouds are usually not adjusted to obtain the sample points sitting right on the sharp features of the given model. In fact, almost none of the sample points acquired lie exactly on such sharp features. Therefore, the information of sharp features of the original surface model are lost during this sample acquisition process. Our goal is to propose a new and effective method that not only gives good reconstruction
to the smooth areas of the given surface, but also recovers the sharp features originally existed in the source surface model provided that a reasonably nice sampling point cloud is given. We present an algorithm that elegantly make use of the polar normals obtained from the co-cone algorithm and the triangles on the smoother areas near the sharp features to estimate the positions of the inserted points
approximating locally sharp features. We have also conducted some preliminaries experimentation on some surface models with different kinds of sharp features, and use the piecewise sharp feature groups to estimate corner or
apex features. Also, the identifying of group of sharp features can be a great information to handle the noised model, by constraining the de-noising process to the surface vertices without crossing sharp features. We find that our algorithm significantly improves the quality of the reconstructed surfaces in most cases.

[1] M. Attene and M. Spagnuolo. Automatic surface reconstruction from point sets in space. Computer Graphics Forum (Procs. EUROGRAPHICS ’00), 19(3), 457–465, 2000.

[2] N. Amenta, S. Choi and R. Kolluri. The power crust. In 6th ACM Symposium on Solid Modeling and Applications, 249–260, 2001.

[3] J. Giesen and M. John. Surface reconstruction based on a dynamical system. Computer Graphics Forum (Procs. EUROGRAPHICS ’02), 21(3),
363–371, 2002.

[4] W. Lorensen and H. Cline. Marching Cubes: a high resolution 3D surface construction algorithm, Computer Graphics (Procs. SIGGRAPH ’87),163–169, 1987.

[5] J. Bloomenthal. Polygonization of implicit surfaces, Computer Aided Geometric Design, 5, 341–355, 1988.

[6] S. W. Cheng and T. K. Dey. Improved construction of Delaunay based contour surfaces. In Proc. ACM Sympos. Solid Modeling and Applications,322–323, 1999.

[7] G. Cong and B. Parving. Robust and Efficient Surface Reconstruction from Contours. The Visual Computer, 17,199–208, 2001.

[8] B. Curless and M. Levoy. A volumetric method for building complex models from range images. SIGGRAPH ’96, 303–312, 1996.

[9] H. Hoppe, T. DePose, T. Duchamp, J. McDonald and W. Stuetzle. Surface reconstruction from unorganized points. SIGGRAPH ’92, 71–78,1992.

[10] L. P. Kobbelt, M. Botsch, U. Schwanecke and H-P. Seidel. Feature Sensitive Surface Extraction from Volume Data. Computer Graphics (Proc.SIGGRAPH ’01, 57–66, 2001.)

[11] J. D. Boissonnat. Geometric structures for three dimensional shape representation. ACM Transact. on Graphics, 3(4), 266–286, 1984.

[12] H. Edelsbrunner and E. P. Mucke. Three-dimensional alpha shapes. ACM Trans. Graphics, 13, 43–72, 1994.

[13] N. Amenta, M. Bern and M. Kamvysselis. A new Voronoi-based surface reconstruction algorithm. SIGGRAPH ’98, 415–421, 1998.

[14] N. Amenta, S. Choi, T. K. Dey and N. Leekha. A simple algorithm for homeomorphic surface reconstruction. Internat. J. Comput. Geom. &Applications, 12, 125–141, 2002.

[15] T.K. Dey, J. Giesen, N. Leekha, and R. Wenger. Detecting Boundaries for Surface Reconstruction Using Co-Cones. Internat. J. Computer Graphics and CAD/CAM, 16, 141–159, 2001.

[16] T.K. Dey and S. Goswami. Tight Co-cone: A Water-Tight Surface Reconstructor. In Proc. 8th ACM Symp. Solid Modeling and Applications,127–134, 2003.

[17] T. K. Dey, J. Giesen and J. Hudson. Delaunay based shape reconstruction from large data. IEEE Symposium in Parallel and Large Data Visualization and Graphics, 19–27, 2001.

[18] H.T. Yau, C.C. Kuo, and C.H. Yeh. Extension of Surface Reconstruction Algorithm to the Global Stitching and Repairing of STL Models.Computer-Aided Design, 35(5), 477–486, 2002.

[19] C.C. Kuo and H.T. Yau. Reconstruction of Virtual Parts from Unorganized Scanned Data for Automated Dimensional Inspection. J. Computing and Information Science in Eng. (JCISE), Trans. ASME, 3(1),76–86, 2003.

[20] C.C. Kuo and H.T. Yau. A New Combinatorial Approach to Surface Reconstruction with Sharp Features. IEEE Transactions on Visualization and Computer Graphics, 12(1), 73–82, 2006.

[21] Charlie C.L. Wang. Bilateral Recovering of Sharp Edges on Feature Insensitive Sampled Meshes. IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 4,JULY/AUGUST 2006.

[22] S.Fleishman, D.Cohen-Or and T. Silva. Robust Moving Least-squares Fitting with Sharp Features. Proceedings of ACM SIGGRAPH, 2005

[23] E. Mencl and H. Muller. Graph-Based Surface Reconstruction Using Structures in Scattered Point Sets. In Proc. CGI ’98 (Computer Graphics International Conference), 298–311, 1998.

[24] F. Bernardini, J. Mittleman, H. Rushmeier, C. Silva, and G. Taubin. The Ball-Pivoting Algorithm for Surface Reconstruction. IEEE Trans.Visualization and Computer Graphics, 5(4), 349–359, 1999.

[25] M. Gopi and S. Krishnan. A Fast and Efficient Projection-Based Approach for Surface Reconstruction. Internat. J. High-Performance Computer Graphics, Multimedia and Visualization, 1(1), 1–12, 2000.

[26] S. Petitjean and E. Boyer. Regular and Non-Regular Point Sets: Properties and Reconstruction. Computational Geometry: Theory and Applications, 19, 101–126, 2001.

[27] J. Huang and C.H. Menq. Combinatorial Manifold Mesh Reconstruc tion and Optimization from Unorganized Points with Arbitrary Topology. Computer-Aided Design, 34(2), 149–165, 2002.

[28] D. Cohen-Steiner. A Greedy Delaunay Based Surface Reconstruction Algorithm. Research report, INRIA, 2002.
[29] C.C. Kuo and H.T. Yau. A Delaunay-Based Region-Growing Approach to Surface Reconstruction from Unorganized Points. Computer-Aided Design, 37,825–835, 2005.

[30] M. Attene, B. Falcidieno, J. Rossignac and M. Spagnulo. Edge Sharpener: Recovering features in triangulation of non-adaptive remeshed surfaces. The Eurographics Association, 2003.

[31] F. Bernardini, C. Bajaj, J. Chen, and D. Schikore. Automatic Reconstruction of 3D CAD Models from Digital Scans. International J. Computational Geometry Theory and Applications, 9(4-5), 327–370, 1999.

[32] J.-D. Boissonnat and F. Cazals. Smooth Surface Reconstruction via Natural Neighbor Interpolation of Distance Functions. In Proc. 16th ACM Symp. Computational Geometry, 223–232, 2000.

[33] J.C. Carr, R.K. Beatson, J.B. Cherrie, T.J. Mitchell, W.R. Fright, B.C. McCallum, and T.R. Evans. Reconstruction and Representation of 3D Objects with Radial Basis Functions. In Proc. SIGGRAPH ’01, 67–76,2001.

[34] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Mesh Optimization. In Proc. SIGGRAPH ’93, 191–226, 1993.

[35] L. Kobbelt, M. Botsch, U. Schwanecke, and H.P. Seidel. Feature Sensitive Surface Extraction from Volume Data. In Proc. SIGGRAPH ’01,57–66, 2001.