On the shape of a set of points and lines in the plane

Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free‐form shape of a p...

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Published in:	Computer graphics forum Vol. 30; no. 5; pp. 1553 - 1562
Main Authors:	van Kreveld, M., van Lankveld, T., Veltkamp, R. C.
Format:	Journal Article
Language:	English
Published:	Oxford, UK Blackwell Publishing Ltd 01.08.2011
Subjects:	and systems Boundaries Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Boundary representations Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Geometric algorithms Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems Construction Geometry languages Lidar Marketing Mathematical models Planes Reconstruction Studies Three dimensional models
ISSN:	0167-7055, 1467-8659
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Abstract	Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free‐form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided α‐shape: an extension of the well known α‐shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided α‐shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided α‐shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided α‐shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α‐shapes of both synthetic and real data and show they are consistently better than α‐shapes for this application.
AbstractList	Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free‐form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided α‐shape: an extension of the well known α‐shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided α‐shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided α‐shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided α‐shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α‐shapes of both synthetic and real data and show they are consistently better than α‐shapes for this application. Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free-form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided alpha -shape: an extension of the well known alpha -shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided alpha -shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided alpha -shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided alpha -shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided alpha -shapes of both synthetic and real data and show they are consistently better than alpha -shapes for this application. Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free-form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided α-shape: an extension of the well known α-shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided α-shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided α-shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided α-shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α-shapes of both synthetic and real data and show they are consistently better than α-shapes for this application. [PUBLICATION ABSTRACT]
Author	Veltkamp, R. C. van Kreveld, M. van Lankveld, T.
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References	Salman N., Yvinec M., Merigot Q.: Feature preserving mesh generation from 3d point clouds. Computer Graphics Forum 29, 5 (2010), 1623-1632. 2. Van Kreveld M., Van Lankveld T., Veltkamp R. C.: On the shape of a set of points and lines in the plane. Tech. rep., Utrecht University, Department of Information and Computing Sciences, 2011. 4. Rottensteiner F.: Automatic generation of high-quality building models from lidar data. IEEE Computer Graphics and Applications 23, 6 (2003), 42-50. 2. Schnabel R., Wahl R., Klein R.: Efficient RANSAC for point-cloud shape detection. Computer Graphics Forum 26, 2 (2007), 214-226. 2, 9. Tseng Y.-H., Tang K.-P., Chou F.-C.: Surface reconstruction from LiDAR data with extended snake theory. In EMMCVPR ' 07 (2007), pp. 479-492. 1, 2. Edelsbrunner H., Kirkpatrick D. G., Seidel R.: On the shape of a set of points in the plane. In IEEE Transactions on Information Theory (1983), vol. 29, pp. 551-559. 2. Ruppert J.: A delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms 18, 3 (1995), 548-585. 3. Veltkamp R. C.: The γ-neighborhood graph. Computational Geometry Theory and Applications 1, 4 (1992), 227-246. 2. Guibas L., Knuth D., Sharir M.: Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7 (1992), 381-413. 7. Vosselman G., Gorte B. G. H., Sithole G., Rabban T.: Recognising structure in laser scanner point clouds. International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences 46 (2004), 33-38. 2. Chazelle, Devillers, Hurtado, Mora, Sacristan, Teillaud: Splitting a Delaunay triangulation in linear time. Algorithmica 34 (2002), 39-46. 7. Schnabel R., Degener P., Klein R.: Completion and reconstruction with primitive shapes. Computer Graphics Forum 28 (2009), 503-512. 2. Chew L. P.: Constrained delaunay triangulations. Algorithmica 4, 1 (1989), 97-108. 2, 6. Amenta N., Bern M., Eppstein D.: The crust and the β-skeleton: combinatorial curve reconstruction. Graphical Models and Image Processing 60, 2 (1998), 125-135. 2. 1992; 7 2007; 07 1989; 4 2001 2011 2010; 29 2002; 34 2004; 46 1998 2008 2007 2005 1992 2003 1998; 60 1995; 18 1983; 29 1992; 1 1979 2007; 26 2003; 23 2009; 28 e_1_2_7_5_2 e_1_2_7_4_2 e_1_2_7_3_2 e_1_2_7_2_2 e_1_2_7_9_2 e_1_2_7_8_2 Vosselman G. (e_1_2_7_21_2) 2004; 46 e_1_2_7_7_2 e_1_2_7_6_2 e_1_2_7_18_2 e_1_2_7_17_2 e_1_2_7_16_2 e_1_2_7_15_2 e_1_2_7_14_2 e_1_2_7_13_2 e_1_2_7_24_2 e_1_2_7_12_2 e_1_2_7_23_2 e_1_2_7_11_2 e_1_2_7_10_2 e_1_2_7_20_2 Van Kreveld M. (e_1_2_7_22_2) 2011 Tseng Y.‐H. (e_1_2_7_19_2) 2007; 07
References_xml	– reference: Schnabel R., Degener P., Klein R.: Completion and reconstruction with primitive shapes. Computer Graphics Forum 28 (2009), 503-512. 2. – reference: Veltkamp R. C.: The γ-neighborhood graph. Computational Geometry Theory and Applications 1, 4 (1992), 227-246. 2. – reference: Schnabel R., Wahl R., Klein R.: Efficient RANSAC for point-cloud shape detection. Computer Graphics Forum 26, 2 (2007), 214-226. 2, 9. – reference: Guibas L., Knuth D., Sharir M.: Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7 (1992), 381-413. 7. – reference: Rottensteiner F.: Automatic generation of high-quality building models from lidar data. IEEE Computer Graphics and Applications 23, 6 (2003), 42-50. 2. – reference: Tseng Y.-H., Tang K.-P., Chou F.-C.: Surface reconstruction from LiDAR data with extended snake theory. In EMMCVPR ' 07 (2007), pp. 479-492. 1, 2. – reference: Edelsbrunner H., Kirkpatrick D. G., Seidel R.: On the shape of a set of points in the plane. In IEEE Transactions on Information Theory (1983), vol. 29, pp. 551-559. 2. – reference: Vosselman G., Gorte B. G. H., Sithole G., Rabban T.: Recognising structure in laser scanner point clouds. International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences 46 (2004), 33-38. 2. – reference: Chew L. P.: Constrained delaunay triangulations. Algorithmica 4, 1 (1989), 97-108. 2, 6. – reference: Ruppert J.: A delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms 18, 3 (1995), 548-585. 3. – reference: Amenta N., Bern M., Eppstein D.: The crust and the β-skeleton: combinatorial curve reconstruction. Graphical Models and Image Processing 60, 2 (1998), 125-135. 2. – reference: Salman N., Yvinec M., Merigot Q.: Feature preserving mesh generation from 3d point clouds. Computer Graphics Forum 29, 5 (2010), 1623-1632. 2. – reference: Van Kreveld M., Van Lankveld T., Veltkamp R. C.: On the shape of a set of points and lines in the plane. Tech. rep., Utrecht University, Department of Information and Computing Sciences, 2011. 4. – reference: Chazelle, Devillers, Hurtado, Mora, Sacristan, Teillaud: Splitting a Delaunay triangulation in linear time. Algorithmica 34 (2002), 39-46. 7. – year: 2011 – volume: 60 start-page: 125 issue: 2 year: 1998 end-page: 135 article-title: The crust and the β‐skeleton: combinatorial curve reconstruction publication-title: Graphical Models and Image Processing – start-page: 579 year: 2003 end-page: 588 – start-page: 1 year: 2008 end-page: 8 – volume: 29 start-page: 551 year: 1983 end-page: 559 article-title: On the shape of a set of points in the plane publication-title: IEEE Transactions on Information Theory – start-page: 67 year: 1998 end-page: 72 – volume: 46 start-page: 33 year: 2004 end-page: 38 article-title: Recognising structure in laser scanner point clouds publication-title: International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences – volume: 34 start-page: 39 year: 2002 end-page: 46 article-title: Splitting a Delaunay triangulation in linear time publication-title: Algorithmica – volume: 29 start-page: 1623 issue: 5 year: 2010 end-page: 1632 article-title: Feature preserving mesh generation from 3d point clouds publication-title: Computer Graphics Forum – start-page: 57 year: 2001 end-page: 66 – volume: 07 start-page: 479 year: 2007 end-page: 492 article-title: Surface reconstruction from LiDAR data with extended snake theory publication-title: EMMCVPR ' – start-page: 39 year: 2007 end-page: 48 – start-page: 53 year: 1992 end-page: 62 – volume: 23 start-page: 42 issue: 6 year: 2003 end-page: 50 article-title: Automatic generation of high‐quality building models from lidar data publication-title: IEEE Computer Graphics and Applications – volume: 4 start-page: 97 issue: 1 year: 1989 end-page: 108 article-title: Constrained delaunay triangulations publication-title: Algorithmica – volume: 7 start-page: 381 year: 1992 end-page: 413 article-title: Randomized incremental construction of Delaunay and Voronoi diagrams publication-title: Algorithmica – volume: 1 start-page: 227 issue: 4 year: 1992 end-page: 246 article-title: The γ‐neighborhood graph publication-title: Computational Geometry Theory and Applications – start-page: 55 year: 2005 end-page: 61 – start-page: 1383 year: 2008 end-page: 1391 – volume: 18 start-page: 548 issue: 3 year: 1995 end-page: 585 article-title: A delaunay refinement algorithm for quality 2‐dimensional mesh generation publication-title: Journal of Algorithms – volume: 26 start-page: 214 issue: 2 year: 2007 end-page: 226 article-title: Efficient RANSAC for point‐cloud shape detection publication-title: Computer Graphics Forum – start-page: 18 year: 1979 end-page: 27 – volume: 28 start-page: 503 year: 2009 end-page: 512 article-title: Completion and reconstruction with primitive shapes publication-title: Computer Graphics Forum – ident: e_1_2_7_5_2 – ident: e_1_2_7_9_2 doi: 10.1007/BF01758770 – ident: e_1_2_7_13_2 doi: 10.1109/MCG.2003.1242381 – ident: e_1_2_7_23_2 doi: 10.1007/3-540-44842-X_59 – ident: e_1_2_7_4_2 doi: 10.1007/s00453-002-0939-8 – ident: e_1_2_7_14_2 doi: 10.1006/jagm.1995.1021 – ident: e_1_2_7_15_2 doi: 10.1111/j.1467-8659.2009.01389.x – ident: e_1_2_7_3_2 – volume-title: On the shape of a set of points and lines in the plane year: 2011 ident: e_1_2_7_22_2 doi: 10.1111/j.1467-8659.2011.02029.x – ident: e_1_2_7_10_2 doi: 10.1145/383259.383265 – ident: e_1_2_7_18_2 – ident: e_1_2_7_24_2 doi: 10.1145/1463434.1463444 – ident: e_1_2_7_7_2 doi: 10.1109/TIT.1983.1056714 – ident: e_1_2_7_16_2 doi: 10.1111/j.1467-8659.2007.01016.x – ident: e_1_2_7_2_2 doi: 10.1006/gmip.1998.0465 – ident: e_1_2_7_8_2 doi: 10.1145/142675.142690 – ident: e_1_2_7_12_2 doi: 10.1111/j.1467-8659.2008.01278.x – volume: 07 start-page: 479 year: 2007 ident: e_1_2_7_19_2 article-title: Surface reconstruction from LiDAR data with extended snake theory publication-title: EMMCVPR ' – ident: e_1_2_7_11_2 doi: 10.1109/SFCS.1979.15 – volume: 46 start-page: 33 year: 2004 ident: e_1_2_7_21_2 article-title: Recognising structure in laser scanner point clouds publication-title: International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences – ident: e_1_2_7_20_2 doi: 10.1016/0925-7721(92)90003-B – ident: e_1_2_7_6_2 doi: 10.1007/BF01553881 – ident: e_1_2_7_17_2 doi: 10.1111/j.1467-8659.2010.01771.x
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SubjectTerms	and systems Boundaries Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Boundary representations Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Geometric algorithms Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems Construction Geometry languages Lidar Marketing Mathematical models Planes Reconstruction Studies Three dimensional models
Title	On the shape of a set of points and lines in the plane
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