Fast Non-Convex Hull Computation

3D surface reconstruction usually begins with a point cloud and aims to build a representation of the object producing that point cloud. There are several algorithms to solve this problem, each with different priors over the point cloud, such as the type of object represented, or the method by which it was obtained. In this work, we focus on an algorithm called Non-Convex Hull (NCH), which reconstructs surfaces through a concept similar to the Medial Axis Transform. A new algorithm called Shrinking Planes is proposed to compute the NCH, based on the Shrinking Ball method with a few improvements. We prove that the new method can approximate surfaces to arbitrarily small error, and evaluate its performance on the surface reconstruction task. The new method maintains the same reconstruction quality as the Naïve Non-Convex Hull method, while achieving a large performance improvement.

[1]  Tamal K. Dey,et al.  Curve and Surface Reconstruction , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[2]  Gabriel Taubin,et al.  Non-convex hull surfaces , 2013, SIGGRAPH ASIA Technical Briefs.

[3]  Ravi Peters,et al.  Robust approximation of the Medial Axis Transform of LiDAR point clouds as a tool for visualisation , 2016, Comput. Geosci..

[4]  Pierre Alliez,et al.  A Survey of Surface Reconstruction from Point Clouds , 2017, Comput. Graph. Forum.

[5]  Cláudio T. Silva,et al.  Medial Kernels , 2012, Comput. Graph. Forum.

[6]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[7]  M. Gross,et al.  Algebraic point set surfaces , 2007, SIGGRAPH 2007.

[8]  Alexandru Telea,et al.  Surface and Curve Skeletonization of Large 3D Models on the GPU , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Filip Biljecki,et al.  Visibility Analysis in a Point Cloud Based on the Medial Axis Transform , 2015, UDMV.

[10]  Gregory M. Nielson,et al.  Dual marching cubes , 2004, IEEE Visualization 2004.

[11]  Dinesh Manocha,et al.  Efficient computation of a simplified medial axis , 2003, SM '03.

[12]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[13]  W. Walthen-Dunn.  A Transformation for Extracting New De scriptors of Shape ' , in , 2017 .

[14]  Gabriel Taubin,et al.  SSD-C: Smooth Signed Distance Colored Surface Reconstruction , 2012, Expanding the Frontiers of Visual Analytics and Visualization.

[15]  Benjamin B. Kimia,et al.  A formal classification of 3D medial axis points and their local geometry , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[16]  Ravi Peters.  Geographical point cloud modelling with the 3D medial axis transform , 2018 .

[17]  Sang Won Bae,et al.  3D medial axis point approximation using nearest neighbors and the normal field , 2011, The Visual Computer.

[18]  Nicholas M. Patrikalakis,et al.  Differential and Topological Properties of Medial Axis Transforms , 1996, CVGIP Graph. Model. Image Process..

[19]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[20]  Michael M. Kazhdan,et al.  Screened poisson surface reconstruction , 2013, TOGS.

[21]  Marc Alexa,et al.  Point set surfaces , 2001, Proceedings Visualization, 2001. VIS '01..

[22]  Kaleem Siddiqi,et al.  Medial Representations: Mathematics, Algorithms and Applications , 2008 .

[23]  Wolfgang Heidrich,et al.  Shape simplification based on the medial axis transform , 2003, IEEE Visualization, 2003. VIS 2003..

[24]  Sunghee Choi,et al.  The power crust, unions of balls, and the medial axis transform , 2001, Comput. Geom..

[25]  Tao Ju,et al.  Dual contouring of hermite data , 2002, ACM Trans. Graph..

[26]  Dominique Faudot,et al.  From A Medial Surface To A Mesh , 2012, Comput. Graph. Forum.

[27]  Gabriel Taubin,et al.  Smooth Signed Distance Surface Reconstruction and Applications , 2012, CIARP.

[28]  Michael M. Kazhdan,et al.  Poisson surface reconstruction , 2006, SGP '06.

[29]  Kaleem Siddiqi,et al.  Medial Spheres for Shape Approximation , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[30]  H. Seidel,et al.  Multi-level partition of unity implicits , 2003 .