Junjie Cao 1, Andrea Tagliasacchi 2, Matt Olson 2, Hao Zhang 2, Zhixun Su 1 1 Dalian University of Technology 2 Simon Fraser University Point Cloud Skeletons via Laplacian-Based Contraction

Curve skeletons and their applications 2 A 1D curve providing a compact representation of the shape [Cornea et al ]

Existing curve skeleton extraction methods 1. Voxel thinning 2. Template skeleton adaption 3. Pruning medial axis 4. Volume contraction 5. Mesh contraction [Baran and Popovic 2007] [Au et al. 2008] [Bucksch and Lindenbergh 2008] [Dey and Sun 2006] [Wang and Lee 2008] 3

Existing curve skeleton extraction methods 1. Reeb graph 2. Geometry snake 3. Generalized rotational symmetry axis [Verroust and Lazarus 2000] [Sharf et al. 2007] [Tagliasacchi et al. 2009] 4

Is extracting skeleton directly from point cloud data necessary? PCD with missing partPoisson reconstruction and skeletonization by mesh contraction [Au et al. 2008] Our method Point cloud Missing data Volume Mesh Skeleton 5

Contributions 1. Directly on point cloud 2. No normal or any strong prior 3. Application of point cloud Laplacian 4. Skeleton-assisted topology-preserving reconstruction 6

Outline Geometry contractionTopological thinning 7

Geometry Contraction  Minimizing the quadratic energy iteratively: Contraction constraint Attraction constraint Laplacian constraint weightsPosition constraint weights 8

Laplacian construction for point cloud  Voronoi-Laplacian, PCD-Laplacian?  Planar Delaunay triangulation of points within a distance R  Assumption: point cloud is smooth enough and well sampled  KNN + 1-ring of local (planar) Delaunay triangulation  Keep the 1-ring during the contraction iterations  Cotangent weights Voronoi-Laplacian: C. Luo, I. Safa, and Y. Wang, “Approximating gradients for meshes and point clouds via diffusion metric”, Computer Graphics Forum, vol. 28, no. 5, pp. 1497–1508, PCD-Laplacian: M. Belkin, J. Sun, and Y. Wang, “Constructing Laplace operator from point clouds in R d ”, in Proc. of ACM Symp. on Discrete Algorithms, pp. 1031–104, ε-sampling (ε,δ)-sampling

Topological thinning Previous approach: MLS projection (line thinning) + Joint identification Our approach: Building connectivity + Edge collapse [Li et al. 2001] [Shapira et al. 2008], [Tagliasacchi et al. 2009] 10

Topological thinning – Farthest point sampling 1.Sample contracted points using farthest-point sampling and a ball of radius r (r=0.02*diag(BBOX|P|) ) 11

Topological thinning – Building connectivity Adjacency matrix 1.Sample contracted points using farthest-point sampling and a ball of radius r (r=0.02*diag(BBOX|P|) ) 2.Connecting two samples if their associated points share common local 1- ring neighbors i j i j skeleton point point on contracted point cloud point on the original point cloud 12

Topological thinning – Edge collapse 1.Sample contracted points using farthest-point sampling and a ball of radius r (r=0.02*diag(BBOX|P|) ) 2.Connecting two samples if their associated points share common local 1- ring neighbors 3.Collapse unnecessary edges until no triangles exist 13

Gallery 14 Spherical region Sheet-like region Close-by structure Missing data Genus Surfaces with boundaries

Insensitive to random noise 1%, 2% and 3% random noise 15

Insensitive to misalignment 0.5%, 1% and 1.5% misalignment noise 16

Insensitive to non-uniform sampling 17

Comparison with [Au et al. 2008] 18 Mesh model Point Cloud model [Au et al. 2008] Our method [Au et al. 2008] Our method

Comparison with four methods in [Cornea_tvcg07] 19

More comparisons ReebDeformable blobROSAOur methodMesh contraction Comparison with ReebComparison with Potential Field 20

Skeleton driven point cloud reconstruction 2. Reconstruction along a skeleton branch 1. Reconstruction on a skeleton cross-section 21

Skeleton driven point cloud reconstruction 22

Limitations and future work 1. Improve neighborhood construction  Handle close-by structures 2. Use the curve skeleton to repair the point clouds directly 23

24 Anonymous Reviewers NSFC (No and No. U ) NSERC (No ) Acknowledgements