Alexander Hornung and Leif Kobbelt RWTH Aachen

1. Alexander Hornung and Leif Kobbelt RWTH Aachen • Robust Reconstruction of • Watertight 3D Models from • Non-uniformly Sampled Point Clouds Without Normal Information

2. Point Cloud Reconstruction

3. Point Cloud Reconstruction • Non-uniform sampling • Holes • Noise • Bad scan alignment • No (reliable) normals

4. Point Cloud Reconstruction • Smooth watertight manifold • No topological artifacts (low genus) • Detail preservation • Robustness to • Non-uniform sampling • Holes • Bad registration and noise • From 3D points only

5. Outline • Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results

6. Related Work • Wrapping and Voronoi-based • Amenta et al., Bernardini et al., Boissonat and Cazals, Dey and Goswami, Mederos et al., Scheidegger et al., … • Deformable models • Esteve et al., Sharf et al., … • Volumetric reconstruction • Hoppe et al., Curless and Levoy, Carr et al., Ohtake et al., Fleishman et al., Kazhdan, …

7. Related Work • Wrapping and Voronoi-based • Amenta et al., Bernardini et al., Boissonat and Cazals, Dey and Goswami, Mederos et al., Scheidegger et al., … • Deformable models • Esteve et al., Sharf et al., … • Volumetric reconstruction • Hoppe et al., Curless and Levoy, Carr et al., Ohtake et al., Fleishman et al., Kazhdan, … • Graph-based energy minimization and surface reconstruction • Boykov and Kolmogorov, Vogiatzis et al., Hornung and Kobbelt

8. Signed vs. Unsigned Distance

9. Signed vs. Unsigned Distance

10. Signed vs. Unsigned Distance

11. Signed vs. Unsigned Distance

12. Overview • Point cloud P

13. Overview • Point cloud P • Surface confidence (unsigned distance)

14. Overview • Point cloud P • Surface confidence (unsigned distance) • Embed weighted graph structure G

15. Overview • Point cloud P • Surface confidence (unsigned distance) • Embed weighted graph structure • Min-Cut of G yields unknown surface

16. Outline • Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results

17. Surface Confidence • Insert 3D samples into volumetric grid • Sparse set of occupied voxels • Compute a confidence map  “Probability” that surface intersects a voxel v

18. Surface Confidence • Insert 3D samples into volumetric grid • Sparse set of occupied voxels • Compute a confidence map  “Probability” that surface intersects a voxel v • Compute “crust” containing the surface • Morphological dilation • Medial axis approximation

19. Surface Confidence • Insert 3D samples into volumetric grid • Sparse set of occupied voxels • Compute a confidence map  “Probability” that surface intersects a voxel v • Compute “crust” containing the surface • Morphological dilation • Medial axis approximation • Estimate by volumetric diffusion

20. Outline • Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results

21. Find Optimal Surface • Minimize energy • Min-Cut of an embedded graph • Global optimum • Highly efficient • Graph structure?

22. Dual Graph Embedding • : Probability that v is intersected by surface s • Intersected voxels are split into 2 components • Interior faces • Exterior faces

23. Dual Graph Embedding Voxel split-edges Graph cut-edges • : Probability that v is intersected by surface s • Intersected voxels are split into 2 components • Interior faces • Exterior faces  Split along a sequence of edges • Octahedral graph structure

24. Min-Cut Surface Extraction • Embed graph into a crust containing the surface

25. Min-Cut Surface Extraction • Embed graph into a crust containing the surface • Edge weights defined per voxel

26. Min-Cut Surface Extraction • Embed graph into a crust containing the surface • Edge weights defined per voxel • Min-cut yields set of intersected surface voxels

27. Min-Cut Surface Extraction • Embed graph into a crust containing the surface • Edge weights defined per voxel • Min-cut yields set of intersected surface voxels • Parameter s to emphasize strong/weak maxima

28. Outline • Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results

29. Hierarchical Approach • Single resolution impractical • High volumetric resolutions • Non-uniform sampling / large holes • Hierarchical framework • Adaptive volumetric grid (Octree) • Proper initial crust at low resolutions • Simple narrow-band approach insufficient • Loss of fine details not contained within crust

30. Hierarchical Approach • Single resolution impractical • High volumetric resolutions • Non-uniform sampling / large holes • Hierarchical framework • Adaptive volumetric grid (Octree) • Proper initial crust at low resolutions • Simple narrow-band approach insufficient • Loss of fine details not contained within crust  Re-insertion of data samples • Merge samples with crust

31. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 643

32. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 1283

33. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 1283

34. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 2563

35. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 2563

36. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 5123

37. Hierarchical Approach • Surface confidence estimation • (Re-)Insert point samples • Dilate and compute • Graph-based surface extraction • Generate octahedral graph • Compute min-cut • Volumetric refinement • Narrow band 5123

38. Outline • Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results

39. Cut Manifold to Triangle Mesh Graph cut-edges Loop of voxel split-edges

40. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners

41. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges

42. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges

43. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges

44. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges

45. Cut Manifold to Triangle Mesh • Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges

46. Cut Manifold to Triangle Mesh • estimated per voxel  Mesh vertices at voxel centers

47. Cut Manifold to Triangle Mesh • estimated per voxel  Mesh vertices at voxel centers

48. Cut Manifold to Triangle Mesh • estimated per voxel  Mesh vertices at voxel centers • Voxel corners correspond to non-planar faces • Cycle over shared split-edges

49. Cut Manifold to Triangle Mesh • estimated per voxel  Mesh vertices at voxel centers • Voxel corners correspond to non-planar faces • Cycle over shared split-edges

50. Cut Manifold to Triangle Mesh • estimated per voxel  Mesh vertices at voxel centers • Voxel corners correspond to non-planar faces • Cycle over shared split-edges