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## Alexander Hornung and Leif Kobbelt RWTH Aachen

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**Alexander Hornung and Leif Kobbelt**RWTH Aachen • Robust Reconstruction of • Watertight 3D Models from • Non-uniformly Sampled Point Clouds Without Normal Information**Point Cloud Reconstruction**• Non-uniform sampling • Holes • Noise • Bad scan alignment • No (reliable) normals**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**Outline**• Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results**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, …**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**Overview**• Point cloud P**Overview**• Point cloud P • Surface confidence (unsigned distance)**Overview**• Point cloud P • Surface confidence (unsigned distance) • Embed weighted graph structure G**Overview**• Point cloud P • Surface confidence (unsigned distance) • Embed weighted graph structure • Min-Cut of G yields unknown surface**Outline**• Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results**Surface Confidence**• Insert 3D samples into volumetric grid • Sparse set of occupied voxels • Compute a confidence map “Probability” that surface intersects a voxel v**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**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**Outline**• Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results**Find Optimal Surface**• Minimize energy • Min-Cut of an embedded graph • Global optimum • Highly efficient • Graph structure?**Dual Graph Embedding**• : Probability that v is intersected by surface s • Intersected voxels are split into 2 components • Interior faces • Exterior faces**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**Min-Cut Surface Extraction**• Embed graph into a crust containing the surface**Min-Cut Surface Extraction**• Embed graph into a crust containing the surface • Edge weights defined per voxel**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**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**Outline**• Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results**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**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**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**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**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**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**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**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**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**Outline**• Introduction • Surface confidence estimation • Graph-based surface extraction • Hole filling and detail preservation • Mesh extraction • Results**Cut Manifold to Triangle Mesh**Graph cut-edges Loop of voxel split-edges**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges**Cut Manifold to Triangle Mesh**• Loops define non-planar polygonal faces • Mesh vertices at voxel corners • Cycle along split-edges**Cut Manifold to Triangle Mesh**• estimated per voxel Mesh vertices at voxel centers**Cut Manifold to Triangle Mesh**• estimated per voxel Mesh vertices at voxel centers**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**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**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