computer graphics group alexander hornung alexander hornung and leif kobbelt rwth aachen robust...
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Computer Graphics GroupAlexander Hornung
Alexander Hornung and Leif KobbeltRWTH Aachen
Robust Reconstruction of Watertight 3D Models from
Non-uniformly Sampled Point Clouds Without Normal Information
Computer Graphics GroupAlexander Hornung
Point Cloud Reconstruction
Computer Graphics GroupAlexander Hornung
Point Cloud Reconstruction
• Non-uniform sampling
• Holes
• Noise
• Bad scan alignment
• No (reliable) normals
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
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, …
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
Signed vs. Unsigned Distance
Computer Graphics GroupAlexander Hornung
Signed vs. Unsigned Distance
Computer Graphics GroupAlexander Hornung
Signed vs. Unsigned Distance
Computer Graphics GroupAlexander Hornung
Signed vs. Unsigned Distance
Computer Graphics GroupAlexander Hornung
Overview
• Point cloud P
Computer Graphics GroupAlexander Hornung
Overview
• Point cloud P
• Surface confidence (unsigned distance)
)(v
Computer Graphics GroupAlexander Hornung
Overview
• Point cloud P
• Surface confidence (unsigned distance)
• Embed weighted graph structure G
)(v
Computer Graphics GroupAlexander Hornung
Overview
• Point cloud P
• Surface confidence (unsigned distance)
• Embed weighted graph structure
• Min-Cut of G yields unknown surface
)(v
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
Surface Confidence
• Insert 3D samples into volumetric grid• Sparse set of occupied voxels
• Compute a confidence map
“Probability” that surface intersects a voxel v
)(v
Computer Graphics GroupAlexander Hornung
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
)(v
Computer Graphics GroupAlexander Hornung
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
)(v
)(v
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
Find Optimal Surface
• Minimize energy
• Min-Cut of an embedded graph• Global optimum
• Highly efficient
• Graph structure?
S S
dSadxxSE )()(
Computer Graphics GroupAlexander Hornung
Dual Graph Embedding
• : Probability that v is intersected by surface s
• Intersected voxels are split into 2 components• Interior faces
• Exterior faces
)(v
Computer Graphics GroupAlexander Hornung
Dual Graph Embedding
• : 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
)(v
Voxel split-edges
Graph cut-edges
Computer Graphics GroupAlexander Hornung
Min-Cut Surface Extraction
• Embed graph into a crust containing the surface
Computer Graphics GroupAlexander Hornung
Min-Cut Surface Extraction
• Embed graph into a crust containing the surface
• Edge weights defined per voxel avvw s )()(
Computer Graphics GroupAlexander Hornung
• 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
avvw s )()(
Computer Graphics GroupAlexander Hornung
• 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
Min-Cut Surface Extraction
avvw s )()(
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
643
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
1283
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
1283
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
2563
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
2563
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
5123
Computer Graphics GroupAlexander Hornung
Hierarchical Approach
1) Surface confidence estimation• (Re-)Insert point samples
• Dilate and compute
2) Graph-based surface extraction• Generate octahedral graph
• Compute min-cut
3) Volumetric refinement• Narrow band
)(v
5123
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
Loop of voxel split-edgesGraph cut-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
• Cycle along split-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
• Cycle along split-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
• Cycle along split-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
• Cycle along split-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Loops define non-planar polygonal faces
• Mesh vertices at voxel corners
• Cycle along split-edges
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• estimated per voxel
Mesh vertices at voxel centers
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• estimated per voxel
Mesh vertices at voxel centers
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
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
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
Computer Graphics GroupAlexander Hornung
Cut Manifold to Triangle Mesh
• Elimination of grid artifactsError controlled Bi-Laplacian smoothing
• Based on surface confidence• Stop smoothing if
svvp )1)((
Computer Graphics GroupAlexander Hornung
Outline
• Introduction
• Surface confidence estimation
• Graph-based surface extraction
• Hole filling and detail preservation
• Mesh extraction
• Results
Computer Graphics GroupAlexander Hornung
Max Planck
Resolution Time Genus Vertices
5123 199s 0 320K
Computer Graphics GroupAlexander Hornung
Statue
Resolution Time Genus Vertices
10243 269s 0 448K
Computer Graphics GroupAlexander Hornung
Rings
Resolution Time Genus Vertices
2563 45s 4 91K
Computer Graphics GroupAlexander Hornung
Rings
Resolution Time Genus Vertices
2563 45s 4 91K
Computer Graphics GroupAlexander Hornung
Leo
Resolution Time Genus Vertices
2563 48s 1 47K
Computer Graphics GroupAlexander Hornung
Monkey
Resolution Time Genus Vertices
2563 82s 0 72K
Computer Graphics GroupAlexander Hornung
Dragon
Resolution Time Genus Vertices
5123 150s 1 (>400) 318K
Computer Graphics GroupAlexander Hornung
Conclusions
• New algorithm for point cloud reconstruction
• Surface confidence map and graph cuts• No normals required• Guaranteed watertight surface• No topological artifacts
• Hierarchical approach • Handles non-uniform sampling and large gaps• Preserves fine details• Reduces number of computed voxels Efficiency
• Conversion of min-cut into triangle mesh
Computer Graphics GroupAlexander Hornung
Future Work
• Voxel representative• Slow smoothing convergence
• Subvoxel precision using input samples
• Performance• No explicit graph generation
• Flow from previous levels
• Graph structure for thin-plate surfaces
• Flux for preferred cut directions
Computer Graphics GroupAlexander Hornung
Thank You