alexander hornung and leif kobbelt rwth aachen

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