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


Point Cloud Reconstruction



• Non-uniform sampling

• Holes

• Noise

• Bad scan alignment

• No (reliable) normals



• 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


Signed vs. Unsigned Distance


Overview

• Point cloud P


Overview

• Point cloud P

• Surface confidence (unsigned distance)

)(v


Overview

• Point cloud P


• Embed weighted graph structure G

)(v


Overview

• Point cloud P


• Embed weighted graph structure

• Min-Cut of G yields unknown surface

)(v


Outline

• Introduction




• 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

)(v


Surface Confidence




• Compute “crust” containing the surface• Morphological dilation

• Medial axis approximation

)(v


Surface Confidence




• Compute “crust” containing the surface• Morphological dilation

• Medial axis approximation

• Estimate by volumetric diffusion

)(v

)(v


Outline

• Introduction




• Mesh extraction

• Results


Find Optimal Surface

• Minimize energy

• Min-Cut of an embedded graph• Global optimum

• Highly efficient

• Graph structure?

S S

dSadxxSE )()(


Dual Graph Embedding

• : Probability that v is intersected by surface s

• Intersected voxels are split into 2 components• Interior faces

• Exterior faces

)(v


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


Min-Cut Surface Extraction

• Embed graph into a crust containing the surface




• Edge weights defined per voxel avvw s )()(



• Edge weights defined per voxel

• Min-cut yields set of intersected surface voxels


avvw s )()(



• Edge weights defined per voxel

• Min-cut yields set of intersected surface voxels

• Parameter s to emphasize strong/weak maxima


avvw s )()(


Outline

• Introduction




• 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



• 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



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






• Compute min-cut


)(v

1283






• Compute min-cut


)(v

2563






• Compute min-cut


)(v

5123


Outline

• Introduction




• Mesh extraction

• Results


Cut Manifold to Triangle Mesh

Loop of voxel split-edgesGraph cut-edges



• Loops define non-planar polygonal faces

• Mesh vertices at voxel corners



• Loops define non-planar polygonal faces

• Mesh vertices at voxel corners

• Cycle along split-edges



• estimated per voxel

Mesh vertices at voxel centers



• estimated per voxel

Mesh vertices at voxel centers

• Voxel corners correspond to non-planar faces

• Cycle over shared split-edges



• Elimination of grid artifactsError controlled Bi-Laplacian smoothing

• Based on surface confidence• Stop smoothing if

svvp )1)((


Outline

• Introduction




• Mesh extraction

• Results


Max Planck

Resolution Time Genus Vertices

5123 199s 0 320K


Statue


10243 269s 0 448K


Rings


2563 45s 4 91K


Leo


2563 48s 1 47K


Monkey


2563 82s 0 72K


Dragon


5123 150s 1 (>400) 318K


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


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


Thank You

alexander hornung and leif kobbelt rwth aachen

Documents

surface extractionembed

confidence map probability

surface reconstructionboykov

unsigned distancesigned

surface sintersected

volumetric gridsparse

d samples

volumetric reconstructionhoppe