Surface Reconstruction
Some figures by Turk, Curless, Amenta, et al.
Two Related Problems
• Given a point cloud, construct a surface
• Given several aligned scans (range images),
construct a surface
Surface Reconstruction from Point Clouds
• Most techniques figure out how to connect up
“nearby” points
• Need sufficiently dense sampling, little noise
• Delaunay triangulation: connect nearest points
– Officially, a triangle is in the Delaunay triangulation
iff its circumcircle does not contain any points
The “Crust” Algorithm
• Amenta et al., 1998
• Medial axis: set of points equidistant from 2
original points
• In 2D:
Medial Axes in 3D
• May contain surfaces as well as edges and
vertices
Voronoi Diagrams
• Partitioning of plane according to closest point
(in a discrete point set)
• A subset of Voronoi vertices is an approximation
to medial axis
The “Crust” Algorithm
• Compute Voronoi
diagram
• Compute Delaunay
triangulation of original
points + Voronoi vertices
Voronoi Cells in 3D
• Some Voronoi vertices lie neither near the
surface nor near the medial axis
• Keep the “poles”
Crust Results
• 36K vertices
• 23 minutes (1998)
Crust Problems
• Problems with sharp corners
– Medial axis touches surface
– Theoretically need infinitely high sampling
– In practice, heuristics to choose poles
• Topological problems
The Ball Pivoting Algorithm
• Bernardini et al., 1999
• Roll ball around surface, connect what it hits
Alpha Shapes
Problems With Reconstruction from
Point Clouds
Surface Reconstruction from Range Images
• Often an easier problem than reconstruction
from arbitrary point clouds
– Implicit information about adjacency, connectivity
– Roughly uniform spacing
Surface Reconstruction From Range Images
• First, construct surface from each range image
• Then, merge resulting surfaces
– Obtain average surface in overlapping regions
– Control point density
Range Image Tesselation
• Given a range image, connect up the neighbors
Range Image Tesselation
• Caveat #1: can’t be too aggressive
– Introduce distance threshold for tesselation
Range Image Tesselation
• Caveat #2: Which way to triangulate?
• Possible heuristics:
–
–
–
–
Shorter diagonal
Dihedral angle closer to 180
Maximize smallest angle in both triangles
Always the same way (best triangle strips)
Scan Merging Using Zippering
• Turk & Levoy, 1994
• Erode geometry in overlapping areas
• Stitch scans together along seam
• Re-introduce all data
– Weighted average
Zippering
Point Weighting
• Higher weights to points facing the camera
– Favor higher sampling rates
Point Weighting
• Lower weights
(tapering to 0)
near boundaries
– Smooth blends
between views
Point Weighting
Consensus Geometry
Zippering Example
Zippering Example