Transcript Surface Reconstruction

A New Voronoi-based
Reconstruction Algorithm
CS 598 MJG
Presented by: Ivan Lee
N. Amenta, M. Bern, and M. Kamvysselis. In Proceedings of
SIGGRAPH 98, pp. 415-422, July 1998.
What is Surface Reconstruction?
• Set of points in 3-d space
• Generate a mesh from the points
http://web.mit.edu/manoli/www/crust/crust.html
What to talk about
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Previous Work
Definitions
The Crust Algorithm
Comparison to Previous Work
Further Research
Previous work
• Alpha shapes
• Zero-set
• Delaunay Sculpting
Alpha Shapes
Dey et al. [5]
• Given a parameter, α,
connect vertices within α
units
• Subset of Delaunay
triangulation
• Generalized convex hull
Zero sets
• Using input points, define implicit signed
distance function
• Distance function is interpolated and
polygonized using marching cubes
• Approximation rather than interpolation
• e.g. Curless and Levoy paper
Delaunay Sculpting
• Remove tetrahedra from Delaunay triangulation
• Associate values to tetrahedra and eliminate
largest valued ones
First, some definitions
• Voronoi cell
– A cell where all points in
the cell are closer to a
given sample point than
any other point
• Voronoi diagram
– A space partitioned into
Voronoi cells
• Voronoi vertex
– A point equidistant to
d+1 sample points in Rd
Amenta et al. [1]
Some more definitions
• Delaunay triangulation
– Dual of Voronoi diagram
– Each triangle’s
circumcircle contains no
other vertices
Amenta et al. [1]
• Medial axis
– Set of points with more
than one closest point
Amenta et al. [1]
And finally…
• Poles
– Farthest Voronoi
vertices for a sample
point that are on
opposite sides
• Crust
– Shell created to
represent the surface
Amenta et al. [1]
On to the algorithm
• Compute the Voronoi diagram of S, where
S is the set of sample points
• For each sample point, find the poles on
opposite sides of the sample point
• Compute Delaunay triangulation of S U P,
where P is the set of all poles
• Keep all triangles in which all three
vertices are sample points
On to the algorithm
• Delete triangles whose
normals differ too much
from the direction vectors
from the triangle vertices
to their poles
• Orient triangles
consistently with its
neighbors and remove
sharp dihedral edges to
create a manifold
Advantages
• No need for experimental parameters in
basic algorithm
• Not sensitive to distribution of points
Disadvantages
• Sampling of points needs to be dense
– Undersampling causes holes
• Does not handle sharp edges
– Can be fixed by picking two farthest vertices
as poles, regardless of being on opposite
sides
• Boundaries cause problems
– But not always
Comparison to Previous Work
• Alpha Shapes
– No need for experimental values
• Zero set
– Essentially low-pass filtering, lose information
• Delaunay sculpting
– Very similar to this algorithm
Hull
• Command line implementation of Voronoi
regions in C
• Downloadable at:
http://cm.bell-labs.com/netlib/voronoi/hull.html
Proposed Future Research in 1998
• Fixing problems with boundaries and
sharp edges
• Using sample points with normals
– Allows for sparser samplings
• Lossless mesh compression
What’s happened since then?
• Co-cones (Amenta et al.
[2])
– Cone with apex at sample
point and aligned with poles
– Algorithm only requires one
Voronoi diagram
computation
– Eliminates normal trimming
step
– Still does not support sharp
edges
What’s happened since then?
• The power crust (Amenta et
al. [3])
– Use polar balls and power
diagrams to separate the
inside and outside of the
surface
– Approximates medial axis
What’s happened since then?
• Detecting Undersampling (Dey
and Giesen [4])
– Fat Voronoi cells or dissimilarly
oriented neighboring Voronoi cells
imply undersampling. Add sample
points to accommodate
– This accounts for sharp edges and
boundaries
• Tight Co-cone
– After detecting undersampling, stitch
up holes
Summary
• “New” Crust Algorithm
• Advantages over previous algorithms
• Advancements to fix original crust
algorithm’s flaws
Thank you
References
• [0] N. Amenta and M. Bern. Surface Reconstruction by
Voronoi Filtering. Annual Symposium on Computational
Geometry, pp. 39-48, 1998.
• [1] N. Amenta, M. Bern, and M. Kamvysselis. A New
Voronoi-Based Surface Reconstruction Algorithm. In
Proceedings of SIGGRAPH 98, pp. 415-422, July 1998.
• [2] N. Amenta, S. Choi, T. Dey, and N. Leekha. A Simple
Algorithm for Homeomorphic Surface Reconstruction.
Internation Journal of Computational Geometry and its
Applications, vol. 12 (1-2), pp. 125-141, 2002.
• [3] N. Amenta, S. Choi, and R. Kolluri. The Power Crust.
ACM Symposium on Solid Modeling and Applications, pp
249-266, 2001.
References
• [4] T. Dey and J. Giesen. Detecting Undersampling in
Surface Reconstruction. In proceedings for 17th ACM
Annual Symposium for Computational Geometry, pp.
257-263, 2001.
• [5] T. Dey and S. Goswami. Tight Cocone: A Water-Tight
Surface Reconstructor. In Proceedings for 8th ACM
Symposium for Solid Modeling Applications, pp. 127134, 2003.
• [6] T. Dey, J. Giesen, and M. John. Alpha-Shapes and
Flow Shapes are Homotopy Equivalent. STOC ’03,
2003.
• [7] H. Edelsbrunner and E. Mücke. Three-dimensional
Alpha Shapes. ACM Transactions on Graphics,
13(1):43-72, 1994.
Questions and Discussion