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Shape Reconstruction From Planar Cross Sections
CAI Hongjie | May 28, 2008
Motivation
S. Hahmann, et al. CAGD 2000, GMP 2008
Reconstruction From Cross Sections
• Triangulated tiling
Reconstruction From Cross Sections
• Smooth skinning
Papers List• T.W. Sederberg, K.S. Klimaszewski, et al
Triangulation of Branching Contours using Area Minimization ,1997
• Jean-Daniel Boissonnat
Shape Reconstruction From Planar Cross-Sections , 1988• L.G. Nonato, A.J. Cuadros-Vargas, et al
Beta-Connection: Generating a Family of Models from Planar Cross Sections, 2005
• N.C. Gabrielides, A.I. Ginnis, et al
G1-Smooth Branching Surface Construction From Cross Sections, 2007
Triangulation of Branching Contours using Area Minimization
T.W. Sederberg, K.S. Klimaszewski Mu Hong, Kazufumi Kaneda
International Journal of Computational Geometry & Applications, 1997
Thomas W. Sederberg
• Main Post Professor of Computer Science,
Brigham Young University
• Research Interests Computer aided geometric design Computer graphics Image morphing
• Publications SIGGRAPH 11; CAGD 26; CAD 6; …
Ambiguous Triangulation
Previous Works
• Graph representation
• Some heuristics Volume maximized (E. Keppel, 1975) Faces area minimized (H. Fuchs et al, 1977)
Branching and Link-Edge
Bad Case and Remedy
Polygonal Bridge & Area Minimization
More Results
Shape Reconstruction From Planar Cross-Sections
Jean-Daneil Boissonat
INRIA
Computer Vision, Graphics, and Image Processing, 1988
Jean-Daneil Boissonat
• Main Posts Research Director at
INRIA Sophia-Antipolis Head of the Geometrica project Chair of the Evaluation Board of INRIA
• Research Interests Discrete and Computational Geometry Voronoi diagrams & Delaunay triangulation Surface reconstruction
3D Delaunay Triangulation
A
B
CD
F
E
a
b
c
Computing 3D Delaunay Triangulation
Subdivision of Contours
Result
Beta-Connection: Generating a Family of Models from
Planar Cross Sections
L.G. Nonato, A.J. Cuadros-Vargas
R. Minghim, M.C. F.DE Oliveira
ACM Transactions on Graphics, 2005
Different Correspondence
β-Connection
Tetrahedron Types
• Internal tetrahedron
• External tetrahedron
• Redundant tetrahedron
• Reverse tetrahedron
Graph Representation
• Graph G from Delaunay triangulation
Sphere node: region Cylinder node: redundant tetrahedron Cone node: external tetrahedron
β-Components
• Definition 1: a,b region nodes in G , dG(a,b) is the length of the shortest path connecting a and b.
• Definition 2: ,region nodes a and b are said to be β-connected, denoted , if
G, where ,such that dG(σi, σi+1) ≤ β, i=1,…,k-1.
• Lemma 1: is an equivalence relation.• Definition 3: β-components are defined by the equi
valence class generated by .
a b
1 { ,..., }k region nodes
1 , ka b
Algorithm
• 3D Delaunay triangulation
• Create graph representation and determine
β-components
• Remove cylinder and cone nodes connecting different β-components
• For each β-component, tackle with the singularity
Examples
More Example
G1-Smooth Branching Surface Construction From Cross Sections
N.C Gabrielides, A.I. Ginnis,
P.D. Kaklis, M.I. Karavelas
Computer-Aided Design, 2007
Menelaos Karavelas
• Curriculum Vitae Ph.D. at Stanford University Post-doc at INRIA Sophia-Antipolis Assistant professor at the Department of
Applied Mathematics of the University of Crete
• Research Interests Voronoi diagrams and Delaunay triangulations Shape-preserving interpolation Shape reconstruction
Transfinite Interpolation Surface
• Given parametric surface f(u,v),0≤u,v ≤1.Let u- and v-isoparametric boundaries be 0v,1v;u0,u1. Then
are transfinite interpolation surfaces.
T1 1
T2 2
1 2
P P ( , ) (1 , )( , ) ,
P P ( , ) ( , )(1 , ) ,
1 1P P ( , ) (1 , ) ( , ) (1 , ) ,
u v u u
u v v v
v vu v u u u u
v v
0 1
0 1
0 00 010 1
1 10 11
f f v v
f f u u
vf u u
v
Coons, MIT Project MAC-TR-41,1967
Coons Bi-cubic Blending Surface
• Let boundaries hodograph of f(u,v) be 0vu,1vu; u0v,u1v (0vu=∂ f(u,v)/ ∂u|u=0), then Coons surface is
where
1 2 1 2 1 2P P ( , ) (P P P P ) ( , )u v u v f f
T1 0 1 0 1
T2 0 1 0 1
0
1 2 0 1 0 1
P ( , ) ( ( ), ( ), ( ), ( ))( , , , ) ,
P ( , ) ( , , , )( ( ), ( ), ( ), ( )) ,
P P ( , ) ( ( ), ( ), ( ), ( ))
u u
v v
v v
v v
u u uv uv
u u uv uv
u v F u F u G u G u
u v F v F v G v G v
F
u v F u F u G u G u
0 1 0 1
0 1 0 1
00 01 00 01
10 11 10 11
00 01 00 01
10 11 10 11
f v v v v
f u u u u
f 1
0
1
( )
( ).
( )
( )
v
F v
G v
G v
Sketch of the Algorithm
• Step 1: tangent vector estimation
• Step 2: planar contours and tangent ribbons
• Step 3: surrounding surface
• Step 4: trimming
• Step 5: hole filling
Step 1: tangent vector estimation
Step 2: Planar Contours and Tangent Ribbons
Step 3: Surrounding Surface
Step 4: Trimming
• Trimming curve Y(t)=S•X(t)
• Cross-tangent of trimming curve
(1) ( , ) ( , )( ) ( ) ( )S S
u v u vt a t b t
u v
S S
Y
Step 5: Hole Filling
• Selecting center point K
xy coordinate
centroid of Mk
z coordinate Kz
K
K
Mk
Mk-1Mk+1
1
z i
i i
K zr
z z
Step 5: Hole Filling
• Guide curves
• Gordon-Coons surface
K
Mk
Mk-1Mk+1
Gk
Gk+1Gk-1
The “many-to-many” Problem
The Separating Strip
Symmetric Data Set
Symmetric Construction
Non-Symmetric Case
Integral Construction
Open Set Case
The Bulbous Hull Example
Failure of the Algorithm
Improvement by Voronoi Diagram
Conclusions
• Ensuring G1 continuity
• Preserving data symmetry
• Automatic solution is available
• Topological instability
• Failure for multiple-connected residuals
Thanks!
Q&A