Transcript Document

ACM Transactions on Graphics 2006, 25
Discrete conformal mappings
via circle patterns
Liliya Kharevych Boris Springborn
Peter Schröder
Speaker: CAI Hongjie
Date: Nov 22, 2007
Authors
• Liliya Kharevych
Graduate student in Multi-Res Modeling group
Computer Science, Caltech
• Peter Schröder
Professor of Computer Science,
Applied and Computational Mathematics
Director of Multi-Res Modeling group
Interested in multiresolution methods
Outline
• Related concepts



Surface parameterization
Isometric & conformal maps
Voronoi Diagram & Delaunay triangulation
• Sketch of paper


Basic algorithm
Mapping to the sphere and the disk
Surface Parameterization
• Definition
A one-to-one mapping from the surface to
a suitable domain,especially to a regoin of
the plane.
S
Bijective map f
(u,v)
Surface Parameterization
• Mesh case
piecewise linear
Surface Parameterization
• Applications



Scattered data fitting
Repair of CAD models
Texture mapping
Texture mapping
Isometric & Conformal maps
• A surface to surface regular map f: S→S* is


Isometric, if it preserves lengths of curves
Conformal, if it preserves intersection angles of
any two curves
• Regular map f: S→S*, I and I* be the first
foundamental forms of S and S* respectively
then
f isometric
f conformal
 I  I*
 I   I ,  0
2 *
Stereographic Projection
Circle-preserving
angle-preserving
f : S2 
2
{}
2u
2v
u 2  v2 1
( 2 2 , 2 2 , 2 2 )  (u, v)
u  v 1 u  v 1 u  v 1
Voronoi Diagram &
Delaunay Triangulation
• Voronoi Diagram
Given points P1, P2,…, Pn in a plane, voronoi region
V (Pi ) : { x 
2
: Pi  x  P j  x , j  i}
Boundaries of V(Pi) form the Voronoi diagram
• Delaunay Triangulation
Dual graph of Voronoi diagram with straight lines
Voronoi Diagram &
Delaunay Triangulation
dualize
Voronoi region
Angles assignment on Edges
• A Delaunay triangulation
(V,E,T) of finite points in
a plane, V={vi}, E={eij},
T={tijk} be the sets of
vertices,edges,
and triangles, for
k
l






for interior edges

ij
ij
eij  E :  eij  
k



for boundary edges

ij

Interpretation of
e
• Intersection angle of circumcircles
k
l






for interior edges

ij
ij
eij  E :  eij  
k



for boundary edges

ij

Restrictions of e
• eij  E : 0  e   (for unique Delaunay triangulation)
• vi  Vint :  ev  e  2
• vi Vbdy :  i  2   ev e
i
i


0  i  
Delaunay triangulation
  i  
Circle patterns
Basic Algorithm of
Discrete Conformal Mappings
• Angles assignment
A similar angle system is needed
• Minimizing the energy
The energy is defined on radiuses of triangles
and minimized by sofeware Mosek
• Generating the layout
Step 1: Angles Assignment
• Given a mesh with E,V,
T, {ijk } for edges, Vertices,
triangles and angles
{ijk }
• Firstly find corresponding
k
ˆ
angles {ij } such that



ˆijk : ˆijk  0;
eij  Eint : ˆijk  ˆijl   ;
tijk  T : ˆijk  ˆ ijk  ˆ kij   ;
Q(ˆ )   ˆ  
k
ij
k 2
ij
vk Vint :  t
is minimized
ijk  vk
ˆijk  2 ;
Step 1: Angles Assignment
• Secondly, for
  ˆijk  ˆijl
for interior edges
eij  E :  e  
k
ˆ



for boundary edges

ij
v1
vˆ2
v2
v3
v5
v4
vˆ5
1
ˆ 23
vˆ3
vˆ1
vˆ4
The right graph is probably not a planar graph, but abstract graph
with assigned angles
Counterexample
• A graph with assigned angles may not be
lay out as a planar graph
Treat the inner
square as a point,
coherent angle
property is satisfied
So more restriction is needed for graph layout
Characterization of Flat Faces
• Below is a kite formed by two triangles
yellow region
magnify
1
x

f
(
x
)

tan
(sin

/(
e
 cos  e )), e  Eint

e
e
k
e  
, x  ln(rijk / rjil )
e  Ebdy

   e ,
Every triangular face is flat iff
t  T : 0  2   2et
et
Step 2: Minimizes the Energy
• Energy is defined as (Bobenko, Springborn [2004])
S ( ) 

eEint
(Im Li 2 (e k  l ie )  Im Li 2 (e l  k ie )  (   e )(  k  l ))

 2(   )
e
eEbdy
k
 2  t
tT
k  ijk  ln rijk , l  ijl  ln rijl
  {t : t T}
x
ln(1   )
0

Li2 ( x)  
• Then
t  T : 0  2   2et

xk
d   2 ,
k 1 k
x 1
 S (  )  0
et
Minimizes energy S
Step 3: Generating Layout
• When radiuses of triangles
are achieved from the above
energy minimized step, we
can obtain
1
x

f
(
x
)

tan
(sin

/(
e
 cos e )), e  Eint
 e
e
k
e  
, x  ln(rijk / rjil )
e  Ebdy
   e ,
eij  2rk sin ek  2rl sin el
go around every triangle, and lay out the graph
Review of the Basic Algorithm
• Step 1: angles assignment on edges
A similar angle system is found {ˆijk }


Angles assigment
  ˆijk  ˆijl
interior edges
eij  E : e  
k
ˆ



boundary edges

ij
• Step 2: minimizes the energy S(ρ)


Energy defined on the log of radiuses of triangles
Radiuses is achieved after this step
• Step 3: generating the layout
Mapping to the Sphere
• Algorithm: input is a triangle mesh of genus 0


Remove one vertex together with incident faces
Generate a parameterization by the basic algorithm
Mapping to the Sphere
• Algorithm: input is a triangle mesh of genus 0


Project it
stereographically
to the sphere
Adding a vertex
at the north pole
and fill the hole
Results for Mapping to Sphere
Mapping to the Disk
• Algorithm: input is a triangle mesh that is
a topological disk

Remove a boundary vertex together with
incident edges (red objects)
Fix the curvature angles of
the other boundary vertices
to be 0 (blue vertices)

Restrictions of e
• eij  E : 0  e   (for unique Delaunay triangulation)
• vi  Vint :  ev  e  2
• vi Vbdy :  i  2   ev e
i
i


0  i  
Delaunay triangulation
  i  
Circle patterns
Mapping to the Disk
• Algorithm: input is a triangle mesh that is
a topological disk

Remove a boundary vertex together with
incident edges (red objects)
Fix the curvature angles of
the other boundary vertices
to be 0 (blue vertices)

Mapping to the disk



Generate a parameterization by basic algorithm
with boundary restrictions
Inversion transformation
Fill back the removed vertex and edges
Other Boundary Controls
More results
Free boundary
Disk boundary
Advantages
• Flexible boundary control
• Angle preserved
• Robustness by Delaunay triangulation
Different sample rates but result in almost the same shape
Disavantage and Remedy
• Large area distortion
• Cone singularities
Reference
• L. Kharevych, B. Springborn, P. Schröder Discrete
conformal mappings via circle patterns
• A. Bobenko, B. Springborn
Variational principles for circle patterns and
Koebe’s theorem
• A. Sheffer, E. de Sturler
Surface paramterization for meshing by
triangulation flattening
• Joseph O’Rourke
Computational geometry in C
Thanks!
Q&A