Download Three Points Make a Triangle… Or a Circle

Three Points Make
a Triangle… Or a
Circle
Peter Schröder
joint work with
Liliya Kharevych, Boris Springborn, Alexander Bobenko
DDG Course SIGGRAPH 2005
1
In This Section
Circles as basic primitive
Q
it’s all about the
underlying geometry!
Q
Q
Q
Euclidean: triangles
conformal: circles
two examples
Q
Q
conformal parameterization
discrete curvature energies
DDG Course SIGGRAPH 2005
2
Geometries
The Erlangen program (1872)
Q
geometry through symmetries
Q
Q
affine, perspective
conformal
|CA||CA’|=R2
C
A
A’
Möbius xform
DDG Course SIGGRAPH 2005
3
Conformal Mappings
Piecewise Linear Surfaces
map to domain
Q discrete conformal
Q preserve angles
Q
Q
Q
as well as
possible
circles as primitives
DDG Course SIGGRAPH 2005
4
Paramaterizations
An old problem…
Q
can’t have it all
Q
Q
keep angles
keep areas
Our setup
Q
USGS Map Projections Site
Mercator (1512-1594);
Frontispiece of Atlas Sive
Cosmographicae (1585-1595)
find discrete conformal map from
triangle mesh to Euclidean domain
DDG Course SIGGRAPH 2005
5
Conformal Maps
Mathematical basis
Q
Riemann mapping
theorem
Q
unique (up to
Möbius xforms)
e*
y’
e
x
x’
y
DDG Course SIGGRAPH 2005
6
Conformal Maps
Mathematical basis
Q
Riemann mapping
theorem
Q
unique (up to
Möbius xforms)
Intrinsic
angles in
original mesh
DDG Course SIGGRAPH 2005
Texture coordinates
7
Sounds Great!
You knew there was a catch…
Q
Laplace problem
Q
Dirichlet or Neumann bndry. cond.
DDG Course SIGGRAPH 2005
8
Sounds Great!
You knew there was a catch…
Q
Laplace problem
Q
Q
Dirichlet or Neumann bndry. cond.
too much control or too little…
DDG Course SIGGRAPH 2005
9
An Old Idea
Riemann mapping theorem
Q
conformal maps map infinitesimal
circles to infinitesimal circles
Thurston (85)
finite circles
Q circle packing
Q
© Ken Stephenson
DDG Course SIGGRAPH 2005
10
An Old Idea
Riemann mapping theorem
Q
conformal maps map infinitesimal
circles to infinitesimal circles
Thurston (85)
finite circles
Q circle packing
Q
© Ken Stephenson
DDG Course SIGGRAPH 2005
11
An Old Idea
Riemann mapping theorem
Q
conformal maps map infinitesimal
circles to infinitesimal circles
Thurston (85)
finite circles
Q circle packing
Q
DDG Course SIGGRAPH 2005
Stephenson’s
CirclePack
© Ken Stephenson
12
History
Theory
early: Koebe 36; Andreev 70
Q modern: Rudin & Sullivan 87 (hex
packing); He & Schramm 98 (C∞
convgce.)
Q variational approaches 91-02
Q
Q
de Verdière, Brägger, Rivin, Leibon,
Bobenko & Springborn
DDG Course SIGGRAPH 2005
13
Basic Setup
Given a mesh
Q
local geometry around an edge
DDG Course SIGGRAPH 2005
14
Circle Pattern Problem
Rivin 94, Bobenko & Springborn 04
given a triangulation K
Q an angle assignment
Q
Q
sum conditions
DDG Course SIGGRAPH 2005
15
Circle Pattern Problem
Rivin 94, Bobenko & Springborn 04
given a triangulation K
Q an angle assignment
Q
Q
sum conditions
DDG Course SIGGRAPH 2005
16
Circle Pattern Problem
Uniquely realizable iff
Q
a coherent angle system exists
Q
linear feasibility problem
DDG Course SIGGRAPH 2005
17
Geometry at an Edge
Relationship between variables
Q
angle and radii
DDG Course SIGGRAPH 2005
18
Energy
Solution is unique minimum!
Q
convex energy in
Q
easy gradients and Hessians!
DDG Course SIGGRAPH 2005
19
Algorithm
Angle assignment
quadratic program
Q boundary curvatures
Q
Q
free, prescribed
Minimize energy
Lay out circles
Q
angles and radii determine layout
DDG Course SIGGRAPH 2005
20
Results
Disk boundary
Prescribed boundary
Free boundary
37.9k F
39.6k F
12.6k F
28+8s
DDG Course SIGGRAPH 2005
26+9s
7+2s
21
Boundary Control
You get to control curvature…
DDG Course SIGGRAPH 2005
22
Quasi-Conformal Distr.
DDG Course SIGGRAPH 2005
23
Quasi-Conformal Distr.
DDG Course SIGGRAPH 2005
24
Robustness
DDG Course SIGGRAPH 2005
25
Problems
The price to pay
want angles
(nearly)
preserved
Q must suffer
large area
distortion
Q
DDG Course SIGGRAPH 2005
26
Piecewise Flat
Back to first principles
Q
what does the mesh give us?
Q
Q
everywhere flat with some exceptions
Euclidean metric with cone
singularities
DDG Course SIGGRAPH 2005
27
Cone Singularities
Circle pattern approach
allows for cone singularities!
Q set cone vertices
Q
Q
rest of machinery works as before
DDG Course SIGGRAPH 2005
28
Examples
DDG Course SIGGRAPH 2005
29
Examples
DDG Course SIGGRAPH 2005
30
Properties
Circle patterns with cone sing.
Q
discrete conformal
DDG Course SIGGRAPH 2005
31
Properties
Circle patterns with cone sing.
discrete conformal
Q low area distortion
Q arbitrary topology
Q no cutting a priori!
Q globally continuous
Q
DDG Course SIGGRAPH 2005
32
Summary
Discrete conformal mappings
formulate as circle pattern problem
Q solution is min. of convex energy
Q
Q
Q
simple gradient and Hessian
cone singularities
Q
no cutting a priori & arbitrary topology
You can have both: area & angle!
DDG Course SIGGRAPH 2005
33
More Fun with Circles
Willmore energy of a surface
vanishes iff
surface is
sphere
DDG Course SIGGRAPH 2005
35
More Fun with Circles
Willmore energy of a surface
Möbius invariant
Q
of interest: minimizers
Q
Conformal Geometry
theory of surfaces
DDG Course SIGGRAPH 2005
36
More Fun with Circles
Willmore energy of a surface
Q
of interest: minimizers
Q
Q
Conformal Geometry
theory of surfaces
geometric modeling
DDG Course SIGGRAPH 2005
37
More Fun with Circles
Willmore energy of a surface
Q
of interest: minimizers
Q
Q
Q
Conformal Geometry
theory of surfaces
geometric modeling
physical modeling
DDG Course SIGGRAPH 2005
38
Discrete Willmore Flow
Approach
simplicial 2-manifolds
Q preserve symmetries
Q
Q
Q
Möbius invariance
non-linear formulation
semi-implicit integration
Q boundary conditions
Q
DDG Course SIGGRAPH 2005
39
Previous Work
Discrete setting
Q
Discrete Differential Geometry
Q
Q
Q
4th order flows [SK01] [XPB05] [YB02]
use existing lower order operators
discretized continuous setting
Q
Q
level set [TWBO03] [DR04]
FEM [DDE03] [HGR01] [CDD*04]
DDG Course SIGGRAPH 2005
40
Disc. Willmore Energy
Definition [B05]
Q
object of conformal geometry…
Q
…use circles and angles
at each vertex
DDG Course SIGGRAPH 2005
41
Properties I
Discrete Willmore energy
Q
vanishes iff co-spherical & convex
DDG Course SIGGRAPH 2005
42
Properties I
Discrete Willmore energy
Q
vanishes iff co-spherical & convex
DDG Course SIGGRAPH 2005
43
Properties I
Discrete Willmore energy
Q
vanishes iff co-spherical & convex
DDG Course SIGGRAPH 2005
44
Properties II
Discrete Willmore energy
Q
smooth limit
discrete
continuous
Q
independent of κ1,2
DDG Course SIGGRAPH 2005
45
Properties II
Discrete Willmore energy
Q
smooth limit
discrete
continuous
Q
Q
independent of κ1,2
R≥1 with equality for curvature line
parameterizations
DDG Course SIGGRAPH 2005
46
Properties III
Evaluation
Q
symmetry
DDG Course SIGGRAPH 2005
47
Properties III
Evaluation
Q
symmetry
Q
Q
derivatives
gradient
Q
N x N linear system in (a,b,c,d)
DDG Course SIGGRAPH 2005
48
Properties III
Evaluation
Q
symmetry
Q
Q
derivatives
gradient
Q
N x N linear system in (a,b,c,d)
DDG Course SIGGRAPH 2005
49
Properties III
Evaluation
Q
symmetry
Q
Q
derivatives
gradient
Q
N x N linear system in (a,b,c,d)
DDG Course SIGGRAPH 2005
50
Properties III
Evaluation
Q
symmetry
Q
Q
derivatives
gradient
Q
N x N linear system in (a,b,c,d)
DDG Course SIGGRAPH 2005
51
Properties III
Evaluation
Q
symmetry
Q
Q
derivatives
gradient
Q
Q
N x N linear system in (a,b,c,d)
exact Hessian is 3N x 3N…
DDG Course SIGGRAPH 2005
52
Properties IV
Gradient singularity
what about csc(β)?
Q direction of decrease
Q
Boundary conditions
Q
fixed: easy
DDG Course SIGGRAPH 2005
53
Properties IV
Gradient singularity
what about csc(β)?
Q direction of decrease
Q
Boundary conditions
fixed: easy
Q free: add vertex at ∞
Q
Q
boundary edge β
DDG Course SIGGRAPH 2005
54
Properties IV
Gradient singularity
what about csc(β)?
Q direction of decrease
Q
Boundary conditions
fixed: easy
Q free: add vertex at ∞
Q
Q
boundary edge β
DDG Course SIGGRAPH 2005
55
Properties IV
Gradient singularity
what about csc(β)?
Q direction of decrease
Q
Boundary conditions
fixed: easy
Q free: add vertex at ∞
Q
Q
boundary edge β
DDG Course SIGGRAPH 2005
56
Results I
Simple tests
Q
sphere
DDG Course SIGGRAPH 2005
57
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
58
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
59
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
60
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
61
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
62
Results I
Simple tests
sphere
Q boundaries
Q
DDG Course SIGGRAPH 2005
63
Results II
Curvature alignment
DDG Course SIGGRAPH 2005
64
Results III
Hole filling
DDG Course SIGGRAPH 2005
65
Results III
Hole filling
DDG Course SIGGRAPH 2005
66
Results III
Hole filling
DDG Course SIGGRAPH 2005
67
Results III
Hole filling
DDG Course SIGGRAPH 2005
68
Results III
Hole filling
DDG Course SIGGRAPH 2005
69
Results III
Hole filling
DDG Course SIGGRAPH 2005
70
Results III
Hole filling
DDG Course SIGGRAPH 2005
Inpainting
71
Results III
Hole filling
DDG Course SIGGRAPH 2005
Inpainting
72
Results III
Hole filling
DDG Course SIGGRAPH 2005
Inpainting
73
Results III
Hole filling
DDG Course SIGGRAPH 2005
Inpainting
74
Results IV
Shrinkage free denoising
DDG Course SIGGRAPH 2005
75
Summary
Discrete Willmore flow
preserve symmetries: Möbius
Q semi-implicit time stepping
Q relevance in many geo. proc. areas
Q
Q
Q
Q
surface theory
variational geometric modeling
physical modeling
DDG Course SIGGRAPH 2005
76
Outlook
Future work
more boundary conditions
Q incorporate reference curvature
Q control of triangle quality
Q variational subdivision
Q numerical robustness
Q
DDG Course SIGGRAPH 2005
77
Circle Summary
Obey the geometry
Q
what geometry do the objects of
interest belong to?
Q
Q
conformal parameterization
curvature energies
circles and the angles they make
with one another
Q complete non-linear treatment
Q
DDG Course SIGGRAPH 2005
78