Mesh Shape Editing

Mesh Shape Editing 
 
Computer Animation and Visualisation 
 
Lecture 14
Taku Komura

Today
– Laplacian mesh editing
– As-rigid-as possible shape interpolation
– Deformation transfer
– Generalized Barycentric Coordinates

Editing shapes
• For animating rigid/articulated objects, we can
use previous techniques like skinning
• Let’s think of animating cloths, clay, rubber, soft
tissues, dolls, etc

Editing shapes
• Usually, it is easier to edit a given shape rather
than modeling it from the beginning
• What is important when editing shapes?
• Keeping the local information unchanged
–For faces, the relative location of the eyes,
nose, and mouth must be similar to the
original mesh
• As-rigid-as possible
• Laplacian coordinates

Surface Editing
• Must
– Be fast (interactive rate)
– Preserving the details
– > Differential coordinates – Laplacian coordinates
– Linear, invariant to scale, rotation [Sorkine ’04]

Laplacian coordinates
• Assuming all polygons are
split into triangles
• Every vertex vi is surrounded
by a group of vertices Ni
• Coordinate i will be
represented by the difference
between vi and the average of
its neighbors
• The transformation between
the Laplacian coordinates and
the original vertices is linear

Laplacian coordinates (2)
• Suppose the new position of the vertices are v'i
• We want to keep the Laplacian coordinates the same after the
deformation
• T is a homogeneous transformation of scaling, rotation, and
translation

Laplacian coordinates (3)
• We also want to constrain the position of some
points (keeping vi’ close to ui)
• After all, we need to minimize the following
function
• This becomes a simple quadratic optimization
problem

9
Solving a quadratic problem
Rewriting this problem, it becomes like
min || c - A x ||
The optimal x can be computed by solving
AA x = A c
2
x
T T

Editing Surfaces by Keeping the Details
• The user specifies the region of interest (ROI)
(the area to be edited)
• The user directly moves some of the vertices
• The rest of ROI is decided by minimizing the
error function

Some more …
http://www.youtube.com/watch?v=Yn3P4EK8sYE&feature=channel_page

It is possible to add more constraints
• Sometimes the shape might shrink as we allow
scaling for the transformation
• It might be better to keep the volume the same

Creating a Volumetric Graph
• Add internal vertices and edges
• Compute and preserve the details for the internal structure

Today
– Laplacian mesh editing
– Deformation Transfer
– Generalized Barycentric Coordinates

What if we want to interpolate
different shapes?
– Not editing the shapes but need to morph it to
the target shape

As-Rigid-As Possible Shape
Interpolation [Alexa ’00]
• Interpolating the shapes of the two polygons
so that each triangle (2D) / tetrahedron (3D)
transformation appears as rigid as possible

Interpolating the shapes of triangles
Represent the interpolation of triangles by
rotation & scaling
• linear interpolation
• rotation & scaling

19
Background Knowledge: 
2x2 matrix
Scaling
Rotation
A
A

20
Transformation by 2x2 Matrix
If the matrix is symmetric, can be decomposed as
A = R Λ RT
, R orthogonal, where Λ scaling matrix
(true if A is symmetric)
The eigenvectors of A are the axes of the ellipse
A

21
Geometric analysis of linear transformations
If the matrix is symmetric, can be decomposed as
A = R Λ RT
, R orthogonal (true if A is symmetric)
The eigenvectors of A are the axes of the ellipse
R
RT
Λ

22
General linear transformations:  
Singular Value Decomposition (SVD)
In general A will also contain rotations, not just scales:
A
1 1
σ2
σ1
T
A U V
= ∑

23
General Transformation : SVD
U
VT
Σ
T
A U V
= ∑

Back to ARAP Deformation 
Least-Distorting Triangle-to-Triangle Morphing 
• Say the three vertices P=(p1,p2,p3) are
morphed to Q=(q1,q2,q3)
We want to compute a transformation that
produces (q2-q1, q3-q1)=A (p2-p1, p3-p1)
• We can apply SVD to compute the scaling and
rotation part.
• A = R S
rotation
scaling
S
R

Interpolation of the transformation matrix
The intermediate vertices will be computed by
V(t)=A(t) P
A is decomposed into the rotation part R(t) and
scaling part and S.

Keeping the transformation similar
to A{i,j,k}
• We cannot just interpolate all the triangles
independently
• We need to move the vertices, not the triangles
• A vertex configuration that minimizes the error
between A and B is computed

3D objects
• The method is applicable to polyhedra
• Tetrahedralization is applied to polyhedra and
the tetrahedra are morphed so that they are as
rigid as possible

As-rigid-as possible shape
manipulation [Igarashi et al. 05]
• interactive manipulation of characters based on
the “as-rigid-as-possible” concept
• http://www-ui.is.s.u-tokyo.ac.jp/~takeo/research/rigid/index.html
• http://www.youtube.com/watch?v=1M_oyUEOHK8

32
Deformation Transfer
A standard approach to apply the deformation of
one object to another
Deformation Transfer fo
Triangle Meshe
a Paper (SIGGRAPH 2004) by Robert W. Sumner & Jovan Pop
presented by Roni Oesch
Source deformed Target deformed
Deformation
Transfer

33
What does it do?
Given a source mesh in a reference pose and
several deformations of it:
Given another mesh, called Target in same reference
pose:
Roni Oeschger, Nov. 2005
3
What is it all about?
• Given a source mesh in a reference pose and
• Given another mesh, called Target in same
reference pose:
3
reference pose:

34
What does it do?
Transfer the deformation to the target
Reuse the deformation that was created with
probably a lot of effort
4
• Transfer the deformations to the target:
• Reuse of deformations which were probably
created with a lot of effort

35
No need of Skeleton
Deformation Transfer is purely
mesh-based
No need for an underlying
skeleton structure
Deformation Transfer
• Deformation Transfer is
purely mesh-based
• No need for an underlying
skeleton structure

36
Not skeletal-driven deformations
– non-rigid or facial deformations etc
8
Wider Area of Application
• Not skeletal-driven deformations
– non-rigid or facial deformations etc.
8
Wider Area of Application
• Not skeletal-driven deformations
– non-rigid or facial deformations etc.

37
Approach
Compute the deformation Q for every
source triangle (orientation, scale, skew)
• Apply Q to the corresponding target
triangles

38
Approach
Deformation based on per-triangle
affine transformation
12
Deformation Details
• Deformation based on per-triangle affine
transformation
Qv1 + d = v1
v1
Source
v1
Deformed Source

39
Deformation Gradient
“Deformation Gradient” Q depends on
– triangle in reference pose 
– triangle in deformed pose
12
Deformation Details
transformation
Qv1 + d = v1
v1
Source
v1
Deformed Source

40
Computing the Deformation
Gradient
• Let vi and vi, i ∈ 1...3, be the undeformed and
deformed vertices of the triangle
• We compute a fourth undeformed vertex as
• The deformation gradient can be then computed by
n these
l lattice
olve for
1992].
ly sized
vertices
locally
global
nds the
terpola-
hape by
ow that
transfer
mendous
mplexity
ons that
n anal-
mesh T,
n T and
form of
curves
exhibited by the source deformation onto the target. We represent
the source deformation as a collection of affine transformations tab-
ulated for each triangle of the source mesh. We use this represen-
tation because the non-translational portion of each affine transfor-
mation encodes the change in orientation, scale, and skew induced
by the deformation on the triangle. However, the three vertices of
a triangle before and after deformation do not fully determine the
affine transformation since they do not establish how the space per-
pendicular to the triangle deforms. To resolve this issue, we add a
fourth vertex in the direction perpendicular to the triangle. Let vi
and ṽi, i ∈ 1...3, be the undeformed and deformed vertices of the
triangle, respectively. We compute a fourth undeformed vertex as
v4 = v1 +(v2 −v1)×(v3 −v1)/
!
|(v2 −v1)×(v3 −v1)| (1)
and perform an analogous computation for ṽ4. We scale the cross-
product by the reciprocal of the square root of its length since
this causes the perpendicular direction to scale proportional to the
length of the triangle edges.
An affine transformation defined by the 3×3 matrix Q and dis-
placement vector d, which, for notational convenience, we write as
Q+d, transforms these four vertices as follows:
Qvi +d = ṽi, i ∈ 1...4. (2)
r d, which, for notational convenience, we write as
ms these four vertices as follows:
Qvi +d = ṽi, i ∈ 1...4. (2)
e first equation from the others to eliminate d and
matrix form treating the vectors as columns, we get
V = [v2 −v1 v3 −v1 v4 −v1]
Ṽ = [ṽ2 −ṽ1 ṽ3 −ṽ1 ṽ4 −ṽ1]
. (3)
xpression for Q is given by
Q = ṼV−1
. (4)
ended shape by
We show that
red for transfer
has tremendous
ical complexity
ify regions that
sed as an anal-
target mesh T,
between T and
S′. This form of
een two curves
plies in the spe-
is a continuous
ry relationship.
f the source de-
mation transfer
source and tar-
et meshes with
ion uses a neu-
the T-pose.
v4 = v1 +(v2 −v1)×(v3 −v1)/
!
|(v2 −v1)×
and perform an analogous computation for ṽ4. We
product by the reciprocal of the square root of
this causes the perpendicular direction to scale pro
length of the triangle edges.
An affine transformation defined by the 3×3 m
placement vector d, which, for notational convenie
Q+d, transforms these four vertices as follows:
Qvi +d = ṽi, i ∈ 1...4.
If we subtract the first equation from the others to
rewrite them in matrix form treating the vectors as
QV = Ṽ where
V = [v2 −v1 v3 −v1 v4 −v1]
Ṽ = [ṽ2 −ṽ1 ṽ3 −ṽ1 ṽ4 −ṽ1]
.
A closed form expression for Q is given by
~

41
What happens if we apply Q to the
target mesh?
3
reference pose:
3
reference pose: Roni Oeschger, Nov. 2005
12
Deformation Details
transformation
Qv1 + d = v1
v1
Source
v1
Deformed Source
?

42
Resulting Meshes
Leads to holes in the resulting mesh(B)
Because each triangle is transformed
independently…
Resulting Meshes
• Leads to holes in the resulting mesh (B)
• Used representation affords too many degrees
of freedom Q

43
Consistency needed
• Preservation of consistency leads to a optimization
problem:
15
Minimization Problem
Preservation of con-
sistency leads to a
optimization problem:

44
Vertex Optimization
• Use the vertices as the variables
• Optimize the vertex locations such that the
• deformation gradient of the source S and
• deformation gradient of the target T are as
similar as possible
• We solve the following quadratic problem:
To a
Given this definition, we rewrite the minimization pro
min
ṽ1...ṽn
|M|
∑
j=1
!
!Ssj −Ttj
!
!2
F
.
Since the target transformations are defined in terms o

45
Vertex Optimization (2)
Remember deformation gradient is computed by
V is known, V is unknown for T
So T is linear with respect to V
Thus the following equation is a quadratic
optimization problem with respect to V
To a
Given this definition, we rewrite the minimization pro
min
ṽ1...ṽn
|M|
∑
j=1
!
!Ssj −Ttj
!
!2
F
.
Since the target transformations are defined in terms o
wo curves
n the spe-
ontinuous
ationship.
ource de-
n transfer
e and tar-
shes with
es a neu-
-pose.
skeleton-
ue known
Qvi +d = ṽi, i ∈ 1...4.
If we subtract the first equation from the others to elimin
rewrite them in matrix form treating the vectors as colum
QV = Ṽ where
V = [v2 −v1 v3 −v1 v4 −v1]
Ṽ = [ṽ2 −ṽ1 ṽ3 −ṽ1 ṽ4 −ṽ1]
.
A closed form expression for Q is given by
Q = ṼV−1
.
2
6)
h
),
-
s
-
et
transformed to the same location.
For each target triangle, we add a fo
(Equation 1) and write the non-translatio
transformation in terms of the undeforme
T = ṼV−1. The elements of V−1 depe
formed vertices of the target shape. The e
ordinates of the unknown deformed vertic
T are linear combinations of the coordin
formed vertices.
3
̃
̃
̃

46
Vertex Optimization (3)
Rewriting this problem, it becomes like
where x are the unknowns (components of v),
the rest are known
We can solve this kind of problem by solving a
linear equation
tices themselves and the continuity constraints are implicitly s
fied. Positional vertex constraints can be enforced by simply t
ing a vertex as a constant rather than as a free variable.
The solution to this optimization problem is the solution
system of linear equations. Rewriting the problem in matrix f
yields
min
ṽ1...ṽn
||c−Ax̃||2
2
where x̃ is a vector of the unknown deformed vertex locations
a vector containing entries from the source transformations, an
is a large, sparse matrix that relates x̃ to c. Setting the gradie
the objective function to zero gives the familiar normal equatio
AT
Ax̃ = AT
c
The entries in A depend only on the target mesh’s undeformed
tex locations. Furthermore, the system is separable in the sp
dimension of the vertices. Thus, for each source/target pair
compute and store the LU factorization of ATA only once. R
m of linear equations. Rewriting the problem
s
min
ṽ1...ṽn
||c−Ax̃||2
2
e x̃ is a vector of the unknown deformed vert
ctor containing entries from the source transfo
arge, sparse matrix that relates x̃ to c. Settin
bjective function to zero gives the familiar no
AT
Ax̃ = AT
c
entries in A depend only on the target mesh’s
ocations. Furthermore, the system is separab
~
̃

47
Deformation Transfer : Results
http://people.csail.mit.edu/sumner/research/deftransfer/
Taku Komura
Designing shapes, edit

Barycentric Coordinates 
Given a polygon with N vertices, the barycentric
coordinates is a N dimensional vector
We can use it to represent a position in the space with
respect to the N points of the polygon
We can also use it to interpolate attribute values defined
at the N vertices of the polygon

Interpolation
Geometric interpolation – derive the global
coordinates for a position in parametric cell
space
Attribute interpolation – derive the attribute
value for a position defined in parametric
cell space
are called barycentric coordinates
Demo
v
p1
p2 p3
p4
p5
v
f1
f2 f3
f4
p5
f(v)
http://www.lidberg.se/math/shapetransforms/barycentric.html

Applications 
Surface Deformation
Control Mesh Surface Computing Weights Deformation
216 triangles 30,000 triangles 1.9 seconds 0.03 seconds

Applications 
Surface Deformation
Control Mesh Surface Computing Weights Deformation
98 triangles 96,966 triangles 3.3 seconds 0.09 seconds

Applications 
Boundary Value Problems

Applications 
Solid Textures
Interpolating the texture coordinates
Extend texture to interior

How to compute the barycentric
coordinates for each point?
x
p0
p1
p2
p3
p4
p5
r
s

Wachspress coordinates
http://www.lidberg.se/math/shapetransforms/wachspress.html

Mean Value Coordinates
A good and smooth barycentric coordinates
that can smoothly interpolate the boundary
values
Also works well for concave polygons
There is also a 3D version

Comparison
[Wachspress 1975]
[Floater 2003, Hormann 2004]

Previous Work
[Wachspress 1975]
[Floater 2003, Hormann 2004]

Application: 
Surface Deformation
pi
v

Application: 
Surface Deformation
̂v=
∑ wi ̂pi
∑ wi
̂pi
pi
v
̂v

Application: 
Surface Deformation
v
pi
̂v=
∑ wi ̂pi
∑ wi
̂v
̂pi

Interpolation : problem polygon
http://www.youtube.com/watch?v=egf4m6zVHUI

Harmonic Coordinates
The problem with Mean Value Coordinates is that
their values are affected by the Euclidean
distance but not the distance that needs to be
travelled
Affected by geometrically close points
Harmonic Scalar Field
– The value depends on the distance travelled
inside the polygon
Not affected by the Euclidean distance but the
geodesic distance

Harmonic Coordinates : 
Procedure
- For each vertex i of the cage, set the potential
value of vi to 1, and the rest to 0
- Compute the potential for all the points inside
the the polygon (pi) by solving a Laplace
equation (as taught in Lecture 12)
(p1, p2, ...., pn) is the harmonic coordinates
for each point after normalization
The global position of the point can be expressed
in the form
P = p1 v1 + p2 v2 + … + pn vn

Harmonic Coordinates: 
comparison

Barycentric Coordinates 
Summary
Triangulation / Tetrahedralization
Only C0 continuous at the boundaries
Wachspress coordinates
Can only handle convex objects
Mean value coordinates
Can handle concave objects well to some extent
Values affected by geometrically close control points
Defined outside the polygon too
Harmonic Coordinates
Can handle concave objects well
Only defined inside the polygon

Summary
Modelling surfaces
• Deforming surfaces
– Laplacian Coordinates
• Generalized Barycentric Coordinates

Readings
• Alexa et al. “As-rigid-as-possible shape interpolation SIGGRAPH ’00
• Deformation Transfer for Triangle Mesh, Sumner et al. SIGGRAPH 2004
• Igarashi et al. “As-rigid-as-possible shape manipulation”, SIGGRAPH ‘05
• O. Sorkine, D. Cohen-Or, Y. Lipman, M. Alexa, C. Rössl and H.-P. Seidel,
“Laplacian Surface Editing”, Eurographics Symposium on Geometry
Processing (2004)
• Large Mesh Deformation Using the Volumetric Graph Laplacian
Kun Zhou, Jin Huang, John Snyder, Xinguo Liu, Hujun Bao, Baining Guo,
Heung-Yeung Shum. ACM SIGGRAPH 2005, 496-503.
• Harmoinc Coordinates for Character Articulation: Joshi et al. SIGGRAPH
2007
• Mean Value Coordinates for Closed Triangular Meshes, Ju et al. SIGGRAPH
2005

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