AS-RIGID-AS-POSSIBLE SHAPE MANIPULATION

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AS-RIGID-AS-POSSIBLE
SHAPE MANIPULATION
TAKEO IGARASHI
TOMER MOSCOVICH
JOHN F. HUGHES
THE UNIVERSITY OF TOKYO
BROWN UNIVERSITY
BROWN UNIVERSITY
INTRODUCTION
INTRODUCTION

RELATED WORK

OVERVIEW
ALGORITHM

EXTENSIONS

RESULTS
FUTURE WORK
We present a two-step closed-form algorithm that
achieves real-time interaction.
The first step finds an appropriate rotation for each
triangle.
The second step adjusts its scale.
Each step uses a quadratic error metric so that the
minimization problem is formulate as a set of
simultaneous linear equations.
RELATED WORK
INTRODUCTION

SHAPE MANIPULATION TECHNIQUES FALL
ROUGHLY INTO TWO CATEGORIES:
RELATED WORK
OVERVIEW

ALGORITHM
EXTENSIONS
Deform the space in which the target shape is embedded.
[Lewis et al. 2000]--using predefined skeleton.
[McCracken and Joy 1996]--each point is associated with a
closed region in a FFD grid.
RESULTS
FUTURE WORK

Deform the shape while taking its structure into account.
[Gibson and Mirtich1997]--mass-spring models.
OVERVIEW
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
a)Triangulation and registration(pre-computed)
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Input a 2D model
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Silhouette : marching squares algorithm
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Triangulation : Delaunay triangulation
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Registration : accelerate the computation during interaction
OVERVIEW
INTRODUCTION
RELATED WORK
b)Compilation(pre-computed)
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User clicks on the shape to place handles.
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So far, user can only place handles at existing mesh vertices.
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
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c)Manipulation
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User drags the handles to make a deformation of the shape.

Also support multiple-point input devices.

During interaction, update the handle configuration to solve the
quadratic error functions.
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
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
Overview of the algorithm
ALGORITHM
INTRODUCTION

Step 1 : scale-free construction(allow rotation and
uniform scaling)
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OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
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v2  v0  x01 v0v1  y01R90 v0v1
v2
desired
0  1
R90  

1
0


 v0 ' x 01 v0 ' v1 '  y01 R90 v0 ' v1 '
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM

Step 1 : scale-free construction(allow rotation and
uniform scaling)
The error between v2desired and v2’ is then represented as
E{v2 }  v2
desired
 v2 '
2
EXTENSIONS
We can define v0desired and v1desired similarly, so the error
RESULTS
associated with the triangle is
FUTURE WORK
E{v0 ,v1 ,v2 } 

i 1, 2 , 3
vi
desired
 vi '
2
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW

Step 1 : scale-free construction(allow rotation and
uniform scaling)
The error for the entire mesh is simply the sum of errors for all
triangles in the mesh. We can express it in matrix form:
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
T
E1{v '}
 u  G00
 v' Gv'    
 q  G10
T
G01  u 
 

G11  q 
The minimization problem is solved by setting the partial
Derivatives of the function E1{v’} with respect to the free
variables u in v’ to zero.
E1
T
T
 (G00  G00 )u  (G01  G10 ) q  0
u
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM

Step 1 : scale-free construction(allow rotation and
uniform scaling)
Rewrite as
G' u  Bq  0
EXTENSIONS
G’ and B are fixed and only q changes during manipulation.
RESULTS
Therefore, we can obtain u by simple matrix multiplication by
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pre-computing G’-1B at the beginning.
ALGORITHM
INTRODUCTION
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Step 2 : scale adjustment
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Fitting the original triangle to the intermediate triangle
OVERVIEW
ALGORITHM
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RESULTS
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v2
fitted
 v0
fitted
 x 01 v0
fitted
v1
fitted
 y01 R90 v0
fitted
v1
fitted
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW

Step 2 : scale adjustment

Fitting the original triangle to the intermediate triangle
Given a triangle{v0’,v1’,v2’}in the intermediate result and
ALGORITHM
corresponding triangle in the rest shape{v0,v1,v2},the first
EXTENSIONS
problem is to find a new triangle{v0fitted,v1fitted,v2fitted}that is
RESULTS
congruent to{v0,v1,v2}and minimizes the following function.
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E f {v
fitted
0
, v1 fitted ,v2 fitted }


i 1, 2 , 3
vi
fitted
 vi '
2
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW

Step 2 : scale adjustment

Fitting the original triangle to the intermediate triangle
We minimize Ef by setting the partial derivatives of Ef to zero.
ALGORITHM
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RESULTS
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E f
w
 Fw  C  0
By solving this equation, we obtain a newly fitted triangle
{v0fitted,v1fitted, v2fitted} that is similar to the original triangle
{v0, v1, v2}. We make it congruent simply by scaling the fitted
triangle by the factor of
v0
fitted
 v1
fitted
/ v0  v1
ALGORITHM
INTRODUCTION
RELATED WORK

Step 2 : scale adjustment
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Generating the final result using the fitted triangles
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
We can define the quadratic error function by
E2{v0 '',v1 '',v2 ''} 

( i , j ){( 0,1),(1, 2 ),( 2, 0 )}
vi ' ' v j ' '  vi
2
fitted
vj
fitted
ALGORITHM
INTRODUCTION
RELATED WORK

Step 2 : scale adjustment

Generating the final result using the fitted triangles
The error for the entire mesh can be represented as :
OVERVIEW
ALGORITHM
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RESULTS
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u 
E2{v ''}  v' ' Hv' ' fv' 'c   
q
T
T
 H 00
H
 10
H 01  u 
u 
   ( f 0 f1 )   c

H11  q 
q
We minimize E2 by setting the partial derivatives of E2 to zero.
E2
T
T
 ( H 00  H 00 )u  ( H 01  H10 )q  f 0  0
u
Rewrite as
H ' u  Dq  f 0  0
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
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Algorithm summary
EXTENSIONS
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
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RESULTS
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Collision detection and depth adjustment
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INTRODUCTION
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OVERVIEW
ALGORITHM
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RESULTS
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Weights for controlling rigidity
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INTRODUCTION
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OVERVIEW
ALGORITHM
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RESULTS
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Animations
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INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
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RESULTS
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As-rigid-as-possible curve editing
EXTENSIONS
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INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
As-rigid-as-possible curve editing
RESULTS
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
FUTURE WORK
INTRODUCTION

Determine the depth order of the overlapping
regions.
RELATED WORK
OVERVIEW
ALGORITHM
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Extend the technique to 3D shapes.
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Allow users to put handles at arbitrary locations.
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Volume preservation.
EXTENSIONS
RESULTS
FUTURE WORK

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