AS-RIGID-AS-POSSIBLE
SHAPE MANIPULATION
TAKEO IGARASHI
TOMER MOSCOVICH
JOHN F. HUGHES
THE UNIVERSITY OF TOKYO
BROWN UNIVERSITY
BROWN UNIVERSITY
INTRODUCTION
INTRODUCTION
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RELATED WORK
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OVERVIEW
ALGORITHM
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EXTENSIONS
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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.
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INTRODUCTION
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SHAPE MANIPULATION TECHNIQUES FALL
ROUGHLY INTO TWO CATEGORIES:
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OVERVIEW
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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.
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Deform the shape while taking its structure into account.
[Gibson and Mirtich1997]--mass-spring models.
OVERVIEW
INTRODUCTION
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OVERVIEW
ALGORITHM
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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
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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.
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c)Manipulation
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User drags the handles to make a deformation of the shape.
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Also support multiple-point input devices.
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During interaction, update the handle configuration to solve the
quadratic error functions.
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INTRODUCTION
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OVERVIEW
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Overview of the algorithm
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INTRODUCTION
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Step 1 : scale-free construction(allow rotation and
uniform scaling)
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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 '
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INTRODUCTION
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OVERVIEW
ALGORITHM
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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
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We can define v0desired and v1desired similarly, so the error
RESULTS
associated with the triangle is
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E{v0 ,v1 ,v2 } 

i 1, 2 , 3
vi
desired
 vi '
2
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INTRODUCTION
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OVERVIEW
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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:
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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
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INTRODUCTION
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OVERVIEW
ALGORITHM
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Step 1 : scale-free construction(allow rotation and
uniform scaling)
Rewrite as
G' u  Bq  0
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G’ and B are fixed and only q changes during manipulation.
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Therefore, we can obtain u by simple matrix multiplication by
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pre-computing G’-1B at the beginning.
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Step 2 : scale adjustment
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Fitting the original triangle to the intermediate triangle
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v2
fitted
 v0
fitted
 x 01 v0
fitted
v1
fitted
 y01 R90 v0
fitted
v1
fitted
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INTRODUCTION
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OVERVIEW
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Step 2 : scale adjustment

Fitting the original triangle to the intermediate triangle
Given a triangle{v0’,v1’,v2’}in the intermediate result and
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corresponding triangle in the rest shape{v0,v1,v2},the first
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problem is to find a new triangle{v0fitted,v1fitted,v2fitted}that is
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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
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OVERVIEW
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Step 2 : scale adjustment

Fitting the original triangle to the intermediate triangle
We minimize Ef by setting the partial derivatives of Ef to zero.
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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
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Step 2 : scale adjustment
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Generating the final result using the fitted triangles
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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
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INTRODUCTION
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Step 2 : scale adjustment

Generating the final result using the fitted triangles
The error for the entire mesh can be represented as :
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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
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Algorithm summary
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INTRODUCTION
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Collision detection and depth adjustment
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INTRODUCTION
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Weights for controlling rigidity
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INTRODUCTION
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Animations
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INTRODUCTION
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As-rigid-as-possible curve editing
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INTRODUCTION
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As-rigid-as-possible curve editing
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INTRODUCTION
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OVERVIEW
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INTRODUCTION
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Determine the depth order of the overlapping
regions.
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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.
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