as-rigid-as-possible shape manipulation
DESCRIPTION
AS-RIGID-AS-POSSIBLE SHAPE MANIPULATION. TAKEO IGARASHI THE UNIVERSITY OF TOKYO TOMER MOSCOVICH BROWN UNIVERSITY JOHN F. HUGHES BROWN UNIVERSITY. INTRODUCTION. INTRODUCTION RELATED WORK OVERVIEW ALGORITHM EXTENSIONS RESULTS FUTURE WORK. - PowerPoint PPT PresentationTRANSCRIPT
TAKEO IGARASHI THE UNIVERSITY OF TOKYO TOMER MOSCOVICH BROWN UNIVERSITYJOHN F. HUGHES BROWN UNIVERSITY
AS-RIGID-AS-POSSIBLE SHAPE MANIPULATION
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
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
SHAPE MANIPULATION TECHNIQUES FALL ROUGHLY INTO TWO CATEGORIES:
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.
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) Input a 2D model Silhouette : marching squares algorithm Triangulation : Delaunay triangulation Registration : accelerate the computation during
interaction
OVERVIEW
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
b)Compilation(pre-computed) User clicks on the shape to place handles. So far, user can only place handles at existing
mesh vertices.
c)Manipulation 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
FUTURE WORK
Overview of the algorithm
ALGORITHM
INTRODUCTION
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OVERVIEW
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RESULTS
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Step 1 : scale-free construction(allow rotation and uniform scaling)
109001100102 vvRyvvxvv
''''' 109001100102 vvRyvvxvv desired
01
1090R
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 1 : scale-free construction(allow rotation and uniform scaling)
The error between v2desired and v2’ is then
represented as
We can define v0desired and v1
desired similarly, so the error
associated with the triangle is
2
22}{ '2
vvE desiredv
2
3,2,1},,{ '210
i
idesiredivvv vvE
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
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:
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.
q
u
GG
GG
q
uGvvE
T
Tv
1110
0100}'{1 ''
0)()( 100100001
qGGuGGu
E TT
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 1 : scale-free construction(allow rotation and uniform scaling)
Rewrite as
G’ and B are fixed and only q changes during manipulation.
Therefore, we can obtain u by simple matrix multiplication by
pre-computing G’-1B at the beginning.
0' BquG
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 2 : scale adjustment Fitting the original triangle to the intermediate
triangle
fittedfittedfittedfittedfittedfitted vvRyvvxvv 109001100102
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 2 : scale adjustment Fitting the original triangle to the intermediate
triangle
Given a triangle{v0’,v1’,v2’}in the intermediate result and
corresponding triangle in the rest shape{v0,v1,v2},the first
problem is to find a new triangle{v0
fitted,v1fitted,v2
fitted}that is
congruent to{v0,v1,v2}and minimizes the following function.
3,2,1
2
},,{'
210i
ifittedivvvf
vvE fittedfittedfitted
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 2 : scale adjustment Fitting the original triangle to the intermediate
triangle
We minimize Ef by setting the partial derivatives of Ef to zero.
By solving this equation, we obtain a newly fitted triangle {v0
fitted,v1fitted, v2
fitted} that is similar to the original triangle
{v0, v1, v2}. We make it congruent simply by scaling the fitted triangle by the factor of
0
CFww
E f
1010 / vvvv fittedfitted
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 2 : scale adjustment Generating the final result using the fitted
triangles
We can define the quadratic error function by2
)}0,2(),2,1(),1,0{(),(}'','',''{2 ''''
210
ji
fittedj
fittedijivvv vvvvE
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Step 2 : scale adjustment Generating the final result using the fitted
triangles
The error for the entire mesh can be represented as :
We minimize E2 by setting the partial derivatives of E2 to zero.
Rewrite as
cq
uff
q
u
HH
HH
q
ucfvHvvE
T
Tv
)('''''' 10
1110
0100}''{2
0)()( 0100100002
fqHHuHHu
E TT
0' 0 fDquH
ALGORITHM
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Algorithm summary
EXTENSIONS
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
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RESULTS
FUTURE WORK
Collision detection and depth adjustment
EXTENSIONS
INTRODUCTION
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ALGORITHM
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RESULTS
FUTURE WORK
Weights for controlling rigidity
EXTENSIONS
INTRODUCTION
RELATED WORK
OVERVIEW
ALGORITHM
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RESULTS
FUTURE WORK
Animations
EXTENSIONS
INTRODUCTION
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RESULTS
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As-rigid-as-possible curve editing
EXTENSIONS
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
RELATED WORK
OVERVIEW
ALGORITHM
EXTENSIONS
RESULTS
FUTURE WORK
Determine the depth order of the overlapping regions.
Extend the technique to 3D shapes.
Allow users to put handles at arbitrary locations.
Volume preservation.