Transcript Mesh Optimization for Polygonized Implicit Surfaces

INFORMATIK
Interpolatory Subdivision Curves
via Diffusion of Normals
Yutaka
Alexander
Hans-Peter
Ohtake
Belyaev
Seidel
Max-Planck-Institut für Informatik, Germany
Subdivision Curves
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• A smooth curve is obtained
as the limit of a sequence of
successive refinements of a polyline.
Interpolatory Subdivision
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• Interpolating the given control points.
Interpolation
Control Polygon
Subdivision level
Approximation
Interpolatory Subdivision
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Subdivision level
Insert vertices
Update
positions
Newly
inserted
Update
fixed
The Four-point Scheme
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4-point
scheme
Vertex
to be inserted
P 2
1

16
P1
P2
9
16
9
16
P

1
16
P1
Averaging vertex positions
1
9
9
1
P   P2  P1  P1  P2
16
16
16
16
Purpose
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Developing a new interpolatory subdivision scheme.
• Round shapes
• Curvature-continuous
4-point scheme
Our method
Basic Idea
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• Smooth variation of curve normals
Our method
4-point scheme
angle of
normal
angle of
normal
arclength
arclength
Contents
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•Basic algorithm
•Non-uniform weighting
for Curvature Continuity
•Generating Corners
Algorithm Overview
1. Insert odd vertices.
2. Produce
smoothed normals.
3. Update
odd vertex positions.
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Averaging
Normals
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• Averaging angles between normals.
• Vector based averaging does not work well.
m
n1
Averaging
1
n2
3
Weight of averaging
(Chaikin’s weight)
Updating Odd
Vertex Positions
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• A straightforward approach is
to place the intersection point.
• However, edges flipping near inflection vertices.
Intersection point
Edge flipping
Updating Odd
Vertex Positions
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• Squared differences of normals are minimized.
• Quick minimization by the conjugate gradient method.
Error to be minimized :
n1
m1
l1
E (x)  l1 n1  m1  l2 n 2  m 2
2
2
4-point scheme
Our method
4-point scheme requires more control points
to generate circular shapes.
Open Polyline
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• Normals on end-edges
are fixed during the averaging.
open
closed
Contents
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•Basic algorithm
•Non-uniform weighting
for Curvature Continuity
•Generating Corners
Curvature Discontinuity
Even intervals of control points
Curvature
discontinuity
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Un-even intervals of control points
angle of normal
curvature
arc-length
Non-uniform Weighting
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• Taking into account the length of edges.
L. Kobbelt and P. Schröder. “A variational approach to
subdivision”. ACM TOG, 1998
4-point scheme
Undesirable
peaks
Uniform
weight
Non-uniform
weight
Non-uniform Weighting
INFORMATIK
• Non-uniform corner cutting.
J. Gregory and R. Qu. “Non-uniform corner cutting”.
CAGD, 1996
n1
m
Averaging
l1
l1
n2
l2
l2
Edge lengths
Curvature Continuous Curves
INFORMATIK
Non-uniform weight
Uniform weight
angle
of normal
angle
of normal
arc-length
arc-length
4-point scheme
Our method
Uniform
weight
Non-uniform
weight
Contents
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•Basic algorithm
•Non-uniform weighting
for Curvature Continuity
•Generating Corners
Sharp Corners
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• Corner points are considered
as end-points.
Sharpened
Rounded
Averaging rules are
linearly interpolated.
Examples
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1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
1
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1 1
Conclusion
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• A new interpolatory subdivision method
• Via averaging normals
• Round shapes
• Curvature-continuous
• A rigorous mathematical study is needed.
Future Work
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• Extension to surfaces (Mesh subdivision).
Control mesh
Butterfly scheme
Our method