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Mesh Smoothing by Adaptive and
Anisotropic Gaussian Filter
Applied to Mesh Normals
Yutaka
Ohtake
Alexander
Belyaev
Hans-Peter
Seidel
Max-Planck-Institut für Informatik
Saarbrücken, Germany
Noise on Meshes
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Meshes obtained from digitalizing real world
objects often contain undesirable noise.
From range image
of Stanford Bunny
Angel model
from shadow scanning
Mesh Smoothing
Mesh smoothing is required
for removing the noise.
Mesh
Smoothing
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Mesh Smoothing Methods
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•Laplacian, Bilaplacian smoothing flows
•Taubin’s signal processing (l | m)
approach
•Mean curvature flow
•Anisotropic
diffusion
Pold
P
new
P
old
D
Pnew
Iteration
Conventional
Smoothing Approaches
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We have to specify several parameters.
• Number of iterations
• A threshold deciding geometric features
Noisy
Best
Over-smoothing
Iterations
Our Objective
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Developing fully automatic smoothing
method
no parameter is required
Taubin’s
l|m smoothing
Developed
method
Key Technique
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Amount of smoothing is decided adaptively.
Noisy mesh
White: large smoothing is needed
Black: small smoothing is needed
Contents
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•Adaptive Gaussian Filter on 2D Image
•Mesh Smoothing via Diffusion of Normals
•Adaptive and Anisotropic
Gaussian Filter on Normal field
Adaptive Gaussian Filter
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Proposed by G.Gómez, 2000
Noisy image
Smoothed image
fully automatic
Local scale map (Size of Gaussian kernel)
Scale Space
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The best smoothing amount
is adaptively found in scale space.
0

How to Choose
Optimal Local Scale
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Constant independent of input
 best  argmin

c


 ( I  I )
kernel size original
Homogeneous
region
2
Minimum is found
in scale space.
smoothed
Near edge region
Only one iteration is required.
Contents
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•Adaptive Gaussian Filter on 2D Image
•Mesh Smoothing via Diffusion of Normals
•Adaptive and Anisotropic
Gaussian Filter on Normal field
Extension to Triangle Meshes
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Instead of the intensity of 2D images,
the field of normals on meshes is smoothed.
2D Image
I ( x, y )
Triangle mesh
n(T )
Works Exploring Similar Idea
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•Karbacher and Häusler, 1998
• Smoothing vertex normals
•Ohtake, Belyaev, and Bogaevski, CAD2000
• Diffusion of face normals for crease enhancement
•Taubin, 2001
• Analysis of integrability of smoothed face normals
•Tasdizen, Whitaker, Burchard,
and Osher, Vis’02
• Anisotropic diffusion of normals for smoothing
implicits (level set approach)
Mesh Smoothing via
Diffusion of Normal Field
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Our mesh smoothing =
smoothing normals + integration of normals
(in a least-square sense)
Smoothing
normals
Adaptive Gaussian filter
Integration
of normals
I will explain first.
Integration of
Face Normal Filed
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Minimizing squared differences
of triangle normals and smoothed normals
Efit ( M ) 
 area (T ) n(T )  m(T )
2
all trianglesT
Conjugate gradient descent method is used.
Result of
Integration of Normals
Original mesh
Flat shaded by
smoothed normals
(100 times averaged)
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Result of integrating
smoothed normals
100K triangle, takes about 10 sec.
Contents
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•Adaptive Gaussian Filter on 2D Image
•Mesh Smoothing via Diffusion of Normals
•Adaptive and Anisotropic
Gaussian Filter on Normal field
Gaussian Filter
on Mesh Normals
wn

Smoothed normal : m 
w n
j
j
j
j
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(weighted average)
Weight : w j  area (T j ) K (d )
Geodesic distance
found via
Dijkstra’s algorithm
4
Primal mesh
Dual mesh
Scale Space
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
0
0.2  (average of edge length )
10 scales
Adaptive Gaussian Filter
on Mesh Normals
Constant
(independent to noise size)
 best  argmin

c e2

 2
Average
of edge length
1
 
 wj
2
Flat
region
High curvature
region
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variance
2
w
(
m

n
)
 j  j
Golf club
(Cyberware)
410K triangles
5 min.
Problem near Sharp Features
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Smoothed
Scale map
Minimum
support size
Under-smoothing
Anisotropy
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Averaging regions should be
adjusted to geometric features.
Desired
Averaging
region
Anisotoropic Neighborhood
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
 | n( P)  n(Q) |2 | n(O)  n( P) |2 | n(O)  n(Q) |2 

d ( P, Q) 1  k 


6
2
2



Penalty of
changing normals
P
O (start point)
Q
d ( P, Q )
Anisotropic
Adaptive Gaussian Filter
Isotropic
Adaptive Gaussian Filter
Large smoothing is achieved
near sharp edges
Noisy mesh
(50K triangles)
Taubin’s
smoothing
Proposed method
(small features are
well preserved)
Error Analysis
Compare
Smoothed after
adding noise
Ideal
Our method, Desbrun’s mean curvature flow,
Taubin’s smoothing with various weights
0.0045
0.079
0.0044
0.077
0.0043
0.075
L2 vertex-based error
L2 normal-based error
30
28
26
24
22
20
18
16
14
0
30
28
24
26
22
18
20
16
14
10
12
0.063
8
0.0036
6
0.065
4
0.0037
2
0.067
0
0.0038
12
0.069
10
0.0039
0.071
8
0.004
0.073
6
0.0041
4
Our method
Hn
taubin
taubin 1/d
taubin cot
2
0.0042
Conclusion
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•Fully automatic smoothing method;
• produces good results
if noise is not so large (natural noise).
• preserves sharp features.
•It is time-consuming in comparison with
conventional mesh smoothing methods.
• Fast averaging normals
on large ring neighorhoods is required.
•It is not capable to remove large noise.
• Noise size is close to sampling interval.