Advanced Computer
Graphics
Mesh Processing
G. Zachmann
University of Bremen, Germany
cgvr.cs.uni-bremen.de
Consistent Normal Orientation for Meshes
§ Problem:
§ Many models (especially, created with modelling tools)
consist of many unconnected patches
§ Patches do not necessarily have consistent orientation of
normals
§ Bad consequences:
§ Two-sided lighting is necessary (slightly slower than
one-sided lighting)
§ BSP representation of polyhedra
is easier to construct with
consistent normals
(see "Computational Geometry for CG")
§ And many more …
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Advanced Computer Graphics
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§ Idea for solution: boundary coherence = patches with common
boundaries should be oriented consistently
or
§ This is fairly straight-forward to implement, provided we have
complete neighborhood information (topology)
§ And assuming the mesh is closed
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§ General procedure:
1. Detect edges incident to only 1 polygon
(boundary edges), or incident to more than 2
polygons (non-manifold edges)
2. Partition mesh into 2-manifold patches
3. Orient normals consistently within each patch
(propagate consistent normal direction from one
polygon to the next throughout a patch)
4. Determine patch-patch boundaries close to each
other (i.e., that "should" be connected)
5. Propagate normal orientations across those
boundaries, too
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§ More results:
Before
After
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Mesh Smoothing
§ Frequent problem: meshes are a bit noisy
Typical output of
marching cubes
Output from laser scanner
after meshing
Desired, smoothed mesh
§ Idea: "convolve" mesh with a filter (kernel), like Gaussian filter for
images
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Recap: Image Smoothing (Blurring)
§ Simple, linear filtering by convolution:
§ I = I (x,y) = input image
J = J(x,y) = output image
J(x, y ) =
X
I(x + i, y + j)H(i, j)
i= k,...,+k
j= k,...,+k
§ H is called a kernel, k = kernel width
§ Sequential algorithm to construct J:
§ Slide a k×k window across I
§ At every pixel of I, compute average of I inside window, weighted by H
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§ Examples:
§ Gaussian kernel
H
1
16
1
2
1
2
4
2
1
2
1
§ Box filter (= simple averaging):
H
1
9
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1
1
1
1
1
1
1
1
1
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Comparison
Original
Box
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Gaussian
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Digression: Edge Extraction
Sobel
1
0
-1
2
0
-2
1
0
-1
Vertical edges (absolute value)
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Sobel
1
2
1
0
0
0
-1
-2
-1
Horizontal edges (absolute value)
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§ Problem: we can't simply apply the convolution idea to meshes!
§ Why not?
§ Meshes don't have a canonical , tensor-structure-like
parameterization!
§ I.e., usually there is no parameterization like x and y in the plane
§ Goal: filter without parameterization
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Advanced Computer Graphics
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Laplacian Smoothing
§ Idea:
§ Consider edges as springs
§ For a vertex v0, determine its position of
least energy within its 1-ring
§ Energy of v0:
1
E =
2
v0
d
X
i =1
kvi
vd
v0 k2
v1
§ Necessary condition for minimum: derivative equals zero
dE
=
dv0
§ Iterative procedure:
G. Zachmann
d
X
(vi
v0 ) = 0
i =1
d
X
1
0
v0 =
vi
d i =1
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Sometimes a.k.a.
"umbrella operator"
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§ Generalization by "influence" of adjacent vertices and "speed":
kk
X
X
P
vv00 =
w(v
with
wi = 1 , wi 0
=
i (v
i i vv
0 )0 ) ,
i=1
i=1
v00 = v0 +
v0
§ Simplest form of the weights:
k
1X
v0 =
(vi
d i =1
v0 )
where d = degree of v0 = number of neighbors
§ Better weights are wi =
or
1
kvi v0 k
wi = e
kvi v0 k2
(see chapter "Object Representations" for more)
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Advanced Computer Graphics
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Comparison with Other Smoothing Operators
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Laplace-Smoothing Causes Shrinking
Original
With noise
After 4 iterations
After 80 iterations
G. Zachmann
Advanced Computer Graphics
After 10 iterations
After 400 iterations
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A Simple Extension to Prevent Shrinking
d
§ Like before, for every vi compute
vj 1
1 X
vi =
(vj
d
vi
vi
vj 2
vi )
j2N (i)
§ Average all neighboring Δ's (including the own Δ):
1 X
di =
d
vj
j2N (i)
§ Push the new vertex towards the 1-ring equilibrium and
outwards away from the local direction of contraction:
vi0 = vi + ↵( vi
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Advanced Computer Graphics
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computer
is digitization.
From
the cloudsurface
of points
his paper
presentsgraphics
a technique
for smoothing
polygonal
meshes that avoids the well-known problem of
scanned
by
digitizing
devices
like
laser
scanners
or
tactile
eformation and shrinkage caused by many smoothing methods, like e.g. the Laplacian algorithm. The basic idea
a surface
description
mayback
be obtained
connectto pushscanners,
the vertices
of the smoothed
mesh
towards by
their
previous locations. This technique can be also used
Comparison
ing
the
points
in
an
appropriate
manner
into
a
surface
n order to smooth unstructured point sets, by reconstructing a mesh,
surface mesh to which the smoothing technique is
e.g.
a
mesh
of
triangles.
pplied. The key observation is that a surface mesh which is not necessarily topologically correct, but which can
ficiently be reconstructed, is sufficient for that purpose.
oduction
l approach to acquire complex geometric models in
er graphics is digitization. From the cloud of points
by digitizing devices like laser scanners or tactile
s, a surface description may be obtained by connectpoints in an appropriate manner into a surface mesh,
esh of triangles.
consequence is that the mesh has to be smoothed, with the
goal to remove the noise.
Smoothing with pushback
Figure 2: From left to right: origin, noisy mesh, thr
tions of our algorithm.
Laplacian smoothing
One approach to surface smoothing is the Lapl
gorithm. Versions of Laplacian smoothing are e.g. k
image processing and finite-element-meshing.1 2 3 4
sic idea in image processing is to replace the grey
Unfortunately, the digitized points often do not reflect the
a pixel with an average of the grey values of pixels
correct location on the real surface, because of physical noise
neighborhood. Similarly, the location of a vertex in
added by the technical scanning device. The effect is that
element mesh is corrected by calculating a new lo
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15 June 2016
G. Zachmann
Advanced Computer Graphics
Mesh Processing
the reconstructed surfaces often do not look satisfactory.
The
the locations
of vertices
in24the neigh
Figure
2: From the
left average
to right:oforigin,
noisy mesh,
three itera-
Figure 1: From left to right: origin, noisy mesh, three iterations of the Laplacian algorithm.
Global Laplacian Smoothing
§ Given: mesh M = (V, E, F), V = {v1, …, vn}, vi = (xi, yi, zi)
§ Sought: mesh M' with vertices vi' such that
§ M' is smoother than M, and
§ M' approximates M
§ If M' was perfectly smooth (i.e., a plane), then
8i :
X
wij (vj0
vi0 ) = 0
(1)
j2N (vi0 )
§ Can be written as 3 systems of linear equations, one for x coords,
one for y coords, one for z
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Advanced Computer Graphics
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0 1
x10
Bx 0 C
2C
§ Consider the x coords; write (1) as LB
B .. C = 0
@.A
xn0
8
>
< 1 ,i = j
where L is a n×n matrix, with Lij = wij , (i, j) 2 E
>
:
0
, else
§ Analogously construct a system of equations of y and z
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§ Avoiding shrinking: introduce the constraint that the barycenters of
the new triangles are the same as the barycenters of the old ones
1 0
1
0
0
8(i, j, k) 2 F : (vi + vj + vk ) = (vi + vj + vk )
3
3
0 1
x10
✓ ◆ B 0C ✓ ◆
x2 C
0
§ Write (1) and (2) as L B
=
B C
B @ ... A
b
xn0
(2)
(3)
where B is a m×n matrix, m = number of triangles, and b is a column
vector with m entries, with the k-th row corresponds to triangle
Fk = (i1, i2, i3) and Bki = 1 for i = i1, i2, i3, 0 elsewhere,
and bk =
G. Zachmann
1
(x
3 i1
3
+ xi2 + xi3 )
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§ Solve (over-determined) system (3), which has the form Ax = c
in the least squares sense:
x = (AT A) 1 AT c
§ In real life, use a sparse solver, e.g., TAUCS or OpenNL
§ Results:
fea
is
su
Original
G. Zachmann
Smoothed
(a )
Advanced Computer Graphics
so
the
po
(b )
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po
§ Further problem: certain points
should be retained
§ Solution: introduce constraints
§ Pick feature points
vi1 , . . . , vik
(a )
- Either by user, or by automatic salient point detectors
(b )
Figure 2. Global Laplacian smoothed mesh
§ Add constraint vi0 = vil , l = 1, . . . , kwith 0
1
barycenter
constraints. (a) the(4)
origil
§ Add equations (4) to system (3):
0
0 nal1birdx1mesh;
0 the
1 smoothed mesh
(b)
L
0
Bx 0 Cconstraints
barycenter
with parameter
2C
B
@B
A B . C = @b A
0=3.
@ .. A
C
c
0
xn
with
=
The positions of the vertices can be found by solving the
where C is a matrix containing
normal equation of Eq. 7, which is equivalent to minimizing
following error function:
in every row l just one 1 at position c
, 1≤l≤k,
il the
P 0
2
0 2
2
kLX
k
+
|v
v
|
min
and c = (xi1 , . . . , xik )
n
n
X0
n5F
¯
¯2
2¯ 0
0
0
¯ =
(v
+
v
+
v
)
(v
+
v
+
v
)
l
m
n
seperately
l
m
n
P
+
§ Again, we do this for x-, y-, and z-coordinates
hl>m>ni5I
G. Zachmann
Advanced Computer Graphics
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Note that the linear system is also defined for each com15ponent
June 2016
MeshThe
Processing
30 the
of the coordinates {> |> and}.
positions of
vertices can be found by solving the linear system in Eq. 7
model of an “8” model with heavily added random noise; (b) the smoothed mesh with parameters = 5 and = 0=1.
§ More results:
Model
Eight (Fig. 3)
Dog (Fig. 4)
Dragon (Fig. 5)
Venus (Fig. 6)
Vertex N
1
1
1
Table 1. Running time
shown in the paper.
6. Conclusions
In this paper a novel tec
noise in arbitrary mesh surf
Noisy original
Smoothed
(a )
(b )
applied successfully. The La
over the mesh surface in a g
Figure 4. Denoising a scanned mesh of a dog
smoothed by solving a spars
model using our global Laplacian smoothing
straints and barycenter cons
approach: (a) the noisy mesh data; (b) the
linear system to keep the fe
smoothed mesh with parameters = 5 and
to avoid the shrinkage and
= 0=03.
ing. The proposed approac
it is non-iterative, fast and ef
non-manifold meshes. We
our approach
over traditiona
Noisy original
Laplacian smoothing
Bilateral smoothing
Global
smoothing
(a)
(b )
(c )
(d )
mental results.
The presented approach
Figure 6. Comparisons with other smoothing approaches. Observe the difference
in details
the
provements
andinextensions.
15 June
2016(b) the result denoised
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Advanced
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area of G.
the
lip and eye:
(a) the
noisy
Venus headSSmesh
data;
byProcessing
the
traditional
Mesh
used
in our
approach
shou
Laplacian smoothing approach; (c) the result using bilateral filtering approach in [11]; (d) the result
Simplification
§ Simplification: Generate a coarse mesh from a fine mesh
§ While maintaining certain critera (will not be discussed further here)
§ Elementary operations:
§ Edge collapse:
- All edges adjacent to the edge are required
§ Vertex removal:
- All edges incident to the vertex are needed
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Advanced Computer Graphics
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