MESH ANALYSIS
Lecture 4
(Discrete differential geometry)
SURFACE APPROXIMATION
• Polygonal
meshes are most often intended as
approximations of smooth surfaces
• Since
meshes have straight-line edges and flat faces, they are
just C0 at their vertices and across edges
• Differential
properties of the surface are estimated either by
fitting a smooth surface, or by an averaging process (discrete
integration)
2
SURFACE APPROXIMATION
• Surface
fitting: compute a differential property at point p analytically, by
fitting a smooth surface to a local neighborhood N(p)
• Discrete
integration: compute a differential property at point p as
spatial average of a local neighborhood N(p)
• For a mesh vertex vi, the neighborhood is usually its k-ring Nk(vi) for
some given value of k
• Rule of thumb: the larger the neighborhood, the more smoothing is
introduced
• more stable to noise
• small details are lost
3
DISCRETE INTEGRATION
• Vertex neighborhoods that partition the mesh into disjoint
regions are often preferred:
• barycentric cells
v5
v4
• Voronoi cells
• mixed Voronoi cells
v6
v
A(v)
v1
4
Figure 5.1:
The Laplace-Beltrami
v3
v2
Sf
(v) of a vertex v ⌅ V
SURFACE NORMAL
• Surface
normal of a triangle t = (vi,vj,vk) is the normal of the plane
containing t , i.e.: !
!
vi ) ⇥ (vk
vi ) ⇥ (vk
(vj
n(t) =
||(vj
vi )
vi )||
• Surface
normal at a vertex v is computed as a weighted average of
normals of its incident faces:
!
!
n(v) =
||
P
t2N1 (v)
↵t n(t)
t2N1 (v)
↵t n(t)||
P
where αt is the area of the averaging region related to t
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AREA AND VOLUME
• Total
area of a mesh: X
v1
v2
Af
f 2M
!
1
• For a triangle mesh: Af = |(v1
2
v0
v0 ) ⇥ (v2
v0 )|
!
• For
the volume we use divergence theorem:
ZZZ
⌦
· F(u)dV =
6
ZZ
@⌦
F(u) · n(u)dA
AREA AND VOLUME
1
• Set F(x) = x then we have divF(x) = 1
3
• Therefore
!
VM
=
!
• For
ZZZ
the volume inside a mesh M is 1dV =
⌦M
ZZZ
1
· ( u)dV =
3
⌦M
a polygonal mesh:
VM
!
where xf is any point inside face f
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ZZ
M
1
u · n(u)dA
3
1 X
=
x f · n f Af
3
f 2M
CURVATURE VIA SURFACE
FITTING
The radius r of the neighborhood of each
point p is used as a scale parameter
1. gather all faces in a local neighborhood
of radius r
2. let n be the surface normal at p and ni
be the surface normal at gathered
vertex vi
3. discard all vertices vi such that ni ⋅ n < 0
8
CURVATURE VIA SURFACE
FITTING
3. set a local frame (u,v,n) where u and v are any two
orthogonal unit vectors lying on the tangent plane at p,
and such that the frame is right-handed 4. express all vertices vi in such a local frame with origin at
p 5. fit to these points a polynomial of degree two through p
(least squares fitting):
f (u, v) = au2 + bv 2 + cuv + du + ev
9
CURVATURE VIA SURFACE
FITTING
• Curvatures at p are computed analytically via first and second
fundamental forms of f at the origin
• Since f is a polynomial, coefficients E, F, G, L, M, N of the
fundamental forms can be computed easily in closed form
(they depend just on first and second derivatives of f )
• The shape operator is obtained with respect to frame (u,v,n)
• SVD decomposition of the shape operator yields principal
curvatures and principal curvature directions
10
CURVATURE VIA SURFACE
FITTING
• Curvatures extracted at different scales
11
CURVATURE VIA DISCRETE
INTEGRATION
• Normal curvature at vi along edge (vi,vj) can be estimated by
fitting an osculating circle through vi and vj tangent to the
plane defined by normal ni :
ni
(vi vj ) · ni
ij = 2
||vi vj ||2
12
vi
1
ij
vj
CURVATURE VIA DISCRETE
INTEGRATION
• Once κij has been computed for all vj‘s neighbors of vi, the
second fundamental form can be estimated by least squares
fitting of equations
!
!
ij =
tTij

L
M
M
N
tij
where tij denotes the unary projection on the tangent plane
at vi of vector vj-vi, expressed in any orthogonal coordinate
system (u,v,w) with origin at vi and w axis along ni
13
CURVATURE VIA DISCRETE
INTEGRATION
• A more accurate estimation of the curvature tensor at v can
be computed by averaging curvature tensors defined along
edges of a neighborhood N(v) • For a single edge e curvature is considered zero along e, and
maximum according to the dihedral edge β(e) formed by the
two incident faces across e
X
1
C(v) =
|N (v)|
e2N (v)
14
||e \ N (v)|| T
(e)
ee
2
||e||
FUNCTIONS DEFINED ON
SURFACE
• Consider a (smooth) surface S and a smooth (C2) scalar field
!
f :S !R
• The concepts of partial derivatives, gradient, Laplacian, etc. can
be generalized from traditional calculus of bivariate functions
!
f :R⇥R !R
to functions defined on S
15
GRADIENT
• Let
ƒ(u,v) be a bivariate smooth (C2) scalar field
• The
gradient of f is defined as:
rf =
• The
✓
fu
fv
◆
gradient ∇ƒ(p) of a function ƒ at a point p is a vector on
the plane pointing to the direction of maximal ascent of ƒ and
having a size proportional to the rate of ascent
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BARYCENTRIC COORDINATES
• Each point p inside a triangle t can be written as a convex
combination of the vertices of t p = Bi (p)vi + Bj (p)vj + Bk (p)vk
!
• Each function Bi(p) is linear with value 1 at vi and 0 along edge vjvk
1
Bj(p)
Bi(p)
vk
vi
Bk(p)
vk
vi
1
vk
vi
1
vj
vj
17
vj
GRADIENT ESTIMATION
Let a function f be defined point-wise at the vertices of triangle
mesh M fi = f (vi)
and extended to all points of M by linear interpolation over each
triangle
!
f (p) = fi Bi (p) + fj Bj (p) + fk Bk (p)
where p is a point of triangle t = (vivjvk) and Bi(p), Bj(p), Bk(p) are
its barycentric coordinates in t
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GRADIENT ESTIMATION
• The gradient of f is defined as
!
rf (p) = fi rBi (p) + fj rBj (p) + fk rBk (p)
• Barycentric coordinates satisfy partition of unity:
Bi (p) + Bj (p) + Bk (p) = 1
!
• Therefore
rf (p) = (fj
fi )rBj (p) + (fk
19
fi )rBk (p)
GRADIENT ESTIMATION
• Each function Bi(p) is linear and has value 1 at vi
and 0 along edge vjvk 1
• Therefore ∇Bi(p) is constant over t, it is
orthogonal to vjvk and directed towards vi
Bi(p)
vi
vj )?
vj
rBi (p) =
where At is the area of t
2At
(vi vk )?
(vj vi )?
• Finally: rf (p) = (fj fi )
+ (fk fi )
2At
2At
(vk
20
vk
LAPLACE OPERATOR
• Let
ƒ(u,v) be a bivariate smooth (C2) scalar field
• The
f :
!
Laplace operator is defined as the divergence of the gradient of
f = r2 f = r · rf = fuu + fvv
• The
Laplacian Δƒ(p) of a function ƒ at a point p is the rate at which
the average value of ƒ over disks centered at p, deviates from ƒ(p)
as the radius of the disk grows
•A
function is said to be harmonic if its Laplacian is null at all points
21
LAPLACE-BELTRAMI
OPERATOR
• The
Laplace operator can be generalized for a function f defined on a surface S:
f :S !R
!
Sf
= divS rS f
where the gradient and divergence need suitable definitions that take into account
the metrics on S (through the first fundamental form) • If
we apply the Laplace-Beltrami operator (element-wise) to the coordinate function
X of S, we obtain: !
SX
=
2Hn
where H denotes the mean curvature and n denotes the surface normal
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ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Uniform
Laplacian: • directly
measure the difference between function at a vertex
with respect to the average of its neighbors
1
f (vi ) =
|N1 (vi )|
!
!
• it
X
(fj
vj 2N1 (vi )
does not take into account mesh geometry
• not
accurate for non-uniform meshing
23
fi )
• Cotangent formula
ESTIMATION OF LAPLACEBELTRAMI2OPERATOR
!
∆S f (v) :=
• Cotangent
formula: A(v)
• we
(cot αi + cot βi )
vi ∈N1 (v)
integrate the value of the Laplacian over the averaging
region A(v) about a vertex v
v
!
1
f (v) =
A(v)
ZZ
v
A(v)
α
f (u)dA
A(v)
vi
vi
24
vi
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
• makes
!
!
formula: use of the divergence theorem:
ZZ
A
r · F(u)dA =
I
@A
F(u) · n(u)ds
where n(u) is the outward pointing unit normal at each
point u of the boundary of region A
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ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
Z!
formula: f (u)dA =
A(v)
!
• we
Z
Z
divrf (u)dA =
A(v)
Z
@A(v)
rf (u) · n(u)ds
split the integral separately for each triangle of A(v) :
@A(v)
rf (u) · n(u)ds =
X
t2star(v)
26
Z
@A(v)\t
rf (u) · n(u)ds
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
Z
@A(v)\t
formula: gradient is constant in each triangle
rf (u) · n(u)ds = rf (u) · (a
v
1
b) = rf (u) · (vj
2
?
b
vk
a
vj
27
vk )?
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
Z
@A(v)\t
formula: plug in gradient formula
b
vk
a
rf (u) · n(u)ds =
= (f (vj )
v
vj
f (v))
(v
vk )? · (vj
4At
+(f (vk )
f (v))
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(vj
vk )?
+
v)? · (vj
4At
vk )?
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
formula: area of triangle
1
At = sin
2
1
= sin
2
cos
j
(vj
=
||vj
v
j ||vj
v||||vj
vk || =
k ||v
vk ||||vj
vk ||
v) · (vj
v||||vj
vk )
vk ||
cos
29
k
γk vk
γj
vj
(v
=
||v
vk ) · (vj
vk ||||vj
vk )
vk ||
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• Cotangent
Z
@A(v)\t
formula: substitute
v
γj
vj
rf (u) · n(u)ds =
1
= (cot
2
k (f (vj )
γk vk
f (v)) + cot
30
j (f (vk )
f (v)))
2
∆S f (v) :=
A(v)
(cot αi + cot βi ) (f (vi ) − f (v
v ∈N (v)
ESTIMATION
OF LAPLACEBELTRAMI OPERATOR
i
1
v
• Cotangent
v
formula: sum on all triangles
A(v)
1
f (u)dA =
2
1
f (v) =
2A(v)
v
αi
βi
vi
Z
A(v)
vi
X
(cot ↵i + cot
i )(f (vi )
f (v))
Mark Pauly
vi 2N1 (v)
X
vi
(cot ↵i + cot
vi 2N1 (v)
31
i )(f (vi )
f (v))
ESTIMATION OF LAPLACEBELTRAMI OPERATOR
• There
exist also other methods that use different weights for
the discrete integral
• General
!
formula:
f (vi ) = wi
• wi
vertex weight
• wij
edge weights
X
wij (f (vj )
vj 2N1 (vi )
32
f (vi ))
LAPLACIAN MATRIX
• We
define the following matrices:
•D
= diag(w1,...,wn) diagonal matrix of vertex weights
!
•M
defined as mi,j
!
8 P
>
<
vj 2N1 (vi ) wij ,
= wij ,
>
:
0,
symmetric matrix of edge weights
•L=
DM the Laplacian matrix
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i = j,
vj 2 N1 (vi ),
otherwise.
LAPLACIAN MATRIX
• Then
we have:
0
B
B
B
B
@
f (v1 )
·
·
·
f (vn )
1
0
C
B
C
B
C = LB
C
B
A
@
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f (v1 )
·
·
·
f (vn )
1
C
C
C
C
A
CURVATURE FROM LAPLACEBELTRAMI OPERATOR
• If
we apply the Laplace-Beltrami operator (element-wise) to
the coordinate function X of S, we have: !
X=
2Hn
where H denotes the mean curvature and n denotes the
surface normal
• Therefore:
H=
1
X·n
2
35
DISCRETE GAUSSIAN
CURVATURE
•A
discrete estimation of Gaussian curvature at a vertex v is
given by the angle defect of its incident triangles
!
!
0
1 @
K(v) =
2⇡
A(v)
X
vj 2N (v)
1
✓j A
where θj denotes the angle between vvj and vvj+1
36
DIFFERENTIAL COORDINATES
vector ΔX(v) defines the differential coordinates of v with
respect to its neighbors, and it approximates the local shape
characteristics of surface S at v
• The
aplacian Mesh Processing
eres
neor
⇤i =
1
di
⇥ j⇤N(i) (vi
v j)
37
1 ⇧
|⌅| v⇤⌅ (vi
v)dl(v)
SUMMARY
• Surface
normal at triangles and at vertices
• Evaluation
of curvature tensor (principal curvatures and curvature
directions), Gaussian and mean curvature • Properties
of a scalar field defined over the surface:
• gradient
• Laplace-Beltrami
• Differential
operator (Laplacian matrix)
coordinates
38