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For example, the
c
k
coecients can be defined by the following recurrence
relation:
c
1
=1
c
k
+1
=
(7)
1
2
c
k
If we consider an image which size is
NxN
pixels, the number of non-null el-
ements in
C
1
is equal to 2
N
(
N
1) is we assume a four pixels connectivity.
The number of non-null elements in
C
2
is proportional to
N
2
.Itisobviousthat
for computational reason, we will not consider connection of higher degree than
two.
−
2.3 Notation
Considering an image with
M
pixels, in the following section,
Z
,
Y
and
Z
will
respectively represent [
x
1
, ..., x
M
]
t
,[
y
1
, ..., y
M
]
t
and [
z
1
, ..., z
M
]
t
.
X
and
Y
are
uniformly distributed (coordinates of the pixels in the plane), while
Z
represents
the value of the pixels. Each pixel in the image is numbered according to its
column and then its rows. For a square image,
M
=
N
2
,
N
being the number
of pixel in a row (or column). When
C
is used, it implicitly represents the first
order node-edge matrix. The notation
C
1
..N
will be used for higher order node
edge matrix.
3 Optimisation-Based Approach to Mesh Smoothing
The present section presents an overview of the method. The idea is to generalize
and reformulate Laplacian smoothing. A detailed approached can be found in
[12].
3.1 General Framework
Hamam and Couprie showed in [12] that mesh smoothing may be reformulated
as a minimisation of the cost function J as defined below:
Z
Z
+
θ
0
Z
t
Z
+
θ
1
Z
t
AZ
+
θ
2
Z
t
A
2
Z
Z
t
Q
Z
1
2
J
=
−
−
(8)
where
-
Q
is a symmetric positive definite weighing matrix,
-
θ
0
,
θ
1
and
θ
2
are weighing scalars,
-
A
=
C
t
ΩC
,and
Ω
is a diagonal matrix of weight associated to each edge,
-
C
is the node-edge matrix of the image,
-
Z
and
Z
are respectively the value of the pixels and their initial value.
The inclusion of initial values in the cost function prevents the smoothing from
shrinking the object. For large size problem, a gradient descent algorithm may
be used to minimise
J
and the convergence is guaranteed.
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