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In-Depth Information
F
X
=
f
i
=
1
,...,
N
(
)
(2)
i
i
There are infinite solutions of equation(2). But the ideal result should make the
interpolation surface smooth and its energy is as little as possible at the same time. In
other word, the real result obtains when the vale of energy function is getting least. So
the general form of minimum energy solution can be written as equation (3)
=
ʻ
N
i
F
(
X
)
=
ˆ
X
−
X
)
+
P
(
X
)
(3)
i
i
1
X
represents any point in interpolation surface;
X
i
is a sample point;|
X
-
X
i
|is
Euclidean norm; Ф is RBF, and
ʻ
i
is the weight factor for every correspondence RBF.
To guarantee the linearity and continuity for interpolation surface in equation (3),
P
(
X
)
is defined as follow:
P
X
=
c
x
+
c
y
+
c
z
+
c
(
)
(4)
1
2
3
4
c
is undetermined coefficient, Ф
ij
=Ф(|
X
i
-
X
j
|), Linear system like equation (5) is
obtained:
ˆ
ˆ
ʻ
x
y
z
f
1
1
11
1N
1
1
1
1
ˆ
ˆ
ʻ
x
y
z
f
1
N
N1
NN
N
N
N
N
0
c
x
x
0
0
0
=
0
(5)
1
1
N
0
0
0
0
c
0
y
y
1
N
2
0
0
0
0
c
0
z
z
1
N
3
0
0
0
0
c
0
1
1
4
Combination coefficient {
ʻ
1
,…,
ʻ
N
} and multinomial coefficient{
c
1
,
c
2
,
c
3
,c
4
} are
calculated by getting resolution of equations (5). Substituting that result into equation,
interpolation function
F
(
X
)is obtained. Further more, the zero level set of
F
(
X
) is the
vary interpolation surface of sample points.
4
Topology Reconstruction of TIN Based on Manifold Study
Manifold study is a novel nonlinear dimensionality reduction method. Compared with
traditional methods, manifold study will reveal the potential key low dimensions from
high dimensionality, and it also has many advantages, such as: adaptive to nonlinear
data structure, smaller parameter selection, and readily understood for its construction.
Base on manifold study, this article takes a dimensionality reduction on the optimized
original 3-D point cloud to generate planar point set. The Delaunay triangulation of the
planar point set is a classic problem in computational geometry. For arbitrary planar
point set, its Delaunay triangulation existed uniquely. So the error between original
point cloud and its low dimensional mapping could be minimized in account of some
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