Image Processing Reference
In-Depth Information
S OLUTION
The parameters t, v, and u are
x x i
x iþ1 x i ¼
1
:
5
1
t ¼
1 ¼
0
:
5
2
y y i
y iþ1
0
:
6
0
v ¼
y i ¼
0 ¼
0
:
6
1
z z i
z iþ1 z i ¼
1
:
85
1
u ¼
¼
0
:
85
2
1
The interpolated values at nodes n 00 n 01 n 10 n 11
½
are
p 00 ¼ p 000 þ tp 100 p 000
ð
Þ ¼
3
þ
0
:
5(2
3)
¼
2
:
5
p 01
¼ p 001
þ tp 101
ð
p 001
Þ ¼
2
þ
0
:
5(5
2)
¼
3
:
5
p 10 ¼ p 010 þ tp 110 p 010
ð
Þ ¼
2
þ
0
:
5(1
2)
¼
1
:
5
p 11 ¼ p 011 þ tp 111 p 011
ð
Þ ¼
4
þ
0
:
5(3
4)
¼
3
:
5
The interpolated values at nodes n 0 n 1
½
are
p 0 ¼ p 00 þvp 10 p 00
ð
Þ ¼
2
:
5
þ
0
:
6(1
:
5
2
:
5)
¼
1
:
9
p 1
¼ p 01
þvp 11
ð
p 01
Þ ¼
3
:
5
þ
0
:
6(3
:
5
3
:
5)
¼
3
:
5
Finally, the interpolated value of the function at point p is
p ¼ p 0 þup 1 p 0
ð
Þ ¼
1
:
9
þ
0
:
85(3
:
5
1
:
9)
¼
3
:
26
6.2.3 T ETRAHEDRAL I NTERPOLATION
Tetrahedral interpolation is another approach for interpolating the regularly sampled
LUTs [2]. Now assume that we have a 3-D LUT of size
N M L
grid
points uniformly spaced in
xyz
3-D space and would like to interpolate the
grid point
(x, y, z). Let
the surrounding eight nodes in the xyz plane be
½
n 000
n 001
n 010
n 011
n 100
n 101
n 110
n 111
and the corresponding points
in the f plane be p 000
, as shown in
Figure 6.8. The tetrahedral interpolation divides this cube into six tetrahedrals, as
shown in Figure 6.9. The interpolated value is the weighted sum of the values of the
function at the four vertices of the tetrahedral enclosing the desired point. That is,
½
p 001
p 010
p 011
p 100
p 101
p 110
p 111
p ¼ p 000 þ p x x x 0
y y 0
y 1 y 0 þ p z
z z 0
z 1 z 0
x 1 x 0 þ p y
(
:
)
6
10
The expressions for p x , p y , and p z depend on the location of p with respect to the six
tetrahedral and are given as follows:
(1) If x x 0 > y y 0 > z z 0 , then p is in tetrahedral 1 and
p x ¼ p 100 p 000 ,
p y ¼ p 110 p 100 ,
and
p z ¼ p 111 p 110
(
6
:
11
)
Search WWH ::




Custom Search