Image Processing Reference
In-Depth Information
with respect to x 2 R n is an n n matrix, which is
The second derivative of f ( x )
de
ned by
2
4
3
5
2
2
f ( x )
q x 1
q f ( x )
q x 1 q x 2
f ( x )
q x 1 q x n
q
q
2 f ( x )
q x 2 q x 1
2 f ( x )
q x 2
2 f ( x )
q x 2 q x n
q
q
q
2
f ( x )
q x 2
q
¼
(
3
:
207
)
.
.
.
.
2
2
2
f ( x )
q x n q x 1
q
f ( x )
q x n q x 2
q
q
f ( x )
q x n
Example 3.51
4. Find q f ( x )
q x
Let x 2 R 3 and f (x)
¼ x 1 þ
3x 1 þ
2x 2
7x 2 x 3 þ
.
S OLUTION
2
4
3
5
q f (x)
q x 1
q f (x)
q x 2
q f (x)
q x 3
2
4
3
5
3x 1 þ
6x 1
q f (x)
q x ¼
¼
2
7x 3
7x 2
The second derivative is
2
4
3
5
6x 1 þ
600
2
f (x)
q x 2
q
¼
0
0
7
0
70
Example 3.52
Let x 2 R n and f (x)
¼ c T x, where c is an n
1 column vector, then
¼ c T x ¼ c 1 x 1 þ c 2 x 2 þþc n x n
f (x)
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