Image Processing Reference
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And the second derivative is
2
q
f (x)
q x 2
¼ q
(Ax þ b)
q x
¼ A
3.12.3 D ERIVATIVE OF A V ECTOR F UNCTION WITH R ESPECT TO A V ECTOR
Let f ( x ) 2 R m be a vector function of vector x 2 R n , then by de
nition
2
4
3
5
q f 1 ( x )
q x 1
q f 2 ( x )
q x 1
q f m ( x )
q x 1
q f 1 ( x )
q x 2
q f 2 ( x )
q x 2
q f m ( x )
q x 2
q f ( x )
q x ¼
(
3
:
215
)
.
.
.
.
q f 1 ( x )
q x n
q f 2 ( x )
q x n
q f m ( x )
q x n
Example 3.55
, then
x 1 x 2 þ
3x 2 þ
4x 3
5
Let x 2 R 3 and f (x)
¼
x 1 x 2 þ x 2
2x 1 x 2 x 3
2
4
3
5
q f 1 (x)
q x 1
q f 2 (x)
q x 1
2
4
3
5
2x 1 x 2
x 2
2x 2 x 3
q f (x)
q x ¼
q f 1 (x)
q x 2
q f 2 (x)
q x 2
x 1 þ
¼
3 x 1
2x 1 x 3 þ
1
4
2x 1 x 2
q f 1 (x)
q x 3
q f 2 (x)
q x 3
PROBLEMS
3.1 (a) Let
x ( n ) ¼ d( n ) þ
d( n
) þ
d( n
)
and
2
1
3
2
h ( n ) ¼ d( n ) þ d( n
1
) þ
2
d( n
2
)
.
(b) Consider the LTI system with impulse response
Find x ( n ) * h ( n )
2 n u ( n )
h ( n ) ¼
1
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