Image Processing Reference
In-Depth Information
and
2
4
3
5
q f (x)
q x 1
q f (x)
q x 2
.
q f (x)
q x n
2
4
3
5
c 1
c 2
.
c n
q f (x)
q x ¼
¼
Hence,
q c T x
q x ¼ c
(3
:
208)
Similarly,
q x T c
q x ¼ c
(3
:
209)
3.12.2 D ERIVATIVES OF Q UADRATIC F UNCTIONS
Let x 2 R n and f ( x ) ¼ x T Ax, where A is an n n symmetric matrix. The multivariate
quadratic function f ( x )
can be written as
X
X
n
n
f ( x ) ¼ x T Ax ¼
a ij x i x j
(
3
:
210
)
i ¼
1
j ¼
1
Consider the first component of vector q f ( x )
q x
, that is,
q x 1 X
X
q x 1 X
X
n
n
n
n
q f ( x )
q x 1 ¼ q
a ij x i x j ¼ q
a ij x i x j þ q
q x 1 a 11 x 1 x 1
i ¼ 1
j ¼ 1
i ¼ 2
j ¼ 2
q x 1 X
q x 1 X
n
n
þ q
a 1j x 1 x j þ q
a i1 x i x 1
j ¼ 2
i ¼ 2
X
X
q x 1 x 1 X
q x 1 x 1 X
n
n
n
n
¼ q
q x 1
a ij x i x j þ q
q x 1 a 11 x 1 þ q
a 1j x j þ q
(
:
)
a i1 x i
3
211
i ¼ 2
j ¼ 2
j ¼ 2
i ¼ 2
The
first term in the above expansion is independent of x 1 , therefore it does not
contribute to the derivative. Hence,
X
n
X
n
X
n
X
n
2 X
n
q f ( x )
q x 1 ¼
0
þ
2a 11 x 1 þ
a 1j x j þ
a i1 x i ¼
a 1j x j þ
a i1 x i ¼
a 1j x j
j ¼ 2
i ¼ 2
j ¼ 1
i ¼ 1
j ¼ 1
(
3
:
212
)
Search WWH ::




Custom Search