Graphics Reference
In-Depth Information
In the previous expressions,
is the gradient of block
B
pr
evaluated
∂
B
∂
B
pr
pr
∇
B
=
,
pr
∂
x
∂
y
at
A(X;
a)
,and
is first computed in the coordinate frame of
B
pr
and then warped
back onto the coordinate frame of
B
pl
using the current estimate of the warp
A(X;
a)
.
∇
B
pr
is the gradient of block
B
nl
evaluated at
S(X;
d
L
)
, and
is
∇
∂
B
∂
B
∇
B
=
nl
,
nl
nl
nl
∂
x
∂
y
first computed in the coordinate frame of
B
nl
and then shifted back onto the
coordinate frame of
B
pl
using the current estimate of the shift
S(X;
d
L
)
.
is the gradient of block
B
nr
evaluated at
(
(
)
)
, and
S
A
X
;
ʱ ʔ
+
ʱ
;
d
∂
B
∂
B
∇
B
=
nr
,
nr
R
nr
∂
x
∂
y
is first computed in the coordinate frame of
B
nr
and then shifted back onto the
coordinate frame of
B
pr
using the current estimate of the shift
d
R
and then warped
back onto the coordinate frame of
B
pl
using the current estimate of
∇
B
nr
(
)
.
A
X
;
ʱ ʔ
+
ʱ
(
)
∂
A
is the Jacobian of the affine, which is defined as
( ) ( )
The term
.
T
A
X
;
ʱ
,
A
X
;
ʱ
x
y
∂
After the first order Taylor expansion of Eq.(1) is performed. The partial derivative
of Eq.(1) with respect to
ʔ
,
and
can be achieved respectively.
can be
ʔ
d
ʔ
ʱ
ʔ
R
L
solved by setting the partial derivative of Eq.(1) with respect to
ʔ
to zero.
T
∂
A
[
]
()
(
( )
)
∇
B
⇅
⇅
B
X
−
B
A
X
;
ʱ
pr
pl
pr
∂
ʱ
T
∂
S
∂
A
∂
A
−
1
ʔ
ʱ
=
H
⇅
+
∇
B
⇅
⇅
−
∇
B
⇅
nr
pr
∂
d
∂
ʱ
∂
ʱ
X
R
∂
S
(
(
)
)
(
(
(
)
)
)
⇅
B
A
X
;
ʱ
−
B
S
A
X
;
ʱ
;
d
−
∇
B
⇅
⇅
ʔ
d
pr
nr
R
nr
R
∂
d
R
(5)
Where
H
is a
matrix:
6
×
6
T
∂
A
∂
A
∇
B
⇅
∇
B
⇅
pr
pr
∂
ʱ
∂
ʱ
T
∂
S
∂
A
∂
A
H
=
X
+
∇
B
⇅
⇅
−
∇
B
⇅
nr
pr
∂
d
∂
ʱ
∂
ʱ
R
∂
S
∂
A
∂
A
⇅
∇
B
⇅
⇅
−
∇
B
⇅
nr
pr
∂
d
∂
ʱ
∂
ʱ
(6)
R
Similarly,
and
can be obtained, as shown in Eq.(7) and Eq.(9):
ʔ
d
ʔ
d
R
L
T
∂
S
ʔ
d
=
H
−
1
⇅
∇
B
⇅
R
R
nr
∂
d
X
R
∂
S
∂
A
∂
A
(
(
)
)
(
(
(
)
)
)
⇅
B
A
X
;
ʱ
−
B
S
A
X
;
ʱ
;
d
−
∇
B
⇅
⇅
−
∇
B
⇅
⇅
ʔ
ʱ
pr
nr
R
nr
pr
∂
d
∂
ʱ
∂
ʱ
(7)
R
Where
H
R
is the
Hessian matrix:
2
×
2
Search WWH ::
Custom Search