The International. Archives of the Photo gramme try. Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
dg
da
j t = g^-g:.J +d yg<
dg
BSx
itik = ik(cos(Rx)x + sin(Rr)y)
dg
dRx
Ir-âr = t Sx(-sin(Æx).x + cos(Rx)y)
dg
Bb
T = g M ,j-gi.j +dx -g d
Jg_
BSy
Tit = |-(^sin(/?y)x + cos (Ry)y)
Bg
BRy
Tit = T s y(~ cos ( R y) x ~ sin (Ry)y)
The second-order derivatives of x,y with respect to
a,Sx,Rx,b,Sy,Ry are given as follows:
1^=3
' = C.AXu. ■&) + 2C„ ¿> 2 & -
c, 2 c„£
A full Newton-Raphson implementation (Chambers, 1977;
Adby and Dempster, 1974) using first and second derivatives
was implemented:
BSxBSx
BSxBRx
BRxBa
BRxBSx
BRxBRx
0
0
0
0
0
0
0
0
=
0
0
- sin(/?x)x + cos(Rx)y
- cos(Ry)x - sin(/?y)y
0
0
- sin(Rx)x + cos(/fr)y
- cos(/?y)x - sin(/?y)y
Sx(-cos(Rx)x - sin(/fa)y)
5y(sin(/?y)x - cos(Ry)y)
[da] = -H' G = -
d 2 R
-1
"a/?“
dada j
da.
where H is the Hessian matrix and G is the Jacobian gradient.
3. THE EQUIVALENCE OF LSM AND GCC
Least squares matching assumes that the left and right image
grey-level values should be identical between two small patches
surrounding the left and right points:
The matrix of second-order (partial) derivatives of the grey-
level value g with respect to a,Sx,Rx,b,Sy,Ry can be
explicitly expressed as follows:
d 2 g
BaBa
d 2 g
BaBSx
BaBRx
d’g
BaBb
d’g
BaBSy
BaBRy
d’g
BSxBa
d‘g
BSxdSx
d’g
BSxBRx
d’g
BSxBh
yg
BSxBSy
BSxBRy
BRxBa
d’g
BRxBSx
^ g
BRxBRx
yg
BRxBb
yg
BRxBSy
BRxBRy
B 2 g
BbBa
d’g
BbBSx
d’g
BbBRx
d’g
BbBb
yg
BbBSy
BbBRy
B'g
BSyBa
B 2 g
BSyBSx
BSyBRx
BSyBb
B’g
BSyBSy
B ! g
BSyBRy
BRyBa
B 2 g
BRyBS-x
BRyBRx
BRyBb
yg
BRyBSy
yg
BRyBRy
0
0
0
gj
& d dSy
g dit
0
0
Bx BSxBRx
g d -t
g _&L A.
ô d dSx BSy
p JilJL.
<5 d dSx BRy
0
j£—£i-
Bx BRxBSx
Bx BRxBRx
g disk
g j£—
«5 d dRx BSy
p -&--ÈL
O d dRx BRy
g«
g jit
g jit
0
0
0
p Jl.
<5 d dSy
p a»
O d BSy BSx
ü
<5 d BSy BRx
0
0
dg d 2 y
By BSyBRy
p -ÈL
<5 d dRy
a -Hz ài-
<5 d dRy BSx
p ■& Js-
O d dRy BRx
0
dg d’y
By BRyBSy
dg d 2 y
By BRyBRy
Assuming or. represents one of parameters ( a,Sx,Rx,b,Sy,Ry )
which need to be solved, the first-order and second-order
derivatives of the cross correlation coefficient R with respect
to each parameter or are given as follows:
M. = !L
d or. M
where
I M=c„c„JcTT,
The second-order derivatives of cross correlation coefficient R
with respect to each parameter or are given as follows:
d 2 R
dadoCj M 2
where
gM>y) = gi( x >y)
A radiometric correction and a geometric correction for the
right images are applied:
g t (x, y) + n, (x, y) = c 0 +c ] g 2 (x 2 ,y 2 ) + n 2 (x 2 , y 2 ) (2)
[ x 2 - x 20 + a + Sx ■ cos(Rx) ■ x + Sx • sin(/fa) • y
\y 2 = y 20 +b-Sy- sin(Ry) -x + Sy- cos(Ry) ■ y
where n,,n 2 are the left and right image random noises, c 0 ,c,
are the radiometric correction coefficients and jc 20 , y 20 are the
starting image locations for the right point.
The least squares observation equation after linearisation (2) is:
v = dc 0 + g 2 dc t +%-da + lfcdSx + -$£dRx + %-db + ^dSy + %£dRy-dg
dg = g t (x,y)-g 2 (x,y)
(3)
Of course, the radiometric correction can be treated as either in
a separate prior step or estimated with other affine parameters
simultaneously.
The matrix version of least squares matching (2) is:
L = AX - V
where X is the unknown vector, L is the observation vector
and A is the design matrix. The least squares normal equation
and its solution are:
A T AX = A T L
X =[A T AY'L = N'A T L
In order to show the equivalence of least squares matching and
gradient cross correlation, firstly, that the correlation coefficient
R is invariant with respect to a linear radiometric correction.
Assume after applying a linear radiometric correction, that the
right image value is:
g 2 — ^0 ^ I gl
The formulation of the new cross correlation coefficient R' is: