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: