The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
registrations and transformations which, taken together, are 
equivalent to a three dimensional solution. 
There have been a number of studies on DEM matching and 3D 
surface matching. The robust estimation was used for detecting 
the change between surface models without the assistance of 
ground control points. (LI et al., 2001; Pilgrim, 1996) A 
“multimatch-mutimosaic” approach was used for matching and 
mosaicing TOPSAR DEM data. The cross-correlation was 
calculated to find the horizontal and vertical offsets. A number 
of offsets were then used to derive 3D affine transformation, 
through which the DEM data were converted and mosaiced 
together. (Lu et al., 2003) The most commonly applied 
approach for matching 3D point clouds is least squares 
registration. (Gruen and Akca, 2005) An approach fusing 
ASTER DEM and SRTM is similar to “multimatch- 
mutimosaic”. The conjugate points were selected through 
valley and ridge lines. Only vertical shift was applied for 
aligning the two DEMs. (Karkee et al., 2006) 
In this study, the cross-correlation through frequency domain is 
applied for searching for 3D conjugate points on InSAR DEM 
and reference DEM. Those 3D control points are then used for 
deriving seven-parameter 3D transformation equations between 
InSAR DEM and reference DEM. After InSAR DEM is 
converted through the seven-parameter transformation 
equations, the resampling is needed to obtain the InSAR DEM 
with regular grids of posts. 
The refined InSAR DEM is then evaluated against the truth 
DEM. The InSAR DEM with GCPs applied is also evaluated 
against the same truth DEM. The errors of those two InSAR 
DEMs are compared to each other. 
2.1 InSAR DEM to reference DEM registration 
Cross-correlation is the most commonly used approach for 
image registration. The algorithm is simple to implement, the 
speed and accuracy are acceptable, and it is not data sensitive 
and can be applied in automatic registration easily. The cross 
correlation is used to search for conjugate points in InSAR 
DEM registration. 
Cross-correlation can be calculated in the space domain as Eq. 
(1), where image patch A has dimensions (Ma, Na) and image 
patch B has dimensions {Mb, Nb). Conj is the complex 
conjugate. It has maximum C(i, j) when two images are aligned 
with each other. (Orfanidis, 1996) 
C(i,j) 
Ma-\ Na-1 
X Y, A ( m ' n ) con j( B ( m+i ’ n+ j)) 
m=0 n=0 
0 < / < Ma + Mb -1 and 0 < j < Na A^¿> — 1 
(i) 
Cross-correlation can also be calculated in frequency domain. 
Convolution in the space domain can be performed as 
multiplication in the spatial frequency domain. Both image 
patches A and B are transformed into the frequency domain 
through two-dimensional (2D) Fourier transformation. Image 
patch A is then multiplied by complex conjugate of B or vise 
versa. The cross-correlation is computed by transforming the 
product back to the spatial domain. The peak of the modulus of 
the transformed product is the location of maximum cross 
correlation (Eq. (2)). Cross-correlation in frequency domain is 
much faster than cross-correlation in the spatial domain. 
= (2) 
In this study, the cross-correlation in the frequency domain was 
calculated to find highly correlated conjugate points. 
Not all maximum values of cross-correlation are used as 
correlation peaks to calculate offsets between conjugate points. 
First, a maximum-to-average ratio (the ratio of maximum cross 
correlation to the average cross-correlation, or MAR (Eq. (3)) is 
computed. If the MAR is lower than certain threshold, this 
maximum value is not considered as correlation peak and this 
pair of conjugate points is not used. 
mR= Afex{e(f.j)} 
Mean{C{t,j)} 
(3) 
Second, if the offsets (icmax,jcmax) between two conjugate points 
are much larger than other conjugate points, this peak will be 
considered as an outlier and is not included either. 
2.2 Solving seven-parameter transformation equations 
Seven-parameter transformation equations express the space 
relationship between two sets of 3D points, which are InSAR 
DEM posts and reference DEM posts. The seven parameters 
include one uniform scale, three rotations, and three translations 
(Eq. (4)). (x,y, hi) are 3D coordinates of InSAR DEM posts, and 
{X, Y, H) are 3D coordinates of reference DEM posts or 3D 
coordinates of transformed InSAR DEM posts. S is the uniform 
scale, co, cp, and k are the rotation angles with regard to x, y and 
z axis, {tx, ty, tz) are the translations from rotated and scaled 
InSAR DEM to reference DEM. (Mikhail et al., 2001) 
X 
cos(Ar) sin(*0 OTcos(p) 0 -Sin(p) 
'10 0 Tjc 
lx 
Y 
= S 
-sin(x-) cos(»r) 0 J 0 1 0 
0 cos(a>) sm(ai) I y 
+ 
<y 
H 
0 0 lj[sin(i>) 0 cos(íí>) 
0 -sin(tt») cos(fi))J[A 
Ih 
A number of conjugate points are acquired through InSAR 
DEM to reference DEM registration. Those conjugate points 
are inserted into Eq. (4). A solution for the seven parameters is 
derived through the least squares approach. 
2.3 Transforming and resampling InSAR DEM 
The InSAR DEM is transformed through those seven-parameter 
transformation equations. However, the posts of the output 
InSAR DEM are not on the regular grids. A step of resampling 
is required to convert irregular InSAR DEM posts into regular 
posts. 
2.4 InSAR DEM Evaluation 
The root mean square error (RMSE, eq. (5)) is commonly used 
for InSAR DEM evaluation (Lin et al., 1994; Miliaresis and 
Paraschou, 2005; Rufino et al., 1996; Rufino et al., 1998; 
Zebkeretal., 1994). 
RMSE = 
4- 4- * • • 4- 
(5)