International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
data) were chosen, respectively. This method estimates the 
transformation parameters of a 3D transformation, which 
satisfies the least squares matching of the search surface to the 
template surface. Akca (2007) reported that in an ideal situation 
one would have 
s (x, y, 2)=f (8 y, 7) (1 
Because of the effects of random errors, Equation (1) is not 
consistent. Therefore, a true error vector v(X, y, z) is added, 
which is expressed as: 
S (x, y. z)-v(x, y, z) = f (x, y. Z) (2) 
Equation (2) is an observation equation, which functionally 
connects the observations s (x, y, z) to the parameters of f (x, y, 
Z). The matching is achieved by least squares minimization of a 
function, minimizing the sum of squares of the Euclidean 
distances between the conjugate points on both surfaces. This 
surface matching technique is a generalization of the least 
squares matching concept and offers high flexibility for any 
type of 3D surface correspondence problem, as well as a 
statistical tools for the analysis of the quality of final matching 
results. 
To express the geometric relationship between the conjugate 
surface patches, a 7-parameter 3D similarity transformation is 
applied: 
Flan 
Y |sSR|Y | 7, (3) 
Z zT 
where S is the uniform scale factor, R = ( P,@,K ) is 
orthogonal rotation matrix, 7.71 is the translation 
vector. Each observation is related to a linear combination of 
the parameters, which are variables of a deterministic unknown 
function. The observation equations of the least squares 
adjustment can be represented in matrix form as Equation (4): 
V=AX-L,P 4) 
where X is the unknown vector to be solved, L is the 
observation vector, A is the coefficient matrix containing the 
partial derivatives from each observation, and P is the a priori 
weight matrix of the observations that reflects measurement 
quality and the contributions of the observations to the final 
result. 
The seven transformation parameters obtained from the above 
least squares process represent the overall differences between 
the two topographic models. By applying the seven 
transformation parameters to one reference topographic model 
(e.g., the one Chang'E-1 data), one set of data can be matched 
to another using the parameters, and finally a detailed 
comparison can be performed. 
3.3 Comparative Analysis of Lunar Topographic Models 
from Chang’E-1 and SELENE Data 
Two typical study areas on the Moon are selected for detailed 
investigation. The first one is the Sinus Iridium, which is the 
primary candidate landing site area for future Chinese robotic 
or human landed missions. The second one is the famous 
Apollo 15 landing site area. Sinus Iridium is located at 44.1? N, 
31.5? W with a diameter of 236 km, which is surrounded from 
the northeast to the southwest by the long range. The Sinus 
Iridium is considered as one of the most beautiful features on 
the Moon, and its bay and surrounding mountains are a 
favourite among lunar observers. The Apollo 15 landing site is 
located at 26.08? N, 3.66? E at the foot of the Apennine 
mountain range. Two typical terrain features can be identified in 
this area, including the winding Hadley Rille and Apennine 
Mountains 
(http://www.nasm.si.edu/collections/imagery/apollo/AS 15/a151 
andsite.htm). 
3.3.1 Experiments at the Sinus Iridium Area 
At the Sinus Iridium area, two DEMs were interpolated using 
the Chang’E-1 laser altimeter data and SELENE laser altimeter 
data with the same resolution 1200 m (see Figure 2). The unit of 
horizontal and vertical axis is degree, while the height 
information is expressed with meter. Figure 3 shows the 
Chang’E-1 and SELENE laser altimetry data directly overlaid 
on the Chang’E-1 images (backward images) at the Sinus 
Iridium area, respectively. Figure 4 shows the 2D grey-scale 
images of the DEMs. They are used to help identifying 
conjugate points on the two surfaces. There is a total of six 
conjugate points selected in this study area for further surface 
matching purpose. Due to the rare texture identified in the 
middle of the relative plat bay, most of the conjugate points 
were selected along the range and the center of the small craters, 
for example, mainly mountain peaks and typical terrain features 
were chosen. A seven parameter transformation was performed 
to match the Chang’E-1 DEM to the SELENE DEM using the 
conjugate points based on the least squares method as discussed 
above. 
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Figure 2. Interpolated DEM with the same resolution of 1200 m 
using Chang’E-1 (a) and SELENE (b) laser altimetry data at the 
Sinus Iridium area, respectively 
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