The International Archives of the Photograrnmetrv, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part B4. Beijing 2008 
estimated 3D disparity shape. Therefore the procedure to extract 
3D craters can be divided into two parts 1) estimation of an 
approximate 3D crater model, 2) image matcher implementation 
applied to the estimated disparities 
2.2.1 Estimation of approximate 3D crater model 
To reconstruct the modelled 3D shape of a crater, the fused 
data, which consist of a detected crater radius and centre as well 
as the DTM of the encircled area, are fitted to a height model. 
Duxbury (1991) suggested a 3D crater model as follows: 
h(D,R) - -kD 2 cos(-^jr) (2) 
(0 < D < 2 R) 
otherwise 
h{D,R) = 0 
where R is the crater radius and D is the distance from the 
centre point. 
However, the outer boundary of this model is not realistic for 
the purpose of stereo matching applications, therefore we 
employed a simple polynomial model in the normalized co 
ordinate as shown below (it should be noted that all units in 
these models are dimensionless): 
(3) 
(0 < r„ < 2R), 
where r n is the normalised distance from the centre point 
(r = D/R) and k„ is the normalised model constants. 
Then, the dimensionless shape of an impact crater can be simply 
transferred into its original shape by multiplying it with a 
normalisation factor which can then be extracted from the radial 
transect of stereo height. Figure 1 shows the process of 
normalised crater model fitting. 
lpixel=20m 
(a) Extracted crater DTM from HRSC stereo DTM (crater size 
=lkm) 
JMO h- 
T 
\ / 
lpixel=20m 
(c) Fitted crater model using 4 th order polynomial and profile 
Figure 1. Model fitting a stereo crater DTM using a polynomial 
model 
Initially, the fitted 3D models computed from the HRSC DTMs 
were compared with these two models using convolution. If the 
convolved values with either of these 3D models is below some 
threshold value, the crater is rejected as being unsuitable. In 
such a case, one of the models whose simulated hill shaded 
image has a higher correlation with the original optical image, is 
used to create the first base DTM model for the disparity 
estimation in the subsequent image matching process. 
2.2.2 Stereo image matcher implementation 
The starting point for the image matcher implementation is a 
spherically shaped y-disparity map in HRSC quasi-epi-polarity 
space as shown in Figure 2. The 3D polynomial crater model 
which was extracted from the stereo HRSC DTM or standard 
model and the boundary information of a 2D crater GIS was 
used to remove the expected image distortion for the ALSC 
(Adaptive Least Squares Correlation, Gruen, 1985) image 
matcher which is employed in this matching system for the 
higher sub-pixel accuracy. The weakness of the ALSC image 
matcher in concave shaped areas can be now successfully 
avoided. 
Figure 2. The stereo disparity of associated with the original 
DTM and the deviation from a model-fitted orthorectified image 
of an impact crater (R=500m) 
The merits of this approach are obvious as shown in Figure 3. 
The crater DTM without a pre-defined 3D model shows a 
coarse shape where the rim and bottom part in the matching 
window are mixed up so some part of the height is 
overestimated or underestimated severely. 
Figure 3. The 3D crater which is extracted from a zero base 
surface (z=0, upper) and the one from a pre-defined 3D crater 
model using the first case of eq (4), lower) 
Normally such crater models are extracted from wide area 
DTMs using 2D crater boundaries. However, it’s not always 
possible to guarantee that HRSC stereo DTMs will have 
sufficiently low noise to calculate a 3D crater approximation 
with sufficient accuracy. Therefore, two kinds of 3D models, 
which represent flat bed centres and concave surfaces, were 
tested against 40 well constructed crater DTMs as follows: 
h(r n ) - 0.11 +1.23r n - 0.70rf +0.11 rf + 0.0lr„ 4 (4) 
h(r n ) - 0.04 + 0.16r„ -1. lr„ 2 + 3.99rf - 4.66r„ 4 + 1.75rf 
The other approach which is employed in the crater specific 
image matching system is to iterate using different matching 
window sizes. The iterative process starts with a maximum 
matching window size which can more easily avoid the local 
image artifacts and proceeds with a smaller patch which can be 
more reliable to reconstruct the detailed shape. At each iteration 
stage, the input images used by the image matchers are rectified 
using a 3D model by 3D intersection and polynomial fitting. 
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