International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
4 connectivity within a fixed edge intensity level, the extent of 
preliminary crater rim edges can be precisely defined as seen in 
Figure 4 
      
PRG | 
(b) Rims detected in 
MOC optical image image 
(MOC WA image 
M0103863) 
(a) Rim edge detection from 
simulated image 
Figure 4. Preliminary crater rim detection results 
2.2 Edge Organization 
The preliminary crater rims are defined in the focusing stage. 
However, it is necessary to organize these edges into more 
useful and general shapes, particularly an optimal ellipse (in the 
case of a geometrically corrected image, it is usually a circle). 
The most well known edge organization method for circle or 
ellipse detection is the Hough transformation and there are a lot 
of modified versions for efficient detection of ellipses or circles 
(Atherton and Kerbyson, 1999, Yuen at al. 1989, Olson 1998). 
However, none of them appear to be sufficiently robust to 
guarantee the reliable detection of impact craters from the 
preliminary crater rim edge according to our experience. Here, 
we address this problem using conic section fitting. 
Among several conic section fitting methods, two algorithms 
are employed in our scheme. One is Pilu's Direct Least Squared 
(DLS) fitting method (Pilu et al, 1999) and the other is 
Kanazawa and Kanantani (1996)'s conic fitting by optimal 
estimation (OE). Both appear to be reliable even with quite 
noisy data but Kanazawa and Kanantani's fitting scheme shows 
much higher accuracy with relatively short arcs, which are 
frequently observed in Martian crater rims. Kanantani and Ohta 
(2004) developed an osculating ellipse (OE) detection algorithm 
by fitting conic sections but their method only appears to work 
if the edge segment covers more than half of the ellipse. This 
assumption is usually not valid for most of the impact craters 
(7100 pixels in optical image), where one connected edge 
frequently covers only a small percentage of an ellipse 
according to our experience. On the other hand, the CPU cost of 
Kanazawa and Kanantani’s covariance tensor approach and 
iterative renormalizztion is quite expensive compared with 
Pilu's DLS fitting scheme. Therefore, we developed an edge 
organization scheme, which employs DLS fitting for an 
intermediate stage and refines organized edges by OE fitting. It 
consists of two sub-stages: primitive arc definition and arc 
organization. 
At first the possible path map is constructed using the following 
condition. 
Max (St Max (SS , +$ , ) lc. CH, | (5) 
where S;,, Sj, : geometrical x size of edge segment i,j, 
C1,Cj : centre location of edge segment I.j 
Then a check is made of fitness for every possible arc pairs by 
(6), which measures the matching ratio between the fitted conic 
and edge points. 
o 
Fitness — S TN E(i, j) (6) 
(i,j)sE (5j) 
E=e®s 
where T(i.j) : thresholded edge image 
e(i,)) : binary image of fitted conic 
s : kernel for binary erosion in size 
n: 0.1*radius of fitted conic, if 0.1*radius < 1, n=1 
If the score of such a fitness function is higher than a given 
threshold value, this is considered a primitive arc. These 
primitive arcs are then organised as optimal ellipses by other 
kinds of path checking procedure. If the overlap ratio between 
two primitive arcs is more than some pre-specified thresholding 
value, that path is considered as a possible one for optimal 
ellipse formation. The method for ellipse formation is a very 
simple process using the cycle detection in an undirected graph, 
which is constructed from the previous path search step. 
  
  
  
       
ARE C um » € t d TE = 
(a) edge and fitted arc (b) optimal 
preliminary ellipse 
  
  
  
818 
Figure 5. Ellipse organization 
2.3 Refinement and verification 
Onto the edge points of the optimal preliminary ellipses, OE 
conic fitting is applied and fitness is evaluated once more. If 
fitness is higher than a threshold value (usually 0.4), it is 
considered as a potential crater boundary. As seen in Figure 6 
(a), the outlines of crater rims are not correctly matched with 
the finally fitted ellipse so that one more refinement step is 
necessary. This step uses a Hough transformation at several 
fixed radii and centre point ranges with different margins. 
The final procedure for crater detection is the verification stage 
by template matching. As we already know the size of the 
detected crater, it is possible to examine the correlation value 
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