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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
2.2 Efficient finding the corresponding point 
6) Let Qj = Q i+X ,/ = / + !, turn to Step 3). 
How to find the intersection point P (Fig 1) according to 
NDCC? It is important issue related to the efficient of the entire 
matching procedure. 
Usually the above procedure will give the correct intersect point 
P . However, it does not always work well. Therefore, some 
remedy for this limitation is required. 
Normal Vector 
Second Surface 
>V 
From the above procedure, the anticipated intersection point P , 
the series of temporal points Q i and q : are all in the 
intersection curve L , where the second surface intersects with 
the plane 7r(XP'Z') , which is determined by the normal 
vector« and the Z-axis through P' (See Figure 1). Therefore, 
the above procedure can be simplified from 3D space to 2D 
space, i.e. to find the intersection point between n and L in 
the place 7r(XP'Z'). So, we will analyze and discuss the 
shortcomings of the original question according to the 
simplified version. 
Figure 1 The intersection point 
As shown in Figure 2, Q t _ x , Qj , Q i+ \ are three successive 
Assume that i is the number of iterations, P'{X r , Y',Z') is the 
point on the transformed second surface, and where the surface 
normal vector is n[n x ,n y ,n z ). n can be calculated either by 
temporal points, and the slope F ; 
7r(X'P'Z') can be described as: 
M 
of Qi-1 , Qi 
in 
F‘ 1 = arctan 
f A 
convolution cross for regular gridded date sets only or 
associated with a local quadratic surface fit. 
(2) 
where AZ/ -1 is the different between (2,_i and Q i along Z-axis, 
d is the distance between Q t _ x and Q t along X' -axis. The 
slope will change with the position of Q t . Whether the 
procedure convergences is determined by the relationship 
The intersection point P can be determined by the following 
steps: 
1) i = 0,X = X' ,Y = Y'; 
2) Project P' to first surface along the Z-axis, an intersection 
point Qi can be determined; 
3) Find the intersection point between the plane through Q t 
and n , its planar coordinate (X, Y) can be determined by 
between F‘ 1 and F n . F n is the slope of the normal vector n . 
When F‘ 1 <n — F n (Figure 2-2a), Q i+l is much closer 
to P than Qj_j , the procedure will converge. When 
F‘ 1 —jt — F n (Figure 2-2b), Q i+l and Q t _j are the same 
point, the procedure will neither converge nor diverge. When 
F/~ ] > 71-F n (Figure 2-2c), Q M is much farther to P than 
Qj_j , the procedure will diverge. 
x q ,= x e, + {z e .-zX ! f 
n z 
\ =Y Qi + ( z a ~ z 'p) x z L 
During the iteration, when F‘ 1 > n — F n , Q t will be replaced 
by Q'j. Then continue the original procedure. With this remedy, 
the entire procedure will converge to the anticipated intersection 
point P . 
4) Project qi back to the first surface along Z-axis, we will get 
another intersection point Q i+l ; 
5) If Q j+l and Qj is sufficient close to each other, Q i+X is 
considered to the anticipated points P , otherwise turn to Step 
6) ; 
(a)convergent (b)neither convergent nor divergent (c)divergent 
Figure 2 Finding the intersection point between n and L in the place 7r(X'P'Z') 
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