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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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