ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
ei Tei 8574 e, (15)
= f LS = e) (16
where weights of all points are assumed to be equal and n is
number of samples.
This computation is repeated until the evaluation function x (D)
converges on the minimum value.
Equations (15) and (16) shows that computation of e; and its
partial differentiation can realize least square optimization. The
value e; can be easily computed by equation (6). The following
discussion gives how partial differentiation can be computed.
Denoting py; = (x, y;) and p,; — (xj^, y^) , and differentiat-
ing equation (6), we have:
ge, 9
al
d à L-hy i.
aD; dy7 y/
dy. 3 on 9h à dB
ox; 0B ! aD; dy; oB'' aD;
(17)
Partial differentiation of I, in x and y direction can be numeri-
cally computed with pixel values of I, .
From Equation (8), and sr can be described as follow-
ing:
[ry] [5:0]
0
y;/G d
1/G x./G
Otis 0 9 vfu
JB 1 0 3B'i y, 8
0 1/G
=x," 2/0 ~y.’-x./G
1 i
-Xi y./G zv y, 8
Er Ë 1 (18)
G = a,x;+a,y;, +1
Differentiation of both side of equation (11) by D; gives:
oM oB oV
I BEM? =, 19
oD. 9D, ::9D; (19)
j j j
This leads:
oB -L/0V OM )
= -B 20
C (55; aD, en
= and — = can be computed with equations (7) and (8). Thus
m l items n equation (17) can be calculated.
6. STEREO PLANE MATCHING UNDER VARIOUS
CONSTRAINTS
One particular merit of stereo plane matching is that various
constraints can be implemented by fixing control pairs.
6.1 Fixation of One/Two Points on Plane in 3-D Space
Fixation of one or two points on the plane in 3-D space means
fixation of one or two of control pairs. One-point fixation can be
implemented simply by setting projection pair of this point as
P1(3) and P2). Fixation of two points can be also imple-
mented by setting projection pairs of these points as Ppy(3)-
P2(3) And P1(2)-P2(2)
6.2 Constraint in the Direction of Plane
There are two types of constraint in the direction of a plane. One
is the specification of direction parallel to the plane, and the
other is the specification of direction perpendicular to the plane.
6.2.1 Constraint in direction parallel to the target plane: In
this case, vanishing points in the specified direction can be used
as a fixed control pair as show in Figure 5. This type of con-
straint decrease the degree of freedom of D to two. For exam-
ple, vertical walls of buildings in an aerial photo have a common
vanishing point: the vertical point. The Vertical points in both
images can be the control pair that is common in all vertical
planes.
Vanishing points in direction v can be calculated by the follow-
ing equation:
Ev, = F,(0,+v)
Ev, = F,(0,+v) QD
Vanishing Point
Vanishing Point
^ 1j Ev,
Figure 5. Matching between Vanishing Points
in Direction v
6.2.2 Constraint in direction perpendicular to the target
plane: in this case, two control pairs can be fixed by two pairs of
vanishing points in directions that are perpendicular to the speci-
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