yj = by + bz, + bliy, (24)
2; and yi from equations 21 and 22 we can rewrite the affine
transformation relating windows w; and w; (equations 16
and 17) as a function of the two sets of parameters which
relate each window to the template.
Proceeding further according to conventional least squares
approach, we have a total of 6(n — 1) statistically indepen-
dent transformation parameters, relating each image win-
dow w; (i — 2,3,...n) to the reference template. Therefore
the dimensions of the associated vector of unknowns
2” = [da}?, da}?,... dbl", dbl] (25)
will be 6(n — 1) x 1. Each set of observation equations
(equation 15) must be linearized as following
Oz;
9(@ wi) —e(2,9) = g3(a5,45) + gs pda +.
1
Oy; ji CH
"Tee . - db. J à e d 13 pat
+ + gj, Obi 3 + 9j. Bal? a; +
Ovis
Each pair of pixels from every pair of windows produces one
observation equation. Among the sed distinct pairs of
conjugate windows, there exist (n — 1) pairs relating each
window vw; (i — 2,3,... n) to the reference template w;. Ob-
servation equations formed by these pairs will only produce
six nonzero elements for each line of the coefficient matrix
A, at the columns which correspond to the parameters of
the f! affine transformation. Observation equations relat-
ing two windows w; and w; (i # j # 1) will produce twelve
nonzero elements per line, at the columns corresponding to
the parameters of both the f" and f? affine transforma-
tions. The sparsity pattern of the design matrix A for the
case of five conjugate windows is shown in Fig. 1. The di-
mensions of each block of nonzero elements (gray square) are
(ni n3) x 6, while the parameters are ordered as fU uf
and the observations as 1-2,1-3,..,.,1=5,2-3, ....4-5
The least squares solution is again
z =A PAY API (27)
and the final solution is obtained after iterations. The nor-
mal matrix ( AT P A) is full but the exploitation of the spar-
sity patterns of matrix A can facilitate computations and
storage requirements.
3.2 Introduction of Geometric Constraints
The previously described technique attempts to match mul-
tiple images using solely the recorded gray values, without
imposing any geometric constraints on the relative posi-
tion of overlapping images in the object space. By simply
using the affine transformation as the geometric relation-
ship between two or more conjugate windows, their geomet-
ric interdependence, as expressed by the satisfaction of the
collinearity condition equations, is not taken into consider-
ation. Therefore, this approach just minimizes gray value
differences without enforcing a geometrically coherent so-
lution. Windows displaying sufficient radiometric similar-
ity can be matched even though their parallax values may
be unacceptable. This problem can be overcome either by
checking the resulting parallax values or, in a more robust
fashion, by introducing geometric constraints within the so-
lution process itself.
804
Figure 1: Sparsity pattern of the design matrix for multiple
image least squares matching without additional constraints
Geometric constraints can be introduced either as additional
equations [Grün & Baltsavias, 1988], or by properly mod-
ifying the expression which relates the coordinate systems
of conjugate windows. The image coordinates (eh, yd) (re-
duced to principal point) of a point P(X»p, Y», Zp) of the
object space in photo j satisfy the collinearity condition
ej, 1 : Xp — XJ
i-um mr (28)
n> P Zp zm ZI
or, in matrix notation
zh = x RO, - Xi) (29)
P
where R; the rotation matrix of image j, X2 the ground
coordinates of the exposure center of photo j and A} the
associated scale factor.
Backsolving the collinearity condition for the image of the
same point P in photo i we obtain
Xp = LR, 4 Xi (30)
and substituting this expression of Xp into equation (29)
gives
j _ Ar T ? Gi ri
r= 3 Fifi 7p + 7 R(X] - Xi) (31)
P P
in which we have two expressions (one for z and and one
for y) relating the (25, v5) image coordinates of point P
in photo j to the (z5,yb) image coordinates of the same
point in photo 7, as a function of the exterior orientation
parameters of both photos. Conceptually, in accordance to
equation 18, by expanding the equation over pairs of window
coordinates (wi, w;) and dropping the index P we have
(2j,9;) = 9" (zi, yi) (32)
with $ being the above described function. This function
should be considered the object space equivalent of equation
20 rather than equation 18 since the relationship between a
pair of windows is described through their relationship to
a reference window, which in this case is the object space
patch.
A4 m) At FE^ eh bed i pu — dt: m be "ao gi
CÓ) um pd £5
© +
