it P.
tself is
s filled
/might
(13)
(14)
(15)
(16)
descent
(17)
(18)
(19)
Helmut Kager
using (17) and since p(P, a) = w yielding
= 4 ; w
He = = with (18) therefore = E (20)
So, we might accept this short-cut using
p(P,a) w
U = p= — = 21
Ve®,all ^ dn e)
instead of 4 since these two normals should not differ too much in len
in LLSQ, that’s certainly fine; for convergency behaviour,
adjustment iteratively, the surface as well as the object point
the future.
gth. For measuring an observation discrepancy
problems may be expected; moreover since doing LLSQ
P is only approximately available; needing investigation in
4 DERIVATIVES
4.1 Derivatives for Spatial Similarity Transformation
At first we have to cite the differentials for the of Spatial Similarity Transformation to be able to set up the linearized
observation equations. Equation (1) contains the variates x, , À , 7 , X , and X,. Differentiation of the observable x in
(1) by these variates yields the already well known differential quotients below. For convenience, we use (1) as follows:
= 3 -23-0-£Z(X,X,r) (22)
Derivatives:
E zug Q3)
5 2 68SR(X-X) (24)
pe = 00 Es Ham (25)
m cemaeek--R (26)
SE = ap ME A @n
With I being the Identity matrix; the latter fraction = being also a. (3, 3)-matrix the derivation of which shall not be
exploited to more detail in this context (see also (Dermanis, 1994)).
4.2 Derivatives for Gestalts
Since in the case of GESTALTS the x, = x,(Z, a, q) are themselves functions of other variates (see (2)), we have - after
substitution to (1) - to differentiate (7) by z, a, and q applying the chain-rule esp. for X, X,, and r. We are taking into
account that for all GESTALTs À = const, and o = const, and especially for implicitly given GESTALTs A — 0 and
also (10); instead of v as in (9) we use w to indicate eventually algebraic error as discussed in section 3.1:
x = (EX, X7)ha,q) + À 0 (X, X,r) = 0 + w (28)
Derivatives ( using the definition AN :— (3% + 2-0.) ):
a = (29)
s x re ro p= EAD R= N-R=N (30)
Ee mB, 4, 5.0, p NEN EP D
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 477
