more 
st) 
(6) 
jents 
5.3): 
f the 
will 
no © 
(7) 
ters. 
with 
(8) 
  
Helmut Kager 
3 MINIMIZATION BY LEAST SQUARED DISTANCES 
At first we have to consider the observation vector x in (1); 
x being a “true” quantity, it has to be split into two components: the observation X and its correction v: 
T=T+V (9) 
An observed GESTALT point yields 
&l 
li 
e 
(10) 
due to the above definition of fictitious observations. 
This and the original concept of GESTALTS according (1), based upon explicit functions describing the surface (4), yields: 
ZEA pr Eny,~2 therefore Vv; = %(Z,a,q9) +0 - Z (11) 
since A := 1,and p > 0 being an arbitrary normalization radius. 
From the LLSQ-point of view this implies that the correction v, is measured along the local z-axis since v; is 
directly related to z = o - 2. This we might call "minimization by least squared coordinate discrepancies". 
It is strict for planes, as given in (3), for Simple Gestalts using a rotation matrix for determining the tiltedness. 
It is weak from a theoretic point of view for planes, the tilt of which is defined by utmost linear terms in (4) since the 
correction is not measured orthogonally to the plane. 
It is weak but acceptable from a practical point of view for street-surfaces which are not much tilted, or, if "height- 
discrepancies" are even desired for minimization in (4). 
But it is really inacceptible for somewhat bent or even closed surfaces. 
Nevertheless, provided a plausible r.m.s. accuracy of the surface's observation, this approach can be easily homogenized 
together with photo-, model- and even polar-coordinates in hybrid adjustment (Kraus, 1996, B3.5.10). 
Closed surfaces are best described using implicitly given formulae. Setting A := 0 in (11), we arrive at 
Zz-%=0=U1-% therefore v, m 2 — 2 (£,0,q) (12) 
with z,(z,a,q) being the formal function (6) vectorized according (8) with GESTALT coefficients a and form parame- 
ters q. The discrepancy v; in (12) appears now as pure functional residual; moreover, as residual of an implicit equation 
since z(Z,a,q) shall represent some closed surface f(Z,7,Z) = f(Z) = 0. An implicit function may be multiplied 
with an arbitrary factor a # 0 letting the geometric appearance of the surface unchanged because this proportionality 
factor a can be absorbed by the coefficients @ — a * a for instance. z(Z,a*a,q) = o* %(Z,a,q) yields v, 
versus a *v, =: w as discrepancy to be LLSQ’d also known as "minimization of algebraic error". This property does 
not matter for a point exactly on the surface since in this case v; — 0 holds true; in hybrid adjustment on the other hand 
- when that point is overdetermined by e.g. rays from photos or it is also measured by polar coordinates - we might bias 
our results neglecting homogenization of weights (see (Kraus, 1996, B3.5.6)). So, the right choice of a is responsible 
for what will be minimized in LLSQ adjustment. One should also confer to those examples of an ellipsoid with semi-axes 
(a,b,c) versus an ellipsoid with reciprocal semi-axes (1/a,1/b,1/c) as form-parameters near the end of section 2.4; 
a = r? - s? -t? illustrates the phenomemon of implicit functions (not only in the case of canonic quadrics). 
Some people don't consider that important enough and do adjustment of a circle with the Observation equation v — 
z? -- y? — r? not worrying about minimizing corrections of r? instead of corrections of r by LLSQ. 
For some time, we used an approximate method to alleviate the mis-homogenization: simply doing some special weight- 
ing modulating the standard weighting due to accuracy. E.g. adjusting a circle, it is simple to multiply the observation 
equation w = z2+y%—r2 with 1/r,, this 7, being an approximationof 7 yielding v = w/r, = z? [ro -y? [r; —r? [r, 
. With this simple measure, circles and also (more or less circular) quadrics may be adjusted with "near-metric" correc- 
tions - eased by the fact that "weighting" an observation equation is also simply multiplying it with a factor. 
This sloppyness being not satisfying, we will try to get a metrical measure for the discrepancy of a point X resp. 
$ used as argument for an implicit function. We consider the euklidean distance measured orthogonally to the surface 
appropriate to serve as such metrical measure. 
  
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 475