ied to each 
dicular di- 
(Eq. 1) 
is a vector 
‘he spatial 
(Eq. 2) 
pect to the 
(Eq. 3) 
est ampli- 
rting from 
pixel. This 
me search 
cted pixel. 
el remains 
rmined by 
| the direc- 
the fitting 
e edge (see 
(Eq. 4) 
es (u) 
es (u) 
ubpixel 
are estimated in a least-squares adjustment. The vector x 
of unknown parameters (Co, C,, c5) is determined by mini- 
mizing the sum of squares of the estimated residuals as 
AT 
A EP 
60 z ; (Eq. 5) 
o> 
  
& 2 (ATPA)-! - ATPy 
The precision of the estimates x can be derived by error 
propagation. 
^r^ 2.2. A T -] 
(Eq. 6) 
As the result should not depend on a user defined window 
size, the determination of the best fitting parabola is re- 
peated with different window sizes of odd number, grow- 
ing from a width of five pixel until the estimated results 
are not improved by the next higher window size. 
The extremum of the best parabola gives the subpixel so- 
lution of the line point (ug): 
  
of(u)/du = 0 ug. (Eq. 7) 
The precision of subpixel solution can be estimated by er- 
ror propagation with the know covarinace matrix of the 
estimates x (parameters of parabola): 
D(y) = A-D(x)-A" (Eq. 8) 
Finally a pixel is accepted as a line pixel, if 
e the extremum of the parabola falls inside the pixel, and 
if 
e C, is negative and differs significantly from zero. 
By applying additional user defined criteria, a pixel can be 
automatically rejected by the algorithm. Such optional cri- 
teria are: 
e if the orientation of the gradient differs more than a 
certain treshold from the mean orientation of all line 
points, 
e if the distance between the estimated line point and the 
approximate line is above a certain treshold, 
e if the standard deviation of the determined subpixel 
line point ug is above a certain treshold, or 
e if the amplitude of the gradient, corresponding to the 
line point, is below a certain treshold. 
Once the line points of all objects within the image will be 
detected, the algorithm converts this raster data into vector 
format by fitting straight lines to the linear boundaries. 
f(x) = ag+a, x 
f(y) = ag+a, y (Eq. 9) 
The estimates for the standard deviation of line points are 
used as weighting functions. The line parameters (ag and 
aj) are estimated by a least-squares adjustment (Eq. 5, 6). 
These vectors will then be used to determine the image 
coordinates of object vertices (xo, yo) by line intersection. 
(Eq. 10) 
yo= a4 t b, : Xo Yo * 85 t b5: xo 
551 
X = — — 
0 Yo 
45—8j 
8b =2 0 (Eq. 11) 
42784 
Here the estimates for the standard deviation of the line 
parameters serve as weighting functions. If more than two 
lines intersect in one point, the intersection is calculated 
as a weigthed mean of all possible intersections. 
3.2.2 Problems to be faced during internal loop 
For the image-based feature measurement not only an 
ideal appearing edge has to be taken into account, often a 
real world scene disturbed by additional phenomena. Such 
phenomena can be occlusions, weak or missing contrast, 
inversion of gradient directions, shadows, radiometric 
interferences, or reflections (see Fig. 4). 
  
“dient direction 
Figure 4: Problematic cases occuring during internal 
loop. 
Shadow edges can easily be rejected with the information 
about the approximate lines. This criterion does not work, 
if the shadow edges run parallel in a short distance to the 
edge one is looking for. Weak contrast, reflections, and 
occlusions can be handled by accepting a line pixel only if 
the parabola fit is significant and if the first parameter is 
negative. Interferences and occlusions tend to mislead the 
measurement of the correct edge. This problem can be 
faced by taking the gradient orientation into account. Line 
parameters can be determined significantly, if just a part 
of the entire line can be observed. Inversion of the gradi- 
ent direction along an object line, typically occurring in 
real world scenes, where the horizon is imaged or an ob- 
ject feature has different reflection properties, can be han- 
dled if the gradient orientation is taken into account. It is 
notable, that the gradient orientation is a strong criterion 
in order to determine the correct line pixels (Hónisch, 
1992); 
3.2.3 External loop and orientation loop 
The internal loop is performed sequentially in all images. 
The object coordinates of a point imaged in two or more 
images are calculated either by a spatial intersection, by a 
bundle adjustment or by a bundle adjustment with self- 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996