International Archives of the Photogrammetry, Remote Sensing
may- be estimated by a non-linear function x,7q(x,) where x, is
the distorted location in the image and x, the location used for
estimation of the homography H,. We used a cubic 2D-
polynome for q and estimated the parameters by forcing
locations that are known to be collinear in the scene to be also
collinear in the images. The effect of this function q is
visualized in Figure 1.
Figure 1. Example for the cubic distortion correction applied to
video Videoll; these images are usually not
calculated, only the positions of the interest points
are corrected.
Calculating pose from such homographies H, estimated from
correspondences (x, x) directly may lead to systematic errors
because the camera distortion (and its non-linear correction)
may also contain an unknown projective part Hy For such
distortions collinearity is an invariant. Hy applies to both
positions of each correspondence: Hy x’. =H Hy x. . If there is a
ground-truth homography G for some images of the scene the
equation HW, G H, may be transformed to Hy H.-G H,;=0,
and used to estimate 77,. But care has to be taken that Hy is
chosen with a sufficient determinant. It should be chosen from a
particular family of mappings like shearing-mapping, rotation
or projective distortions. Systematic errors for a particular set-
up can be measured if ground truth is given with a calibration
run.
Many contemporary thermal cameras feature focal plane array
sensors and can thus be handled like ordinary CCD TV-
cameras: For normal or long focal lengths pin-hole camera
models will usually be precise enough and the distortion of
wide angel lens may sufficiently well be treated by a single
quadratic term. An example for data from such a modern device
is Video III in Sect. 4.
and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
3. SEARCHING FOR PROPER CORRESPONDENCE
SETS WITH PRODUCTION SYSTEMS
A planar homography h(p)=Hp can be linearly calculated from
four correspondences c=(p',p) by direct linear transformation
(DLT) (Hartley; Zisserman, 2000), but they must not be in
special configurations (like if three of the four points are
collinear). If there are more than four correspondences available
a squared residual error sum R is minimized.
R 2 M e(p Hp)
There is again a DLT solution to this, where £ is an algebraic
error that approximates the inhomogeneous 2-d error provided
the point coordinates are given in proper normalization
(Hartley; Zisserman, 2000). However, this process being a least
squares method is very sensitive to the inclusion of outliers in
the calculation. Therefore a robust estimation method is
required ‘ that can detect and eliminate the false
correspondences.
3.1 Robust Estimation
Several proposals have been made to minimize the influence of
such gross errors. One example is the iterative re-weighting
approach (Holland; Welsch, 1977). This method avoids hard
decisions on the set of measurements. However, the most
popular approach today is the well known random sample
consensus (RANSAC) approach (Fischler; Bolles, 1981).
The Random Sample Consensus Method: Let C=/c,, in GUibe
the set of correspondence cues obtained from the images. It is
expected that C is the disjoint union of a set of correct
correspondences C, and outliers C,. The goal is to identify these
sets by automatic means and to minimize the goal function
Hu. = US min(R(H,C,))
H
This minimization varying the homogenous matrix 77 while
keeping C, fixed is a straight forward linear computation using
DLT. However, searching C for the proper subset C;. poses an
exponential challenge (in m). The RANSAC-proposal
recommends probing the power-set of C by drawing random
minimal subsets. Here these are quadruples s=ft;, 14} from
{1,....,n}. Each of these samples leads to a hypothesis for H, (at
least if the calculation succeeds). And for every such hypothesis
the residual error for all elements in C is determined.
, 2
FT ep, Hp)
Using a global threshold à the consensus set of the sample is
defined as /c; in C: r,; < À 7. The largest consensus set is
supposed to be a good approximation for C,. Usually it is too
time consuming to check all (73) samples. There are decision
theoretic considerations that give hints on how many samples
should be drawn given an expected outlier-rate, a variance for
the positioning of correct correspondences and a significance
level [Hartley]. It is also possible to continue probing until a
predefined minimal consensus is reached, or — in an any-time
manner — until a solution is demanded by exterior time
constraints.
If the set C is not equally distributed in the image the method
will adapt the transform with more weight on densely populated
regions. An isolated important correct correspondence in an
otherwise homogenous image region may either end up as
“outlier” or it will have equal weight like any other single
correspondence in the calculation.
International Arch
Good Sample Co
an improvement
random samples |
assessment criteri
implemented the f:
1. Locations are pi
structure to allow
(Foerstner, 1994)
priority.
2. Each sample c
Samples with high
3. Pairs of corres
according to their
far apart gain high
4. Two pairs form
is assessed accordi
four triangles form
property will be ze
with large minimal
5. Each quadruple
space of homograr
vote for close trans
cluster of homogre
DLT that minimize
preceding it. We c
the cluster. It is ass
and also to the as:
geometric propertic
Good Sample Co
discussed in (Mich
assessments are c
production system.
using a data-drivi
capabilities (Stilla,
4. EXPE
4.1 Experiments y
Three video sequen
been used to ver
estimation and pos
homographies. All
Fig. 2 shows examp
Video I: Oblique si
city of Karlsruhe (b
camera from an airy
Such cameras giv
camera was zoomed
X- and y-direction
perspective.
Video II: Oblique
region (including a |
same TICAM camei
3000m height. The
still giving a small fi
Video III: Forward-
little creek (trees, b
camera from a hel;
cameras give almost
has fixed standard 1
spacing ratio is appre
