Part B3. Istanbul 2004
1 fh) (1)
Wil
hs. |
ective elements 7/5; and
minish errors on the
same factor.
ge coordinates x and x
m through f and go into
ries. This has a similar
be balanced. Entrances
h;5 ho, hj; in the affine
tries for the unknown
ipproximately the same
ve elements 7/5; and fi;
rances (factor f^).
lanar homography H we
1¢ the decomposition
ix, / is the translation of
| (Faugeras, 1995). This
d system into the centre
degrees of freedom that
1 angles or as normalized
it. The vectors n and /
‚dom because n will be
je second camera to the
1.
"mined from the image
ial information e.g. from
od approximation for the
sible structure will result
average roof height over
1 good approximation to
formation on the north-
ve rely on shadow and
) geo-reference from the
mized or matched objects
of the air-craft. This 18
of homographies needs to
se from mappings with
mera will be mounted on a
ation will be measured by
more precision than the
ield. This known rotation
he coordinates of the first
9 be the identity. Then the
ntral collineation with real
nar homology or elation
Considering the homology
hat the double eigenvalue
eigenspace is the horizon
d-points (the image of the
plane at infinity) and n also
other cigenspace is |-d and
tion f The eigenvalue
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
corresponding to this eigenvector is /-//». And since n is
normalized we get the proper length of / from this equation.
This solution is unique up to change of sign of » and /.
The eigenvalue calculation of a homography estimated from
correspondences may also result in a pair of conjugated
complex eigenvalues and a single real eigenvalue. In the
rotation free case such result cannot be used for pose
estimation. A homology has to be searched for that is closest to
the estimated homography.
The elation case: The rotation free homography becomes an
elation if the epipole lies on the horizon line i.e. //^n-0. This is
not an exception but common for many standard flight
manoeuvres (keeping level). Such mappings have a triple eig
value with a corresponding eigen-space of rank two (the
horizon line). The epipole cannot be stably estimated from the
eigen-spaces of the homography. Instead it can be estimated
from pairs of correspondences directly by intersecting the
correspondence straight lines. We used the correspondence-
pairs that are part of the best solution for homography
estimation (see Sect. 3) and iterative re-weighting to minimize
the influence of outliers in this estimation. Given an epipole
estimation / and setting the Rotation A to identity / equation (2)
becomes linear in the plane parameters n. To cope for the
unknown scaling of / we set a forth scalar parameter and get
en-
t
Hz LI un). (3)
Dividing this equation by u we get a set of nine homogenous
equations in the four unknowns nm and //u. It is solved by
singular value decomposition. In fact this method is applicable
for all central colineations, i.e. not only for elations but also for
homologies. It may replace the eigen-space construction as
well.
With rotation: Taking non-trivial rotations A into account the
homography is transformed into diagonal form using singular
value decomposition H-UH'V with orthogonal rotations U and
V. Equation (2) is transformed to H’=R —'n" where R-UR Vl,
(=Ut" and n=Vn'. This new equation can be solved analytically.
Assuming the singular values /;, /; and /; in the diagonal
matrix 77' to be sorted and /», scaled to one the rotation may be
restricted to the Y-axis and the Y-components of / and n set to
Zero:
[x |
BR 020) fcos A. 04 -sin ^ CUR. 0 Io (4)
0 1 0 = 0 | 0 = 0 0 0
0-95; sinZ- 0: cos ) ":n., Q fu
| ( 6
Hn ors | Ü zx (ht - 1h ) } Lam, =(h}- Mn,
1, ,
(
m
where all four combinations of signs s,=+/ and s,=+/ are
permitted. Transforming these solutions back to equation (2) we
obtain four solution sets for A, t and n.
A critical situation occurs where the solutions branch, i.e. where
the value in one of the roots becomes small or two of the
singular values are nearly equal. This is the case if / and n are
parallel i.e. the craft is directly going nadir — an unusual flight-
manoeuvre. The method will completely break down for three
equal singular values i.e. 7=0. We exclude this case because it is
physically impossible for aircrafts.
Special forms: The different applications lead to different
structures of the homography matrices. Equation (6) lists some
important cases. A camera looking directly nadir down to the
surface and the craft moving in X-direction will produce
something close to the form FH, Side-looking obliquely
mounted cameras will give almost A. A forward-looking
geometry will result in something close to 77;.
Lov 0)
{1.0 ~» ] vuv —u tem | | (6)
#10 L 0 | W=10 1 ola =o ; ol
0 0 1] 0 0 ud ur]
—uv |
0714
l-u J
u is a velocity parameter and v is a parameter for the tilt of the
plane. Note that /, is an example for the elation case discussed
above.
Error Propagation: These frequent special forms are used to
propagate small displacement errors in the correspondences into
the estimated pose parameters. For this purpose we use the
following set of four correspondences
| —1) {1 nd
f
|
(BL P PPS BP KA (8)
PA FL |)
We set motion to #=0.0/ for the matrix H, and H, (H, for a
angle of 45°) and computed corresponding points from this.
Then we put an error € to the last point computed the disordered
homography and decomposed it again. Displacement results for
the vector / are listed in Table 1.
el=0.0005 e2=0.001 £3=0.0025
Rotation-included
f=10 11% 23% 61%
{=100 166% 281% 665%
Rotation-free
f=10 1.7% 3.2% 7.4%
F-100 1.3% 2.5% 12%
Table 1. Sensitivity of translation 7 to errors at different focal
lengths. Deviations are given in ratio to the length of 7.
These are fairly small errors (3 being some half pixel). They
can only be reached by using more than four correspondences
and a robust estimation method. Also a ratio of 10 or 100 of the
focal length to half of the image size is common for IR-
cameras.
2.5 Non-projective and projective Distortions
Particularly thermal IR cameras of older construction type often
show strong non-projective distortions. They only have a small
number of sensors that are used to scan the image successively.
The rotating mirrors that are used to map the image to the
sensors cause a non projective mapping. Examples for such data
are Videos I and Il from Sect. 4. If the construction details of
camera are not known the non-projective part of the distortion
