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3. METHODS
3.1 Estimating the orientation parameters
The relative orientation of two images is described as a
rigid movement of the bundles of rays by a translation and
a rotation
v=Ru+T
u and v being the projective coordinates [u, uy, 1] and
[Ve Vy, I] of an model point, R a 3x3 rotation matrix and
[t, 5, t] a translation vector with known direction but
unknown length.
In photogrammetric notation the equation is usually
written as
x" x' b.
VERF b,
—C —C b
Z
Five parameters have to be estimated in order to solve the
equation system with a minimum of five corresponding
point pairs, e.g., ®, ©, K, by and b, in the photogrammetric
notation. This involves a linearisation of the non-linear
equations and an iterative estimation technique that
requires approximate values of the unknowns, making it
unsuitable for general cases with arbitrary orientations.
A linear solution has been formulated in the photo-
grammetric society by, eg. [Stefanovic, 1973],
[Thompson, 1968] and later in the computer vision
society by, e.g, [Tsai & Huang, 1984]. The linear
solution is based on the coplanarity condition stating that
the vectors v, Ru and T are in the same plane.
yo Rus,
v, Ru, T,|=v(Rux T)=0
Vz Ru, T.
This can be written as [Stefanovic]
v‘CRu =0
or
v'Eu=0
where C is the skew-symmetric matrix
o^ rb
C = ia 0 fi
[5 =; 0
and E-CR is called the essential matrix. The essential
matrix can be characterised in different ways, but using
singular values is the most common in literature [Tsai &
Huang, 1984]. Let E-USV, be the singular value
decomposition, SVD, of E, where S is a diagonal matrix
S-diag(s, s» $3). A matrix is an essential matrix if and
only if s,=s2 and 5,20 ! This also implies that the matrix
is of rank (2) and that
EE'E - Lirace(EE'E)E
2
The decomposition of E into R and C is a non-trivial task,
but has been solved by, e.g., [Brandstitter, 1991] and in a
more general form using SVD by [Tsai&Huang, 1984].
Since there are eight unknowns in the essential matrix but
only five parameters needed for the relative orientation,
linear dependencies might exist in the linear solution that
will give a biased or singular solution. This can be seen
when writing the explicit equation of the projective
coordinates (x', y', 1) and (x^, y”, 1).
xx'eitxXy'ep +x'e,3 + y'X" 6; + V'y" e9, + Y'e33 +
X"es1 + y'ep +1=0
For image pairs with almost parallel optical axes and
small rotations, as in the case of aerial images, the
coordinates of y’=y” and ez; and es, will be dependent. In
some extreme, but not trivial, cases the number of
unknowns will reduce to five. These dependencies give
rise both to numerical problems in the estimation process
and to a bias in the estimated parameters.
The fact that the solution is sensitive to noise is well
known and has been addressed by, e.g., [Hartley, 1995]
and for the fundamental matrix in the uncalibrated camera
case and by [Hahn, 1995] and [Philip, 1996] for the
essential matrix and calibrated cameras. To ensure that
the estimated E matrix is of rank (2) and that the singular
values s,7$? and 55-0 is not sufficient to remove the bias
in the estimate.
In the used implementations, the numerical problems with
singularities are handled using the SVD algorithm when
solving the equation system. The algorithm is numerically
more stable than, e.g., the Cholesky algorithm and
singularities can be treated and eliminated during the
estimation. The bias is removed by minimising the X(p?),
where p is the distance to the epipolar line, i.e., the y-
parallax for aerial images, using the conjugate-gradient
algorithm.
The difference between the results from the original linear
solution and the improved solution can be seen in table 7.
The residual's from the linear solution show a clear bias.
In the improved solution the bias is removed and the
standard error reduced. It is only the improved linear
solution that has been used in the experiments.
"These requirements hold for a calibrated camera, i.e., with
known principal point and principal distance. For the un-
calibrated case, the matrix is called the fundamental matrix and
the requirements on the singular values are that s; 2s, and s3=0.
43
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
