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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
A
Figure 3: Three different views onto the west facade of Alexiuskapelle. Superimposed on the middle image is the given
outline polygon of the facade.
where A is composed from known x; and X;, and p are
the coefficients of the unknown matrix P written as a vec-
tor. The solution p is the right null vector of A. Refer to
(Hartley and Zisserman, 2000) for details.
An initial solution of P is computed seperately for each
image. There are some disadvantages to the set of camera
parameters gained this way. First, parameters that usually
are constant for all images, like e. g. the principal point,
are calculated individually and will have slightly different
results due to noise in the coordinates. Second, as this is
a linear solution and therefore only linear parameters can
be estimated. Nonlinear parameters such as radial lens dis-
tortion can not be computed. Finally, only the algebraic
error || Ap|| is minimized which in general is not geomet-
rically meaningful (Hartley and Zisserman, 2000).
(b) Bundle adjustment To overcome all disadvantages,
bundle adjustment is a feasible choice. It is based on an
equation system of nonlinear versions of Eq. 1 where a
term for radial distortion has been added. Parameters that
are constant for all images have been included only once.
The equation system is solved simultaneously for all un-
known parameters. The geometric error minimized is the
sum of the squared distances between projected and mea-
sured points in the images.
The result of these two steps are a set of camera parame-
ters — camera calibration as well as pose — that define the
relationship between the vertices of the coarse model and
the image points best.
2.3 Rectification of Image Pairs
Rectification of image pairs is to transform both images
such that all epipolar lines are parallel to image scan lines
and that corresponding epipolar lines have the same y-
coordinate (Koch, 1997). This is an important preprocess-
ing step for dense stereo matching because it simplifies
the search for corresponding pixels along the (normally
slanted) epipolar lines to a search along the scan lines.
First the epipoles are projected to infinity. Then the irn-
ages are rotated such that the epipoles lie in the direction
of the z-axis. Finally, one of the images is shifted in the
y-direction so that corresponding lines of the pair coin-
cide. There are still two degrees of freedom left to improve
the rectification without destroying its properties: a shear
along the z-axis and scale factors for both axes. These have
been exploited such that the coordinates in one of the im-
ages are closest possible to the original coordinates.
The rectifying 2D homographies are left multiplied to the
camera matrices which causes the actual camera calibra-
tions to change. This must be taken into account when
computing depth from pixel disparities.
2.4 Guided Correlation
Given images in epipolar geometry, the main task is to
compute a dense disparity map, i.e. a map containing the
relative distances between corresponding pixels, that de-
scribes the surface relief. In a later step the disparity map
will be upgraded to a depth map containing metric units
instead of pixel units. Corresponding pixel locations are
found via cross correlation — a local maximum hints at
a possible match, but repetitive patterns will also deliver
false hints and in homogeneous regions there are no clear
maxima. A blind search for maximum correlation only
does not give satisfactory results.
2.4.1 Dynamic Programming A dynamic program-
ming scheme that allows to incorporate some constraints
to guide the matching has been used (Falkenhagen, 1997).
The general idea is to define a cost function plus some con-
straints and to find its global minimum. This is feasible
whenever all decisions can be broken up into a sequential
scheme.
For each scan line, we can put up a two dimensional cost
matrix where columns and rows correspond to pixel posi-
tions and disparity values respectively (see Fig. 7). The se-
quential scheme is implemented by filling in the costs
