ZM
4
Yo
Ag
x
Yo.
fp
» XM
si "5
AN
Figure 3: Transformation pixel image to normalized image
images. The rotation M about the X-axis influences the
nonquadratic shape when computing the normalized images.
The normalized rotation matrix Ry is a multiplication of
two rotation matrices: the rotation from pixel image to true
vertical position and the rotation from true vertical to nor-
malized position.
Ry = RgRT (6)
The Ry is an orthogonal rotation matrix which transforms
the pixel image to the normalized image. Since in Eq.(6)
RT is the transposed rotation matrix of exterior orientation
elements, the Ry matrix must be computed for both images
in stereo. We may use one of two transformations from pixel
image to normalized image: transformation using collinear-
ity condition or projective transformation.
3.2.1 Transformation using collinearity The co-
llinearity condition equations can be used for the transfor-
mation of the pixel image to normalized image. The trans-
formation is represented in the following equation and is
illustrated in Fig. 3.
7112p + T12Yp — Ta fp (7)
Ta10p + T32YP — TasfP
Ta10p + T22Yp — T23 fp
T312p + Ta2yp — Tas fp’
VN — —JN
YN = JN
where 711 - - - 733 are the elements of the Ry rotation matrix.
3.2.2 Projective transformation The projective
transformation can be applied since both pixel image and
normalized image are planar.
C€112p cC +c
NS 11TP 12YP 13 (8)
C312p + c32Yyp +1
UN €31*p t C22yP t C23
€312p + c32yp +1
By comparing the coefficients in the projective transforma-
tion with those in the collinearity equations, we find the
406
following identities:
fNT11 fra
Cu = Ca = (9)
fpras fPT33
ÎNT12 fNT22
Cia = C22 =
fpras fpras
fNT13 fNT23
Cis zm Cas =
T33 T33
T31 T32
Cu =-—+— Cg -—
fpras fpras
When performing the transformation pixel image to nor-
malized image, the quadratic tesselation of the pixel image
results in nonquadratic tesselation of the normalized image.
In order to avoid interpolation into quadratic tesselation, it
is recommended to project the tesselation of the normalized
image back to the pixel image (see also Fig. 3). The co-
efficients for backward projection are obtained in the same
fashion by RT, if the focal lengths of the pixel and normal-
ized image are the same (fp = fw).
Cy =Cn Cy =0Cn (10)
Ci = Cn Ch — C
Ca = CafpívN e = Cz2frp fN
1 1
/ a C / = C
31 M fu In 32 23 fof
For the more general case of different focal lengths (fp
fn), the backward projection is obtained by inverting Ry
because Ry" # RY.
3.2.3 Resampling After applying a geometric trans-
formation from the normalized image to pixel image, the
problem now is to determine the gray value of the new pixel
location in the normalized image, because the projected po-
sition in the pixel image is not identical to the center of the
pixel. Therefore, gray values must be interpolated. This
procedure is usually referred to as resampling. Several in-
terpolation methods may be used.
e zero-order interpolation: the gray value of the nearest
neighbor is chosen. This is identical to rounding the
projected position to the integer, corresponding to the
tesselation of the pixel image system. This simplest
process may lead to unacceptable blurring effects.
e bilinear interpolation: the gray values of the four sur-
rounding pixels contribute to the gray value of the
projected point depending on the distance between the
projected and four neighboring pixels.
3.3 Normalized image
The procedure discussed in the previous section establishes
the transformation between pixel image and normalized im-
age. The distortion parameters are determined during cam-
era calibration. When resampling the gray values for the
normalized image, we also apply the correction. Thus, the
computation of the normalized image proceeds in four steps
(see Fig. 4).
T,: Transformation between pixel image and original pho-
tograph (diapositive). The transformation parameters are
determined during camera calibration. Common references
for these transformation parameters are fiducial marks, re-
seau points, and distinct ground features.
