The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008
(j) — Üq + (1 + £2j ) • (f) + Cl2 • A
A = bQ + b x • (j) + (1 + b 2 ) * A
h' = c 0 + c l ■ (j) + c 2 ■ A + h
(/,$) = RPC {(/)', A' ,h')
where
s and / are sample and line coordinates of the GCP in
the image ;
(j), A and h are the longitude, latitude and height of GCP,
respectively;
(/)' , A' and h’ are the refined longitude, latitude and
height of GCP, respectively, consistent with the RPC
transformation;
a 0 , a x , a 2 , b 0 , b x , b 2 , c 0 , c x and c 2 are the
adjustable coefficients.
Under appropriate conditions, some of the coefficients may be
held fixed. The refined RPC can achieve an accuracy of less
than lm. A new set of refined RPC can be obtained from the
coefficients.
4. HEIGHT OF AN OBJECT
To obtain the height of a building from a single IKONOS image,
most of the methods measure the distance from the top to the
base of an object (Willneff, et al., 2005), or from the top to the
shadow (Irvin et al., 1989; Lin et al., 1998; Bouguet et al., 1999;
Hui et al., 2000; Hinz et al., 2001).
Figure 1 shows the geometry of the systems of rays, from the
Sun and the satellite position when the 3D building is imaged.
Figure 2 shows an example of 2D satellite image of a building
with its top, base and shadow clearly located. From the
geometry, we can see that the top point of the building in 3D
geometry will be shown as p(l,s) in a 2D satellite image,
while its base will be located in b(l b ,s b ), and its shadow will
be formed in point s(l s ,s s ) . By measuring the length
L pb from p to b, or the length L ps from p to s in the 2D image,
the height h of the building can be easily derived with (3), or
with (4) to (6):
h = tan A • L pb
(3)
h = tan A' ■ L sb
L 2 ps = L 2 p b + L] b - 2 • L pb ■ L sb • cos (a - a')
1 1
1
2-cos(«-a')
A r-
- + -
^ —
tan 2 A' + tan 2 A tan A'- tan 1
where a = azimuth angle of the satellite
A = elevation angle of the satellite
a'= azimuth angle of the Sun
A'- elevation angle of the Sun
(4)
(5)
(6)
The azimuth and elevation angles of the satellites and the sun
are available from the image metadata file. The accuracy of the
height derived from the distance between top and base, is
directly proportional to tan A , while for method of measuring
distance of shadow, it is relative to both A and A', and the
difference between azimuth angles of the sun and satellite. It is
obvious that the smaller the elevation angle of the satellite or
the longer the shadow formed in the image, the more accurate
the height obtained. For lm resolution image, L pb and L ps can
be measured within lm accuracy if the base and shadow can be
shown clearly in the image.
While the sun angles are consistent during the image acquisition
duration of a few seconds, the satellite angles actually vary over
the imaging period, thus the set of angles given in the IKONOS
metadata file is usually not sufficient to derive the heights
accurately for all buildings over the scene.
We develop a semi-automated method in a reverse way using
the top, base and shadow information. Instead of locating the
base or shadow in the 2D image, and measuring the distance
from top to base or shadow edge, the new approach reduces
two-dimensional problem into one dimension. The new
approach is to change the height of the object, and predict the
locations of the base using RPC and the shadow edge using
both RPC and the angles of the azimuth and elevation of the sun.
The sensor model RPC provides accurate 3D map coordinates
(x,y,h + h b ) to 2D image point p(l,s) (7) through the entire
scene. The base point p h (l b ,s b ) has the same map
coordinates as the top point p(l,s) but with different height,
its 3D coordinates are (x,y,h b ) (8). The shadow point
p s (l s , s s ) can be expressed in terms of /, s, h and angles of
the sun (9-10).
(x, y,h + h b ) >(/, s)
(7)
(x, y,h b ) >(l b ,s b )
(8)
i s =i-t dl -dh-h msa '
■o dh tan A'
(9)
Figure3. The height of the building derived from shadow is
31m.
