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fit into its counterpart 3D polygon. For reviews of the various 
texture mapping techniques, see Haeberli and Segal, 1993, 
Lansdale, 1991 and Weinhaus and Devarjan, 1997. 
In principle, the following algorithm could be used for texture 
mapping: 
For each 3D triangle t: 
1. select one image i from the set of images taken from the 
scene in which triangle ¢ appears, 
2. using exterior orientation, determine the correspondence 
between 3D triangle vertex coordinates in space and 2D 
coordinates in image i, 
3. specify 3D and texture coordinates in some modeling 
language such as VRML, and 
4. view the scene using a standard viewer. 
However, due to the following considerations, this simple 
approach is not feasible in most cases: 
e The correct mapping between the plane triangle ¢ lies in 
and the image plane of image i is given by a projective 
transform. Since viewers do not use this transform, 
distortion arises at triangle edges. 
e When standard lenses are used for the cameras, lens 
distortion parameters have to be applied, else distortions 
will be visible at common edges of adjacent triangles 
mapped from different images. 
e Usually, it is desirable to have a constant texel-size on the 
object. This results in a more uniform appearance and also 
makes it possible to control file size and rendering speed 
more precisely. 
  
  
visible at 
error source triangle | type technique used 
edges 
wrong mapping | all Geome | warping according to 
(viewer) tric collinearity equations 
    
  
lens distortion | mapped | Geome | application of 
  
from tric additional parameters 
different 
images 
radiometric mapped | Radio | global gray-value 
differences from metric | adaptation, blending 
between different 
cameras images 
  
non-uniform mapped | Radio | local gray-value 
radiometry from metric | adaptation, blending 
across single different 
camera images | images 
  
  
large deviations | mapped | Geome | local triangle re- 
  
  
  
  
of triangle mesh | from tric assignment, blending 
from true different 
surface images 
  
Table 1. Error sources for visual discontinuities in mapped 
scenes and techniques used to to minimize their visual impact. 
Thus, it is obvious that image warping has to be done 
independently of what the viewer does to render the scene. Even 
when correct modeling of exterior, interior and additional 
camera parameters is used, however, there are still problems in 
practice that may lead to geometric and radiometric 
discontinuities which can easily disturb the impression of 
looking at a “real” scene. For example, radiometric differences 
between the cameras lead to radiometric differences along 
triangle edges; too large deviations of the underlying triangle 
mesh from the true object surface give rise to geometric errors 
(e.g. parts of the object’s surface appear in more than one 
335 
triangle texture). Table 1 summarizes possible error sources and 
the techniques we adopted to minimize their visual impact. We 
address each of these problems in the following sections. 
6.1 Proper Geometric Fit 
As discussed above, image warping has to be done 
independently of the transformation applied by the viewer. To 
that end, the employed method defines a local texel coordinate 
system for each 3D triangle. The texel size (in object 
coordinates) can be set to the desired resolution. Each texel is 
then computed using exterior and interior orientation, including 
lens distortion parameters obtained from camera calibration. As 
seen in figure 5, there is a clearly discernible difference between 
triangles mapped with and without distortion parameters. 
    
(a) no distortion parameters ^ (b) with distortion parameters 
Figure 5: Ensuring geometric fit by using distortion parameters 
6.2 Radiometric Differences 
Usually, radiometric discontinuities result along common edges 
of adjacent triangles mapped from different images (see e.g. 
figure 7(a)). The main reasons for this are 
1. radiometric differences between cameras, 
2. non-uniform response of each camera across the image 
plane, and 
3. different sensed brightness due to different camera 
positions (i.e. different orientation relative to surface 
normal vector). 
(1) can result from different aperture settings; however, since in 
our case video cameras with automatic gain control are used, 
the radiometric differences have to be modeled on a per-image 
basis rather than per camera. We address this problem by a 
method termed "global gray-value adaptation". (2) is most often 
caused by a brightness decrease from the image center to image 
borders. Both (2) and (3) can be tackled by a radiometric 
correction on a per-triangle basis (termed "local gray-value 
adaptation" in the following). 
The global gray-value adaptation estimates gray-value offsets 
between images. The gray-value differences along the border of 
adjacent regions (triangle sets) are minimized by least-squares 
adjustment (figure 6). 
  
Figure 6: Global gray-value adaptation. Left: regions and 
borders formed by triangles mapped from the same image. 
Right: corresponding observations dj; and unknowns h;.