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2. METHODS 
Non-ideal panoramic camera rigs cause offset to projection 
centres causing perspective errors. The amount of perspective 
error due to such offset is proportional to the distance between 
camera system and targets but also to the depth of a target. 
In this work, we use developed Matlab simulation program to 
calculate the perspective error introduced by a projection centre 
offset. We use one camera as a reference with an arbitrary 
projection centre location, which we can give as an input into our 
simulation program. We can also specify two target planes. One 
is closer to the camera and the second is further away. The closer 
plane is also smaller so it does not occlude the second plane. The 
amount of perspective error is calculated by placing another 
camera beside the reference camera with slightly shifted 
projection centre location (Figure 1). Furthermore, we are able to 
specify any shooting distance, object size and projection centre 
offset in all three dimensions of the object coordinate system. 
Once we have established the geometry of the system, we can 
rotate the camera in order to illustrate the panoramic rotation. 
The ideal panoramic image is achieved when there is no offset in 
projection centre locations. In such case, the perspective is 
constant because the coordinates of projection centres are fixed 
during the rotation. If some projection centre offset does exist, 
the rotation in relation to the reference projection centre causes 
the projection centre locations of sub-images to form a circular 
path around the reference projection centre. This creates a set of 
unique perspectives which leads to varying perspective errors. 
  
Side Top Perspective 
  
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Figure 1. Matlab simulation program showing a geometry with 5 
unit offset in X-direction, 20 unit distance to first target plane 
and 30 unit distance to second target plane. The first target plane 
is illustrated with orange and the second target plane with non- 
color grid in the perspective view. Image planes are shown in 
blue color. 
The simulation program calculates rays starting from the 
projection centres of camera locations, going through the corners 
of our first target plane and reaching onto our second target 
plane. Corner points of the first target plane are named "Point 1" 
and "Point 2" in all cases of this article. However, the offset of 
the projection centre causes the rays to not reach the same 
coordinates on the second plane. This gives us an estimate of the 
perspective error at the object space, simulating a real world 
situation where we know the offset of the projection centre, 
Shooting distance and the target depth. By target depth we mean 
the distance between two target planes. In Figure l, we 
distinguish the rays that pass through the same point at the first 
  
    
  
  
  
  
  
  
  
  
  
   
   
   
   
   
    
  
    
   
   
   
   
   
   
   
  
  
  
   
  
   
    
  
  
  
  
  
  
  
  
  
   
  
  
   
  
   
   
   
    
    
   
   
   
   
   
  
    
  
   
     
target plane with different colours (Point 1, red lines; Point 2, 
green lines). The distance between two corresponding rays in the 
second target plane indicates the amount of shadowing due the 
perspective error. This error also causes physical shifts of image 
features at the image plane that are detected as misalignment 
when sub-images are stitched into a panoramic image. 
3. RESULTS 
In our simulation, we specify the coordinates as project units 
without any prefix but they can be considered as metric units. In 
this particular simulation example, our first target plane is 20 
units away from the panoramic rotation axis and the distance to 
the second target plane is 30 units. A camera constant is four 
units and the image plane size is 4x4 units. This translates to 
53.13° horizontal and vertical FoV for each sub-image. The Z- 
axis is set parallel to the initial attitude of the imaging axis of the 
reference camera and the X-axis is perpendicular to the Z-axis 
and parallel to the width of the image plane. In addition, the Y- 
axis is perpendicular to the X-axis and points up. 
A concentric camera rotation forms geometry where there is only 
a single projection centre offering common perspective to all 
sub-images. Figure 2 illustrates an ideal concentric panoramic 
rotation with sub-images taken with 20° increments. In such a 
case, there is no perspective error in the object space. The 
program plots red and green observation lines for every image 
rotation. Concentric image acquisition leads to single lines 
(Figure 2) and eccentric image acquisition to a bundle of lines 
(Figures 3-6). Orange plane in Figures 1-5 is the first target 
plane and the white grid on the background is the second target 
plane where we calculate the perspective error. In addition, 
simulation always shows the reference camera image plane. This 
can be best seen in Figures 5 and 6 and should not be confused to 
the image planes belonging to a group formed by an eccentric 
rotation. 
Point 1 
Point 2 
Figure 2. Concentric panoramic sub-image acquisition with the 
frame camera geometry using 20° rotation increments. Because 
this camera setup do not cause perspective errors, all rotation 
positions lead to the same simulation rays (red and green lines).