ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
parts of the scanner. In this section different mathematical 
models are used to handle the geometric errors introduced by 
the scanners, the performance of each model is evaluated and 
the best model is remarked. 
2.1 Two-Dimensions Coordinate Transformations 
In order to study and compensate the introduced geometrical 
distortions in the scanning process we employed four 2D 
coordinate transformation models. 
In the first model, Equation (1), three parameters are used to 
represent two shifts and a rotation angle between the 
hardcopy coordinate system and the scanner coordinate 
system. In the second model one more parameter is added to 
consider the uniform scale between the two systems, 
Equation (2). The third model represents the 4paramter 
transformation model. Six parameters are used to represent 
two translations, the rotation angle between the two systems, 
two different scaling factors, and a skew factor, Equation (3). 
The fourth model is the 8paramter transformation model, 
Equation (4). A complete study on the 2D transformation 
models is discussed in (Mikhail ez. al., 2001). 
  
  
[X | [cos@ -sin0[.X; X, 
e * (D 
[Y | |sinO cos0 | Y, 
x; cos0 —sin6 || X, Xo 
fm SH + (2) 
LY» J sind  cos0 ||Y Y, 
[X,| [ess6 -sime]t o]s. o[x, NES 
| Y, le sing cos0 |«& 1110 5, LY Y 
(3) 
a,X, T Y c 
X; aX) T5 Y t1 
lr (4) 
? a Xi +h Yi +c) 
a, X, tb,Y, +1 
Where X, Y, — hardcopy coordinates, 
XY | = scanner coordinates, 
X , Y = shift parameters, 
o o 
0 = rotation angle, 
S — uniform scale factor, 
S ,S = non uniform scale factors 
Ny 
a = skew factor, and 
a, a, a, b, b, b, Cp C, = the ¶mter 
transformation model parameters. 
2.2 Experiment Setup 
Some experiments are performed to evaluate the scanning 
process as well as to choose the most appropriate model that 
minimizes the produced geometric distortions. The process is 
based on scanning a precise hardcopy image of known and 
accurate point coordinates, Figure 1. The measured 
coordinates are used as the reference coordinates for the 
hardcopy points. The experiment is performed with a 
4400CSE Howled Packard scanner. Two 
resolutions were used (2400dpi, and 600dpi). 
scanning 
  
qot 
++ 
++ 
++ 
++ 
++ 
++ 
++ 
++ 
++ 
++++++++++ 
++++++++++ 
++++++++++ 
++++++++++ 
++++++++++ 
+++++++ +++ 
++++++++++ 
++++++++ ++ 
  
  
  
Figure 1. Configuration of Reference Image 
The calibration process is summarized in three steps. First 
the scanned coordinates of all points are measured digitally. 
The points are divided to control points and check points. 
The control point coordinates are used to estimate the 2D 
transformation parameters for each 2D transformation model. 
Then the estimated parameters are used to convert he 
scanned coordinates of the check points to the hardcopy 
coordinate system. The RMS errors are calculated for each 
model and are shown in Figures 2. The Y direction 
represents the scanning direction. The results show that the 
6-parameter transformation model is the best model to 
recompense the geometric errors introduced by the scanners. 
  
  
  
Model(1) Model(2) Model(3) 
—8— RMS (X), 2400dpi 
—$— RMS (X), 600dpi 
Figure 2. RMS Errors for the Scanning Process (Pixels) 
—&— RMS (Y), 2400dpi 
—#— RMS (YY), 600dpi 
3. MATHEMATICAL MODELS FOR CLOSE 
RANGE PHOTOGRAMMETRY 
Due to the wider use of mn-metric cameras for different 
photogrammetric purposes such as in registering 
archeological buildings, documenting historical writings, and 
recording historical buildings, there has been an increasing 
demand to calibrate non-metric cameras. Using non-metric 
cameras incorporate many problems, including: 
1-Non-metric cameras lack a stable inner orientation 
technique and irregular methods of calibration. 
2-Most non-metric cameras have neither a reseau plate nor 
fiducial coordinate system to act as a reference for 
photographic measurements. 
Model(4)