C. Vincent Tao 
  
Where X, Y, Z are the ground coordinates and height respectively. 
Remarks: 
The model is simple and computationally fast. 
It is independent of the sensor geometry and platform. 
Applicable when the relief displacement does not influence the result significantly. 
Polynomial expressions are characterized by a great capability for absorption of accidental distortions. 
Accuracy is less than rigorous models in general. 
For the inexperienced user a high-order polynomial may seem to provide a perfect fit at the reference points, as the 
residuals are small. However, undulations between the reference points occurred could create large errors. 
2.3  Projective Transformation 
The projective transform describes the relationship between two planes. It is defined by eight parameters, which can be 
derived from four object points lying in a plane and their corresponding image coordinates. 
a,x a, ya, 
mt 213 zt z mJ x.v) (5) 
Cc xrc, y 4-1 
bx b, y b, A ' ' 
p=—— = f. x Y^) 
CX e yl ^ 
Where x, y are coordinates of the original image; x”, y' are coordinates of the rectification; and ay, a», as, by, bs, bs, Cy, Ca 
are projective parameters. 
For the images of pushbroom sensors the projective transformation can be modified as: 
axa,y+a : 
IS rum fy (6) 
exe v+l 
y=bx4b,yib =f (x,y) 
where y is given in the flying direction and x represents the pixel in a scan line. Compared to (1), the elements of 
exterior and interior orientations are implicit in these parameters 
Remarks: 
® This method is typically used to rectify aerial photographs of flat terrain or image of facades of buildings (Novak, 
1992). 
The elements of exterior and interior orientation are not required, as they are implicit in the eight parameters. 
This method has little practical significance for satellite sensors, but could be applied for airborne line scanners. 
2.4 Extended Direct Linear Transformation (DLT) Model: 3D 
The mode was investigated by Okamoto, et al. (1999) and has been compared with other models for SPOT rectification. 
„AA Y 3 aZ a, (7) 
: + Ay X), 
a,X +anY +ayZ+1 T 
a X 3 a.Y4 a,Z4 a, 2 
VE ET DER Chau 
aX tayY tajZ-l 
JC 
Where x,, y, are the coordinates on the image, and X, Y, Z are the coordinates on the ground. 
Remarks: 
® With the extended DLT model, the acquisition of approximate values is quite straightforward. 
® Individual orientation parameters are not required in many instances. 
  
876 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.