C. Vincent Tao 
ORIs can be used as a ground reference “map” and the “ground” control points can be selected directly from the ORIs. 
It can be seen that an accurate and generalized sensor model is of particular importance to leverage the commercial 
applicability and enlarge the market for Intermap’s products, DEMs and ORIs. We, therefore, investigated RFM under 
this context. 
2 GEOMETRIC MODELS FOR IMAGE RECTIFICATION 
2.1 Collinear Equations based Differential Rectification 
Collinear equations are rigorous models available for frame and pushbroom sensors. For frame images, it is described as 
  
-t n(X- X,- n(Q- Xx) n(Z- 2) = 
X=X 
: dX ~ Xd iad ro) nl —Z,) 
f x.y") (1) 
iX - AtnmQ- X) nm(0-Z) . 
y=y,-c 2 32 = f(x.) 
1X =X) +0 AY = YY +r AZ ~2Z,) : 
  
Where X, Y, Z are three-dimensional coordinates defined by a DEM pixel; x, y are the corresponded position in the 
image which are transformed by the collinear equation; and x’, v’ are equivalent to the map (DEM) coordinates X, Y. 
For line scanning sensors (e.g. SPOT), the mathematical model has to be modified to line-perspective geometry: 
r' (X-X,)* r (Y -Y,)* r (Z— Zi) 
C : == 
VA) it = Ze) 
  
x=- 
Lan) 2) 
r (X =. Xo)t r (y a Ya) + y' (Z = Lin) s 
0=-c-4 A ; Ef. 
r (X “x 0 + y (y -Yo)* r'(Z din) 
  
Where x; is the coordinate in scan-line i, orthogonal to the direction of travel, and x,, y, = 0. 
Remarks: 
e [mage domain corrections can be utilized for removing camera distortions 
e Triangulation is possible so that both systematic and random errors can be reduced to a great deal. 
e This is a rigorous model with both relief displacements and camera distortions corrected. 
e 
Specified information about interior orientation, exterior orientation of the sensor and other related orbital data are 
required. 
Software must be changed for each different image sensor. 
It is mathematically complex with long execution time. 
e The imaging parameters are not always available, e.g., IKONOS, for the use of this model. 
22 Polynomial Rectification: 2D and 3D 
When the image area is flat, polynomials with a low order can serve accurate results. In this case the model can be 
represented as a 2D model: 
XF 09. aX + ay (3) 
v= bg + BX + bY 
Where x, y are the pixel position in the image; X, Y are the coordinates on ground (map); and a, ai, a», bo, bi, b» are 
polynomial coefficients. 
Some studies have shown that the use of low-order polynomial 3D models for rectifying images in hilly and 
mountainous areas can reach the accuracy level that is close to the rigorous models (Palà and Pons 1995, Okamoto et al. 
1999). For example, a typical formula is: 
x = ay + a,X + a,Ÿ + a3Z + a4XZ + asYZ (4) 
y= bo + bX + b, y + b3Z + b,XZ + b;YZ 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 875