International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
also supply RFM-enabled imagery products in the near future. 
This paper reviews the developments in the use of the RFM 
mainly over the past five years to summarize the essential 
progresses in this field. The methodology of developing the 
RFM is summarized in Figure 1, where the individual processes 
and their interrelations are explained in following sections. 
Imaging sensors 
  
    
  
      
  
   
  
Physical senso No 
model available? 
Terrain-independent Terrain-dependent | 
approach | approach | 
+ 
  
  
  
  
1 Y frit == le eta 
RFM solution | K ZZ» 
| 
Y ad oe J 
| RFM parameters | 
  
SIOPUSA Autore 
  
  
  
  
  
  
  
   
| 
[E e 
Yes 
  
RFM refinement 
  
  
  
  
  
Y 
Refined RFMs 
9oAZIS Ty SIS 
  
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| 
  
Mapping applications: 
Ortho-rectification, 3-D feature collection, 
DSM generation, multi-sensor integration 
  
  
  
  
  
  
  
  
  
Figure 1. The strategy of developing the RFM 
1.1 The Rational Function Model 
The RFM relates object point coordinates (X, Y, Z) to image 
pixel coordinates (/, s) or vice visa, as physical sensor models, 
but in the form of rational functions that are ratios of 
polynomials. The RFM is essentially a generic form of the 
rigorous collinearity equations and the generalized sensor 
models including the 2-D and 3-D polynomial models, the 
projective transformation model and the (extended) direct linear 
transformation model. For the ground-to-image transformation, 
the defined ratios have the forward form (OGC, 1999a): 
L (x, Y» Zu) / pXX,, Y. Zn) 
Sn Di V Zn) / DXX, Y» Zn) (1) 
where (/,, s,) are the normalized line (row) and sample (column) 
index of pixels in image space; (X,, Y, Zn) are normalized 
coordinate values of object points in ground space; polynomial 
coefficients aj, bj cj dij, are called rational function 
coefficients (RFCs). The normalization (i.e., offset and scale) 
minimizes the introduction of errors during computation 
(NIMA, 2000). The total power of all ground coordinates is 
usually limited to three. In such a case, each numerator or 
denominator is of twenty-term cubic form, and the order defined 
in NIMA (2000) has become the de facto industry standard. 
The polynomial coefficients are also called RPCs, namely, rapid 
positioning capability, rational polynomial coefficients or 
rational polynomial camera data. In this paper, we refer the 
RFM as general rational functions with some variations, such as 
subsets of polynomial coefficients, equal or unequal 
denominators and two transformation directions (i.e., forward 
and inverse equations). Nine configurations of the RFM have 
been analyzed in Tao and Hu (2001a, 2001b). An inverse form 
of the RFM is described in Yang (2000), but is seldom used. 
The term - RPC model - often refers to a specific case of the 
RFM that is in forward form, has third-order polynomials, and 
is usually solved by the terrain-independent scenario. The RPC 
model is more concentrated because it is transferred with Space 
Imaging and Digital Globe imagery products. 
1.2 RFM Solution 
The unknown RFCs can be solved by least-squares adjustment. 
The normal equation is given by Eq. 2 (Tao and Hu, 2001a), 
where / is the vector of RFCs; T is the design matrix of the 
linearized observation equations (Eq. 1); W is the weight matrix 
for image pixel coordinates G. The covariance matrix associated 
with / is given by Eq. 3 (Hu and Tao, 2002), where R is the 
covariance associated with the measured image positions. 
) 
T'WTI - T'WG 0 ( 
(3) 
P -(TWTy! -R 
La No 
To tackle the possible ill-conditioning problem during the 
adjustment, the Tikhonov regularization technique was 
suggested to turn the normal equation into a regularized one. 
Then the RFCs may be solved iteratively as follows: 
L=l + WET EEE TI Wa ok 1,2, ... (4) 
with | /,20, Wo —- W(Iy) = E 
where A is the regularization parameter; k is the iteration 
number; JW, ^ W(I;) is the weight matrix; w, ^ G — TI, is the 
misclosures at GCPs. 
2. APPROACHES OF DETERMINING RFCS 
The RFCs can be solved by terrain-independent scenario using 
known physical sensor models or by terrain-dependent scenario 
without using physical sensor models. 
2.4 Terrain-independent Approach 
For the terrain-independent scenario, the RFM performs as a 
fitting function between the image grid and the object grid 
(Yang, 2000; Tao and Hu, 200la). In detail, an image grid 
covering the full extent of the image is established and its 
corresponding 3-D object grid with several layers (¢.g., four or 
more layers for the third-order case) slicing the entire elevation 
range is generated. The horizontal coordinates (X, Y) of a point 
of the 3-D object grid are calculated from a point (/, s) of the 
image grid using the physical sensor model with specified 
elevation Z. Then the RFCs are estimated using a direct least- 
squares solution with an input of the object grid points and the 
image grid points. The regularization technique is not needed 
because the linearized observation equations are well 
conditioned. However, the regularization may help produce 
well-structured RFCs, among which the 2" and 3"-order 
coefficients will be constrained to be close to 0, and the 
constant and 1%-order components represent the optical 
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