International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
The main characteristics of the GEORIS geometric correction 
model are: 
• Global correction (using a classical polynomial function) 
• Local correction 
• Capability to use linear features 
2.1. Global correction 
A first global model is calculated using an empirical approach 
based on tie points measured by the operator in the set of 
images. The tie points are used to determine the coefficients of a 
polynomial transformation (least squares minimisation). This 
model allows performing the main geometric correction. 
A polynomial function of order one handles similarity and 
affine transformations (rotation, scale, translation). A higher 
order polynomial permits modelling of more complex 
distortions, e.g. due to attitude variations. However, the use of 
order higher than two is dangerous; if the number of tie points 
is limited and/or their spatial distribution poor, it usually leads 
to wrong corrections. 
The quality of the model is usually evaluated through the use of 
the residuals at each tie point (error between the location of the 
tie point and the location given by the model). When a tie point 
has a larger residual than the other ones, it is usually re-visited 
by the operator to check whether it is due to a pointing error. 
However, in most of the cases, this error does not come from 
the tie point measurement but from local deformation (for 
instance, due to the relief). This type of problem can usually not 
be solved using this classical approach. A larger set of tie points 
will minimise the problem locally but will generally introduce a 
larger error around this area. To limit this risk, especially 
because the operator does not know the mathematical approach 
used, control tie points are required. Then, residuals are 
computed but the control tie points are not used in the least 
squares estimation. 
In the GEORIS system, there are in fact four categories of tie 
points available: 
• Control tie point: not considered in the minimisation 
process 
• Normal tie point: this is the usual case 
• Questionable tie point: the user is not confident in the 
location of the tie point (because of difficulties to point it 
in the image with a good accuracy) 
• Super tie point: when the user is very confident in the 
location of the tie point 
The last three categories of tie points are introduced in the 
minimisation process using different weights. This approach has 
a small overhead for the user but permits during this step the 
analysis of residual errors and leads to higher accuracy. 
At this step, a pixel of co-ordinates (Xj,Yj) is transformed in the 
coordinates (x i ’,y i ? ) in the other image using a polynomial 
function f . 
-> (X i \y i ’) 
For each tie point, the coordinates (xi, yi) in the other image are 
known. This permits to compute residuals defined by Eq. 1: 
dx I = x i -x i ’ (1) 
dyi = Yi-yi’ 
2.2. Local correction 
A specific characteristic of GEORIS is the use of a local model. 
This model guarantees that each of the selected tie point, 
measured by the operator, will be perfectly fitted by the model 
(no positioning errors on selected tie points). This local 
correction is applied taking into account the tie point 
neighbourhood area, whose size depends of the surrounding tie 
point context (i.e., road crossing, bridge over a river, shoreline, 
etc.). Obviously, this local model permits to correct for local 
deformation errors such as introduced by the terrain relief. If a 
digital elevation model is also available, the local correction can 
be automatically linked to a 3-D function (horizontal position 
and altitude). In this case, the local model and correction are 
achieved taking into account the third dimension. 
When no DEM is available, the pixel coordinates (X^Y;) are 
transformed to (x;”, y^ in the other image using successively a 
polynomial f plus a function (called local correction): 
Xi,Yj —*■» (xr,y^ — l i-> (xj-yfl 
The function /. depends on the location of pixel X^Y,. 
For a tie point Xj.Yj, it is a translation corresponding to the 
residuals (dx^dyO, see Eq. 2. 
Xi” = Xi’+ dxi (2) 
yi”=yi’+dyi 
For another point (X^YO of the image and its transformed 
coordinates (x i ’,y i ’), the translation dx^dy, is defined by: 
dx i =^Pk^ dx k / Pk (3) 
k 
<>»= Zp* Z^* 7 Pt 
k 
where: 
k: number of tie points 
p k : distance between (x i , ,y i ’) and the tie point (x k ,y k ) 
dx k : residual value (along x axis) of the tie point k 
dy k : residual value (along y axis) of the tie point k 
This means, that the translation depends on existing 
surrounding tie points, each with a weight depending on its 
distance from the currently processed tie point. 
This approach has been shown to be interesting in many cases. 
Table 1 shows the results of two typical cases where this 
approach is especially useful. The first one corresponds to 
results obtained at a mountainous area (using SPOT images 
(Xi,Yj) 
L