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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
are recovered using the previous equations. R is given by the 
column number j the resolution step d R and the Nadir range Ro, 
by R = j x ôR + Ro. Thus the 3D point M is the intersection 
of a sphere with radius R, the Doppler cone of angle Op and a 
plane with altitude A. The coordinates are given as solutions ofa 
system with 3 equations and 2 unknowns (since the height must 
be given). 
Inversely, knowing the 3D point M allows to recover the (3, j) 
pixel image coordinates, by computing the sensor position for the 
corresponding Doppler angle (which provides the line number) 
and then deducing the sensor - point distance, which permits to 
define the column number, since j = Sag 
  
The geometrical model for optical image acquisition is completely 
different and is based on the optical center. Each point of the im- 
age is obtained by the intersection of the image plan and the line 
joining the 3D point M and the optical center C. The collinear 
equations between the image coordinates (z;,, ,,) and the 3D 
point M (Xm, Ym, Zm ) are given by: 
Qii-Xa t a12YMu + A132GM + Q14 
a31 XM + A32YM + 033 ,M + 034 
  
Um = 
_ az XM + A22YM + A23M + Q24 
a31XM + A32YM + 033 ,M + A34 
  
where the a;; coefficients include internal and external parame- 
ters. Once again, a height hypothesis is necessary to obtain M 
from an image point (z5, ym). 
3.2 Processing of the optical image 
The SAR features are supposed to correspond to radiometric dis- 
continuities in the optical image. Therefore an gradient operator 
is applied to the optical data. The operator proposed by Deriche 
(Deriche, 1987), which is a RII filter built using the formalism 
of Canny (Canny, 1986) has been chosen. The two outputs of 
the filter (magnitude and direction of the gradient) are used in the 
following. An example is shown on figure 2. 
3.3 Matching 
The real position of the SAR feature f in S; U S, is searched by 
projecting it on the optical image for a set of height hypotheses. 
Let us denote by Z(f, h) the set of pixels in the optical image 
corresponding to the projection of f using the height hypothesis 
h. For a segment, it is done by projecting both extremities on the 
optical data and linking them. This is thus an approximation but 
it is valid for short segments. Figure 3 illustrates the projection 
of one point of the SAR image in the optical image for a set of 
heights. 
The tested height set Sp = [Hin; Hmaz] is chosen to contain 
the true height of the primitive (typically in urban area, the maxi- 
mal size of the buildings can be used as upper bound of the inter- 
val). Figure 4 shows the variation interval of the SAR primitive. 
For each tested height h € Sh, a score s(f, h) is computed as 
the average of the gradient responses. In the case of the linear 
features, a constraint on the gradient direction is also introduced. 
Denoting by e(i, j) and d(i, j) the gradient responses (magnitude 
and direction) for pixel (2, j), s( f, h) is then given by: 
s(f,h) = Y eli.s)A(d(i, 3), O(f) 
(4,5)EZ(f,h) 
1 
card(Z(f, h)) 
  
Figure 2: Part of the optical image (on the top), magnitude (left 
bottom) and direction (right bottom) of the gradient. 
with O( f) the direction of the primtive f, and: 
A(d(i, 5), O(f)) 
Il 
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For a point feature, s( f, h) is the gradient magnitude of the pro- 
jected point. 
Optical image center 
      
  
Radar antenna i. 
Optical image plane 
One radar pixel 
  
Hmax 
  
  
Hmin 
Figure 3: A point of the SAR image is projected in the optical 
image for different heights. 
For each primitive f € S; U Sp, the three best scores s(f, h) and 
the associated heights h are kept, with the condition to be higher 
than a fixed threshold ths. Figure 5 shows the three best matches 
for a small test area. In point of fact, in the remaining of the 
article the only best match will be considered.