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IAPRS & SIS, Vol.34, Part 7, ^Resource and Environmental Monitoring", Hyderabad, India,2002 
  
The iterative least-squares technique was then used to identify 
the false matches and compute the 'affine transformation 
between the images. The affine transformation was found to be 
a translation of 7 pixels in the range direction and 3 pixels in 
the azimuth direction. 
4. VIEWING GEOMETRY AND THE RESULTING 
DISPLACEMENT BETWEEN THE IMAGES 
In radar imaging, an electromagnetic pulse illuminates the 
terrain and creates an image line. Objects on the ground are 
imaged onto locations on the image line as a function of their. 
distance from the antenna (Fig. 2). Thus, the projection lines 
are concentric circles around the antenna. This slant-range 
projection is a circular projection as opposed to the perspective 
projection in conventional camera photography. 
  
: Fig2 The imaging process 
S! [Bom S2 
Ih | and 
B 
* Ar = Bsin® 
Bs” 025 Da27® 
Ar 58m 70m  Gim 
Number cf pixels 74h 897 10:38 
Fig 3 Displacement between the mages due to ground range 
Let us consider a point in the datum or reference plane, above 
which the altitude of the satellite is measured. The satellites S1 
and S2 are positioned as shown in Fig. 3a. The path-difference 
between the radar signals corresponding to such a point has a 
variation of about 3 pixels from near-range to far-range. The 
results that we have obtained using  cross-correlation for 
matching are in accordance with this (the span of the images is 
1250 pixels in the range direction). 
Thus, the relative differences in the displacements between the 
image points corresponding to the same point in the scene are 
small. However, they are significant for interferometry. 
Therefore, sub-pixel accuracy in registering the images is 
necessary. 
5. MATCHING ACCURATE TO THE SUB-PIXEL 
LEVEL 
Sub-pixel accuracy in registration can be achieved by 
interpolating the cross-correlation matrix. It suffices to 
interpolate a single cell of the cross-correlation matrix. 
159 
Interpolation was performed at intervals of 1/10". of a cell in 
both the azimuth and range directions. Cubic B-splines were 
used for the interpolation. 
Cubic B-splines are defined as 
| 3/6x — x? 4 4/6 0«x«l 
h(x) = |-1/6x*+x"-2x+8/6 1<$x<2 
| 0 25x 
and the interpolated image is 
f(x,y) = Z Z c(xk.y1) h(|x-xx|) h(ly-y1|) 
k.] 
where C E! F E'! (F is the image and E the matrix 
| 41 | 
| 141 | 
Paar 4102 sd 
| : | 
| | 
| 2 ot] 
|- "Quse d441j 
| 
14| 
A fast algorithm for interpolation which exploits the local 
support property of cubic B-splines is given in Hou, 1987. We 
found an error in this reference - the value of p; which is given 
as 2 on page 513 should be 4 for EE to be the identity matrix. 
- 
6. COMPUTING THE MAPPING FUNCTION FOR 
REGISTRATION 
We first computed pairs of corresponding points in the images 
(for points at intervals of 100 pixels in the azimuth and range 
directions in the master image) with an accuracy of 1/20". of a 
pixel by interpolating the cross-correlation matrix . The cross- 
correlation function needs to be computed only in the vicinity 
of the possible range and azimuth displacements between the 
images. This speeds up the computation significantly. 
The co-ordinates of the pairs of matching points, when 
subjected to an iterative least-squares procedure, yielded the 
transformation 
[xa |1.000231 -0.000002| |x ,! [| 7.080685 | 
| = + 
|y 21 [0.000000 1.000000| |y; |-2.800045 | 
where (x 2, y 2) is the point in the slave image corresponding to 
the point (x ;,y ;) in the master. 
Then, the residuals in the co-ordinates of the points in the slave 
image, corresponding to points in the master image, were 
computed. These residuals were interpolated using cubic B- 
splines for the interpolation. The interpolated residuals were 
then used to compute the position of a point in the slave image 
corresponding to each pixel of the master image. 
7. RESAMPLING THE SLAVE IMAGE 
The slave image was then resampled at the computed points 
using quadratic interpolation.