The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
bicubic polynomial equations for ERS tandem data were used 
(Rufino et al., 1996; Rufino et al., 1998). 
In this research, all these transformation equations (4, 6, and 12 
parameters) were tested with real data. Through comparison, 
the most effective transformation equations were investigated. 
All polynomial equations are solved by least squares. 
If the sine length S is an even number (Hanssen and Bamler, 
1999), the sine function is the same as Eq. (3), but n = - 
SI2+1, ..., 0, ..., S/2. A becomes a fractional number 
representing the coordinate difference between the nearest 
original point on the left side and the interpolated point, so A 
belongs to [0, 1). 
2.2.3 Resampling Slave Image 
After transformation equations are set up, one can resample the 
slave image according to the subpixel transformation. 
Interpolators commonly used for resampling optical images, 
such as bilinear and cubic convolution, are also used for SAR 
complex images (Kwoh et al., 1994; Liao et al., 2004). 
However, SAR images are complex data, which contain not 
only intensity information, but also phase information. Each 
degree error of this phase data is directly related to the InSAR 
DEM error. Moreover, most SAR images have none-zero 
Doppler centroid. The interpolation error due to repeated 
spectrum overlap aliasing and spectrum comer cutoff should be 
avoided. The interpolator must therefore be selected carefully 
for resampling SAR images. Hanssen and Bamler investigated 
the theory and simulation of nearest neighbor, bilinear, four- 
and six-point cubic convolution, and truncated sine kernels 
applied in SAR image resampling. (Hanssen and Bamler, 1999) 
In SAR image resampling, the tradeoff between accuracy and 
computational effort must be considered when selecting 
interpolation kernels. Hanssen and Bamler (Hanssen and 
Bamler, 1999) performed both a comprehensive theoretical 
analysis for these most commonly used interpolators and a 
simulation study to evaluate these interpolators with coherence 
and phase error as criteria. The authors listed the following 
interpolators and their spectra: nearest neighbor, bilinear, four- 
point cubic convolution, and truncated sine. 
Due to the tmneation of the sine function, a Gibbs phenomenon 
appears. Gibbs phenomenon is also called ringing artifacts, the 
oscillations of sine spectrum near the jump. Usually a 
windowing filter should have been applied to eliminate the 
oscillations. Also, SAR images normally have a non-zero 
Doppler centroid, so the band pass sine function should have 
been modulated to work better on SAR images (ESA, 1999). 
F(n) = Sinc\n{n + A)], 
A g [0,1) 
(4) 
If the original discrete signal is x(n), S is an odd number, and 
the interpolated point is at (m-A), the value of the interpolated 
point is 
x(w-A)- x(k)Sinc \n{k - m + A)] 
(5) 
A Harm Window W(n) can be added to reduce Gibbs 
phenomenon. 
(6) 
Not only Hann window, many other windows, such as Kaiser 
window, can also reduce Gibbs phenomenon. 
In order to make the resampled image independent of the length 
of the sine, a normalization coefficient A is required for the 
output image: 
A band limited continuous signal, if sampled without aliasing, 
can be reconstructed by convolving with a sampled infinite sine 
function. However, an infinite sine function is not possible and 
one always has to truncate the sine kernel. The sine length S is 
preferred to be an odd number in the ESA manual (ESA, 1999). 
A typical truncated discrete sine function for image resampling 
could be: 
F(n) = Sinc[7t(n + A)], 
„=-£±2 
A g [-0.5,0.5) 
(3) 
In Eq. (3), A is a fractional number representing the coordinate 
difference between the nearest original point and the 
interpolated point. 
A= £F(k)W(k) 
(7) 
SAR images normally have non-zero Doppler centroid, though 
they are band limited data. The sine function is a band pass 
filter, so it is better to modulate the sine function to fit the SAR 
image spectrum. The modulation function is: 
M( w ) = Éf' 2 * [ " +A+(S - 1,/2! 
f - 
Jc PRF 
(8) 
435 
f c is called the central frequency of the SAR image, normalized 
at [-0.5, 0.5]. f dc is the Doppler Centroid Frequency and PRF is 
the Pulse Response Frequency of the SAR image.