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.