International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
dielectric and geometric properties of the scattering medium (Lee
and Pottier, 2009).
The true covariance matrix X, which contains sufficient statis-
tics to characterize the acquisition vector k, is not known, and is
estimated using a maximum likelihood method by n-sample (n-
look) spatial coherent averaging: A — 1 Sm d k; Ki. The true
covariance matrix X, as well as its n-sample estimate A, can be
portioned as:
|
which summarize all the information (the joint and marginal)
from temporal multi-channel SAR acquisitions. Ai» = A}, isa
m x m dimensional cross correlation matrix between the acqui-
sition vectors Kr and E» which characterizes the interferometric
and polarimetric information. Since the variables fall naturally
into two sets, they can be correlated or uncorrelated processes
(0 < P? 2 X1 X125] X31 X Im) Over time depending on the
monitored objects. The matrices A11 and A2» are the standard
n-look and m x m dimensional polarimetric covariance matrices
of separate temporal images. Note that, for m — 1, matrix A;;
reduces to the single intensity scalar a;.
3u
321
31
3122
An
Azı
A12
| and A= | A
| ()
2.2 Proposed Algorithm:
The MI is well known technique for co-registration of remotely
sensed images such as medical, optical and radar images (Maes et
al., 1997, Reinartz et al., 2011, Chen et al., 2003). In this paper,
the MI is adopted for glacier monitoring in terms of temporal
multi-channel images. The major requirement to compute the
MI between two images is the accurate estimation of the joint
histogram from the samples. However, in the case of polarimetric
images, it is really time consuming work due to the requirement
of 6 dimensional joint histogram. To overcome this problem joint
distribution between temporal polarimetric covariance matrices
derived by Erten et al., 2012 is used.
In probability and information theory, the mutual information
is a commutative measure of the difference between the joint
probability distribution px y (x, y) and the marginal probability
distributions px (x) and py (y) of the random variables X and
Y, respectively (Papoulis, 2002). Taking the joint distribution
p(A11, A22) of temporal polarimetric covariance matrices A11
and A2», the MI between them due to the Wishart process in time
is (Erten et al., 2012):
Dmr(A11; A22) E{log(oF1(n, C)}
2nP?
- we(h ple uo
n log ( ) L-p ©
C = XügEnYXgAuAnYEgYnYgua
Mu = Yh = Ti,
where o F1(n, C) is the complex hypergeometric function of ma-
trix C. This function can be calculated with the help of the posi-
tive eigenvalues ofthe m x m Hermitian matrix C by (Smith and
Garth, 2007).
In terms of tracking applications the best match between tempo-
ral images is found by maximizing a tracking algorithm between
a reference image at time ¢; and a second image at time ta. In
practice, the reference image is fixed, and the second image is
shifted by a factor of resolution (subpixel). At every subpixel
shift the value of the tracking algorithm is stored, and the sub-
pixel shift at the peak is assumed to be the correct displacement
42
between images. In this work, the maximization of the MI is used
as a tracking algorithm.
Let's assume that a reference block Ai;, — [Ai1,, A115,"
T
Asi, 00 An] matches a second block A»»,, = [A22,; , A22,2,
da A22, a» 42
ment (shift) vector v; , where i indicates the block which is shifted
i subpixels, and s is the number of pixels in block A11, and
A»22,,. Then, the displacement vector v between the images is
obtained by maximizing the MI(A11,, A22,, ) for each block i
This matching results in a displace-
is
V — argmax f (MI (Au,, A22.) , Vi) - Q)
Newer eer’
Vi
3 EXPERIMENTAL RESULTS
This section contains a concise presentation of the proposed po-
larimetric tracking method. It is not intended to give an in-depth
geophysical analysis, but merely an example of motion estima-
tion using the MI. To validate the proposed approach and to show
its performances in different acquisition circumstances, two dif-
ferent sensors having different system parameters have been used.
It is worth mentioning that the precise/coherent geometric coreg-
istration of temporal SAR images is a strict requirement for track-
ing algorithms. In order to perform a precise coregistration for
spaceborne image, backward geocoding was employed (Sansosti
et al., 2006). In the backward geocoding approach, for each DEM
element, the image pixel with the nearest range-Doppler coordi-
nate is calculated. For airborne images, which is more challenge
than the spaceborne ones due to the absence of precise orbit in-
formation, the multi-squint approach has been applied (Prats et
al., 2009).
The first pair considers airborne images acquired by the Exper-
imental SAR (E-SAR) system of the German Aerospace Cen-
ter (DLR) in the frame of the SWISAR campaign, 2006 (Prats et
al., 2009). Table 1 summarizes the acquisitions over the Aletsch
glacier, located in Swiss Alps including their system parameters.
This pair, having one day temporal resolution, provides an exam-
ple of monitoring the fast glacier surface velocity in the presence
of relatively correlated speckle patterns.
To address the application potential of the proposed method in
single channel SAR images, spaceborne sensor ASAR Envisat
images acquired over Inyltshik glacier located in Kyrgyzstan are
processed. In case of spaceborne monitoring, a very fast flowing
glaciers deserves a special attention. In particular, the speckle
patterns of the temporal images are no longer correlated due to
the satellite fixed temporal resolution (35 days for Envisat) and
the shorter wavelength.
Fig. 1 shows the histograms of the coherences of both pairs in
the glacier area. The decorrelation effect in the Envisat pair com-
pared to the E-SAR pair is already visible in the figure. In case of
fast moving glacier, a larger wavelength like the L-band (24.3
cm) one, compared to a shorter wavelength X-band (5.6 cm),
tackles some of the decorrelation problems because of its proper-
ties to penetrate more into the snow and the firn.
3.1 Aletsch glacier (coherent) monitoring with L-band
This section presents the results to show the performances of the
proposed fully polarimetric MI approach with L-band data having
one day temporal resolution. In particular, a detailed analysis of
polarimetric tracking based on MI over glacier having temporal
(relatively) correlated speckle patterns is handled. Fig. 2 plots the
