FRINGE 96
Presentation of an improved Phase Unwrapping Algorithm based on Kalman
filters combined with local slope estimation
Rainer Krämer 

Zentrum für Sensorsysteme (ZESS), UniversitätGHSiegen,
PaulBonatzStr. 911, D57068 Siegen, Germany
kraemer@nv.etinf..unisiegen.de www.nv.etinf.unisiegen.de/pb2/www_pb2

Otmar Loffeld 

Zentrum für Sensorsysteme (ZESS), UniversitätGHSiegen,
PaulBonatzStr. 911, D57068 Siegen, Germany
loffeld@wiener.zess.unisiegen.de www.nv.etinf.unisiegen.de/pb2/www_pb2




Abstract

 The paper presents a phase unwrapping algorithm based on an Extended
Kalman filter. The Kalman filter exploits a so called „Basic  Slope Model"
enabling the filter to incorporate additional local slope information obtained
from the sample frequency spectrum of the interferogram by a local slope
estimator. The local slope information is then optimally fused with the
information directly obtained from real and imaginary part of the
interferogram. The paper outlines the principle operation of the phase
unwrapping algorithm and explains the cooperation of the Extended Kalman
filter with the local slope estimator. At last the efficiency of this phase
unwrapping algorithm will be shown by simulations and real InSAR images.

 Keywords: Phase unwrapping techniques, phase slope estimation, SAR 
interferometry
Introduction
The main problem in calculating digital terrain elevation maps from a SAR
interferogram is the unwrapping of the phases. The "measured" phases, calculated
directly with the arctanfunction from the complex interferogram, are all mapped
into the same „baseband" interval (e.g. ,), while any absolute phase offset (an
integer multiple of 2) is lost. The searched unambiguous "height" phase, which
is a geometrical function of the height, must be generated from the measured
phase by phase unwrapping.
In the year 1994 we firstly demonstrated the basic possibility of doing phase
unwrapping with Extended Kalman filters. Since then we are working on the
improvement of this kind of phase unwrapping algorithms. Based on a new
optimised model we were able to remarkably improve the performance of the Kalman
filter. The main idea of the new model is to apply a "Basic  Slope  Model",
incorporating information which is obtained with an algorithm we would call
"Local  Slope  Estimator". With this model, being inexact and partly
incorrect, the Kalman filter 'fuses' two kinds of information  the local slope
information obtained from a local slope estimator  and the information directly
gained from exploiting inphase and quadrature components of the complex
interferogram.
The phase unwrapping algorithm
The concept of our phase unwrapping algorithm is shown in figure 1. Starting
points are the two coregistered SAR images from which the interferogram, a
coherence map and the measured (interferometric) phase is calculated.
In the next step the algorithm computes the necessary parameters for the
Kalman filter. The measurement noise variance is calculated from the data of the
interferometric amplitude and the coherence map. The measurement noise variance
is needed in the observation model of the Kalman filter. Running in parallel the
Local  Slope Estimation, which will be described later, calculates the local
phase slopes as well as their error variances. With these parameters a very
robust and efficient state space model can be built.
In the next step the two dimensional extended linearized Kalman filter
optimally combines the information from the local slope estimation given in the
state space model, with the interferometric phase observations. The principle
operation of a two dimensional Kalman filter for phase unwrapping has been
published in (Krämer et. al.,1996a), (Loffeld et. al.,1996) and
(Loffeld et. al.,1994) and will not be described in this paper. The state
space model will be outlined in the next chapter.
Figure 1: Concept of the phase unwrapping algorithm
The state space model utilised by the Kalman Filter
Due to the Local  Slope Estimation the state vector of the state space model
can be simplified to only one dimension and we get following state space model:
where x(n,m) is the searched unwrapped phase and s_{h}(n,m) and
s_{v}(n,m) are the local phase slopes in horizontal and vertical
direction calculated by the local slope estimator. The phase x(n,m) is
calculated as the mean of the preceding phase value plus the local phase slope
in horizontal direction and the preceding phase value plus the phase slope in
vertical direction. (See figure 2)
Figure 2: Calculation of a new phase value x(n,m)
The Local  Slope Estimator
The local slope estimation
In the following the local slope estimator will be described in the one
dimensional case. The two dimensional case is straight forward.
The searched unwrapped phase can be written as follows:
where
In the discrete case with t=(k+1)T and t_{0}=kT we obtain
where is the unknown phase variation.
As we can see the phase variation can be decomposed into two parts the mean
and dynamic phase variation. The goal of the local slope estimator is to
calculate the mean variation of the phase, respectively the frequency
f_{0,} and also the variance of the dynamic phase variation
Ew(k)^{2}.
The estimation of the unknown mean phase variation:
The complex interferogram can be written as:
We notice that the mean phase variation can be observed as a spectral shift f_{0} in the
interferogram. The complex autocorrelation can be written as:
With this equation the power spectral density is:
If the power spectral density is unimodal and approximately symmetric around
the mode, then
and the spectral shift f_{0} can be estimated with the relation
which means, that the spectral shift can be found by seeking the spectral
mode of the power spectral density.
Estimating of the variance of the dynamic phase variation
The variance E{w(k)^{2}}, where w(k) denotes the difference between
nominal (mean) phase variation and total phase variation, is identical with the
driving noise covariance Q(k) which is needed by the Kalman filter.
The variance _{f0}^{2}(k) of the variation between the
estimated mean slope variation and the true mean slope variation can be obtained
from the spectral bandwidth of the interferogram, by calculating the squared
spectral bandwidth as the second central moment
where
is the normalised power spectral density from a section of y(k) bounded by
the interval [kN/2,k+N/2].
We are now able to calculate the driving noise variance. Starting with
where is the estimated mean frequency in the interval [kN/2,k+N/2] we
get under the assumption of the following solution for the error variance of the spectral
shift:
where is the ensemble average of all individual variances within the
interval [kN/2,k+N/2]. Finally we get the desired driving noise variance:
Results
The capability of the filter will be shown by a fractally simulated phase
image and an ERS1/2 scene from Egypt.
We will begin with the fractally simulated phase image, which is shown in
figure 3. Starting with this phase a measured phase is generated by
superimposing white Gaussian noise onto the complex image and wrapping the
result. The measured phase, which we got for a signal to noise ratio of 7.2dB
corresponding to a coherence value of 0.4, is shown in figure 4. The result of
the phase unwrapping is depicted in figure 5. If we rewrap this result again
(figure 6), we can compare the result with the measured phase. We see that the
noise has been cancelled completely and that neither additional fringelines
occur nor any fringelines are missing, which is an important requirement for
error free phase unwrapping.
Figure 3: Original phase Figure 4: Measured phase
Figure 5: Unwrapped phase Figure 6: Unwrapped phase
In the following pictures we see the phase unwrapping result of a real ERS1/2
interferogram of a part of Egypt. Figure 7 shows the measured phase and the
images 8 and 9 the coherence of the interferogram. As we can see there are large
regions of very low coherences in that interferogram.
Figure 7: Measured phase
Figure 8: Places with coherence Figure 9: Places with coherence
lower than 0.4 lower than 0.1
Figure 10 presents the result we got from the phase unwrapping algorithm. To
examine the errors we have rewrapped this result again (fig. 11). We see that
some little error propagations occurr, but only in very large regions of low
coherence and even there the errors do not always occur. As we can see this kind
of phase unwrapping algorithm works very well even there are regions of
coherences near zero if this regions are not to large.
Figure 10: Unwrapped phase Figure 11: Unwrapped phase
Conclusions
A method to calculate local slope variations has been presented. The results
of this local slope estimation was used to improve the state space model of a
phase unwrapping algorithm based on an extended Kalman filter. The results show
that this combination yields a very robust phase unwrapping algorithm which
works down to a coherence value of 0.4 without error propagation, but also it is
able to cross limited regions of coherence down to zero, if these regions are
not too large.
Further work will be concentrated to improve this phase unwrapper in a way,
that should the occasion of a error propagation arise, this error propagation
should be limited on a small area in the image.
The work herein has been motivated and funded by the DARA (project „Rapid",
grand number 50 EE 9431) which is greatly appreciated.
References
 Krämer, Rainer, Loffeld, Otmar, 1996a:
 Phase Unwrapping for SAR Interferometry with Kalman Filters. In:
Proceedings of the EUSAR'96 Conference, pp. 165169.
 Krämer, Rainer, Loffeld, Otmar, 1996b:
 A Novel Procedure for Cutline Detection. In: Proceedings of the
EUSAR'96 Conference, pp. 253256.
 Krämer, Rainer, Loffeld, Otmar, 1996c:
 A Novel Procedure for Cutline Detection. In: International Journal of
Electronics and Communications, Vol. 50, No. 2, Hirzel Verlag, Stuttgart,
pp. 112116.
 Loffeld, Otmar, Krämer, Rainer, 1996:
 Local Slope Estimation and Kalman Filtering. PIERS'96, Innsbruck.
 Loffeld, Otmar, Krämer, Rainer, 1994:
 Phase Unwrapping for SAR Interferometry. In: Proceedings of the
IGARSS'94 Conference, pp. 22822284.
Keywords: ESA European
Space Agency  Agence spatiale europeenne,
observation de la terre, earth observation,
satellite remote sensing,
teledetection, geophysique, altimetrie, radar,
chimique atmospherique, geophysics, altimetry, radar,
atmospheric chemistry
