On phase unwrapping based on minimum cost flow networks

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					      On phase unwrapping based on minimum cost flow networks

                                     Rüdiger Gens
             Delft Institute for Earth-Oriented Space Research (DEOS),
   Delft University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlands



ABSTRACT: Phase unwrapping is a              since it can provide continuous
key step in the SAR interferometric          information about surface changes over
processing chain as it converts the          large areas.
phase information derived from an
interferometric image pair into valuable     In the interferometric processing chain,
height information. Many algorithms          phase unwrapping plays a key role in
have been developed to solve the phase       deriving      the     valuable      height
unwrapping problem. None of the              information from SAR interferometric
algorithms implemented so far has met        data. Since none of the algorithms
with all the       requirements for an       implemented so far has met all the
optimal solution. Recently, a very           requirements for an optimal solution,
promising      approach     has     been     this complex task in still in the focus of
introduced by Costantini (1998). The         research. This paper deals with the
new method formulates the phase              investigation of a recently established
unwrapping problem as a global               approach using minium cost flow
minimisation problem which can be            (MCF) networks for solving the phase
solved by using minimum cost flow            unwrapping problem and pays special
(MCF) networks. These MCF networks           attention to the optimal definition of the
in general have been well studied and        cost function used for this approach.
efficient algorithms exist. However,
application of the MCF for phase             Phase unwrapping using minimum
unwrapping is a new approach and             cost flow networks
requires further research. This paper        Over the years, numerous phase
deals with the investigation of this         unwrapping algorithms have been
particular algorithm and focuses on the      proposed but none of the implemented
optimisation of the cost function used to    methods has been able to provide an
define the MCF network.                      optimal solution for the phase
                                             ambiguity problem. Recently, a
                                             completely different approach from the
                                             existing techniques has been proposed
Introduction                                 by Costantini (1998). Using the fact
Three-dimensional movements of the           that phase differences of neighbouring
Earth's surface caused by the sea-level      pixels can be estimated with a potential
rise and land subsidences have serious       error that is an integer multiple of 2π,
consequences for countries near sea-         he formulated the phase unwrapping
level such as the Netherlands. Hence,        problem as a global minimisation
monitoring and prediction of these           problem with integer variables. The
motions is of vital importance, e.g., for    underlying      global     minimisation
the     coastal   management.      SAR       problem can be efficiently solved by
interferometry serves as an important        minimum cost flow networks. MCF
data source for these kind of studies        networks in general have been well
studied and efficient algorithms are       The design of the minimum cost flow
available. The RelaxIV algorithm for       network for phase unwrapping is
the minimum cost flow problem was          depicted in Figure 1. The size of the
originally developed as a FORTRAN          phase image in rows and columns
code by Bertsekas and Tseng (1988).        defines the number of nodes which are
The code used for the implementation       required in the MCF network. Since the
was entirely rewritten and implemented     network only works with positive flows
in C++ by Frangioni and Gentile from       two neighbouring nodes are connected
the University of Pisa, Italy. For a       via two arcs to enable positive and
complete mathematical description of       negative charges. One demand for the
the formulation of phase unwrapping as     optimisation of the MCf network is that
a global minimisation problem the          the flows within the network are in
reader is referred to Costantini (1998).   balance. This requires the definition of
                                           a grounding node which is connected to
                                           all boundary nodes.


                                                         Figure 1. Design of the
                                                         minimum       cost   flow
                                                         network      for    phase
                                                         unwrapping. In the centre
                                                         of four phase values
                                                         (indicated with crosses) a
                                                         node (represented by a
                                                         circle) is defined. Each
                                                         node is connected its
                                                         neighbouring node by
                                                         two arcs. The boundary
                                                         arcs are connected to the
                                                         grounding node. The
                                                         energy is flowing from
                                                         the positive residue (plus
                                                         node) to the negative
                                                         residue (minus node).



For the input of the MCF network the       cost serves as a weighting factor for the
number of nodes and arcs are needed.       optimisation of the flows and can be
Furthermore, a capacity and a cost is      derived from various sources. This is
assigned to each arc, defined by its       described in the next chapter in further
starting and ending node. The capacity     detail.
on an arc is the maximum flow that is
allowed. A value of 20 is larger than      As output the MCF network determines
any likely flow and allows to connect      an optimal objective function value,
several residue pairs along a common       optimal flows and optimal reduced
path. Reducing the capacity would          costs. For the phase unwrapping
prevent more than two cuts from lying      problem only the optimal flows are
on top of each other and increase the      relevant as they define the position of
probability to lie side by side on         the branch cuts which are used during
parallel arcs (Wilkinson, 1999). The       the actual phase unwrapping.
Cost function in the minimum cost
flow network
The     processing      of    the    phase     The optimisation using the MCF
unwrapping approach is shown in                network provides the positions of the
figure 2. In a first step, the residues are    branch cuts. For the actual phase
located in the interferogram. The costs        unwrapping a flood-fill algorithm is
for each flow can be derived from              used, unwrapping the interferogram
various     sources      such     as    the    pixel by pixel circumnavigating all
interferometric coherence, the image           defined branch cuts.
amplitudes, the phase gradients and the
residue density.



                                    Coherence
                                 Image amplitudes
      Interferogram                                          Figure 2. Schematic
                                  Phase gradients
                                  Residue density            processing chain for
                                                             phase unwrapping using a
                                                             minimum       cost    flow
                                       Costs
                                                             network. The input for
   Position of residues
                                                             the MCF network is
                                                             defined by the position of
                                                             the residues found in the
                  Minimum cost flow
                       network                               interferogram and the
                                                             costs     derived    from
                                                             various sources. The
                                                             optimisation of the MCF
                      Position of
                      branch cuts                            network determines the
                                                             position of the branch
                                                             cuts used during the
                                                             phase unwrapping.
                  Phase unwrapping
               using flood-fill algorithm




For the definition of the costs several        used for this approach were calculated
methods have been proposed in the              by minimising the absolute value of the
literature.                                    sum of absolute distances of the
                                               unwrapped phase. Their second
Eineder et al. (1998) used two different       approach tried to maximise the
approaches. First, they defined the costs      smoothness of the unwrapped phase
from amplitude, residue density and            that is defined as the sum of the
flatness. High costs were assigned to          absolute values of the phase gradients.
low amplitude values (to force the             The absolute difference of the values of
branch cut definition in foreshortening        the phase estimator is directely used as
and layover areas) and to low residue          the cost function.
density (to prevent branch cuts in flat
terrain areas). Both values were               Carballo and Fieguth (1999) formulated
combined by a maximum operation to             phase unwrapping as a Maximum
obtain the final costs. The thresholds         Likelihood estimation problem. This
approach is based on the knowledge of         coherence values, rescaling the
the probability of phase discontinuities      coherence value to the power of four to
that can be derived as a function of          the appropriate value range.
coherence and topographic slope from
known statistical properties of the SAR       Proposed method
phase.                                        The approaches found in the literature
                                              are only partly convincing. The
Refice et al. (1999) proposed an              approach of Eineder et al. (1999) is
automatic     inference    methodology        based on the empirical estimation of
considering       coherence,     image        some threshold values. Carballo and
intensities, residue density, phase           Fieguth (1999) as well as Refice et al.
gradients and an indicative image based       (1999) have used methods based on
on mean-field annealing and relying on        mathematical     approaches    without
some a priori assumptions about the           proving that the theory behind these
regularity of the absolute phase              approaches is applicable to the phase
function.                                     unwrapping     problem.    Wilkinson's
                                              method (1997) already leads to
Wilkinson (1997) achieved satisfactory        promising results but makes only use of
results by defining the cost by enhanced      the coherence values.




Figure 3. Branch cut images from different phase unwrapping algorithms. Left: The
classical Goldstein algorithm shows one part completely isolated by branch cuts which
leads to an incorrect unwrapping result. Centre: Branch cuts from MCF network with
constant costs also show some erronous connections. Right: Branch cuts from MCF
network with costs derived from enhanced coherence values pass the visual inspection.
Figure 3 shows branch cut images using      The first results look very promising.
different algorithms. The images are        Nevertheless, it needs a lot more
calculated from an interferogram that       processing of a variety of scenes to gain
has been simulated from a digital           the experience required for finding a
elevation model of the mountainous          better definition of the cost function. It
terrain around Long's Peak, Colorado,       appears quite logical that the result is
provided by Ghiglia and Pritt (1998) via    supposed to become better the more
the Wiley ftp server. The left image        relevant information derived from
shows the result of the classical           various sources is considered for the
Goldstein algorithm. It clearly shows       definition of the cost function. Since the
problems in layover regions where even      phase unwrapping problem is an
one part is completely isolated by          ambiguity problem there is generally no
branch cuts. The centre image shows         true solution to it. The way to find an
the MCF solution using constant costs.      optimal approximation to it leads to the
It clearly indicates that minimising the    use of interferograms simulated from
total geometric length of the branch        real elevation models since they offer a
cuts does not always lead to plausible      reference for comparison.
results. The MCF solution with costs
derived from coherence image by             Acknowledgements
rescaling the fourth power of the           This research forms part of the project
coherence values offers visually better     on "Determination and prediction of the
results.                                    three-dimensional movement of th
                                            earth's surface (land and sea)" of the
The last result needs to be further         Delft Interfacultary Research Center
enhanced. The major problem areas for       (DIOC).
any phase unwrapping algorithm are in
the foreshortening and layover areas.       The author would like to thank Andrew
Based on the local phase gradient a         Wilkinson for valuable discussions and
smooth terrain surface needs to be          Antonio Frangioni for his technical
reconstructed. This simulated terrain       support concerning the minimum cost
surface can serve as the basis for          flow network.
defining the weighting of the costs.
                                            For the calculations of the MCF
Conclusions                                 network the C++ implementation of the
The implementation using the RelaxIV        RelaxIV algorithm provided by Antonio
algorithm for the minimum cost flow         Frangioni    and    Claudio    Gentile
network is computationally not              (University of Pisa, Italy) have been
practical for larger data sets since it     used. For the implementation of the
requires hugh amount of memory. Part        phase unwrapping algorithm partly C
of the current research is focusing on      codes provided by Dennis Ghiglia and
designing MCF networks more adapted         Mark Pritt (1998) via the Wiley ftp
to the specific constraints of the phase    server have been adapted.
unwrapping problem instead of being
operational for general purpose. It can     References
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