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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 be anticipated that MCF network more Bertsekas, D.P. and Tseng, P., 1988: suitable for the specific task of phase RELAX: A computer code for unwrapping are available soon but this minimum cost network flow problems. does not affect the search for an optimal Annals of Operations Research, 13, definition of the cost function used for 127-190. this approach. Carballo, G.F. and Fieguth, P.W., 1999: Probabilistic cost functions for network flow phase unwrapping. Proceedings of IGARSS'99, Hamburg, Germany. Costantini, M., 1998: A novel phase unwrapping method based on network programming. IEEE Transactions on Geoscience and Remote Sensing, 36, 813-821. Eineder, M., Hubig, M. and Milcke, B., 1998: Unwrapping large interferograms using minimum cost flow algorithm. Proceedings of IGARSS'98, Seattle, Washington. Ghiglia and Pritt, 1998: Two- dimensional phase unwrapping: Theory, algorithms, and software. New York: Wiley. Refice, A., Satalino, G., Stramaglia, Chiaradia, M.T. and Veneziani, N., 1999: Weights determination for minimum cost flow InSAR phase unwrapping. Proceedings of IGARSS'99, Hamburg, Germany. Wilkinson, A., 1997: Techniques for 3- D Surface Reconstruction using Radar Interferometry. Doctoral thesis, University College London, United Kingdom. Wilkinson, A., 1999: Personal communication

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phase unwrapping, journal of the optical society of america, sar interferometry, no. 3, absolute phase, configuration file, branch cut, interferometric sar, network flow, ieee trans, remote sensing, pairing algorithm, ieee transactions on geoscience and remote sensing, phase diﬀerences, electromagnetics research

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