IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 3, MAY 2011 527

A Markovian Approach for InSAR PhaseReconstruction With Mixed Discrete and

Continuous OptimizationA. Shabou, J. Darbon, and F. Tupin

Abstract—In this letter, we propose a Markovian approach forinterferometric synthetic aperture radar (InSAR) phase recon-struction. Recently, Markovian models based on multichannelInSAR likelihood statistics and total variation prior have beenproposed to reconstruct the noisy and wrapped phase. Efficientdiscrete optimization algorithms based on the graph-cut techniqueare used to efficiently minimize the energy. Our contributionconsists in extending these works to cope with continuous labelsets providing more precise and accurate reconstructed profiles.The proposed approach also provides a good way to estimatelocal hyperparameters to adjust the prior model and preserve welldiscontinuities in profiles. This task is useful when working withreal InSAR data where the quantization of the continuous label setleads to a loss of some physical information. The proposed methodis compared to other Markovian approaches with discrete multi-label optimization algorithms. Experiments show better qualityresults both on simulated and real InSAR data.

Index Terms—Continuous optimization, graph-cut, Markovrandom field, multichannel phase unwrapping (MCPU).

I. INTRODUCTION

THREE-DIMENSIONAL (3-D) reconstruction of theEarth’s surface is becoming a task of increasing impor-

tance for the last years due to the high resolution of newsynthetic aperture radar (SAR) sensors (TerraSAR-X, ALOS,CSK, RadarSat-2). The 3-D reconstruction is mainly performedby estimating the absolute phase from the interferograms gen-erated by interferometric SAR (InSAR) systems. First, the ob-served phase is known in the principal interval [−π, π] and hasto be unwrapped and denoised. Then, knowing the parameteracquisition of the InSAR system, the Digital Elevation Model(DEM) of the observed surface can be computed.

Unwrapping and denoising the InSAR phase is a difficultinverse problem due to both the noise corrupting the observeddata and the wrapping operation. More precisely, this problemis known to be ill-posed if the so-called Itoh condition is notsatisfied [1], i.e., there are jumps of more than π in the estimatedphase profile.

Although many methods have been proposed to address thisproblem, robust phase unwrapping and denoising remain a

Manuscript received March 22, 2010; revised June 10, 2010 andSeptember 30, 2010; accepted October 25, 2010. Date of publicationDecember 9, 2010; date of current version April 22, 2011. This work wassupported under Grant ONR 000140710810.

A. Shabou and F. Tupin are with Institut TELECOM, TELECOM ParisTech,CNRS LTCI, 75634 Paris, France (e-mail: aymen.shabou@telecom-paristech.fr; florence.tupin@telecom-paristech.fr).

J. Darbon is with CMLA, ENS Cachan, CNRS, PRES UniverSud, 94235Cachan, France (e-mail: darbon@cmla.ens-cachan.fr).

Digital Object Identifier 10.1109/LGRS.2010.2090336

challenge, particularly when dealing with highly noisy interfer-ograms with high discontinuity rate profiles. One of the possiblemethods to solve this problem is the use of the multichannelMaximum a posteriori (MAP) estimation method introducedin [2]. In this approach, the multichannel likelihood statisticsare exploited to overcome the problem of phase unwrapping,and a Markovian a priori is used to regularize the phase. Then,the problem is resolved by optimizing the MAP criterion usingoptimization algorithms like the simulated annealing one.

Recently, similar multichannel phase unwrapping (MCPU)approaches have been proposed [3], [4], where new a priorimodels are introduced with adapted discrete optimization algo-rithms, like the efficient graph-cut-based algorithms [5].

These MAP-MCPU approaches work by transforming theoriginal problem of phase reconstruction into a labeling one. Tofix notations, let us assume that the estimated phase is definedon a lattice denoted by V . The value of the unwrapped phase xat the site p will be referred to as xp and is supposed to takeits value in a linearly ordered finite set of phase labels L ={l1, l2, . . . , lk}. This quantization of the phase values is nec-essary for most graph-cut optimization algorithms. This pointwill be discussed in this letter. Using the Markovian formalism,the lattice is endowed with a neighborhood system and pairwiseinteractions are considered, where two sites p and q that are ininteraction with each other are denoted by (p, q). The set of allconsidered pairwise interactions is referred to as E . Then, thelabeling problem is solved by minimizing an objective functiondefined as the following first-order Markovian energy:

E(x) =∑p∈V

Ep(xp) +∑

(p,q)∈EEp,q(xp − xq) (1)

where terms Ep encode the multichannel likelihood statistics,while terms Ep,q correspond to the prior model.

A popular model to regularize the phase while preservingthe discontinuities in the estimated profile is the total variation(TV). Some results of this kind of regularization model onheight estimation in case of urban area are presented in [6].Furthermore, this a priori model is well adapted to the graph-cut-based optimization algorithms.

The graph-cut optimization approach is a well-known tech-nique in computer vision for Markov random field minimiza-tion. A specific graph is constructed, where topology andedge capacities are defined from the energy terms, in sucha manner that an s,t-minimum-cut on this graph gives theoptimal configuration x (i.e., with the minimal energy value).The s,t-minimum-cut is obtained by computing the maximumflow on the graph [7]. A necessary and sufficient condition

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528 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 3, MAY 2011

for minimizing a first-order Markovian energy with the graph-cut approach is the submodularity of the energy [8], [9]. Forenergies of the form of (1), the latter relies on the convexity ofEp,q [9], [10].

Our contributions in this letter are twofold. First, weimprove the discrete optimization step for this class ofMarkovian models by exploiting the new optimization algo-rithms introduced in [11] that requires low memory comparedto the original ones [9], [10]. Second, we increase the accuracyof the reconstructed profile by developping a new approach thatimproves the efficiency of the optimization step while allowinggenerating height construction with high precision. This is donedue to a combination between discrete- and continuous-basedoptimization algorithms of the MAP criterion that also providesa robust method for local prior hyperparameter estimation. Thelatter leads to a better discontinuity preservation and a reducedloss of contrast due to the TV a priori.

The remainder of this letter is as follows. In Section II, webriefly review the MAP-MCPU approach. Section III describesthe discrete optimization-based algorithm used to compute adiscrete 3-D reconstruction of the surface height. Some draw-backs of this approach are discussed. Then in Section IV, asecond step is added to the algorithm allowing DEM generationwith high precision. Finally, some experiments on synthetic andreal InSAR data are presented in Section V.

II. MAP-MCPU WITH TV PRIOR

With InSAR multichannel systems, independent multifre-quency or multibaseline interferograms related to the samescene are provided [2]. The phase reconstruction problem fromthese noisy and wrapped phase data can be solved well usingthe MAP-MCPU approach.

The Markovian energy function proposed in [3] is defined,using the multichannel log-likelihood function for data fidelityterm and TV for the prior term, as the following:

E(x|y, γ)=∑p∈V

M∑c=1

Ep(yp,c, γp,c|xp)+∑

(p,q)∈Ewp,q|xp−xq| (2)

where x refers to the regularized phase (unwrapped and de-noised); and y.,c and γ.,c respectively encode the observedphase and coherence map of the cth channel, one of the Mavailable channels. Quantities w are some estimated disconti-nuity map which can be constant or varying spatially, this pointwill be described in Section IV-B, and

Ep(yp,c, γp,c|xp)=−log

(1− γ2

p,c

2π(1− γ2

p,c cos(yp,c − xp)2)

×

⎛⎝1 +γp,c cos(yp,c − xp)Arccos (γp,c cos(yp,c − xp))√

1− γ2p,c cos(yp,c − xp)2

⎞⎠⎞⎠ .

(3)

In [3], the authors exploit two different graph-cut-baseddiscrete minimization algorithms, namely, the binary move-based approximate optimization algorithm α-expansion [5] andthe multilabel exact optimization algorithm [10]. Neverthe-less, these two algorithms present some limits when copingwith low coherent interferograms with high discontinuities.

To overcome this problem, a more refined quantization of thecontinuous phase labels is needed to define L. This fact leadsto a huge graph that is prohibitive for the exact minimization,while binary partition move minimization algorithms, like theα-expansion one, reaches no good enough local minima [11].

One possible solution to get results with good precision isproposed in [4]. The idea is to perform binary jump partitionmove-based optimization algorithm with a multiresolution ap-proach, i.e., by suggesting smaller quantified intervals from oneiteration to the other. The algorithm is fast due to the use ofbinary moves. However, in case of high discontinuities, no goodenough local minima may be reached.

Another solution is given in [11], where new multilabeland large partition move-based optimization algorithms areproposed. In fact, the continuous label set could be quantifiedwith high precision, and to overcome the need of large memoryfor the exact minimization, multilabel moves (i.e., implyinga subset of labels from L) are performed. These moves haveshown to be able to reach good local minima (in practice, globalminima are reached) while keeping memory use low. However,using this optimization method is usually not efficient for timecomputation. A large time is needed to reach a good solution interm of precision and optimum quality.

We propose in the next section a brief review of this opti-mization approach, as well as its limits.

III. MULTILABEL OPTIMIZATION ALGORITHM

In [11], the proposed optimization algorithm provides goodapproximate solutions while maintaining a low memory re-quirement. This is done by performing large and multilabelpartition moves (LMPMs) of the estimated solution.

The concept of partition move consists of changing theestimated configuration x iteratively until convergence to alocal minimum x̃. Each change is obtained by minimizing theenergy function Em defined on a set of labels Lm, where mis the size of Lm. A move is called large and multilabel ifa large set of sites can change their labels in a large set oflabels, i.e., Card(x) � 1 and Card(Lm) � 1. An optimal moveis obtained if an exact minimization of the energy function isperformed on the considered set of labels. This is possible usinggraph-cut technique when the prior function Ep,q(·) is convex[10]. If the convexity property is satisfied, a specific graph Gm

with specific capacities on arcs is constructed and a maximumflow is computed on the graph to get the s,t-minimum-cut.This cut gives the configuration with the minimum energywith respect to the considered label set. We show in Fig. 1an example of such a construction. For more details about thegraph topography, we refer the reader to [11]. In Fig. 2, theLMPM algorithm is presented.

Such an optimization technique is very useful for SAR appli-cations. Indeed, in case of high-dimensional data with a highrange of labels, good approximate minima could be reachedwith a low memory use. In practice, with a good choice of labelset size, a global minimum of the energy is reached. However,there is still a problem of computational time if label sets areof large size. The algorithm has to iterate on all subsets Lm

of the label set L, where at each iteration a maximum flow iscomputed on the constructed graph Gm.

In the next section, a new algorithm is proposed to overcomethese problems, while providing more accurate reconstruction.

SHABOU et al.: MARKOVIAN APPROACH FOR INSAR PHASE RECONSTRUCTION 529

Fig. 1. Graph construction for an optimal multilabel move. On the left, a partof the graph Gm defined on three pixels. For each pixel, the column of nodescorrespond to the possible label values chosen in Lm. A cut is depicted, andarcs in the cut are dotted, whereas continuous ones are not. A part of the graphis highlighted on the right. Capacities on the edges are defined in [11].

Fig. 2. Optimization algorithm based on the large and multilabel move.

IV. PROPOSED APPROACH

The proposed approach is based on three steps, mixing dis-crete and continuous optimization algorithms, with an efficientlocal hyperparameter estimation method.

A. Discrete Optimization

The first step consists in a discrete multilabel optimization ofthe multilabel energy function (2), where a reduced size of theoriginal label set L, through a quantization of the continuousset [lmin, lmax], is exploited. The multilabel optimization is per-formed using the LMPM approach described previously. Sincea reduced size of label set is used, an efficient optimization, bothin terms of computational time and memory use, is performed.However, the obtained solution, that we note by x̃1, couldnot be good enough in term of precision due to the label setquantization. We note that, for the regularization terms, a globalhyperparameter β = wp,q, ∀(p, q) is used. This scalar couldbe adjusted manually or automatically through the use of theL-curve method [3].

B. Local Hyperparameter Estimation

The second step consists in estimating discontinuity mapsfrom x̃1. Indeed, the latter, as it represents an unwrapped andregularized phase, could bring us a prior knowledge about thetrue profile discontinuities. Two kinds of discontinuities couldbe identified. The first one is the set of weak discontinuitiesrelated to the underprecision of the reconstructed profile. Thesecond one is the set of sharp discontinuities that are relatedto the true profile. Hence, by thresholding the existing dis-continuities, we are able to define the local hyperparameters{wp,q}(p,q)∈E to adjust the energy function of (2), i.e.,{

wp,q = 0 if∣∣x̃1p − x̃1q

∣∣ > μwp,q = 1 otherwise

where μ is a prescribed threshold. Note that in practice, fixing μcould be performed easily for our case, since x̃1 is a regularizedimage. A low phase threshold on the latter is good enough topreserve true profile discontinuities while smoothing the falseones. Local hyperparameter estimation is a necessary task toovercome the known loss of contrast problem of TV-basedregularization approaches. Then, by adjusting the prior energymodel of (2) using these hyperparameters and starting fromx̃1, a continuous minimization of the new energy function isperformed allowing higher precise reconstruction.

C. Continuous Optimization

The third step, which is a continuous-based minimizationalgorithm, is based on the gradient-descent technique. As wellknown, different continuous optimization algorithms of suchobjective function are possible, and for a convex minimization,the gradient-descent one is efficient and sufficient. In our case,the first-step optimization algorithm gives usually in practice anexact discrete minimum if the label set size m is well chosen.Thus, only a continuous minimization on a convex set is neededto reach the nearest continuous minimum. This explains theuse of the gradient-descent as a second-step minimization pro-cess. Note also that the continuous optimization may convergequickly since a good initialization is given. Other continuousminimization algorithms could be applied and present moreefficient convergence propertie; however, this is not the aim ofthis letter.

Let us remind the gradient-descent-based optimization algo-rithm. We first denote by x(0) the initial guess for the estimatedsolution, which is in our case the discrete minimum obtainedat the end of the first step optimization algorithm. x(i) is theintermediate solution at the ith iteration of the gradient descentalgorithm. At each iteration, x(i) is given by

x(i) = x(i−1) − λ(i)∇E(x(i−1)

)(4)

where λ(i) designs the step size of the gradient descent. Inpractice, a line search [12] could be applied to select the stepsize that gives sufficient decrease in the objective functional.This leads to the method of the steepest descent. ∇ is thegradient operator, i.e., derivative with respect to x. Becauseof the differentiation problem of the TV function in (2), an

530 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 3, MAY 2011

Fig. 3. Simulated data sets. The 2-D view of the first (a) original profile and(b) one generated interferogram and the second (c) original profile and (b) onegenerated interferogram.

approximate model needs to be used instead. A direct approxi-mation of this function could be

TV (x) =∑(p,q)

wp,q

√|xp − xq|2 + ε (5)

where ε is a precision constant close to zero. In this letter, wehave rather chosen the l2 form of TV, given by

TV (x) =∑

(p,q)(p,r)

√wp,q(xp − xq)2 + wp,r(xp − xr)2 + ε

(6)

where pixels q, r are neighbors to p such that if p = (k, l), thenq = (k + 1, l) and r = (k, l + 1). (k, l) encodes the coordinateof p in the image grid. This form is more isotropic and appropri-ate to the gradient descent technique than (5). However, it couldnot be used in the first-step discrete optimization algorithm(LMPM) since it does not satisfy the submodularity conditionneeded by the graph-cut technique.

In the next section, some experimental results are performedto highlight the efficiency and accuracy of this algorithm.

V. EXPERIMENTS AND DISCUSSION

In this section, several experiments are proposed to showthe effectiveness and robustness of the proposed approach thatwe call later (MCPU-DC). First, our method has been testedon simulated InSAR data of different profile scenarios, suchas urban structures and mountain elevations. Quantitative andqualitative evaluations are presented with comparisons to otherapproaches. Then, the method is tested on real InSAR data.

A. Simulated InSAR Data

In the first experiment, we consider a synthetic height profileFig. 3(a), of size (200 × 400), with a maximum height of 14 radexhibiting both smooth and discontinuous areas. We used fourfrequencies ({5,6.33,7.66,9} GHz) to generate four noisy in-dependent interferograms with a constant coherence of {γp,c =

Fig. 4. Results on the simulated data 3(a). 3-D views of the (a) original profileand the (b) reconstructed phases obtained, respectively, using the MCPU-GCapproach at the (c) end of the first-step optimization algorithm (x̃1) and at the(d) end of the continuous minimization (x̃2).

Fig. 5. Results on the simulated area 3(c). 3-D views of the (a) original profileand the (b) reconstructed phases obtained, respectively, using the MCPU-GCapproach at the (c) end of the first-step optimization algorithm (LMPM) (x̃1)

and at the (d) end of the continuous minimization (x̃2).

0.6; ∀p, c}. In Fig. 3(b), we show one noisy interferogram. Notethat the profile is ambiguous for these frequencies. In fact, thereare phase jumps greater than π violating the Itoh condition.

In the second experiment, we consider a more representativeheight profile of natural image Fig. 3(c) with the same sizes asthe previous one and a maximum height of 19.1 rad exhibitingarea characterizing highland scenes. The same InSAR parame-ters as the ones used in the previous experiment are applied onthis profile to generate independent interferograms.

In Figs. 4 and 5, we show the reconstruction results obtained,respectively, on the two scenarios using the proposed approach.Results given by the multichannel approach (MCPU-GC) in [3]are also presented in Figs. 4(b) and 5(b), where the continuouslabel set is quantified into 200 labels, and an exact minimizationalgorithm is applied based on the Ishikawa graph construction.We note that a huge graph has to be build to provide an accuratesolution.

SHABOU et al.: MARKOVIAN APPROACH FOR INSAR PHASE RECONSTRUCTION 531

TABLE IQUANTITATIVE RESULTS OF THE TWO EXPERIMENTS

Fig. 6. Three-dimensional reconstruction of real InSAR data. (a) The firstnoisy interferogram, (b) the 2-D view of the reconstructed profile with theMCPU-GC approach, (c) the 2-D view of the reconstructed profile with theproposed approach, (d) the 3-D view of the reconstruction.

Reconstructions obtained at the end of the first-step opti-mization approach are shown in Figs. 4(c) and 5(c), where thecontinuous label set is quantified into 64 labels for both imagesand the LMPM optimization algorithm is applied using labelsets of sizes, respectively, m = 8 and m = 16, to converge to agood local minimum of the energy.

Starting from these results, local hyperparameters are esti-mated by thresholding the existing discontinuities using thedescribed method in Section IV-B. Note that the thresholdparameter μ is fixed to π/8 in these experiments. Then, thecontinuous optimization algorithm is performed. Results onboth data sets are shown in Figs. 4(d) and 5(d). We show thatsmooth areas are well reconstructed, while discontinuities arealso preserved. This is not the case in Figs. 4(b) and 5(b),where discontinuities are well preserved due to the TV prior,but smooth areas are noisy, even when a large label set isconsidered.

Quantitative measures, in terms of reconstruction qualityusing the mean and root mean square deviation errors (MSEand RMSE, respectively) and computational time and memoryneeded for graph allocation, are presented in Table I. Theseresults show both the accuracy and the efficiency of the two-stepoptimization algorithm and its effectiveness compared to otherapproaches. We also note that improvements in terms of MSEand RMSE are less important for the second profile comparedto the first one, since less smooth regions are characterizingthis profile. However, we can clearly see that in terms ofcomputational time and qualitative visual results, our approachis always more interesting compared to the MCPU-GC one.

B. Real InSAR Data

We have tested the new algorithm on a real data set of an ur-ban scenario. A set of eight L-Band E-SAR interferograms (twointerferograms for each of the four polarizations) are acquiredon the city of Dresden. The smallest orthogonal baseline is ofabout 8.42 m, and the biggest is of about 28.34 m. Due to thepresence of noise, the Itoh condition is violated in some areas.

We can see in Fig. 6(c) and (d) the good quality of thereconstructed phase, compared to the result obtained using theMCPU-GC Fig. 6(b). In fact, it is clearly seen that discontinu-ities and smooth or homogeneous areas are well preserved usingthe mixed discrete and continuous optimization approach, whileusing only a discrete optimization algorithm provides resultswith low precision. We have to note that only topographicfringes are present in the processed data. Thus, to deal with dif-ferential interferograms, other problems need to be addressed,such as the atmospheric phase contributions. Improvements inthis direction will be proposed in a future work.

VI. CONCLUSION

In this letter, we have developed a new 3-D height re-construction methodology based on an MCPU approach andefficient and accurate optimization algorithm. The proposedalgorithm overcomes the limits that characterize other MAP-MCPU approaches both in terms of computational complexityand solution accuracy. We have tested this approach on sim-ulated and real InSAR data. Good quantitative and qualitativeresults are obtained. Future work will focus on the exploitationof these approaches for the differential InSAR applications.

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