IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007 3599

Polarimetric and Interferometric SAR Image PartitionInto Statistically Homogeneous Regions Based on

the Minimization of the Stochastic ComplexityJérôme Morio, François Goudail, Xavier Dupuis, Pascale C. Dubois-Fernandez, and Philippe Réfrégier

Abstract—In this paper, we show that polarimetric and inter-ferometric SAR (PolInSAR) images can be efficiently partitionedinto homogeneous regions with a statistical technique based onminimization of a parameter-free criterion. This technique con-sists of finding a polygonal partition of the image that minimizesthe stochastic complexity, assuming that the image is made ofa tessellation of statistically homogeneous regions. The obtainedresults demonstrate that a global partition in statistically homoge-neous regions of PolInSAR images can provide better results thana partition based on a single characteristic such as polarimetry orinterferometry only.

Index Terms—Image segmentation, polarimetric interferome-try, SAR interferometry, SAR polarimetry, synthetic apertureradar (SAR).

I. INTRODUCTION

POLARIMETRIC and interferometric synthetic apertureradar (SAR) (PolInSAR) images have been a real advance

in the SAR imaging domain since they contain complete infor-mation about polarimetry and interferometry. They can revealthe nature of the wave interaction with the ground and the soiltopography, respectively [1]–[3]. Major works have contributedto a better understanding of PolInSAR data. Indeed, Cloude andPapathanassiou have derived a formulation of coherent inter-ferometry between polarized waves [4]. Forest and vegetationanalysis [5] is one of the main applications; the estimation of

Manuscript received March 22, 2006; revised February 7, 2007. The work ofJ. Morio was supported in part by the Office National d’Etudes et RecherchesAérospatiales and in part by the region Provence-Alpes-Côte d’Azur(PACA).

J. Morio is with the Office National d’Etudes et Recherches Aérospa-tiales, 13661 Salon Cedex Air, France, and also with the Physics andImage Processing Group, Fresnel Institute, Unité Mixte de Recherche Cen-tre National de la Recherche Scientifique 6133 Marseille, France (e-mail:jerome.morio@fresnel.fr; jerome.morio@onera.fr).

F. Goudail is with the Laboratoire Charles Fabry de l’Institut d’Optique,Unités Mixtes de Recherche Centre National de la Recherche Scientifique 8501,Centre Scientifique, 91403 Orsay, France (e-mail: francois.goudail@iota.u-psud.fr).

X. Dupuis and P. C. Dubois-Fernandez are with the Electromagnetismand Radar Department, Office National d’Etudes et Recherches Aérospatiales(ONERA), 13661 Salon Cedex Air, France (e-mail: xavier.dupuis@onera.fr).

P. Réfrégier is with the Physics and Image Processing Group, Fresnel Insti-tute, Unité Mixte de Recherche Centre National de la Recherche Scientifique6133, École Centrale de Marseille, Domaine Universitaire de Saint-Jérôme,13397 Marseille, France (e-mail: philippe.refregier@fresnel.fr).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2007.903689

forest height using PolInSAR data and the Random Volumeover Ground model (RVoG) [6] have notably been given muchattention. The RVoG describes the forest as a two-layer coherentscattering model and enables one to estimate the height and thecanopy extinction of a forest.

Image-processing techniques such as image segmentation,partition, or classification also enhance our knowledge of SARimages. Many unsupervised or supervised classification meth-ods have been applied on polarimetric image [polarimetricSAR (PolSAR)] since they provide information on the mediumimaged by the radar. Indeed, Cloude and Pottier [7] proposed anunsupervised classification based on their target decompositiontheorem. More statistical approaches have also been studied,notably by Lee et al. [8]–[10] who used multivariate complexWishart distribution and integrated the estimation of the co-herency matrices in the classification procedure for multilookdata. The same methods have also been applied on PolInSARdata [11], [12]. Hoeckman [13], [14] proposed a supervisedmethod that leads to interesting results with a great accuracyand without a large amount of training data, but an accurateground truth is needed. Image segmentation is also a source ofinterest for the analysis of SAR image. In the case of PolSARimages, edge detection filters based on likelihood ratio andthe Wishart distribution can be applied with constant falsealarm rate [15]. Other approaches that estimate the coherencymatrices with region-growing techniques [16] and filtering [17]have also been evaluated.

In this paper, we address image partition into statisticallyhomogeneous regions, i.e., regions which have the same po-larimetric and interferometric statistical properties. Partitioninto statistically homogeneous regions can be, for example,the first stage of a segmentation method that will partitionan image into semantically coherent regions. We present analgorithm based on a statistical active grid that can be appliedto PolInSAR, PolSAR, and InSAR images. It consists of theestimation of the number of regions, the number of nodes, andthe segments of the polygonal grid that partitions the image intostatistically homogeneous regions. For this purpose, we applythe stochastic complexity minimization principle [18]. Thestochastic complexity depends on the grid model that has beenchosen to partition the image and on a statistical description ofthe data. It is composed of two terms: the first one correspondsto the code length needed to describe the data once the modelis known (estimated with the log likelihood), and the secondterm is the code length of the parameters of the model that

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3600 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007

depends on the number of nodes, segments, and on the numberof parameters of the probability density functions (PDFs). Thisstatistical grid-based approach has already been applied onscalar [19] and on vectorial images whose channels are assumedstatistically independent [20]. In this paper, we will extend it tothe case of PolInSAR images modeled by correlated Gaussiancircular processes. The fundamental question that appears inthat case is to determine if this technique is sensitive to thecurse of dimensionality which may appear when the numberof unknown parameters (here, the 6 × 6 covariance matrix)increases.

We present in Section II-A the image models that we useto describe the InSAR, PolSAR, and PolInSAR data. Then, inSection II-B, we define the information-theory-based criterionthat will be used to perform partition of multicomponent SARimages. Application of this principle to such high-dimensionaldata as PolInSAR images involves specific issues that areaddressed in Section III. Namely, we address in Section III-APolInSAR matrix estimation and region merging in the specialcase of very small regions. Then, we describe in Section III-Bhow the polarimetric contrast and the interferometric contrastcan be extracted from PolInSAR data. Section IV is devotedto the application of the statistical grid to the partition instatistically homogeneous regions of real PolInSAR images.We conclude in the last section, and we discuss the perspectivesof this method for partitioning PolInSAR images in statisti-cally homogeneous regions based on the minimization of aparameter-free criterion.

II. HOMOGENEOUS POLINSAR PARTITION

In this section, we present the model of multicomponent SARdata, then the criterion used to obtain partitions in statisticallyhomogeneous regions, and, finally, the proposed optimizationprocedure.

A. PolInSAR, InSAR, and PolSAR Models

The models generally used to partition the different SARdata (PolInSAR, PolSAR, and InSAR) are quite similar sincethey comprise a multicomponent Gaussian circular process.Indeed, scalar interferometry (InSAR) is based upon thecoherency between two complex signals S1 and S2 with thesame polarization. A complex 2-D vector represents the back-scattered signal measured by two antennas: Kinterf =(S1, S2)T. In fully PolSAR images, the coefficients of thescattering matrix in linear horizontal and vertical basis enableus to define the target vector Kpol: It is defined by threeparameters considering the reciprocity theorem and is repre-sented by a 3-D vector Kpol = (1/

√2)(SHH+VV, SHH−VV,

2SHV)T. PolInSAR combines the principles of polarimetry andinterferometry since two fully PolSAR images Kpol1 and Kpol2

are measured. It is defined as a complex 6-D vectorcorresponding to the polarimetric response of both antennas:Kpolint = ( Kpol1, Kpol2)T = (1/

√2)(S1HH+VV, S1HH−VV,

2S1HV, S2HH+VV, S2HH−VV, 2S2HV)T. Classically, the vec-tors Kpolint, Kinterf , and Kpol are measured at each pixel and

Fig. 1. Polygonal grid with R = 9 regions, 43 nodes, and p = 49 segments.

are assumed circular Gaussian with zero mean and covariancematrices defined as follows:

Υpol = 〈 KpolK†

pol〉

Υinterf = 〈 KinterfK†

interf〉

Υpolint = 〈 KpolintK†

polint〉

where † and < > denote complex conjugate transpose andensemble averaging, respectively. In the case of statisticallyhomogeneous, this ensemble averaging can be approximatedby spatial averaging. It is not any more the case when theregions are not statistically homogeneous, which appears, forinstance, when there is a linear variation of the phase due to alinear variation of the height. In the next section, we use thecircular Gaussian-correlated model to determine the stochasticcomplexity associated to the partition of a multicomponentSAR image.

B. Stochastic Complexity

In order to partition a SAR image into statistically homoge-neous regions, we use a recently proposed method based on theminimization of the stochastic complexity [19]. It consists ofpartitioning the image with a polygonal grid that is composed ofnodes joined together by segments. Fig. 1 shows the definitionof regions, nodes, and segments of a polygonal grid. This griddefines the different regions Ωr, r ∈ [1, R] of the image. Theauthors proposed minimizing the stochastic complexity ∆ ofthe image to determine the number of regions R, the numberof nodes, and their locations in three different steps: mergingregions, moving nodes, and removing nodes. Figs. 2 and 3present these three different optimization steps to partition asynthetic image. The stochastic minimization principle is a gen-eral principle which has been introduced by Rissanen in 1978[21]. The main idea is to find the image description that hasthe lowest complexity. For that purpose, Risannen proposed thestochastic complexity which corresponds to the mean numberof bits needed to describe the image with an entropic code [18],[22] in the meaning of Shannon. The number of bits to describethe image is a sum of two terms. The first one corresponds to thenumber of bits needed to describe the image model, and it thuscharacterizes the complexity of this model. The second one isthe number of bits needed to encode the gray levels of the imagewith an entropic code once the model is known. In the proposed

MORIO et al.: POLINSAR IMAGE PARTITION INTO STATISTICALLY HOMOGENEOUS REGIONS 3601

Fig. 2. Partition of a synthetic PolInSAR image. (a) |HH|2 image of syntheticPolInSAR. (b) Initial grid consisting of 8 × 8 pixel regions. (c) Polygonal gridafter a GLRT-merging stage. (d) After a move of the grid and a node removal.(e) After a merging stage. (f) Final partition obtained after three iterations of themerging/moving/removing sequence. All the partition results are superposed tothe |HH|2 image.

Fig. 3. Zooms of Fig. 2(a) and (b) initial grid is an 8 × 8 pixel brick wallshape. (c) Merging two regions consists of removing their common segment.Two regions are merged if their merging decreases the stochastic complexity.(d) In the moving stage, the node locations are tested in eight directions ofthe plane and with different amplitudes. The chosen position of a node is theone that minimizes the proposed criterion. During the removing stage, a nodeis removed if its suppression implies a decrease of the stochastic complexity.(e) and (f) Each of the three optimization stages is applied iteratively to find thepartition that minimizes the stochastic complexity. The optimization processis finished when no operation (merging, moving, and removing nodes) can beapplied without increasing the criterion.

algorithm, the image model is composed of the polygonal gridthat partitions the image and of a parametric PDF model forthe data. The first term of the stochastic complexity is thuscomposed of two parts: ∆G is the code length for the grid,and ∆P is the code length for the PDF parameters. A completeexpression of these terms is given in the following.

1) ∆G is the number of bits required to encode the grid. Itnotably depends on the number of nodes and segments of

the partition w. The complete expression [19] of ∆G isgiven by

∆G(w) = n (log(N) + log(p)) + log(p)+ p (2 + log(2mx) + log(2my))

where N is the total number of pixels of the image, n isthe minimum number of starting nodes that are necessaryto draw the grid by passing one and only one time byall the segments, p is the number of segments of thegrid, and mx (respectively, my) is the horizontal (verti-cal) mean length of the segments of the grid measuredin pixels.

2) ∆P is the number of bits necessary to encode the para-meters θ (in the case of PolInSAR, PolSAR, or InSAR, θis related to the covariance matrix) of the parametric PDFin each region of the grid. This term depends on the typeof PDF and on the partition w of the image [19], [23].An approximation of the number of bits associated to thisterm is [18]

∆P(θ, w) =α

2

R∑r=1

log(Nr)

where α is the dimension of the parameter vector θ, andNr is the number of pixels in the regions r. In the case ofPolInSAR image, θ contains different real-valued coef-ficients of the covariance matrix Υpolint. Since Υpolint isan Hermitian 6 × 6 matrix, θ is a 36-D vector, and thus, αis equal to 36 (the six real coefficients from the diagonalof the matrix and the 15 real and imaginary parts from itstriangular superior part). In the same way, we have α = 9for PolSAR and α = 4 for InSAR.

3) ∆L is the number of bits needed to encode the backscat-tering coefficients of the image once the partition in statis-tically homogeneous regions (i.e., the grid) and the PDFparameters have been described. ∆L can be approximated[19], [23], [24] by the opposite sum of the likelihoods Lr

of the data in each region

∆L(w) = −R∑

r=1

Lr.

If the parameter θ is estimated with the maximum-likelihood method, Lr is the profile likelihood in regionr. InSAR, PolSAR, and PolInSAR data are classicallymodeled as an n-component (n = 2 for InSAR, n = 3for PolSAR, and n = 6 for PolInSAR) complex vectorwith Gaussian circular PDF with covariance matrix Υ.This model of PDF enables us to determine the profilelikelihood of a sample that is given by Conradsen et al.[15], Goudail and Réfrégier [25], and Chesnaud et al. [26]

Lr = Nr log[det Υr

]+ 6Nr [1 + log(π)] (1)

where Nr is the number of pixels in region Ωr and

Υr =1

Nr

∑(x,y)∈Ωr

K(x, y) K(x, y)T. (2)

3602 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007

K(x, y) denotes the measurement vector at pixel (x, y).The matrix Υr is the maximum-likelihood estimate of Υin region Ωr. Thus, ∆L has the following expression:

∆L(w) =R∑

r=1

Nr log[det Υr

]+ 6N [1 + log(π)] .

The stochastic complexity is thus equal to

∆ = ∆G(w) + ∆P(w) + ∆L(w). (3)

The optimal partition of an image is thus defined as the onethat minimizes the stochastic complexity ∆. We can noticethat ∆ does not depend on any parameter that has to be tunedby the user but only on the data and on the grid model. Aweighted version of the stochastic complexity will be discussedin Section II-D. The statistically homogeneous regions based onstochastic complexity are thus the result of a tradeoff betweenthe cost in bits to describe the image model and the cost todescribe the image data knowing the image model. Indeed, thegeneral purpose of this algorithm is to determine if it costsmore bits to describe neighboring pixels with two statisticallyhomogeneous regions and thus two different PDFs or withone statistically homogeneous region and one PDF. In the firstcase, the image data are well-characterized but it increases thenumber of bits to describe the image model. In the latter case,the image model is simple but the number of bits to describethe image data is important. Let us now detail the optimizationstrategy to find the minimum of the stochastic complexity ∆.

C. Three-Stage Optimization Process

The aim of the proposed partitioning method is to minimizethe stochastic complexity ∆ that depends on the number ofregions R, the number of nodes, and their locations in thegrid. Starting from an initial grid (generally 8 × 8 or 4 ×4 pixel brick wall shape), the optimization technique consists ofmoving, removing, and merging regions (removing segments)so that the deformations make the stochastic complexity de-crease. These three operations are applied until convergence,i.e., until the stochastic complexity reaches its local minimum[19], [24]. Fig. 2 gives an example of partition in statisticallyhomogeneous regions with the statistical active grid.1) Merging Regions: The region-merging stage is very sim-

ple; two neighboring regions are merged if and only if theirfusion leads to the decrease of the stochastic complexity ofthe image. Let w be the partition before the fusion of twoneighboring regions A and B and w′ the partition where A andB are merged to form the region A ∪ B. The fusion of theregions A and B is consequently accepted if ∆(w′) < ∆(w).If we develop this inequality with the different terms of thestochastic complexity, we obtain

∆G(w′) + ∆P(w′) + ∆L(w′) < ∆G(w) + ∆P(w) + ∆L(w)

i.e.,

∆L(w′)−∆L(w) < ∆G(w) + ∆P(w)−∆G(w′)−∆P(w′)

and consequently

LA + LB − LAB < ∆G(w) + ∆P(w)−∆G(w′)−∆P(w′).(4)

We can remark that it is similar to a generalized likelihood ratiotest (GLRT) compared to a threshold S(w,w′) = ∆G(w) +∆P(w)−∆G(w′)−∆P(w′). This threshold is not constantbut depends on the partitions w and w′. The merging stageapplied with the threshold S(w,w′) considerably decreases thenumber of regions of the partition, and since it is an irreversibleprocess, it could lead to an underpartition of the image. Thus,to obtain a more gradual convergence, we decide, for the firstmerging stage, to arbitrarily choose a GLRT threshold S equalto 30. The other merging stages are performed by minimizingthe stochastic complexity. Each couple of neighboring regionsis merged if they satisfy the inequality in (4). The stage isended when any fusion of two neighboring regions increasesthe stochastic complexity.2) Moving Nodes: We also have to estimate the location

of the nodes. Each node of the grid is successively moved,and its new location is accepted if it decreases the stochasticcomplexity. If it is not the case, the node is restored to itsinitial location. The new locations of a node are tested in eightdirections with constant distance. When no moving of a nodeis possible, then the new location of the nodes are tested butwith a smaller moving distance than before. The end of thisstage is reached when node moving with a weak amplitude isnot possible without increasing the criterion ∆.3) Removing Nodes: A partition with a too large number of

nodes does not provide a satisfactory description of an image. Itis thus necessary to introduce a node-removing stage. We onlyconsider in this process the nodes that have a multiplicity equalto two (nodes that are only linked to two other nodes). A node isremoved if its suppression involves a decrease in the stochasticcomplexity. The stage is finished when no node can be removedwithout increasing the stochastic complexity.

Fig. 2 shows the partition of a synthetic 256 × 256 PolInSARimage. It illustrates the different stages of the algorithm from aninitial 8 × 8 pixel brick-wall-shaped partition to the final one,when no node can be moved or removed and no region can bemerged.

As in any optimization technique, the final partition dependson some initialization parameters such as the initial grid size orthe value of the GLRT threshold (30 in our case). Nevertheless,it is possible to test different initializations and choose theone that minimizes the stochastic complexity. Fig. 4 showspartitions in statistically homogeneous regions of a synthetic256 × 256 PolInSAR image with different sizes of initial grid.The best partition is obtained with a 6 × 6 initial grid sinceit leads to the lowest stochastic complexity ∆. However, itcan be noticed that all the partitions obtained with differentgrid initializations are similar, which shows that the proposedalgorithm does not strongly depend on the initialization. Thesame type of experiment can be performed to determine themost efficient threshold S. In conclusion, it can be said thatinitial grids between 4 × 4 and 8 × 8 pixels and thresholdvalue S = 30 are experimental parameters that often lead to

MORIO et al.: POLINSAR IMAGE PARTITION INTO STATISTICALLY HOMOGENEOUS REGIONS 3603

Fig. 4. Partitions in statistically homogeneous regions of a synthetic 256 ×256 PolInSAR image with different initial brick-wall-shape grid size. For thisimage, the initial grid that leads to a minimal value of the criterion is a 6 × 6initial brick grid.

Fig. 5. Partitions of the synthetic 256 × 256 image of Fig. 4 obtained for threedifferent values of parameter λ.

valuable partition, but not necessarily to the best one, and theseparameters are relative to the optimization technique but not tothe optimized criterion.

D. Stochastic Complexity Analysis

We have shown that the stochastic complexity leads to acriterion that is free from parameter adjustment by a user. Inorder to analyze more precisely the relevance of this approach,it is possible to introduce an adjustment parameter λ such asclassically

∆ = ∆L(w) + λ∆M(w) (5)

where ∆M(w) is a regularization term equal to ∆G(w) +∆P(w). The Risannen’s stochastic complexity used in theproposed algorithm corresponds to λ = 1.

In Fig. 5, we show three partitions on a 256 × 256 syntheticimage that have been obtained for λ = 0.8, λ = 1, and λ = 4.For λ = 4, the statistical active grid does not detect all theregions of the image, and for λ = 0.8, it is not regularized

Fig. 6. Evolution of the ANMP estimated on 20 independent realizations of asynthetic image as a function of the parameter λ. The minimum number of mis-classified pixels is obtained for λ = 1, i.e., when likelihood and regularizationterms have the same weight.

enough and the number of regions is clearly too large. λ = 1corresponds to an equal weight between the likelihood andregularization terms and seems to be the most efficient. Inorder to confirm this qualitative result, we have performed thesimulation in Fig. 6. Knowing the true partition of a 256 × 256synthetic image, we generate 20 independent realizations ofthis image, then partition them, and finally estimate the averagenumber of misclassified pixels (ANMP) [23] of the partitionsobtained for different values of λ. It is shown in Fig. 6 thatλ = 1 leads to the minimal value of misclassified pixels. Thus,on synthetic data, the value λ = 1 given by Risannen’s theoryactually leads to the best partition results in terms of ANMP.Of course, real data may not exactly correspond to the imagemodel (varying mean reflectance over regions, etc.), and whenworking on thematic segmentation, semantic regions may notbe statistically homogeneous. In such cases, better results maybe obtained for values of λ different from one. As shown in (5),the proposed method can be easily adapted to such supervisedapplications.

III. ANALYSIS OF THE PARTITIONING METHOD

The statistical active grid algorithm has been validated onscalar and vectorial images whose channels are assumed inde-pendent [19], [20]. Its application on PolInSAR raises specificissues due to the dimension of the data vector and to the factthat its components are correlated. In this section, we first focuson the partition of small regions in PolInSAR data and thenanalyze how the contrast in a PolInSAR image can be dividedinto a polarimetric and an interferometric contribution.

A. Change in Encoding Due to Covariance Matrix Estimation

The application of the grid on real PolInSAR image can leadto covariance estimation problems in the case of regions witha few pixels. To solve this problem, it is necessary to partiallymodify the criterion ∆. The region-merging stage is an essentialstep in the method, since region splitting is not implemented.One term of the stochastic complexity is the opposite sumof the profile likelihoods of each region, and consequently, it

3604 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007

involves the estimation of a covariance matrix [see (1)]. If thisestimation is not performed on a sufficient number of pixels,then the resulting matrix may be singular, and problems canoccur when very small regions appear. A necessary conditionfor an estimated covariance matrix Υr of a region r to beinvertible is that it is estimated on a sufficient number of pixels.This number of pixels Nr has to be larger than the dimensiond of the covariance matrix to be estimated [27], i.e., d = 6for PolInSAR, d = 3 for PolSAR, and d = 2 for InSAR. Evenif Nr is a little larger than these limit values, the estimationof a covariance matrix may not be accurate. Indeed, the de-terminant Υr may tend to zero and the profile log likelihoodto −∞ (1). Consequently, a small region r cannot be mergedto its neighboring regions because a fusion will automaticallyincrease the stochastic complexity. We have thus defined anew expression of the profile likelihood that depends on thesize of the region. If the number Nr of pixels in the region ris greater than d, the backscattering coefficients are modeledby the correlated Gaussian circular process. Otherwise, if thenumber of pixels Nr is smaller than d, the pixels are modeled byuncorrelated Gaussian circular process, i.e., we only consider adiagonal covariance matrix for the region r. The diagonal termsof the covariance matrix represent the intensities in the differentchannels. To summarize, the likelihood in the region r has thefollowing expression. If Nr > d

Lr = Nr log[det(Υr)

](6)

and if Nr ≤ d

Lr = Nr log[det(diag(Υr)

)](7)

where diag(Υr) is a diagonal matrix whose coefficients are thediagonal coefficients of Υr. This choice avoids the problem dueto singular matrices in the case of small regions.

B. Polarimetric and Interferometric Partition onSynthetic Images

Since PolInSAR combines polarimetry and interferometry,the resulting partition is able to detect both polarimetric andinterferometric variations between the regions. Indeed, twoneighboring regions in a PolInSAR image are partitioned if theyhave a different polarimetric and/or interferometric behavior. Inthis section, we show how we can define a polarimetric and aninterferometric contrast from PolInSAR data and illustrate thisresult on synthetic images. For that purpose, we use the PolIn-SAR matrix decomposition from Cloude and Papathanassiou’scoherence optimization algorithm [28]. It is based on the selec-tion of the projection vectors that maximizes the interferometriccoherence and leads to a 3 × 3 complex eigenvector problem.We examine here the results of this method reformulated withmatrix products.

As seen in Section II-A, PolInSAR data are modeled by a 6-Dcomplex target vector K with K = [ K1, K2]T, where K1 andK2 correspond to the 3-D polarimetric target vectors from the

two antennas. They follow the Gaussian PDF with zero meanand covariance matrix Υ

Υ =(〈 K1

K†1〉 〈 K1

K†2〉

〈 K2K†

1〉 〈 K2K†

2〉

)=(

Γ1 Ω12

Ω†12 Γ2

).

Let us detail the different 3 × 3 blocks that compose thematrix Υ.

1) Γ1 and Γ2 correspond to the polarimetric measurementsof each of the two antennas and, consequently, containthe polarimetric information from the matrix Υ. Thesematrices have notably been studied in [7] and [29]. Theauthors defined useful parameters that characterize therandomness of the scattering process and the interactionbetween the wave and the lightened medium.

2) Ω12 represents the polarimetric interferometry part of Υ.To remove the dependence of Ω12 on polarimetry, it hasbeen proposed to define the normalized coherence matrixM with M = Γ−1/2

1 Ω12Γ−1/22 . It has been shown in [28]

that the larger singular value of M corresponds to themaximum generalized coherence [28] that can be reachedby varying the polarization states of the signals from thetwo antennas.

Considering the matrices Γ1 and Γ2 and the normalizedcoherence matrix M , we propose to introduce the followingnotation in order to decompose the matrix Υ and separatepolarimetry and interferometry characteristics from this matrix.Let us define the joint polarization matrix G that contains thepolarimetric information of Υ

G =(Γ1 00 Γ2

).

If we assume that G is invertible (neither Γ1 nor Γ2 representsa totally polarized wave), we can also introduce the matrix Φ

Φ = G− 12ΥG− 1

2

where G−(1/2) = V D−(1/2)V † with D as a diagonal matrixthat contains the eigenvalues of G, and V contains the eigen-vectors of G. We can develop this matrix in the following way:

Φ =G− 12ΥG− 1

2

=

(Γ− 1

21 0

0 Γ− 12

2

)(Γ1 Ω12

Ω†12 Γ2

)(Γ− 1

21 0

0 Γ− 12

2

)

=

(I3∗3 Γ− 1

21 Ω12Γ

− 12

2

Γ− 12

2 Ω†12Γ

− 12

1 I3∗3

)

where I3∗3 represents the 3 × 3 identity matrix. Following its

definition introduced in [30], one has Φ =(

I3∗3 MM † I3∗3

)and

Υ = G1/2 Φ G1/2. The polarimetric and interferometric matrixΥ can thus be decomposed into an interferometric matrix Φand a polarimetric matrix G. We can use this decomposition tosynthesize PolInSAR images with well-defined polarimetric orinterferometric contrast between neighboring regions. Indeed,if two neighboring regions do not have the same matrix Φ, they

MORIO et al.: POLINSAR IMAGE PARTITION INTO STATISTICALLY HOMOGENEOUS REGIONS 3605

Fig. 7. (a) Label of the 248 × 248 synthetic image and (b) its Pauli vector image. Partition of synthetic (c) PolSAR, (d) InSAR, and (e) PolInSAR images. Thestatistical active grid partitions all the regions of interest of the PolInSAR image, whereas it is not the case on PolSAR and InSAR. (f) and (g) PolInSAR partitionswith the weighted version of the algorithm with λ = 0.2 and λ = 10, respectively.

will have an interferometric contrast, and if they do not have thesame matrix G, they will have a polarimetric contrast.

In Fig. 7, we present the label image that has been used tosimulate the data. It is composed of six regions. In the simulateddata, the pixels of the image follow multidimensional GaussianPDF with different covariance matrices in the six regions of thelabel image. To define these covariance matrices, we considertwo polarimetric matrices G1 and G2 and two interferomet-ric matrices Φ1 and Φ2, which are defined in the Appendix.We make combinations between these four matrices to obtainPolInSAR covariance matrices. Namely, the covariance matrixin region 1 is described with Υ1 = G2Φ1G2; in region 2, onehas Υ2 = G2Φ2G2; in region 3, Υ3 = G1Φ1G1; in region 4,Υ4 = G2Φ1G2; in region 5, Υ5 = G1Φ1G1; and finally, inregion 6, Υ6 = G1Φ2G1. Moreover, we impose that there isno contrast in intensity: The diagonal terms of each matrixΥi with i = 1, 6 are equal to one. Our purpose is to compareon this image the partitions obtained with the statistical gridalgorithm based on PolSAR, InSAR, and PolInSAR models.The PolSAR and the InSAR images are computed as followsfrom the PolInSAR image.

1) The target vectors of a PolInSAR image for each pixelhave the following form: Kpolint = (S1HH;S1VV;√2S1HV;S2HH;S2VV;

√2S2HV)T. With this six-

component vector, we can construct an InSAR image con-sidering only the signal from the HH polarization leadingto the following vector: Kinterf = (S1HH, S2HH)T.In the same way, it is possible to extract a PolSAR imageby choosing the full polarimetric response of only oneantenna, i.e., Kpol = (S1HH, S1VV,

√2S1HV)T.

2) For each image, we apply the statistical active grid withits corresponding model. Thus, we obtain three differentgrid partitions in statistically homogeneous regions forPolSAR, InSAR, and PolInSAR image. The results ap-pear in Fig. 7. We can notice that the number of regionsand the number and the location of the grid nodes varyfrom one image to another.

The partition on PolInSAR data enables us to detect all theregions of interest in the image, whereas it is not the case withPolSAR and InSAR. Indeed, in PolSAR image, we only par-

tition the neighboring regions whose polarimetric components(i.e., matrices G) are different (for instance, regions 1 and 5)and, in the case of InSAR data, the neighboring regions whoseinterferometric components (i.e., matrices Φ) are different (re-gions 1 and 2). This simple example illustrates that the proposedPolInSAR statistical grid method is sensitive to both polarimet-ric and interferometric contrast. The proposed decomposition ofthe PolInSAR matrix can also be used to interpret the contrastof regions obtained by partitioning real images.

IV. APPLICATIONS OF THE STATISTICAL ACTIVE GRID

In this section, we focus on the partition of real PolIn-SAR data with the statistical active grid. Finally, we showthat PolInSAR leads to more precise partition in statisticallyhomogeneous regions than PolSAR and InSAR. In this section,we present partitions on real PolInSAR images. For this pur-pose, we consider two Office National d’Etudes et RecherchesAérospatiales (ONERA) RAMSES SAR [31] data sets acquiredin P- and X-bands. The first one is a RAMSES X-band imageacquired in March 2002 over the Institut National de RechercheAgronomique (INRA)1 site of Avignon. This PolInSAR imagehas a dimension of 6492 × 1152, the incidence angle is 30,and the slant range and azimuth resolution are 0.9 and 0.94 m,respectively. It is a single-pass interferometry image with a63-cm baseline, and the altitude of ambiguity is about 250 m.Trihedral corners and dihedral reflectors were deployed on thescene for radiometric and polarimetric calibration.

INRA’s site is composed of small fields delimited by cypresshedges and planted with clover, wheat, and orchards like peachtrees, pear trees, and apricot trees. The rest of the imagedzone contains some other types of vegetation, like apple trees,pine forest, plane trees, and, notably, poplars (Fig. 8). Theground truth is known for most of INRA’s parcels, which willallow us to evaluate the quality of parameter estimation andclassification. We show the partition on this PolInSAR imageand the corresponding aerial photography in Fig. 9. We can

1French Agronomic Research Center.

3606 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007

Fig. 8. Extract (1670 × 647) of the PolInSAR X-band Avignon image notablycomposed of small agricultural parcels, buildings, and roads. (a) Pauli vectorimage (R: HH + VV; G:HH − VV; B:2HV). (b) Interferometric phase HH(black: −π to white: π). (c) Coherence HH (black: 0 to white: 1). The redpoints in image (a) make the correspondence between the SAR image and theaerial image of Fig. 9.

Fig. 9. (a) Final partition obtained with an initial 8 × 8 pixel grid on thePolInSAR data. Since the scene is complex, the final partition contains a largenumber of regions. (b) Aerial image taken over the corresponding testing areaFig. 8 on the X-band Avignon image.

notice that the final partition contains many regions becauseof the complexity of the image. The statistical active gridhas detected regions which are homogeneous with respect topolarimetry and interferometry. We can remark that very finedetails have been partitioned, notably in the building and forestareas, which is important for the analysis of the image. On

Fig. 10. 1095 × 586 extract of PolInSAR Nezer P-band image. (a) Paulivector image (R: HH + VV; G: HH − VV; B: 2HV). (b) Interferometric phaseHH (black: −π to white: π). (c) Coherence HH (black: 0 to white: 1).

the other hand, larger areas have been partitioned into severalregions by the grid. Indeed, the grid is a partition method that isdesigned to find the statistically homogeneous regions and notthe regions of semantic interest of the image.

The second image has been acquired in P-band by RAMSESover the region of Nezer, France. The campaign was carriedout in January 2004. The image size is 3500 × 1550 pixels,and the slant range and azimuth resolution are 2.5 and 3.2 m,respectively. It is a double-pass interferometry image; the alti-tude of ambiguity is about 45 m, and the incidence angle is 30.This site is composed of major part of rectangle-shaped parcels(Fig. 10) containing different types of trees. The age and thetype of trees are known in most of the parcels. Fig. 11 shows thepartition into statistically homogeneous regions of this imageon PolSAR and PolInSAR data. We can remark that there areseveral regions of the grid inside a parcel of forest. This meansthat a parcel does not constitute a homogeneous region in thesense of polarimetry and interferometry. It is probably due to achange in the canopy density that modifies the polarimetric orinterferometric properties of the region. Thus, the grid providesa region mapping that can be used to estimate covariance matrixand polarimetric and interferometric parameters. In the future,it will be interesting to compare the proposed technique of thispaper to more refined methods [17], [32] than with a boxcarthat take into account the edges between regions and that yieldsa precise estimation of the covariance matrix. Indeed, thesetechniques usually involve determination of the suitable sizeof the window, and it will be interesting to analyze if theproposed technique can be a relevant alternative for PolInSARdata estimation.

V. CONCLUSION

We have proposed a new algorithm for partitioning PolIn-SAR images into statistically homogeneous regions. It consistsof optimizing a polygonal grid and is based on a parameter-freecriterion which relies on stochastic complexity. The partitiontechnique contains nevertheless parameters in its optimizationsuch as the size of the initial grid. This algorithm leads toa partition of the image into regions which have homoge-neous PolInSAR characteristics, although the 36 parameters of

MORIO et al.: POLINSAR IMAGE PARTITION INTO STATISTICALLY HOMOGENEOUS REGIONS 3607

Fig. 11. (a) and (b) Partitions of a managed forest on a 1095 × 586 extract of PolInSAR and PolSAR Nezer P-band image, respectively. A parcel of forest doesnot constitute a statistically homogeneous region of the PolInSAR measurement since the canopy density layer and the tree height are variable for same kind oftrees. (c) Image of the managed forest taken on the lightened zone on the P-band Nezer image.

PolInSAR covariance matrix have to be estimated. The perfor-mances of this method have been demonstrated on syntheticand real PolInSAR images. It can thus provide an alternativeto simple boxcar estimation and to spatially adaptive filtering,notably for agricultural and forest area mapping. Monitoring ofcrops and planted forests thus constitutes the main applicationof this technique, whose practical advantages have been illus-trated on real PolInSAR images.

This work has many interesting perspectives. The partitionobtained with the proposed method of this paper can be a goodstarting point for a thematic classification task, as shown, forexample, in [33], where the authors start from a partition intostatistically homogeneous regions in order to obtain an oil slickdetection method. We thus intend to study the application of

the proposed method to classification and segmentation in thefuture. Further studies should also be made on interferometry-based partition of terrains with significant slope since, in thiscase, the phase is not constant in a partition region. Finally, acomparison with refined filters such as Lee’s filter [17] wouldbe useful in assessing the advantages and drawbacks of thedifferent methods for PolInSAR covariance matrix estimation.

APPENDIX

The matrices shown at the bottom of the page have been usedin Section III-B to synthesize regions that have a polarimetricand/or an interferometric contrast. The values of these differentmatrices have been chosen in order to simulate polarimetric

G1 =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

G2 =

1 0.1 0.4 + 0.4i 0 0 00.1 1 0 0 0 0

0.4− 0.4i 0 1 0 0 00 0 0 1 0.1 0.4 + 0.4i0 0 0 0.1 1 00 0 0 0.4− 0.4i 0 1

Φ1 =

1 0 0 0.7 + 0.1i 0 −0.06 + 0.07i0 1 0 0.03− 0.02i 0.6 + 0.1i 00 0 1 0 0 0.5 + 0.02i

0.7− 0.1i 0.03 + 0.02i 0 1 0 00 0.6− 0.1i 0 0 1 0

−0.06− 0.07i 0 0.5− 0.02i 0 0 1

Φ2 =

1 0 0 0.7 + 0.6i 0.02− 0.005i 00 1 0 0 0.1 + 0.1i 00 0 1 0 0 0.5− 0.5i

0.7− 0.6i 0 0 1 0 00.02 + 0.005i 0.1− 0.1i 0 0 1 0

0 0 0.5 + 0.5i 0 0 1

3608 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 11, NOVEMBER 2007

and interferometric contrasts between the six regions in Fig. 7.Indeed, the simulated covariance matrices are all different inthe PolInSAR case but not in the simulated PolSAR case sinceregions 1, 2, and 4, on the one hand, and regions 3, 5, and 6, onthe other hand, have the same covariance matrix. For regions 1,2, and 4, one has

Υpol =

1 0 00 1 00 0 1

and for regions 3, 5, and 6

Υpol =

1.33 0.2 0.8 + 0.8i0 1.01 0.04 + 0.04i

0.8− 0.8i 0.04− 0.04i 1.32

.

The regions 1, 2, and 4 and 3 and 6 are thus fused on thePolSAR data with the proposed algorithm. In the same way,regions 1, 3, 4, and 5 and 2 and 6 have the same covariancematrix on the simulated InSAR data in Fig. 7. For regions 1, 3,4, and 5, one has

Υinterf =(

1 0.7 + 0.1i0.7− 0.1i 1

)and for regions 2 and 6

Υinterf =(

1 0.7 + 0.6i0.7− 0.6i 1

).

The couple of regions 1 and 5, 3 and 4, and 2 and 6 are thusmerged on the InSAR data with the proposed algorithm.

ACKNOWLEDGMENT

The authors would like to thank ONERA/DépartementElectromagnétisme et Radar (DEMR)/Radar Imageur et Ex-périmentation (RIM) team for their important effort in provid-ing us calibrated SAR polarimetric and interferometric data,the Physics and Image Processing Group at the Fresnel Insti-tute, particularly G. Delyon, for their support on segmentationmethods, the Programme National de Télédétection Spatiale(PNTS), the Radar Image pour les Thématiques Agricoles etde Sol (RITAS) working group, and Direction Général del’Armement (DGA)/Service des Techniques et des Technolo-gies Communes (STTC) (French Department of Defense).

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[31] P. Dubois-Fernandez, O. du Plessis, D. le Coz, J. Dupas, B. Vaizan,X. Dupuis, H. Cantalloube, C. Coulombeix, C. Titin-Schnaider,P. Dreuillet, J. Boutry, J. Canny, L. Kaisersmertz, J. Peyret, P. Martineau,M. Chanteclerc, L. Pastore, and J. Bruyant, “The ONERA RAMSES SARsystem,” in Proc. IGARSS, Toronto, ON, Canada, 2002, pp. 1723–1725.

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Jérôme Morio graduated from the École NationaleSupérieure de Physique de Marseille, Marseille,France , in 2004. He is currently working toward thePh.D. degree at the French Aerospace Laboratory,Office National d’Etudes et Recherches Aérospa-tiales, (ONERA), Salon Cedex Air, France, and inthe Physics and Image Processing Group, Fresnel In-stitute, Unité Mixte de Recherche Centre National dela Recherche Scientifique 6133, Marseille, France.

His main research interests include polarimet-ric and interferometric imaging, segmentation tech-niques, and statistical image processing.

François Goudail received the Optical Engineeringdegree from the Institute of Optics, Orsay, Paris,France, and the Ph.D. degree in image processingfrom the Université of Aix-Marseille III, Marseille,France in 1997.

He has been an Assistant Professor with the Fres-nel Institute, Unité Mixte de Recherche Centre Na-tional de la Recherche Scientifique 6133, Marseille,France, and a Professor with the Institute of OpticsGraduate School, Palaiseau, France. His researchinterests include image processing in relation withthe physics of image formation.

Xavier Dupuis received the Engineer diploma from the Ecole Supérieure enSciences Informatiques, Nice, France, the Master’s degree from the Universityof Nice Sophia-Antipolis, Nice, in 1995, and the Ph.D. degree in imageprocessing from the University of Nice Sophia-Antipolis in 1999.

He has been with the Electromagnetism and Radar Department, FrenchAerospace Laboratory, Office National d’Etudes et Recherches Aérospatiales(ONERA), Salon, France, since 1999. He has been involved in the ONERASAR airborne platform, Radar Aéroporté Multi-spectral d’Etude des Signatures(RAMSES), developing the science applications.

Pascale C. Dubois-Fernandez received the Diplômed’Ingenieur degree from the Ecole NationaleSupérieure d’Ingénieur en Constructions Aéro-nautiques, Toulouse, France, in 1983, and the Mas-ter’s and Engineer’s degrees from the CaliforniaInstitute of Technology, Pasadena, in 1984 and 1986,respectively.

She was with the Radar Science and Technol-ogy Group, Jet Propulsion Laboratory, Pasadena,CA, where she stayed ten years, participating innumerous programs like Magellan, AIRborne SAR,

and Shuttle Imaging Radar-C. She then moved back to France, where sheworked on cartographic applications of satellite data. Since 2000, she has beenwith the Electromagnetism and Radar Department, Office National d’Etudeset Recherches Aérospatiales (ONERA), Salon, France, where she has beeninvolved in the ONERA SAR airborne platform, RAMSES, and developmentof the science applications.

Philippe Réfrégier graduated from the ÉcoleSupérieure de Physique et Chimie Industrielles de laVille de Paris, Paris, France, in 1984 and received thePh.D. degree in solid-state physics in 1987 from theUniversity of Paris, Orsay, France.

He is currently a Full Professor of signal process-ing with the Physics and Image Processing Group,Fresnel Institute, Unité Mixte de Recherche CentreNational de la Recherche Scientifique 6133, ÉcoleCentrale de Marseille, Domaine Universitaire deSaint-Jérôme, Marseille, France. His current re-

search interests include digital image processing and statistical optics. He isthe author/editor of several books, proceedings, and different refereed articlesin international scientific journals. He has also organized and participated indifferent international conferences.

Dr. Réfrégier was a member of the Laboratoire Central de Recherches ofThomson—CSF, Orsay from 1987 to 1994.