IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008 2185
Information Theory-Based Approach forContrast Analysis in Polarimetric and/or
Interferometric SAR ImagesJérôme Morio, Philippe Réfrégier, François Goudail, Pascale C. Dubois-Fernandez, and Xavier Dupuis
Abstract—We propose a new approach for evaluating the con-tribution of the different channels of polarimetric and interfer-ometric synthetic aperture radar (PolInSAR) images. For thatpurpose, we demonstrate that the Bhattacharyya distance betweenthe probability density functions of neighboring regions in theimage provides an efficient scalar contrast measure. We show thatthe analysis of this contrast measure allows one to precisely char-acterize the contribution of each channel for different system con-figurations, including intensity, polarimetric, and interferometricimages. We illustrate this approach using a real synthetic apertureradar image to compare several polarimetric system architectures.Since PolInSAR imaging configurations can correspond to com-plex and expensive systems, the proposed method can be helpfulin system imaging optimization.
Index Terms—Contrast definition, image processing, image sys-tem analysis, radar polarimetry, SAR interferometry, syntheticaperture radar (SAR).
I. INTRODUCTION
SYNTHETIC aperture radar (SAR) imaging covers a largespectrum of data from simple intensity images to polari-
metric and interferometric SAR (PolInSAR) images [1], [2] andis still a major domain of research for remote sensing applica-tions. Indeed, polarimetric and interferometric measurementsbring useful information for scene analysis that can reveal thenature of wave interaction [3], [4] and soil topography [5].However, since such systems can be expensive to build andoperate, it is important to analyze whether it is possible forsome particular applications to reduce the complexity of theimaging system without experiencing large performance loss.
Manuscript received July 23, 2007; revised January 19, 2007. The work ofJ. Morio was supported in part by the Office National d’Etudes et RecherchesAérospatiales and in part by the Provence-Alpes-Côte d’Azur region.
J. Morio is with Long-Term Design and Systems Integration Department-Spatial and Defence Systems Team, ONERA, 92322 Chatillon, France (e-mail:[email protected]).
P. Réfrégier is with the Physics and Image Processing Group, Institut Fresnel,Unité Mixte de Recherche, Centre National de la Recherche Scientifique, 6133École Centrale de Marseille, Domaine Universitaire de Saint-Jérôme, 13397Marseille, France (e-mail: [email protected]).
F. Goudail is with the Laboratoire Charles Fabry de l’Institut d’Optique,Centre National de la Recherche Scientifique, Université Paris-Sud, Cam-pus Polytechnique, 91127 Palaiseau, France (e-mail: [email protected]).
P. C. Dubois-Fernandez and X. Dupuis are with Electromagnetism and RadarDepartment-Radar Imaging and Experiment Team, ONERA, 13661 SalonCedex Air, France (e-mail: [email protected]).
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.2008.926115
Defining new system configurations has thus been the subjectof recent studies [6]–[9], and new radar architectures such asπ/4 mode [7] and circular polarization [8], [10] have beenproposed. These considerations show that it appears importantto be able to quantitatively characterize the difference betweenseveral radar system configurations. This analysis is, of course,dependent on the final purpose of the SAR image analysis.In this paper, we concentrate on the ability of different imagesystem configurations to discriminate different regions. We thuspropose to determine an efficient contrast measure betweenregions in the image and demonstrate that this contrast measurecan be helpful in the local or global comparison of differentimaging system configurations.
Contrast enhancement and noise cleaning have been a subjectof interest in the image processing community. Indeed, a lot oftechniques [11]–[17] have been proposed to improve the visualappearance of an image. Due to the large variety of noises thatcan be present in an image, there is no general definition ofstandard image quality that is useful for image enhancement.On video images, some techniques [18], [19] based on edgecrispening have notably been applied to scalar images to definea difference between distribution models. These techniquesforce the enhanced image to have desired first- and second-order moments but do not enable the estimation of a contrastparameter. Such an approach has not been applied to multicom-ponent images since defining a contrast measure between dif-ferent regions when the number of components of multichannelimages can vary is not an easy task. However, quantifying theamount of information [20] of statistical samples is the purposeof information theory, and different measures of proximitybetween random variables distributions have been proposed.More precisely, different theoretical results establish severalbounds for detection and discrimination performances. Thesetheories demonstrate that the Kullback–Leibler divergence andthe Chernoff distance are relevant measures of proximity be-tween probability density functions (pdfs). Moreover, it canbe shown [21] that these measures of proximity have simplephysical interpretations. Since the Chernoff distance can bedifficult to determine, it has been demonstrated that, in gen-eral, this measure can be satisfactorily approximated with theBhattacharyya distance, which is easier to analyze. Further-more, it has already been demonstrated that the Bhattacharyyadistance is a valuable contrast parameter between homogeneousregions for polarimetric images in [22].
In this paper, we propose to demonstrate that theBhattacharyya distance still constitutes a relevant contrastmeasure for PolInSAR images. It is also demonstrated that
2186 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
Bhattacharyya distance is efficient in comparing imaging sys-tems with different channel numbers. This new result allowsone to compare SAR systems that can acquire different infor-mation (intensity and/or polarimetry and/or interferometry) byanalyzing neighboring regions of the image. The proposedmethod is based on a partition of the most informative image,i.e., the full PolInSAR one, into homogeneous regions. Theanalysis of the information content of the multichannel imageswith different numbers of components is made on the basis ofthis initial partition. An interesting advantage of this approachis that the results only depend on the intrinsic informationcontent of the image and not on the application of particular anddifferent image processing algorithms to the partial PolInSARimages. These new results thus provide a rigorous approach thatcan be useful for analysis of PolInSAR imaging systems. InSection II, we first describe PolInSAR image models and showthat the Bhattacharyya distance is a valuable contrast measureon these images. We then propose in Section III a method basedon the contrast to analyze this information content of multi-channel images and apply it on real situations in Section IV.
II. CONTRAST MEASURE IN
MULTICOMPONENT SAR IMAGES
A. Multicomponent SAR Image Model
Multicomponent SAR images are generally considered asrealizations of multicomponent Gaussian circular processes,whose number of components n is equal to the number ofpolarimetric and interferometric channels of the system. Onewill consider in this paper SAR images whose number ofcomponents ranges between 1 and 6. The first case correspondsto standard intensity SAR images, whereas the latter case corre-sponds to PolInSAR images. The backscattered signal can thusbe represented by a complex n-dimensional Gaussian circularvector �K, with a null mean and an n × n covariance matrix Υand pdf, i.e.,
PΥ( �K) =1
πn det(Υ)exp(− �K†Υ−1 �K) (1)
where the symbol † denotes conjugate transpose, and det is thedeterminant. The covariance matrix Υ is Hermitian and can bewritten as Υ = 〈 �K �K†〉, where 〈 〉 denotes statistical averaging.
The general model above can describe different SAR imageconfigurations. The simplest case corresponds to intensity im-ages for which only the backscattered intensity of the signalin a given polarization is recorded. The backscattered signalis thus a Gaussian circular scalar S, with a null mean andwith the covariance matrix reduced to a scalar positive number.Polarimetric SAR (PolSAR) images yield information aboutterrain scattering, and they correspond to three-componentimages obtained with one emission antenna that alternativelyemits vertical V and horizontal H polarization. Let S
(j)XY be the
measured field on the Y polarization of the antenna numberj when the emitted field has polarization X . The PolSARbackscattered signal vector on the antenna number j can bewritten as
�k(j) =(S
(j)HH , S
(j)V V ,
√2S
(j)HV
)T
and its pdf will be written as P(j)k (�k(j)). Partial PolSAR
measurements simply consist of measuring a subset of(S(j)
HH , S(j)V V ,
√2S
(j)HV ). It can be interesting to minimize the
number of emitted polarizations, which is clearly the casewith, for example, (S(j)
HH ,√
2S(j)HV ). Intensity images can thus
correspond independently to S(j)HH , S
(j)V V , or S
(j)HV , although the
latter measurement is not often used. In that case, the pdf willbe written as P
(j)U (S(j)
U ), where U corresponds to the differentpolarizations HH , V V , or HV . Interferometric SAR (InSAR)images are obtained with two reception antennas shifted inspace that emit the same polarization. They provide informationabout the coherence and the phase of the zone lightened bythe radar. The backscattered signal is a 2-D Gaussian circularvector that can be written as
�I(U) =(S
(1)U , S
(2)U
)T
with U = HH , V V , or HV . The pdf of InSAR measurementswill be written as P
(U)I (�I(U)). Finally, PolInSAR images are
six-component images corresponding to the interferometricresponse between two polarimetric antennas. This last con-figuration leads to the most complete SAR mode, and thebackscattered signal vector is thus
�K = (S1HH , S1V V ,√
2S1HV , S2HH , S2V V ,√
2S2HV )T
=(
�k(1)
�k(2)
).
The pdf on intensity, PolSAR, and InSAR images are obtainedby determining the marginal pdfs of the PolInSAR image pdf.Indeed, one has, for example,
P(�k(1)
)=
∫P
(�k(1),�k(2)
)d�k(2) (2)
where P (�k(1),�k(2)) = P ( �K). One also gets
P(1)HH
(S
(1)HH
)=
∫P
(S
(1)HH ,K2,K3,K4,K5,K6
)× dK2 dK3 dK4 dK5 dK6 (3)
that can also be generalized for any component. Finally, forInSar images, one has, for example,
P(HH)I
(�I(HH)
)=
∫P
(S
(1)HH ,K2,K3, S
(2)HH ,K5,K6
)× dK2 dK3 dK5 dK6. (4)
B. Rigorous Invariant Contrast Parameters onMultidimensional Gaussian Fields
The basic concept behind the definition of invariant contrastparameters is that different statistical situations may lead tothe same performance when one applies an optimal algorithm,which is evaluated on a given performance criterion. A statisti-cal situation is characterized by the pdfs in the different regions.
MORIO et al.: INFORMATION THEORY-BASED APPROACH FOR CONTRAST ANALYSIS 2187
When one considers a given family for describing the pdf, suchas Gaussian for example, each pdf is then characterized by aset of parameters of finite dimension. If P functions of theseparameters are given such that the different statistical situationswith the same value for these functions necessarily lead to thesame optimal performance, then the values of these functionscan be considered as rigorous contrast parameters. Rigorouscontrast parameters are then a set of parameters that necessarilylead to the same performance when one applies an optimalalgorithm. Rigorous invariant contrast parameters are rigorouscontrast parameters that are obtained using statistical invarianceproperties of the pdf of the data.
In the case of two Gaussian pdfs of covariance matricesΥA and ΥB , the statistical situation for regions A and B isdescribed by the 2n2 real coefficients of the covariance matricesΥA and ΥB . Let DAB denote the diagonal matrix that isobtained with the eigenvalues of Υ−1/2
A ΥBΥ−1/2A when they
are ordered by decreasing values on the diagonal. Then, ithas been demonstrated [23], [24] that two regions A and Bwith covariance matrices ΥA and ΥB or with covariancematrices DAB and In×n (where In×n is the matrix identity)necessarily lead to the same performance when an optimalprocessing algorithm is applied. Consequently, to evaluate theperformance of a processing algorithm, one does not need totest all the statistical situations described by (ΥA,ΥB) but onlythe statistical situations that correspond to (DAB , In×n). Tocharacterize an optimal processing technique, one thus needs toexplore an n-dimensional space instead of a 2n2-dimensionalspace. Applied to PolInSAR images, this result allows one toexplore a parameter space of dimension 6 instead of 72. This isthus a very convenient property to evaluate the performance ofimage processing techniques for PolInSAR images.
Furthermore, the result above shows that there exist rigorousinvariant contrast parameters that can be chosen equal to theeigenvalues of Υ−1/2
A ΥBΥ−1/2A . Although very useful in evalu-
ating image processing techniques, this approach does not allowone to obtain a scalar contrast measure that is necessary forsystem evaluation. Indeed, vectorial rigorous invariant contrastparameters do not allow one to define an order relation and,thus, compare two couples of regions with different contrasts.One only defines equivalence classes that lead to the sameperformance. In the following section, it is thus proposed toanalyze how information theory-based measures can allow oneto get such a scalar contrast parameter.
C. Scalar Contrast Measure on MultidimensionalGaussian Fields
To define a convenient scalar contrast parameter, one appliesthe following principle that a couple of regions of fixed pixelnumbers with the same contrast parameter must necessarilylead to the same performance for any image processing task,such as detection, localization, or discrimination. Obviously,any arbitrary mathematical measure between the covariancematrices of a couple of regions will not necessarily satisfythe above principle. A simple method to evaluate a contrastmeasure is thus to represent the performance of discriminationalgorithm as a function of the contrast measure for the differentpossible statistical situations. One thus expects that the pointswhose coordinate are the values of the scalar contrast measure
and the performance are located on a narrow monotonic curve.Indeed, getting a narrow curve means that the difficulty of theprocessing task is correctly summarized by the value of thecontrast measure.
For such an analysis, one considers the following discrim-ination problem that consists of detecting whether or not tworegions with NA and NB pixels have gray levels distributedwith the same pdf. It is well known [25], [26] that a lower boundof the probability of error for such a discrimination problem canbe approximated by the Chernoff bound. This lower bound is adecreasing function of the Chernoff distance between the pdfsPΥA
( �K) and PΥB( �K), which is defined [25] by
C(so) = − log[∫ [
PΥA( �K)
]1−so[PΥB
( �K)]so
d �K
]where so is the value of s that maximizes C(s) for 0 ≤ s ≤ 1,and
∫·d �K stands for complex 6-D integration. It can be shown
that C(s) ≥ 0, and in the particular case of Gaussian fields [27],one gets
To get the Chernoff distance, one has to determine the valueof s that maximizes C(s), which can be a difficult task. Itis thus more convenient to consider an approximation of theChernoff distance with the Bhattacharyya distance [28], whichis equal to C(1/2). For the sake of clarity, the Bhattacharyyadistance B(ΥA,ΥB) between two pdfs PΥA
( �K) and PΥB( �K)
is defined in the general case by
B(ΥA,ΥB) = − log(∫ [
PΥA( �K)PΥB
( �K)] 1
2d �K
)where
∫·d �K stands for 6-D integration. In the case of
Gaussian circular fields with zero mean, the expression of theBhattacharyya distance is given by
B(ΥA,ΥB) = log
[det
(ΥA+ΥB
2
)det(ΥA)
12 det(ΥB)
12
].
It can be shown [25] that the Chernoff distance and, thus,the Bhattacharyya distance are obtained from Kullback–Leiblerdivergence, which is defined by
D(PB‖PA) =∫
log
[PΥA
( �K)
PΥB( �K)
]PΥA
( �K)d �K. (6)
This measure plays a central role in Stein’s Lemma [25], whichshows that the asymptotic behavior for large pixel numbers ofthe probability of detection or of false alarm is a simple functionof D(PΥA
‖PΥB). Since it is also important in information the-
ory [21], the Kullback–Leibler divergence is a standard measureof proximity between pdfs. Furthermore, for the distributionsPΥA
( �K) and PΥB( �K) to play symmetric roles, one considers,
in general, the symmetrized Kullback divergence, which isdefined by
K = D(PΥA‖PΥB
) + D(PΥB‖PΥA
). (7)
2188 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
In the case of Gaussian random vectors, one gets
K(ΥA,ΥB) = Tr(ΥAΥ−1
B
)+ Tr
(ΥBΥ−1
A
)− 2n
where Tr is the trace of a matrix.The symmetrized Kullback divergence and the
Bhattacharyya distance thus appear as two possible contrastmeasures on multidimensional fields [27]. One, however, ex-pects that a convenient contrast measure should decrease whenone considers a subset of the measurements. More precisely, thecontrast measure between two regions should be smaller if oneconsiders only PolSAR or InSAR measurements instead of fullPolInSAR data. Analogously, the contrast measure between tworegions should be smaller if one considers only one componentof PolSAR or InSAR measurements (i.e., an intensity measure-ment) instead of PolSAR or InSAR data. Since we have seenabove that the pdf of PolSAR or InSAR measures are marginalpdfs of PolInSAR data, the aforementioned requirements aresatisfied if the contrast measure using any marginal pdf issmaller than that obtained with the full pdf. We show in theAppendix that it is indeed the case for both the symmetrizedKullback divergence and the Bhattacharyya distance.
D. Simulations on Synthetic Images
In this section, we propose to determine with numericalsimulations which between the Bhattacharyya and the Kullbackdistances is the better contrast measure. As mentioned above,one considers the discrimination problem that consists of de-tecting whether or not two regions with NA and NB pixels havegray levels distributed with the same pdf. The algorithm thatoptimizes detection probability when the false alarm probabil-ity is fixed corresponds to the likelihood ratio test. However,this algorithm is not usually applicable since it assumed thatthe parameters of the pdf in both regions are known. Since itis not the case in practical situations, we implemented the ef-ficient approximation obtained with the generalized likelihoodratio test (GLRT), which consists of substituting the value ofthe parameter of the pdf in both regions by their maximumlikelihood estimations. More precisely, in the case of Gaussianpdfs, in the hypothesis that both regions A and B have the samepdf, one considers the maximum likelihood covariance matrixΥ̂A∪B in the region A ∪ B. Otherwise, in the hypothesis thatboth regions A and B have different pdfs, one considers thecovariance matrices Υ̂A and Υ̂B that correspond to the max-imum likelihood estimate in each region, respectively. In thecase of Gaussian pdfs, the maximum likelihood estimation ofthe covariance matrix [29] simply corresponds to the empiricalestimator on the considered region. The GLRT can thus bewritten as
NA log(det(Υ̂A)
)+ NB log
(det(Υ̂B)
)−(NA + NB) log
(det(Υ̂A∪B)
)> α
where α is a threshold. The detection performance is generallyanalyzed with the receiver operating characteristic (ROC) thatrepresents the curve of the probability of detection as a functionof the probability of false alarm when the detection threshold αis varied. Analyzing the performance with this figure of merit
is difficult since one does not obtain a scalar number and sincea standard approach consists of analyzing the area under thecurve (AUC) of the ROC. The AUC is a scalar number that isequal to 1 when one has a probability of detection equal to 1 fora null probability of false alarm. For a random choice betweenthe two hypotheses of the test, the AUC is equal to 1/2.
The obtained AUC for different pdf values (i.e., for dif-ferent values of ΥA and ΥB) of the regions A and B with,respectively, NA and NB pixels are shown in Fig. 1. For eachvalue of the couple (ΥA,ΥB), the AUC is estimated with 104
independent realizations. The existence of rigorous invariantcontrast parameters has been used to simplify numerical sim-ulations. More precisely, if DAB denotes the diagonal matrixthat is obtained with the eigenvalues of Υ−1/2
A ΥBΥ−1/2A , it has
been mentioned above that the couple of covariance matrices(ΥA,ΥB) and (DAB , In×n) must lead the same performancewith optimal algorithms. It is thus sufficient to analyze theAUC obtained with the GLRT applied to statistical samplesthat have been generated with covariance matrices of the form(DAB , In×n). This property greatly simplifies the numericalsimulations by drastically reducing the number of differentstatistical situations (i.e., of couple of covariance matrices) thathave to be tested.
In Fig. 1, the AUC is represented for three different valuesof the number of pixels (i.e., NA = NB = 6, 36, and 500) andwhen it is plotted as a function of the Bhattacharyya distance orof the symmetrized Kullback divergence. Thus, for each valueof the Bhattacharyya distance (respectively of the symmetrizedKullback divergence), the AUC has been determined for 104
different and randomly chosen statistical situation describedwith (DAB , In×n) that lead to this Bhattacharyya distance(respectively of the symmetrized Kullback divergence). In eachgraph, the different multidimensional field configurations thatcorrespond to partial PolSAR, PolSAR, and PolInSAR havealso been tested. One first remarks that the AUC values areless dispersed when they are represented as a function of theBhattacharyya distance than as a function of the symmetrizedKullback divergence. One also sees that the dispersion of thepoints decreases when NA = NB increases. This is easilyunderstandable since the GLRT is not an optimal algorithmbut becomes a better approximation of the optimal likelihoodratio test when the pixel number increases since it allows oneto better estimate the covariance matrices. A careful analysisof the different results allows one to see that the AUC isless dependent on the considered SAR configuration such aspartial PolSAR, PolSAR, and PolInSAR when it is representedas a function of the Bhattacharyya distance than when it isrepresented as a function of the symmetrized Kullback diver-gence. We have checked that similar results are obtained withintensity images and InSAR images. As discussed above, theseresults clearly demonstrate that the Bhattacharyya distanceis a better contrast measure than the symmetrized Kullbackdivergence.
In conclusion, it has been shown in this section that theBhattacharyya distance is an appropriate contrast parameter forthe comparison of two regions in PolInSAR images. It wasalready shown in [22] that the Bhattacharyya distance was anefficient contrast parameter for PolSAR images with a fixednumber of channels. However, to our knowledge, this is thefirst demonstration that it also provides a unique scalar contrast
MORIO et al.: INFORMATION THEORY-BASED APPROACH FOR CONTRAST ANALYSIS 2189
Fig. 1. AUC obtained with ROC curves estimated with 104 random experiments for different NA’s and NB’s and different multidimensional fields as a functionof the Bhattacharyya distance and the symmetrized Kullback divergence for different SAR systems.
parameter that allows one to compare several PolInSAR imagesthat can have different numbers of channels. This property issupported by two main results. First, it has been shown theo-retically that for a fixed image configuration, the Bhattacharyyadistance is a decreasing function of the number of channels.Second, it has been shown by simulations that it provides anapproximately bijective relation with a performance criterionthat characterizes region discrimination. It is shown in thefollowing sections that this new result is very convenient forsystem performance analysis.
III. APPLICATION TO POLINSAR SYSTEM ANALYSIS
A. Analysis of the Contrast Between Two Regions in aPolInSAR Image
The analysis of the variation of contrast when PolSAR orInSAR images are considered instead of the full PolInSARimage requires that the image be partitioned into homogeneousregions. Indeed, as discussed above, the contrast has to be eval-uated between regions, and to perform an efficient estimation
of the covariance matrix, each region has to be homogeneous.Different image partition techniques or different covariance ma-trix estimation methods can be considered [4], [17], [30]–[36].Since it is not the purpose of this paper to analyze imagepartitions, one will consider a simple technique of multidimen-sional image partition into statistically homogeneous regionsthat is based on the minimization of the stochastic complexity.A complete description of this algorithm is beyond the scopeof this paper, and we refer the reader to [37]–[40] for a morecomplete analysis of this approach. The main advantage ofthis technique for our present study is that it is based on theminimization of a criterion that does not contain parameters thathave to be tuned by the user. The contrast variation of a PolIn-SAR image when one considers only intensity, PolSAR, orInSAR configurations is thus simple to determine. Indeed, onecomputes the Bhattacharyya distances between the differentneighboring regions of a PolInSAR-partitioned image for thedifferent analyzed intensity, PolSAR, or InSAR configurations.
Fig. 2 presents three couples of regions of the partitionin homogeneous regions of the X-band Avignon image. Weestimate for these three couples of regions the Bhattacharyya
2190 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
Fig. 2. Different regions considered in Table I and the intensity, phase, and polarimetric image associated to them.
TABLE IBHATTACHARYYA DISTANCE BETWEEN THE DIFFERENT
REGIONS OF POLINSAR PARTITIONS IN FIG. 2
distance for different SAR systems in Table I to analyze whatthe radar image components are that enable us to determinetheir boundaries. Region A is a hedge of bush that is very high,and region B is composed of sunflowers. The contrast betweenthe intensity components of the image is very important sincethe hedge does not backscatter much of the incident wave,unlike the sunflower region. A PolSAR system is thus notnecessary to find the boundary between A and B since it doesnot increase the contrast significantly. Interferometry is alsonot a discriminating factor between regions A and B since thephase difference between the two regions is weak. A scalarintensity radar system is thus sufficient for the detection of theboundary between A and B.
Region C is a bare soil, and region D is a wheat field.One remarks that the contrast is weak when one considers theintensity components of the covariance matrices. In the sameway, the interferometric components are not efficient to dis-criminate C and D. However, the contrast on PolSAR systemis high because the bare soil and the wheat field have different
polarimetric properties. The incident waves undergo surfacescattering on the bare soil that does not depolarize the electricfields, whereas the incident waves undergo volume scatteringon the wheat field that depolarizes a lot of the backscatteredelectric fields. The PolInSAR system does not give much moreinformation for the discrimination of C and D than does thePolSAR system. Consequently, the best tradeoff in this situationbetween SAR system complexity and detection performances isobtained for PolSAR.
Region E is a low-height vegetation region, whereas regionF is composed of a tree hedge. The intensity and polarimetriccontrast is small and is not useful to detect E and F . However,there is a phase difference between these two regions, and itexplains why the contrast on InSAR and PolInSAR is impor-tant. An interferometric system is thus necessary to detect theboundary between E and F . The visual contrast observed onFig. 2 is confirmed by the results of Table I.
The same kind of analysis can be performed for any couple ofregions of the partition. For a global comparison of the differentSAR configurations, it can be useful to introduce a globalanalysis of the image. This is the purpose of the followingsection to analyze this situation.
B. Global Analysis in a PolInSAR Image
A global description of the contrast between neighboringregions of a PolInSAR image when one considers a partialPolInSAR image such as a PolSAR or an InSAR image can beeasily obtained. Let us illustrate the proposed methodology on a1671 × 648 ONERA RAMSES X-band PolInSAR image (slantrange and azimuth resolution are 0.9 and 0.94 m, respectively)acquired with full polarimetric and interferometric channels.The partition of this PolInSAR image leads to 2561 homo-geneous regions. The mean size of these regions is equal to
MORIO et al.: INFORMATION THEORY-BASED APPROACH FOR CONTRAST ANALYSIS 2191
Fig. 3. Bhattacharyya distance comparisons between PolSAR, InSAR HH , and PolInSAR. The red ellipses contain a couple of regions for which partialcomponents are sufficient to separate the two regions, and the green ellipses represent a couple of regions where PolInSAR is necessary.
422 pixels. With this partition, it is possible to determine theBhattacharyya distance between the neighboring regions fordifferent partial PolInSAR configurations. For each couple ofneighboring regions, its Bhattacharyya distance in the partialPolInSAR image is plotted as a function of its Bhattacharyyadistance in the full PolInSAR image. These results havebeen reported in Fig. 3(a) for the comparison between thePolSAR and the PolInSAR configurations and in Fig. 3(b)for the comparison between the InSAR HH and PolInSARconfigurations. In these figures, the results of the 14 438 couplesof neighboring regions have been plotted. As expected, thePolInSAR Bhattacharyya distance is always greater than thepartial PolInSAR Bhattacharyya distances. This is a directconsequence of the property discussed above—that consideringmarginal pdf instead of the full pdf leads to a decrease in theBhattacharyya distance. In Fig. 3(a), the red ellipse correspondsto couples of regions for which the contrast is mainly due topolarization properties since there is no significant loss betweenPolSAR and PolInSAR configurations. In Fig. 3(b), an analo-gous analysis is reported between InSAR HH and PolInSARconfigurations. Couples of regions for which there is an im-portant loss of contrast when one considers a partial PolInSARconfiguration instead of the full PolInSAR image correspondto the points in the green ellipses in Fig. 3(a) and (b).These results allow one to see that most of the regions thatare easily detected because of strong contrast in PolInSARwill also be detected in PolSAR images. However, this isnot the case with InSAR images. This conclusion could havebeen expected since this image contains fields and bare soilsthat have different polarimetric characteristics and for whichinterferometry does introduce relevant information.
From this graphical representation, several global scalar in-dicators that quantify the loss of contrast due to the partialPolInSAR configurations are extracted. Let us illustrate thatpossibility by analyzing a global figure of merit that can beobtained from the Bhattacharyya distance histograms of thedifferent partial PolInSAR images. For that purpose, for a givenpartial PolInSAR image, the histogram of the Bhattacharyyadistances between the neighboring regions can be compared tothe histogram of the Bhattacharyya distances of the full PolIn-SAR image. Such histograms are reported in Fig. 4. On thatparticular image, the histogram of the Bhattacharyya distanceon intensity HH is shifted toward the very low contrast values,whereas the histogram of the PolSAR image is less degraded.
Fig. 4. Histograms of the Bhattacharyya distance with different SAR systems.
Since PolInSAR is the most effective system, it is interestingto compare the other partial PolInSAR configurations to it.For that purpose, any standard measure of proximity betweenpdfs can be used. However, since the Kullback divergence hasa simple probabilistic interpretation [21], one will considerthat measure in the following. More precisely, let Hp(B) bethe histogram of the Bhattacharyya distances of the couplesof regions in the partial PolInSAR image, and let HF (B) bethe histogram of the Bhattacharyya distance of the couple ofregions in the full PolInSAR image. One considers the fol-lowing Kullback divergence to measure the proximity of thesehistograms:
K =∑B
HP (B) log[HP (B)HF (B)
].
In Section IV, we propose to show that this scalar indicatorprovides simple and relevant information for the comparisonof different partial PolSAR imagery configurations.
IV. APPLICATION TO THE COMPARISON OF
DIFFERENT POLINSAR ARCHITECTURES
In this section, the proposed methodology is illustrated usingthe 1671 × 648 ONERA RAMSES X-band PolInSAR imageconsidered above.
A. Application to PolInSAR Imagery Configurations
We have reported in Table II the Kullback divergences be-tween the PolInSAR histogram of Fig. 4 for different partial
2192 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
TABLE IIKULLBACK DIVERGENCE OF THE CONTRAST HISTOGRAM FOR DIFFERENT
SYSTEMS WITH THE POLINSAR BHATTACHARYYA DISTANCE HISTOGRAM
AND THE NUMBER OF REGIONS OF THEIR PARTITION
PolInSAR configurations. The PolInSAR configuration leads toa null Kullback divergence since it corresponds to the referencehistogram. In Table II, it is shown that the Kullback divergencetends to increase when the number of components in the Bhat-tacharyya distance decreases. In particular, the worst results areobtained from intensity images that have only one component.Furthermore, the results not only depend on the number ofantennas and the number of polarimetric and interferometricchannels of the imagery system but also on the consideredpolarization configurations. Indeed, in Table II, configurationswith the HV polarization couples appear to be less efficientthan HH or V V polarizations. Moreover, one also observesthat the couple of polarizations HH,V V is more valuable thanV V,HV and HH,HV . These observations correspond to thewell-known fact [41] that the cross polarization channel HVhas lower power than the copolarized channels HH or V V .
As an illustration, we have also reported in Table II thenumber of regions that are obtained when one partitions thedifferent partial PolInSAR images with the partition techniquementioned above [37], [38], and that is based on the mini-mization of the stochastic complexity. One clearly observes
a good correlation between the Kullback divergence and thenumber of regions of the different partitions. Indeed, low valuesof the Kullback divergence are related to a large number ofregions in the partition for the configuration, and conversely.This result illustrates the efficiency of the proposed methodbased on the Bhattacharyya distances between a couple ofregions to characterize the global contrast in partial PolInSARconfigurations. In Section IV-B, we propose to demonstratehow this methodology can be used in comparing differentimagery system configurations.
B. Illustration on Different PolSAR Architectures
We propose in this section to illustrate the approach proposedin this paper for the comparison of four different possiblePolSAR imagery systems. These imagery systems include fullpolarimetry, partial polarimetry with the HH and HV chan-nels, partial polarimetry with the V V and HV channels, and,finally, the π/4 mode proposed in [7]. The π/4 mode SARsystem consists of one emitting antenna with 45◦ polarizationand a second antenna that receives the backscattered signalin horizontal H and vertical V polarization. The cost of thiskind of system is approximately the same as for partial polari-metric ones, and a 2-D target vector is obtained by �Kπ/4 =(1/
√2)(S1HH + S1HV , S1V V + S1HV )T .
By using physical assumptions, it has been shown [7] thatone estimates a 3 × 3 covariance matrix from this 2 × 2 matrixthat is an approximation of the true 3 × 3 covariance matrix.Indeed, it has been shown [7] that if reflection symmetry isassumed, the 3 × 3 covariance matrix C1 can be estimated usingthe equation, shown at the bottom of the page, and by
J = �Kπ4
�K†π4
=(
j11 j12j∗12 j22
)where ∗ denotes complex conjugation, and R and I denote thereal and imaginary parts of a complex number, respectively.
In the case of rotation invariance assumption [7], anothercovariance matrix C2 can be estimated using the equationshown at the bottom of the page.
To compare the different SAR polarimetric systems de-scribed above, the full PolSAR image has been partitionedinto homogeneous regions. The Kullback divergences betweenthe full PolSAR histogram of Bhattacharyya distances and theconsidered partial PolSAR histogram of Bhattacharyya dis-tances have been determined and are presented in Table III.
MORIO et al.: INFORMATION THEORY-BASED APPROACH FOR CONTRAST ANALYSIS 2193
TABLE IIIKULLBACK DIVERGENCE WITH THE POLSAR BHATTACHARYYA
DISTANCE HISTOGRAM FOR DIFFERENT POLARIMETRIC SYSTEMS
One observes that the covariance matrices C1 and C2 leadto lower loss of contrast than do the other partial polarizationconfigurations. One can also remark that the π/4 mode aloneleads to the best contrast among the modes using 2 × 2covariance matrices.
Considering C1 and C2 instead of matrix J improves the con-trast values between regions, although C1, C2, and J containa priori the same information that is contained in the coef-ficients j11, j22, and j12. In fact, it is worth noting that itis possible to find transformations that increase the contrast,and thus, the Bhattacharyya distance between neighbor regions,when the dimension of the space is increased by using aparametric model. This is what happens when using the C1 orthe C2 matrix.
V. CONCLUSION
It has been demonstrated in this paper that the Bhattacharyyadistance is an appropriate contrast parameter for couples ofregions in PolInSAR images. To our knowledge, this is the firstdemonstration that the Bhattacharyya distance can provide aunique scalar contrast parameter that allows one to compareseveral PolInSAR images that can have different numbers ofchannels. On the one hand, it has been shown theoreticallythat for a fixed image configuration, the Bhattacharyya distanceis a decreasing function of the number of channels. On theother hand, it has been demonstrated that the performance ofregion discrimination is well described as a function of thisdistance. This result has been illustrated on the comparison ofseveral PolSAR architectures, including the π/4 configurationthat leads to good tradeoff between system cost and perfor-mance. In particular, it has been shown that 2-D diagrams canefficiently summarize the loss of contrast in a PolInSAR image.Furthermore, it has been shown that a unique scalar parametercan be used to simplify the analysis.
The choice of the polarimetric and/or interferometric compo-nents can be an important problem in SAR system optimization.Indeed, the tradeoff between the cost and the measurement ofthe most complete possible information can be a difficult taskfor some practical applications. We believe that the proposedresults constitute a progress in this direction.
This paper opens different perspectives. It has been shownthat the 2-D diagrams that plot the Bhattacharyya distance inthe full PolInSAR image versus that obtained in the degradedmodes is a valuable characterization tool. Further investigations
would be useful to extract more information from it. Theproposed approach could also be used to evaluate the impactof other parameters such as image resolution.
APPENDIX
Property A: If PA(x1, x2, . . . , xn, xn+1, . . . , xp) andPB(x1, x2, . . . , xn, xn+1, . . . , xp) define two multidimen-sional pdfs that depend on p variables x1, . . . , xp and iftheir marginal pdfs on their n first variables are denoted byQA(x1, x2, . . . , xn) and QB(x1, x2, . . . , xn), then
B(PA, PB) ≥ B(QA, QB)
where B(PA, PB) and B(QA, QB) are the Bhattacharyya dis-tances between the considered pdfs.
Proof: One has
B(PA, PB)= −log(∫ √
[PA(x1, . . . , xp)PB(x1, . . . , xp)]
× dx1, . . . , dxp
)B(QA, QB)= −log
(∫ √[QA(x1, . . . , xn) QB(x1, . . . , xn)]
× dx1, . . . , dxn
).
However, for U = A,B, one has
QU (x1, . . . , xn) =∫
PU (x1, . . . , xp)dxn+1, . . . , dxp.
Using the equations, shown at the bottom of the next page, onesees that
B(PA, PB) = −log(∫
G(x1, . . . , xn)dx1, . . . , dxn
)B(QA, QB) = −log
(∫H(x1, . . . , xn)dx1, . . . , dxn
).
Furthermore, the Cauchy Schwartz inequality between(PA(x1, . . . , xp))1/2 and (PB(x1, . . . , xp))1/2, considered asa function of xn+1, . . . , xp, allows one to see that
G(x1, . . . , xn) ≤ H(x1, . . . , xn)
and thus
B(PA, PB) ≥ B(QA, QB).
�Property B: If PA(x1, x2, . . . , xn, xn+1, . . . , xp) and
PB(x1, x2, . . . , xn, xn+1, . . . , xp) define two multidimen-sional pdfs that depend on p variables x1, . . . , xp and iftheir marginal pdfs on their n first variables are denoted byQA(x1, x2, . . . , xn) and QB(x1, x2, . . . , xn), then
D(PB‖PA) ≥ D(QB‖QA)
where D(PB‖PA) and D(QB‖QA) are the Kullback diver-gences between the considered pdfs.
2194 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
If one introduces the conditional Kullback divergenceDC(PB‖PA) between the conditional pdfs PA(xn+1, . . . , xp |x1, . . . , xn) and PB(xn+1, . . . , xp | x1, . . . , xn), i.e.,
Since QA(x1, . . . , xn) and DC(PB‖PA) are positive func-tions of (x1, . . . , xn), one has∫
QA(x1, . . . , xn)DC(PB‖PA)dx1, . . . , dxn ≥ 0
and thus
D(PB‖PA) ≥ D(QB‖QA). (14)
�
ACKNOWLEDGMENT
The authors would like to thank the team from DEMR-RIM,ONERA, for providing us with calibrated PolInSAR data andthe Physics and Image Processing Group, Institut Fresnel, fortheir support on statistical image processing techniques.
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Jérôme Morio received the Dipl.Ing. degree fromthe École Nationale Supérieure de Physique deMarseille, Marseille, France, in 2004 and the Ph.D.degree in polarimetric and interferometric SARimaging from the Université of Aix-Marseille PaulCézanne, Marseille, in 2007.
He is currently a Research Engineer with theDPRS-SSD, Office National d’Etudes et RecherchesAérospatiales, Chatillon, France, a French Aero-space Laboratory, where he is involved in studieson statistical techniques for the analysis of defense
complex systems. His main research interests include statistical techniques, po-larimetric and interferometric imaging, segmentation techniques, and complexdefense system analysis.
Philippe Réfrégier received the Dipl.Ing. degreefrom the École Supérieure de Physique et ChimieIndustrielles de la ville de Paris, Paris, France, in1984 and the Ph.D. degree in solid-state physics fromthe University of Paris Orsay, Paris, France, in 1987.
From 1987 to 1994, he was a member of theLaboratoire Central de Recherches of Thomson-CSF, Orsay, France. He is currently a Full Professorof signal processing with the Physics and ImageProcessing Group, Institut Fresnel, Unité Mixtede Recherche, Centre National de la Recherche
Scientifique, 6133 École Centrale de Marseille, Domaine Universitaire deSaint-Jérôme, Marseille, France. His research interests include digital imageprocessing and statistical optics. He is the author or editor of several books,proceedings, and different refereed articles in international scientific journals.
Dr. Réfrégier was an organizer and a participant in various internationalconferences.
François Goudail received the Dipl.Ing. degree inoptical engineering from the Institute of Optics,Orsay, France, and the Ph.D. degree in imageprocessing from the Université of Aix-Marseille III,Marseille, France, in 1997.
He has been an Assistant Professor with the In-stitut Fresnel, Marseille, France, and is currentlya Professor with the Laboratoire Charles Fabry del’Institut d’Optique, Centre National de la RechercheScientifique, Université Paris-Sud, Campus Poly-technique, Palaiseau, France. His research interests
include image processing in relation to the physics of image formation.
2196 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 8, AUGUST 2008
Pascale C. Dubois-Fernandez received theDipl.Ing. degree from the École Nationale Supér-ieure d’Ingénieur en Constructions Aéronautiques,Toulouse, France, in 1983 and the M.S. and Engine-er’s degrees from the California Institute of Techno-logy, Pasadena, in 1984 and 1986, respectively.
She joined the Radar Science and TechnologyGroup, Jet Propulsion Laboratory, Pasadena, whereshe stayed for ten years, participating in numerousprograms like Magellan, AIRSAR, and SIR-C. Shethen moved back to France, where she worked on
cartographic applications of satellite data. In 2000, she joined the Electro-magnetism and Radar Department, Office National d’Etudes et RecherchesAérospatiales (ONERA), where she has been involved in the ONERA SARairborne platform RAMSES, developing applications in science.
Xavier Dupuis received the M.S. degree and Ph.D.degree in image processing from the University ofNice Sophia-Antipolis, Nice, France, in 1995 and1999, respectively, and an Dipl.Ing. degree from theÉcole Supérieure en Sciences Informatiques, France.
In 1999, he joined the Electromagnetism andRadar Department, Office National d’Etudes etRecherches Aérospatiales (ONERA), Salon CedexAir, France, a French Aerospace Laboratory. He hasbeen involved in the ONERA SAR airborne platformRAMSES, developing applications in science.
