IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006 167

An Interferometric Coherence Optimization Methodin Radar Polarimetry for High-Resolution Imagery

Elise Colin, Cécile Titin-Schnaider, and Walid Tabbara, Member, IEEE

Abstract—This paper investigates to what extent a new inter-ferometric coherence optimization in radar polarimetry allowsthe separation of point scatterers located in the same resolutioncell according to their interferometric phases. An interferometriccoherence definition called the single-mechanism coherence isintroduced, and the corresponding optimization method is brieflydiscussed. This method was first validated theoretically when novolume decorrelation occurs. Then, it has been applied to simpletarget measurements acquired in an anechoic chamber, and toan X-band polarimetric and interferometric synthetic apertureradar image containing man-made targets. In both cases, thesingle-mechanism coherence optimization enables to resolve theinterferometric phases of several scattering centers inside thesame resolution cell.

Index Terms—Coherence optimization, interferometry,polarimetry.

I. INTRODUCTION

I NTERFEROMETRY is an efficient approach used to recon-struct the topography of a given region [1]. It is based on the

measurement of the phase difference between two paired pixelsof two complex synthetic aperture radar (SAR) images obtainedfrom the data collected by two antennas. The elevation of theterrain is proportional to this phase difference, known as inter-ferometric phase.

PolInSAR processing (combining polarimetric and interfer-ometric data) has been first investigated in forest, which is arandom media [2], [3]. One of the key ideas is that it is possibleto obtain interferograms from all possible linear combinationsof polarization states. One benefit that results from the use ofthese combinations is the possible improvement of the coher-ence level using the polarimetric information, and consequentlyan increase in the accuracy of the reconstructed elevation pro-files for each scatterer.

In this context, the generalized coherence has been introducedin [3] as a stochastic variable which measures the resemblancebetween the response of an electromagnetic mechanism at thefirst antenna, and another electromagnetic mechanism at thesecond antenna. The optimization problem has been introducedand solved to obtain the optimum scattering mechanism whichleads to the highest interferometric coherence, and therefore,to the best phase estimates. In this paper, it will be called thetwo-mechanism coherence (2MC) optimization. The efficiencyof this method to obtain accurate estimation of interferometric

Manuscript received April 13, 2005; revised June 13, 2005.E. Colin and C. Titin-Schnaider are with the Electromagnetism and Radar

Department (DEMR), French Aerospace Research Center (ONERA), Palaiseau91761, France (e-mail: elise.colin@onera.fr).

W. Tabbara is with the University of Paris VI, 75005 Paris, France.Digital Object Identifier 10.1109/TGRS.2005.859357

phases has been proven. For example in [4], the impact ofthis polarimetric coherence optimization technique on phaseunwrapping is assessed. The results show that the polarimetriccoherence optimization reduces the density of noise-inducedresidues and, as a consequence, alleviates the resulting bias inphase unwrapping.

However, there are other possible definitions of a generalizedcoherence for polarimetry, and therefore other possible methodsto perform an optimization. For example, the so-called polariza-tion subspace method (PSM) described in [5] is based on findinglocal maxima of the copolar or crosspolar coherence functions.Physically, the mechanisms must be represented as an ellipticpolarization transformation. The approach of the polarizationstate conformation (PSC) algorithm in [6] is very similar: it isbased on the knowledge of the polarimetric basis transformationalong with the polarization signatures of both interferometricimages.

Another aspect investigated in PolInSAR is the possibilityof separating the relative positions of points located inside thesame resolution cell. Thus, the ESPRIT algorithm has been usedin order to separate and retrieve the interferometric phase cen-ters of local dominant scatterers in a target [7]. In [8], the sameapproach is applied to estimate the height of buildings.

The aim of this paper is first to propose another coherenceoptimization, which performs the optimization of the coher-ence using the same complex unitary vector for both antennas.This coherence will be called the single-mechanism coherence(1MC). In the interferometric configuration, both antennas aswell as the incidence angles are very close, and it is sensible tocalculate a coherence for two equal mechanisms [9]. It is evenmore suitable in a single-pass acquisition since signals are notaffected by temporal decorrelation effects.

Using this 1MC coherence observable, a study about the pos-sibility of point scatterers discrimination inside a resolution cellusing single-baseline PolInSAR data was made in [10], and firstnumerical results for a typical urban scenario were provided.The analysis showed that two scatterers can be separated fromsingle baseline dual-polarization interferometric SAR data. Onthe other hand, three scatterers can be separated from fully po-larimetric interferometric data provided to make assumptionsabout the scattering properties of two scatterers.

In this paper, we investigate the use of a coherence optimiza-tion algorithm to separate several individual scattering centers.The results of the optimization obtained with this 1MC defi-nition are always mathematically suboptima of the 2MC opti-mization. However, the choice of the method (1MC coherenceoptimization or 2MC coherence optimization) depends on thecircumstances of the problem. In the context of high-resolutionand high-frequency imagery, ionospheric or orientation effects

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168 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006

need not be taken into account. The 1MC coherence optimiza-tion results can be considered as physically optima, since usingdifferent vectors introduces polarimetric decorrelation betweenthe two scattering mechanisms. Moreover, the 1MC coherenceis easier to interpret, due to the intrinsic assumption that bothimages are transformed in the same way.

Gomez-Danz and Quegan [11] have recently undertaken thestudy of layered targets in the context of polarimetric interfer-ometry, and the use of coherence optimizations was examinedfor layer information retrieval. The 2MC optimization resultswere poor unless a large number of independent looks werecombined. The 1MC optimization on the other hand was provedto be a more robust technique. Moreover, it was also able to re-trieve interferometric heights of two individual layers in a lay-ered model of a crop scattering application.

An important point is the physical interpretation of the op-timum polarization states in terms of scatterer characteristics.Section II presents the general interpretation of such a scat-tering mechanism . Then, the 1MC coherence is introduced, anda numerically efficient optimization method is proposed in Sec-tion III. In Section IV it is mathematically proven that for a res-olution cell containing few independent scattering centers, theoptimum polarization states lead to the estimation of the inter-ferometric phases of these point scatterers. Results and investi-gations obtained from the application of the optimization tech-niques to measures in an anechoic chamber and to a PolInSARimage are discussed in Section V. In the context of high-res-olution imagery, this technique is able to separate up to threedominant point scatterers in a resolution cell, and the optimiza-tion can also be efficiently used in order to obtain better phaseestimates on a SAR image of an urban area at X-band.

II. HOW TO CHOOSE AN ELECTROMAGNETIC MECHANISM ?

A. Data model

With a priori knowledge of the polarimetric behavior of thepoint scatterers located in the same resolution cell, we will dis-cuss how to choose a mechanism to select these points. Weuse the same data model as with the ESPRIT algorithm [7],[8] which can be applied for resolution cells containing severalpoint scatterers without taking into account the interactions be-tween scatterers or the volume effects. The signal acquiredby the first antenna in polarization xy consists of a sum ofdifferent elementary scattering contributions

(1)

and so does the electric field for the second antenna

(2)

is the total power (span) of the point , is the distancebetween point and the first antenna, and is the complex re-flectivity coefficient for polarization . For simplicity, we canassume that the complex diffusion vectorcan be normalized. The points are in the same res-olution cell defined by the first antenna, so that one may assume

. Moreover, let us write

(3)

These expressions can be rewritten in a matrix form as

(4)

is a complex matrix whose columns contain the normal-ized polarimetric diffusion vectors for each point, is the realcolumn vector of length containing the total power (span) ofeach point, and is the -diagonal matrix containing the in-terferometric phases

. . .(5)

where the 4 are the interferometric phases.

B. Choosing Knowing the Polarization States ofEach Points

The coherency matrices are expressed in terms of the matricesand without applying any averaging window

(6)

where indicates the matrix conjugate transposition. Let be amatrix such as , where is the identity matrix.

is the inverse of only in the particular case where is asquare matrix . We will discuss further the existence ofthe matrix .

Let be the th column vector of . Then

(7)

where is an length basis vector. The interferometric coher-ence defined with one mechanism is expressed in terms of thecoherency matrices as

(8)

Without considering any averaging to compute the coherencymatrices, the absolute value of this cross correlation alwaysequals 1. But in the context of statistical fluctuations, thiscoefficient becomes important since it can provide the measureof the loss of coherence and then of the phase noise of theinterferogram. Let us now consider the phase of the coherencewhich is obtained with the mechanism

(9)

is the th column vector of , i.e., . Moreover,is the th element of diagonal of , i.e., . We thus

obtain

(10)

The mechanism enables the calculation of the interferometricphase of the point because

(11)

Let us now discuss the existence of matrix .

COLIN et al.: INTERFEROMETRIC COHERENCE OPTIMIZATION METHOD IN RADAR POLARIMETRY 169

• If 3, is a 3 3 matrix, and is the conjugatetranspose of the inverse. exists if and only if the threecolumns of are linearly independent, which means thethree points have three different polarimetric responses. Inthe particular case where the polarimetric responses areorthogonal, is a unitary matrix and then S.

• If , finding is an overdetermined problem.exists except if two points have similar polarimetric re-sponses. Besides this particular case, it is important to notethat is not unique. For example, if is

(12)

then

(13)

where can be replaced by any value. Therefore, thedifferent interferometric phases can be estimated by themechanisms

(14)

• If , finding is an underestimated problem. ex-ists if and only if several points have the same polarimetricbehavior. This is a well-known result: we cannot separatemore than three points with only three independent polar-ization couples.

To conclude, points having polarimetric responses which aredifferent but not orthogonally represented by a matrix will beselected using mechanisms in the space orthogonal to the spacespanned by .

C. Nonuniqueness of the Matrix

Unfortunately, cannot be obtained from the knowledge ofthe measurements and without any other assumptions.Even worse, is not uniquely defined by and when nostatistical average is performed. If and are known, thetwo matrix relations given in four are equivalent to six complexequations. is a real vector of components, is acomplex matrices ( real unknowns). contains real un-knowns: the interferometric phases. We have then 12 realequations and unknowns.

The normalization of the column vectors of gives three ad-ditional equations. But even in this case the problem is generallyunderdetermined for .

The conclusion here is that separation of point scattererswill be impossible using only the two measurements and

. Therefore, any polarimetric and interferometric methodwill require a statistical analysis with the computation ofsecond-order parameters, like the coherency matrices. Besides,this inability to separate coherent centers has already beentreated in polarimetry [12] studying coherent decomposition ofthe scattering matrix.

III. 1MC COHERENCE OPTIMIZATION SCHEME

Since the measurements of and are not sufficient todetermine the matrices and , we need to compute second-order statistical parameters. One of these possible parameters isthe interferometric coherence.

A. General Scheme

If the same projection vector is chosen for both images, thegeneralized coherence can be written as follows:

(15)

In practice, matrices and are very similar becausethey are both coherency matrices of the target seen under veryclose incidence angles. Provided this assumption is valid, themean average on the denominator is very close to the geometricaverage. It is then possible to replace the definition of by thefollowing:

(16)

where matrix is defined as 2 [13]. Since2 , the modified co-herence is lower than the generalized coherence , and thenalways lies between 0 and 1. Moreover, the argument is not mod-ified by this definition change

(17)

will be called the 1MC coherence. The set of all complexcoherences can be plotted in the complex plane. It will be calledthe coherence set and written .

It is mathematically proved in [13] that

(18)

The set is called field of values of matrix or numericalrange of matrix , and it is defined by

(19)

Mathematically, the optimization of is then equiv-alent to calculating the numerical radius of the matrix

[13]. By definition, the numerical ra-dius of a matrix is .There is no analytical method to find the numerical radius ofa matrix . However, a numerical algorithm exists which isdescribed in [14]. A block diagram of which is shown in Fig. 1.

A local optimum is thus reached associated with a vector .The associated optimal vector for the coherence is given by

[13].Although the definition of the 1MC coherence is satisfactory,

we have to keep in mind that this iteration approach leads to alocal maximum and not to a global one.

B. First Method: Three Independent Optimizations

This optimization results in a local maximum associated withonly one mechanism . However the workspace is of dimen-sion 3. In order to obtain the other mechanisms and , one

170 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006

Fig. 1. Block diagram of an estimation method for the numerical radius of amatrix A.

can apply again the optimization procedure with another ini-tial angle. The choice of this initial angle is delicate if one doesnot want to try too many optimizations. If the coherence set canbe delimited by an angular sector, this angular sector is definedby its angles and , and we can compute these angles.After that, the algorithm will be started using for the initialangle, then , and finally the last initial angle will be chosenby . Otherwise, we can test the optimizationprocedure with other initial angles depending on the configura-tion. The procedure is summarized as follows.

• Compute the matrices , , ,and .

• Represent the numerical range of and choose three dif-ferent initial angles , , .

• Compute three times the algorithm of the numerical radiuson matrix using , , and as initial angles. You ob-tain three interferometric phases , , and three vec-tors , , and .

• Compute the three vectors ,, and . Then form the matrix

and compute . The nextsection shows that matrix contains the diffusion vectorsof each individual scatterer whose interferometric phasesare , , .

C. Second Method: Optimization in an Orthogonal Basis

After having found a first optimal coherence and the associ-ated vector , the second optimization can be achieved in thetwo-dimensional subspace orthogonal to . The procedure issummarized as follows.

• Compute the matrices , , ,and .

• Compute the algorithm of the numerical radius on matrixusing . You will find a first interferometric phaseand the corresponding vector . This vector is found by

an eigenvalue problem. It belongs to a basis change matrix.Let be this matrix whose first column is .

• Then consider the restriction of to the subspace orthog-onal to : compute . The restriction of isdefined by the submatrix of where the first line and thefirst column of are replaced by zeros. Let us call thisnew matrix.

• Compute the numerical radius on this matrix . You willfind two interferometric angles and , associated withtwo vectors and .

• Compute and . Vectors , , andare orthogonal. The corresponding mechanisms can be

computed according to , ,and . , , and are orthogonal, butsince is not a unitary matrix, the mechanisms , ,and are not necessarily orthogonal.

IV. SEPARATING UP TO THREE SCATTERING CENTERS

IN A RESOLUTION CELL USING THE

1MC COHERENCE OPTIMIZATION

This section describes when and why the previous optimiza-tion coherence methods can estimate correct interferometricphases of isolated point scatterers in a resolution cell.

A. Theoretical Approach With a Resolution Cell ContainingExactly Three Point Scatterers

With the same conventions as above we will first study thecase of three point scatterers. The coherency matrices are givenby

(20)If no statistical average is applied to compute the coherence,

all these matrices have a determinant equal to zero becauseis of rank 1. In the context of high-resolution imagery, our modelis a set of point-like scatterers measured over large syntheticapertures. The scatterers are fixed in location, so spatial decor-relation is not as important as in the presence of volume decor-relation for lower frequencies and natural targets. We can thenconsider a decorrelation caused by additive noise of SAR systemand introduce a statistical fluctuation on . Therefore, the co-herency matrices become

(21)

with

(22)

is the correlation matrix of the vector ; it is a real positivesymmetric matrix which can be considered of full rank 3. The1MC coherence is then written

(23)

COLIN et al.: INTERFEROMETRIC COHERENCE OPTIMIZATION METHOD IN RADAR POLARIMETRY 171

Let in this expression, and so that

(24)

Changing by , where is a scalar, does not change theresulting coherence. We can then suppose that is normalized.Moreover, we can quickly check that 1. Let us indicatethe scalar product of two vectors with

The first inequality is the Schwarz’s inequality, whereas thesecond one means that the geometrical mean is always less thanthe arithmetic mean.

Let us consider the numerical range of . Since is a di-agonal matrix, it is mathematically proved in [15] that its nu-merical range is the triangle whose vertices are the diagonal el-ements. We can check that this numerical range is included inthe coherence set. Let be . can be obtained thanks to

(25)

This proves that the three complex values belong to thecoherence set. Moreover this set is convex, so it contains theconvex hull whose vertices are ; in other words it contains thewhole numerical range of . A theoretical graphical exampleis given in Fig. 2. The numerical range of is represented inlight gray; an associated coherence set lies in dark gray. It con-tains the numerical range, and it is bounded by the unitarycircle.

We can immediately deduce from (25) that the coherence ismaximum and is equal to 1 at three points, which are obtainedwith , i.e., .

Theoretically, the first method (the general optimization withthree different initial angles) allows to find exactly the threemechanisms associated with coherencesequal to 1. Because this optimization method does not providea unique solutions and that the choice of initial parameters canbe difficult, this method is intended for measurements of simpletargets, for example in an anechoic chamber.

With the second method, the optimization in an orthogonalbasis and will not give exactly the second and third inter-ferometric phases but will find a good approximation of theseangles in a faster way. Moreover, it can be a better estimate ofthese interferometric angles when one takes into account otherstochastic components, as explained in the next section. Thedrawback is that it will be difficult to interpret physically

Fig. 2. 1MC coherence optimization results on simulated data. (Dark gray)The coherence set. (Light gray) The numerical range of D . (Black dots) Thecoherences in the optimal basis.

and , but we should have this method in reserve in case anaccurate estimation of phase is required rather than a physicalinterpretation of different polarimetric mechanisms.

B. When the Resolution Cell Contains Less Than ThreePoint Scatterers

When the resolution cell contains only one or two main inde-pendent scatterers, the optimization algorithm will still providethe estimation of their interferometric phase. The three interfer-ometric phases found in the optimum basis will be equal.

C. When the Resolution Cell Contains More Than ThreePoint Scatterers

When the resolution cell contains more than three scatteringcenters, the conclusions about the coherence set are no morevalid (see Fig. 3). However the following statements may beargued.

• All points with similar polarimetric responses will not beseparated. They will be regarded as a single point locatedon a weighted-average phase center. No polarimetricmethod is able to separate local scatterers with the samescattering mechanisms.

• If a point has a polarimetric diffusion vector which isorthogonal to all the other ones, it can be estimated by theoptimum scattering vector.

• In the other cases, points have polarimetric vectors whichare not orthogonal but are linearly independent. Prelimi-nary investigations show that the optimization procedureis not always able to separate them. However, the differentresults obtained in this configuration need further inves-tigations using the properties of the numerical range de-scribed in [15].

D. Stochastic Contributions Other Than

The optimization capabilities have been assessed assumingthat the stochastic components of the data model come fromthe amplitude vector . The study can be generalized assumingother sources of noise, as follows.

172 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006

Fig. 3. (Gray) Two different simulated coherence sets obtained with fourscattering centers. (Black dots) Theoretical interferometric phases of the pointscatterers.

Fig. 4. (Gray) Simulated coherence set obtained with statistical variations ofD. (Black stars) Interferometric phases obtained in the orthogonal optimal basis.(Black dots) Theoretical interferometric phases of the point scatterers.

• If the interferometric phase measurement contains a sto-chastic component, a numerical simulation of the effectsof a variation of on the optimization procedures is givenin Fig. 4. On this simulation, the interferometric phasesare supposed to be normally distributed:where 20 , and the coherency matrices are computedusing 25 samples.

• The optimization method of the 1MC coherence in an or-thogonal basis will extract a good approximation of themean interferometric phases. We see that averaging on sev-eral matrices leads to lower coherence observations.This trend involves some biases in the estimation of theinterferometric phases which do not exceed 5 in all oursimulations. The second optimization performed in an or-thogonal basis leads to better estimates than the first opti-mization method (three independent optimizations).

• Let us suppose that contains stochastic signal compo-nents. Many simulations have been conducted. Here wepresent a case where the stochastic components of thecolumn vectors of are Gaussian vectors with an additive

Fig. 5. (Light gray) Simulated coherence set obtained with statisticalvariations of S. (Black stars) Interferometric phases obtained in the orthogonaloptimal basis. (Black diamonds) Interferometric phases obtained in the generalindependent basis. (Black dots) Theoretical interferometric phases of the pointscatterers.

noise of relative standard deviation 0.15: ,. The matrices and are the same ones

as previously, and the average on has been made on25 samples. A resulting coherence set is shown in Fig. 5.Once again, the optimization procedure corresponds tothe extremities of this set, which are plotted in black. Thenumerical range of is drawn in dark gray. Same con-clusions as with statistical variations of can be drawn:the optimization performed in an orthogonal basis leads tobetter estimates than the three independent optimizations,as shown in Fig. 5.

V. EXPERIMENTAL RESULTS

The aim of the section is to verify the theoretical results of theprevious section, i.e., the interferometric phases calculated withthe optimal basis give convincing results for theseparation of interferometric phases of scattering centers. In thefollowing, results are presented using the second optimizationmethod (in an orthogonal basis).

A. In an Anechoic Chamber

Previous results have been obtained using data acquired inthe anechoic chamber CAMERA at the French Aerospace Re-search Center (ONERA). Full experiment description and re-sults can be found in [16]. The optimization methods have beentested on a simple metallic target. This target is considered asa collection of a few scattering centers located inside a singleresolution cell. In this context, both ESPRIT or the 1MC coher-ence optimization methods are able to retrieve the theoreticalinterferometric phases of bright scatters. The model developedin the previous section is well adapted in this particular case be-cause coherences are very high, and because there is no volumedecorrelation. It is then possible to consider statistical fluctua-tion coming from measurement noise. This is confirmed by thefact that the coherence sets have been found to be identical tothose predicted by this theoretical approach. Fig. 6 presents oneof these coherence sets and the coherences found in the opti-mization basis.

COLIN et al.: INTERFEROMETRIC COHERENCE OPTIMIZATION METHOD IN RADAR POLARIMETRY 173

Fig. 6. 1MC coherence optimization results on coherence set computed usingREAL EXPERIMENTAL data. The black dots are the coherence found in theoptimal polarization basis.

In this case, the coherence optimization makes it possible toseparate the different phase centers, but also to obtain polari-metric information contained in th e mechanism vector . Usingthe first optimization method we are able to find the estimationof matrix and therefore to have the polarimetricanswer of each selected bright point. Thus, the polarimetric re-sponse of the dihedral point expressed in the Pauli basis has beenfound to be close to , 1 which is the classicaldouble-bounce response. The polarimetric answer of an edge ofa metallic plate has been found to be close to 0.9where lies between 0 and 0.4, increasing with the incidenceangle. The polarimetric answer of a cylinder roof has been foundto be close to 0.9 0.3 . All these diffusion vectors are similarto those obtained with an image of the target constructed using abidimensional Fourier transform of the hologram. In this imagethe different bright points are located in different cells whosediffusion vectors can be extracted.

B. On Real SAR Data

In this section, the approach is validated on a PolInSARimage. We will see in this case how the 1MC coherence op-timization can give a better estimation of the interferometricphase of the target, which can be considered as one or twoscattering points in a noisy environment.

C. How to Compute the Coherence

The calculation of the interferometric coherence requires av-eraging over many samples from the same distribution. Compu-tationally, it is generally estimated using a boxcar filter wheresamples are in a window. The coherence estimation im-proves as increases. The drawback of this is that samples usedto compute the coherence can be finally chosen in heteroge-neous areas and do not belong to the same statistical population.

In order to avoid this problem, another idea is to use as astatistical population of the pixel, the complex backscatteringcoefficients obtained using a subaperture and a subset of thefrequency bandwidth of the emitted wave. These complex co-efficients are computed thanks to a multidimensional wavelettransform of the backscattering signal registered for a pixel lo-cated at position . This coefficient computed for this pixel

Fig. 7. Urban SAR image at X-band.

and centered around the frequency and the aspect angle iswritten . Detailed information about how to computethis coefficient can be found in [17]. The coherence is com-puted for each polarization using an average performed on allwavelet coefficients of the pixel, which belong to the same dis-tribution. Then the 1MC coherence optimization has been per-formed using this coherence value. After examination of coher-ence sets, it has been found that secondary maxima are often notsignificant. For this reason, we applied the second optimizationmethod in an orthogonal basis which has been found to providebetter estimates when working with important noise sources dif-ferent from strictly amplitude fluctuations.

D. Results on a X-Band Urban SAR Image

A SAR image (see Fig. 7) of 500 500 pixels has been se-lected in the airborne RAMSES X-band data on Bretigny. It con-tains industrial buildings, trees, a parking lot, and four canon-ical trihedrals used for calibration. Wavelet coefficientshave been calculated for ten angles , ten frequencies . Thatmeans that 100 looks are available to compute the coherence.The parameters of the wavelets are chosen in order to obtainspread in frequency and aspect angle domains representing both10% of the angular and frequency bandwidth.

The height corresponding to an interferometric phase canbe estimated by

2(26)

where is called the ambiguity height and depends on thegeometrical parameters of the radar

(27)

is the distance between the radar and the target, is thebaseline component perpendicular to the incidence direction,is the wavelength, and is the incidence angle.

The resulting digital elevation model obtained with the in-terferometric phase corresponding to the optimum scatteringmechanism is shown in Fig. 8.

For most of the pixels of the image, the three optimum vec-tors lead to similar elevations. That means that the target canbe considered as a single point in a noisy environment. Theoptimization procedure leads then to an accurate estimation ofthe interferometric phases of the target with a significant noise

174 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006

Fig. 8. Digital elevation model after 1MC coherence optimization.

Fig. 9. Zoom on edge of building 1. Digital elevation model different betweenthe first optimum mechanism and the second optimum mechanism.

amplitude. The standard deviations of the optimal elevation onhomogeneous areas is improved in comparison with the oneswhich would be obtained with a single polarization.

A more detailed study allows us to highlight some pointswhere the heights are not similar for the three optimal mech-anisms: for example the points located at the edge of building 1.The average process leads to high-resolution degradation. Thus,there are pixels which contain the building roof and the buildingshadow together. These correspond to two different interfero-metric phases: the one of the roof and the one of the ground. Wewill see now that these two different heights can be separated inthe optimal polarimetric basis.

Thedifferencebetweentheheightsobtainedusingtheoptimumvector and the heights obtained using the second optimumvector are thus shown in Fig. 9. We can distinguish two sorts ofpoints: some of them have a height computed using higherthan the height computed using . These heights are selectedand plotted in Fig. 10. The other ones have a height computedusing lower than the one computed using . These heightsare plotted in Fig. 11. We can guess that the optimizationprocedure has separated the heights of two different pointslocated in the same resolution cell and having two differentpolarization mechanisms: one point belonging to the groundand one point belonging to the roof, whose height is thenrelated to the top of the building, which has been evaluated

Fig. 10. Heights of points located on the edge of building 1. (Dark) Using ! .(Gray) Using ! .

Fig. 11. Heights of points located on the edge of building 1. (Dark) Using ! .(Gray) Using ! .

according to ground truth to 10 m. However, the remainingpoor accuracy of the measured heights is due to the geometricalconfiguration of the radar which lead to a very high ambiguityheight (around 200 m). With this configuration, an errorof 1 on the interferometric phase corresponds to an error of55 cm on the corresponding height.

VI. CONCLUSION

The choice of a polarimetric coherence optimization controlsthe resulting interferometric phases. The 2MC coherence op-timization produces the highest possible coherence level, butit does not necessarily generate more accurate interferometricphases. With a resolution cell consisting of several point scat-terers, the 1MC coherence optimization gives convincing re-sults: we have shown for the first time that this optimizationmakes it possible to determine the vertical position of differentphase centers associated with different mechanisms.

However, it is not possible with this technique to separate thecontributions of the same scattering mechanism distributed overdifferent heights. In this case, the method measures the meanphase center of the respective mechanism. Moreover, interactioneffects are not taken into account.

COLIN et al.: INTERFEROMETRIC COHERENCE OPTIMIZATION METHOD IN RADAR POLARIMETRY 175

This approach can be applied to real SAR data in order toretrieve both accurate interferometric phases of dominant scat-tering mechanisms and the polarization state of the main scat-terer. Applications to X-band SAR data show that results aremore accurate using 1MC coherence optimization, even whenonly one significant interferometric phase can be estimated.

Further studies should be devoted to a better statistical char-acterization of the noise effects on the coherence set as well ason the method. Moreover the method should be applied to otherSAR images of man-made targets with better height accuracy.

ACKNOWLEDGMENT

The authors are thankful to the anonymous reviewers, whohave greatly contributed by their remarks to improve the qualityof this paper, and to J. Delille for her careful reading of themanuscript.

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Elise Colin was born on November 15, 1979. Shereceived jointly the engineering degree from Supelecand the Diplôme d’Etudes Approfondies degree intheoretical physics (fields, particles, and matter) fromthe University of Orsay (Paris XI), Orsay, France,and the Ph.D.degree from the University of Paris VI,Paris, France, in 2002 and 2005, respectively.

She joined the Electromagnetic and Radar Divi-sion, French Aerospace Research Center (ONERA),Palaiseau, France, in 2005. She has been working onpolarimetric interferometry with airborne systems.

Her other research interests are SAR signal processing, time-frequency analysisby means wavelets, and vegetation modeling.

Cécile Titin-Schnaider was born in Paris, France,in 1946. She received the Ph.D. degree in nuclearphysics from the University of Orsay (Paris XI),Orsay, France, in 1974.

She joined SNECMA and worked on mathemat-ical methods in thermics. Then, she joined the FrenchAerospace Research Center (ONERA), Palaiseau,France, in 1975. Her research interests cover varioussubjects in electromagnetism and radar. Her currentinterest is polarimetric radar calibration and polari-metric images analysis.

Walid Tabbara (M’83) was born in Beirut, Lebanon,in 1946. He received the Doctorat ès Sciences (Ph.D.)degree from the University of Paris VI, Paris, France,in 1976.

He is currently a Full Professor at the Universityof Paris VI, since 1981. His research is done at theDépartement de Recherche en Electromagnétisme(DRE), Supélec. From 1987 to 1996, he acted asa Deputy Director of the Laboratoire des Signauxet Systémes (Supélec-CNRS). After working for along period in the field of inverse scattering in elec-

tromagnetics and acoustics, his present fields of interest are ElectromagneticCompatibility (EM, remote sensing, indoor propagation and high-power mi-crowaves (HPM). His present and passed researches has led to the publicationof 48 papers in international journals and to 158 presentations at internationaland national conferences. He is on the editorial boards of the Journal of Elec-tromagnetic Waves and Applications and the Journal of Computer Applicationsin Engineering Education.

Dr. Tabbara has organized and/or chaired sessions at national and interna-tional conferences. He was the French delegate to Commission B of URSI from1984 to 1990. He is a member of URSI and the Electromagnetics Academy.