Assessment and Estimation of the RVoG Model in ... - IEEE Xplore

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 52, NO. 6, JUNE 2014

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Assessment and Estimation of the RVoG Model in Polarimetric SAR Interferometry Carlos López-Martínez, Senior Member, IEEE, and Alberto Alonso-González, Student Member, IEEE

Abstract—This paper investigates the validity of the randomvolume-over-ground (RVoG) scattering model assumption for forest scattering on polarimetric interferometric synthetic aperture radar (PolInSAR) data. The model makes some assumptions about the data and the structure of coherency matrices, namely, the equality of the polarimetric covariance matrices and the affine equivalence of the contracted polarimetric interferometric covariance matrix with a Hermitian matrix. The proposed methodology is divided into two main steps. First, invertible affine transforms (ATs) are studied and proposed as a tool to operate with coherence regions. Based on this analysis, the concept of the trace matrix is introduced as its rank depends on the RVoG model assumption validity. Then, with the objective to consider the effects of speckle noise, we consider a maximum-likelihood (ML) framework, on the hypothesis of data distributed according to the complex Gaussian distribution. Hence, we define the ML estimator (MLE) of the PolInSAR coherency matrix according to the RVoG model assumption and the generalized likelihood ratio test of the model. The validity tests and the MLE are analyzed in terms of simulated and real PolInSAR data, considering P-band and L-band data over tropical and boreal forests. Index Terms—Affine transform (AT), forestry, maximum likelihood (ML), polarimetric synthetic aperture radar interferometry (PolInSAR), SAR interferometry, SAR polarimetry.

I. I NTRODUCTION

F

ORESTS study is a crucial field of research, as well as a generator of economical activity in a wide range of applications. Considering that forest cover approximately 30% of the Earth’s solid surface [1], any attempt to provide forest information needs to consider remote sensing techniques based on airborne or spaceborne platforms. Due to the capability of microwaves to penetrate vegetation, synthetic aperture radar (SAR) represents an established technology to provide quantitative information about forests. Polarimetric SAR interferometry (PolInSAR) has been demonstrated to be a convenient remote sensing technique for the study of the 3-D forest structure. PolInSAR allows, for instance, to estimate the forest height or to estimate the complete

Manuscript received January 17, 2012; revised October 24, 2012, March 15, 2013, and May 14, 2013; accepted June 11, 2013. Date of publication July 11, 2013; date of current version March 3, 2014. This work was supported in part by the Spanish Ministry of Science and Innovation (MICINN) under Project MUSEO TEC2011-28201-C02-01, by the Commission for Universities and Research of the Department of Innovations, Universities, and Enterprises of the Autonomous Government of Catalonia, and by the European Social Fund. The authors are with the Remote Sensing Laboratory, Department of Signal Theory and Communications, Technical University of Catalonia (UPC), 08034 Barcelona, Spain. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2013.2269614

3-D scattering profile, considering tomographic analysis. The potential of PolInSAR to perform quantitative remote sensing in forest areas has been already demonstrated, experimentally, based on airborne SAR data at a regional scale. Nevertheless, with the objective to perform global forest mapping, spaceborne PolInSAR measurements are necessary. Currently, the TanDEM-X mission presents a PolInSAR experimental mode at X-band. There exist also future mission concepts under investigation, such as Biomass (P-band) and Tandem-L (L-band), which focus on the study of forests by means of PolInSAR data. The accurate quantitative estimation of forest parameters needs to be based on the inversion of coherent scattering models. The use of these, which is focused on a simplification of the forest structure, makes it possible to fit the number of parameters to the number of radar observables. In microwave forest scattering, a useful model, which is named random volume over ground (RVoG), was introduced in [2]–[4], which considers PolInSAR data. It was employed later on for the quantitative estimation of forest parameters, e.g., forest height [5]–[7], based on the linear behavior of the complex interferometric coherence as a function of polarimetry. Despite the fact that forest height inversion has been demonstrated for different types of forests and for different frequencies [8], [9], the question regarding the validity of the RVoG model assumption still remains. The estimation of the forest height has been employed as an indirect demonstration of this validity. Nevertheless, the lack of a direct mechanism to prove the validity of this assumption generates uncertainty about the applicability of the RVoG model and any parameter derived from its use. The objective of this paper is to answer the previous question by a direct analysis of the validity of the RVoG hypothesis on PolInSAR data. Thus, this paper is structured as follows. Section II introduces PolInSAR. Section III focuses on the analysis of the coherence linearity (CL) assumed by the RVoG model. In order to consider speckle, Section IV makes use of a maximum-likelihood (ML) framework to derive the ML estimator (MLE) of the coherency matrix according to the RVoG model and the generalized likelihood ratio test (GLRT). The tests, as well as the estimator, are analyzed on simulated and real PolInSAR data in Section V. The conclusions are presented in Section VI.

II. P OL I N SAR A. PolInSAR Data Description A monostatic polarimetric interferometer acquires polarimetric data from different spatial positions. Assuming a Pauli

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formalism [10] under the backscattering alignment assumption, the polarimetric target vector of each acquisition is

 m (n) = π Γ

m(m−1) 2

m

Γ(n − i + 1).

(7)

i=1

1 k = √ [Shh + Svv , Shh − Svv , 2Shv ]T 2

(1)

where T represents the vector transposition, and the SAR images Spq represent the scattering coefficients, considering p, q ∈ {h, v}, i.e., linear horizontal and vertical polarizations, respectively. Considering (1), the PolInSAR extension of the target vector, assuming two acquisitions ki for i = 1, 2, is   k1 k6 = . (2) k2 For distributed scatterers, (2) is a random vector due to the complexity of the scattering process. In the case of stationary data, (2) is characterized by a 6-D zero-mean complex Gaussian probability density function (pdf) [11] pk6 (k6 ) =

1 π m |T|

   etr −T−1 k6 kH 6

(3)

where, in this case m = 6, etr(·) indicates the exponential of the matrix trace, H is the transpose conjugate, | · | is the matrix determinant, −1 is the matrix inversion operator, and T corresponds to the coherency matrix     T11 Ω12 H T = E k6 k6 = (4) ΩH T22 12 where E{·} is the expectation operator. Matrices T11 and T22 represent the polarimetric coherency matrices of each acquisition, and Ω12 is the polarimetric interferometric coherency matrix. The analysis of the properties of T, particularly those assumed by the RVoG model, is better performed considering T to be decomposed as [5], [12]  1/2     1/2  12 I Ω 0 0 T11 T11 T= (5) 1/2 1/2 H I Ω 0 T22 0 T22 12  12 = T where Ω is a contraction matrix, and I 11 Ω12 T22 is the identity matrix. The unique positive semi-definite square roots of T11 and T22 exist since both matrices are positive semi-definite; therefore, the inverses of these unique square root matrices exist as well. Due to the stochastic nature of SAR data, i.e., due to speckle noise, T needs to be estimated. Under the assumption of statistical ergodicity and local stationarity, the expectation in (4) may be estimated by means of a spatial averaging, i.e., −1/2

−1/2

n

1 = k6, q kH Z = k6 kH 6 6, q . n q=1

(6)

This spatial averaging, which is referred to as a multilook filtering, corresponds to the MLE of T (see Appendix B). Considering (3), Z follows a Wishart pdf [11], [13], [14] pZ (Z) =

nmn |Z|n−m etr(−nT−1 Z) n  |T| Γm (n)

The interferometric complex coherence is defined as [15] ρ = |ρ|ejφx =

E {S1 S2∗ }

(8)

E {|S1 |2 } E {|S2 |2 }

considering two SAR images S1 and S2 . The importance of this observable lies on its sensitivity to the phase scattering center location in the vertical direction. Considering two different unitary vectors w1 and w2 , which represent polarization states, one may project the original Pauli vectors onto them, i.e., Si = wiH kp, i for i = 1, 2. Therefore, (8) may consider the polarimetric dependence as [5] ρ(w1 , w2 ) =

w1H Ω12 w2 w1H T11 w1 · w2H T22 w2

.

(9)

Equation (9) represents the general form of ρ, in which different polarization states are considered for the interferometric SAR images. Sometimes, it might be more convenient to consider equal polarization states, i.e., w = w1 = w2 . This restricted form, which is usually referred to as the one scattering mechanism, equal scattering mechanism, or single scattering mechanism coherence [16], [17], limits its variability in the complex plane. Physically, this constraint is necessary to avoid the interferometric phase to be biased by polarimetric contributions. B. Coherent Modeling and Physical Information Estimation Equation (9) may be decomposed into different decorrelation contributions [18]. In the case of forests, assuming no decorrelation due to the system or temporal effects and assuming the elimination of the noncommon parts of the range spectra, decorrelation is due to the different projections of the vertical component of the scatterer under observation into the two interferometric data sets. Under this assumption, ρ, considering a vertical distribution of scatterers F (z), may be obtained as  hv jkz z dz jkz z0 0 F (z)e (10) ρ=e  hv 0 F (z)dz where kz is the effective vertical interferometric wavenumber. Equation (10) depends on a reference height, which is represented by z0 , and on the height of the vertical layer of scatterers hv with respect to z0 . F (z) contains all the additional information to characterize the volume. In the case of the RVoG model [7], [19], it accounts for the wave extinction process at the natural volume and the effects of the surface beneath the vegetation −2σ

F (z) = mv e cos θ0 (hv −z) + mg (w)δ(z)

(11)

where w indicates the polarization state vector, and z is in the range of [0, hv ]. In (11), σ is the mean wave extinction coefficient, and delta function δ(z) represents the scattering center accounting for the direct ground and the possible dihedral scattering. Parameters mv and mg are the volume and

LÓPEZ-MARTÍNEZ AND ALONSO-GONZÁLEZ: ASSESSMENT AND ESTIMATION OF RVOG MODEL IN POLINSAR

the ground scattering amplitudes at the phase scattering center, respectively. By inserting (11) into (10), we have ρ(w) = ejkz z0

ρvol + μ(w) 1 + μ(w)

(12)

where μ(w) represents the ground-to-volume ratio, i.e., μ(w) =

2σ cosθ0

(e2σhv / cos θ0

w H Tg w − 1) wH Tv w

(13)

where Tg is the coherency matrix for the reflection symmetric ground scattering contribution, and Tv is the diagonal coherency matrix for the volume scattering. Finally, ρvol corresponds to the volume decorrelation in the absence of the ground component. As given by (12), there is a dependence of the decorrelation on the polarization, which is represented by w. The RVoG predicts a variation of the decorrelation with polarization through μ(w), i.e., this variation is induced by the surface beneath the vegetation. Hence, (12) is   μ(w) (1 − ρvol ) . (14) ρ(w) = ejφ0 ρvol + 1 + μ(w) The interferometric complex coherence ρ may take any value in the closed disk with unit radius of the complex plane as a function of w. Nevertheless, the variation of ρ(w) with respect to w is normally limited to a particular area, which is known as the coherence region. In the case of the RVoG model, the variation of (14) with respect to w is restricted to a line. Therefore, the RVoG is usually referred to as the line model. The position and the line length in the complex plane depend on the different parameters of the RVoG describing the forest and on the characteristics of the SAR system and the interferometric geometry. Thus, this line may be considered for the quantitative estimation of physical forest parameters. For real data, variations of the line model may be found for temporal decorrelation or oriented volume effects. On the former, a different line from the original one is predicted [7], whereas in the latter, the model does not yield a line [20]. At this point, it is worth to relate (14) with (9), for w = w1 = w2 , to observe how the RVoG model assumes certain properties on T. The following list details them. 1) The derivation of (14) from (10) is based on the assumption of T11 = T22 [7], which is referred to as the polarimetric stationarity (PS) hypothesis [21], [22]. Under the previous assumption, (9) reduces to ρ(w) =

wH Ω12 w . wH T11 w

(15)

 12 = UΛUH Ω

(16)

where U is a unitary matrix, and Λ is a diagonal matrix.  12 being a normal matrix, i.e., This is also equivalent to Ω H Ω  H  Ω 12 12 = Ω12 Ω12 .

This property follows from the assumptions that w1 = w2 and T11 = T22 . A different demonstration of this property is done in Section III. The condition expressed in (16) is referred to as the interferometrically PS (IPS) hypothesis [21], [22]. 3) The previous mathematical restrictions do not lead at all to the most relevant property of the RVoG, i.e., the coherence linearity (CL) as a function of w. III. C OHERENCE L INEARITY IN THE RVO G M ODEL When confronted to real PolInSAR data, the coherence region is not a line but a 2-D region in the case of forest areas. During the last decade, the study of the coherence region has been addressed from different perspectives. The variability of ρ was first considered in [19]. In [23], the authors proposed the extraction of the coherence region boundary, whereas in [24] and [25], they analyzed the boundary shape. Neumann et al. investigated the relation of the coherence region with the numerical range or field of values [26] after being introduced by Tabb [24] and studied by Colin et al. [16], its density in the complex plane [27], and also PolInSAR data classification based on the coherence region shape [28]. Complete studies are presented in [29] and [21]. All these works indicate that the coherence region boundary may depend on the parameters of the coherent scattering model; therefore, it may be considered for the extraction of physical quantities. Nevertheless, despite being less studied, it is also shown that the shape itself depends on speckle noise, i.e., it depends on the data statistics [30]. In order to conclude that physical information may be extracted from the coherence region shape, it would be first necessary to decouple the effects of the scattering model from the effects of speckle noise over the coherence region shape. This coupling makes it difficult, for instance, to assess the validity of the RVoG assumption on PolInSAR data, since it is not possible to determine that the absence of CL is due to speckle and filtering or due to inaccuracies of the coherent scattering model itself. Therefore, it is necessary to determine how speckle and filtering affect the estimation of the coherence region. A. Numerical Range As demonstrated in [16] and [21], the exploration of the complex coherence can be simplified based on the concept of numerical range. The numerical range of m × m complex matrix A is defined as the set of complex numbers C that W (A) = {xH Ax, x ∈ Cm , xH x = 1}.

2) The RVoG scattering model assumes also the polarimetric and interferometric coherency matrix to be decomposed as

(17)

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(18)

The use of (18) in PolInSAR needs a simple reformulation of (15). Considering the modified polarization states v = 1/2 (T11 w/ wH T11 w), where vH v = 1, ρ(w) can be repre 12 , as follows: sented, in terms of a numerical range of Ω  12 ) = {vH Ω  12 v, v ∈ C3 , vH v = 1}. W (Ω

(19)

The importance of the numerical range is that the study of the properties of matrix A can be performed in terms of its

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numerical range W (A) as these may induce particular geometries on W (A). In an opposite way, the geometry of W (A) might also impose certain properties on A. Previously, the assumptions made by the RVoG model on the interferometric coherence ρ(w) and on the coherency matrix T were considered. It is worth to analyze them from the point  12 ), i.e., based on the of view of the numerical range W (Ω following theorems [31]. Theorem 1: If W (A) is a line segment, then A is a normal matrix. Theorem 2: If A is a normal matrix with eigenvalues λ1 , . . . , λm , then W (A) is the convex hull of the points in the complex plane corresponding to the eigenvalues. In terms of the RVoG model, Theorem 1 demonstrates that  12 to be a normal matrix. Nevertheless, as CL imposes Ω  12 to be a normal matrix Theorem 2 indicates, to impose Ω does not assure CL. The previous theorems indicate that, when considering the RVoG model, the properties of CL and IPS are closely related in the sense that to impose directly CL results into IPS. In the case of the RVoG, as given by (14), ρ(w) is a line in  12 to be, at least, a the complex plane, imposing the matrix Ω normal matrix. Nevertheless, it is necessary to investigate and  12 and on T, to determine which properties CL induces on Ω beyond the indication that coherence describes a line.

AT and IAT of a matrix.

Given complex matrix A, the following Hermitian matrices may be defined: H1 =

A + AH 2

H2 =

A − AH 2j

(21)

such that A = H1 + jH2 . Then, the AT of matrix A is simply defined as [35]–[37] τabc (A) = aH1 + jbH2 + cIm .

(22)

Fig. 1 shows a scheme of the AT and its IAT when considering matrices. Based on (21), the following theorem applies. Theorem 4: The numerical range W (τabc (A)) of matrix τabc (A) is given by τabc (W (A)). Considering vector x ∈ C3 such that xH x = 1, we have W (τabc (A)) = W (aH1 + jbH2 + cIm )

B. Affine Transforms  12 to be a normal matrix, Theorem 1 shows that CL induces Ω but this is not a sufficient condition to assure CL. One of the  12 [see forms to exploit CL have been unitary transforms of Ω (16)]. Nevertheless, these transforms present serious limitations when considering numerical ranges as these are invariant under unitary transforms, i.e., W (UAUH ) = W (A) [32]. We are concerned with the CL property induced by the RVoG  12 , it is important model. Before seeing how to relate it with Ω to recall the following theorem [31], [33], [34]. Theorem 3: If A is an Hermitian matrix which eigenvalues are λm ≥ · · · ≥ λ2 ≥ λ1 , W (A) is a closed interval of the real axis, which is limited by the extreme eigenvalues of A. This follows directly from Theorem 2. Despite the importance of this theorem, as it relates CL with the Hermitian property of the matrix under analysis, its practical use in terms of the RVoG is seriously limited as CL should be restricted to  12 is to ρ(w) ∈ R. The way to exploit Theorem 3 in terms of Ω consider affine transforms (ATs) in the complex plane. An AT of C is map τabc : C → C with parameters a, b, and c ∈ C such that ρ ∈ C is transformed to τabc (ρ) = aR{ρ} + jbI{ρ} + c

Fig. 1.

(20)

where ab = 0, and ab−1 is not purely imaginary, i.e., the −1 = τa b c with coefficients a , b , and inverse AT (IAT) τabc  c ∈ C exists, and R{·} and I{·} are the real and imaginary parts, respectively [31], [35]–[37]. Among its properties, ATs preserve the collinearity relation between points and ratios of vectors along a line. Hence, this transformation transforms lines into lines in the complex plane.

= xH (aH1 + jbH2 + cIm )x = axH H1 x + jbxH H2 x + c = τabc (W (A)) .

(23)

Similarly, one may demonstrate that W (A) = xH H1 x + jxH H2 x

(24)

where the real and imaginary parts of W (A) are obtained as R {W (A)} = xH H1 x

(25)

I {W (A)} = x H2 x.

(26)

H

The previous theorem relates different complex matrices by considering their numerical ranges through the use of ATs. Thus, considering Theorems 3 and 4, the following result applies [31], [33], [34]. Theorem 5: The numerical range W (A) of matrix A is a line segment in the complex plane if and only if matrix A is normal and has collinear eigenvalues; or equivalently, if and only if A = aH + cIm for some a, c ∈ C and an Hermitian matrix H. The previous theorem results are important in the context of PolInSAR as it allows to relate the CL property induced by the RVoG model with certain properties of the contraction matrix  12 . The first part of Theorem 5, regarding normality, was Ω already considered in [26] but restricted to reflection symmetric scattering. Finally, it must be noted that Theorem 5 does not consider the length of the line; therefore, it may include the case of a matrix representing a single point in the complex plane.

LÓPEZ-MARTÍNEZ AND ALONSO-GONZÁLEZ: ASSESSMENT AND ESTIMATION OF RVOG MODEL IN POLINSAR

C. Affine Transforms Applied to PolInSAR Data ATs make it possible to study the coherence region of a matrix. In particular, one may conclude that the coherence region of a matrix is a line if this matrix is affine equivalent with a Hermitian matrix. In the case of PolInSAR data, the complex coherence (15) may be obtained as the numerical range of the  12 [see (19)]. Thus, since the most relevant contraction matrix Ω property of the RVoG is to impose the complex coherence to be  12 . It a line, one may analyze this property in terms of AT of Ω is important to note that this paper will also need to consider the presence of speckle noise. In the presence of speckle, the IPS and the CL restrictions do not apply. Some previous works have addressed the imposition of these restrictions. The exploitation of the IPS has been reported in [22] and [38], whereas a restricted application of the CL has been employed in [30]. Despite the fact that speckle effects have been mitigated, they have not offered a clear description of the speckle noise effects on the coherence region shape.  12 , under Based on Theorem 5, the numerical range of Ω the hypothesis of the RVoG model, shall be a line in the complex plane if and only if this matrix is affine equivalent with an Hermitian matrix, considering the AT definition in Section III-B. Considering Fig. 1, this can be expressed as  12 = a Hτ + jb Hτ + c I3 Ω 1 2

with Hτ2 = 0

(27)

where Hτ2

 τH τ − Ω Ω 12 12 = 2j (aH1 + jbH2 + cI3 ) − (aH1 + jbH2 + cI3 )H = 0. = 2j (28)

The solution of (28) allows to find the values of a, b, and c that  12 , determine the AT, as well as the affine equivalent matrix of Ω i.e., Hτ1 . The solution is presented in Appendix A. As proven,  12 is a line, i.e., in the absence when the numerical range of Ω of speckle noise, the AT is only determined by a and c, i.e., the transform is determined by two real parameters [22], [31], [33], [38]. On the contrary, when considering real PolInSAR data af 12 is not affine  12 ) is not a line, Ω fected by speckle, where W (Ω τ equivalent to an Hermitian matrix, i.e., H2 = 0. Thus, under the hypothesis of the RVoG model, the effect of speckle noise  12 with a Hermitian is to eliminate the affine equivalence of Ω matrix. In terms of the numerical range, considering Theorem 4 and (26), vH Hτ2 v is also not zero. In the presence of speckle, Hτ2 and vH Hτ2 v are not zero, as indicated. Nevertheless, one may think about the invertible AT that minimizes them. In terms of coherence, this minimization corresponds to   a, b, c = arg min vH Hτ2 v = arg min |I {τ (ρ)}| (29) a, b, c

a, b, c

where v ∈ C3 , such that vH v = 1. That is, it corresponds to the minimization of the imaginary part of the AT coherence. Equation (29) represents a generalization of the line estimation

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procedure proposed in [38]. Equally, the previous minimization may be expressed in terms of matrix Hτ2 by considering a matrix norm. In the case of the Frobenius norm, we have a, b, c = arg min Hτ2 2 .

(30)

a, b, c

Both minimization problems are equivalent as follows:  H τ 2   v H2 v ≤ Hτ2 = tr Hτ2 H Hτ2 . 2

(31)

Appendix A presents the solution, where it is demonstrated that the minimum does not depend on R{a}, I{b}, and R{c}. The solution may be written in a matrix form, i.e., ⎡ ⎤⎡ ⎤ ⎡ ⎤ tr(H1 ) I{a} 0 tr(H2 ) 3 ⎣ tr(H1 H2 ) tr(H2 H2 ) tr(H2 ) ⎦ ⎣ R{b} ⎦ = ⎣ 0 ⎦ (32) tr(H1 H1 ) tr(H2 H1 ) tr(H1 ) I{c} 0 subjected to ab = 0 and ab−1 not to be purely imaginary. Equation (32) can be written as Υt = 0, where Υ shall be referred to as the trace matrix. We will consider now the analysis of (32) from two points of view: the solution of the constrained equation system and, related with it, the interpretation of Υ. First of all, it is necessary to consider the constraints imposed by the AT to be invertible, together with the fact that the solution does not depend on R{a}, I{b}, and R{c}, as indicated in Appendix A. There exist different possibilities to fix them while accomplishing the transform restrictions. Nevertheless, we propose the following for simplicity and due to the fact that they fulfill the constraints independently of the values found by (32). Since the constraints do not involve c, we may set R{c} = 0. With respect to a and b, the constraints are fulfilled by considering I{b} = 0 and R{a} = 1. The solution of (32), assuming the previous constraints together with R{b} = 1, results in a and c to be the line slope and offset parameters proposed in [38]. As demonstrated in Appendix A, in the case of CL, rank{Υ} = 2, or equivalently, det{Υ} = 0; then, the solution set t to the homogeneous system is the same as the null space of the corresponding matrix Υ. In the case that CL does no apply, rank{Υ} = 3, or equivalently, det{Υ} = 0, and the system presents no solution but the trivial one t = 0. Despite the absence of a solution, we are still interested in finding the AT that minimizes Hτ2 and vH Hτ2 v. In this case, we may consider the following minimization problem: min Υt t

subjected to t = 1

(33)

where the restriction is imposed to avoid the trivial solution. The minimization problem in (33) represents a total least squares problem, which solution is found through the singular value decomposition (SVD) of Υ. Given this decomposition, i.e., Υ = UΣVH , where U is a complex unitary matrix, Σ is a diagonal matrix with nonnegative real numbers on the diagonal, and V is a complex unitary matrix, the total squares solution t corresponds to the last column of V, which is associated with the lowest singular value of Σ.

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The rank of Υ depends on whether the RVoG hypothesis applies or not to the PolInSAR data. If rank{Υ} = 2, the RVoG model hypothesis may be considered valid, whereas if rank{Υ} = 3, the interferometric coherence does not describe a line as a function of polarization; therefore, the hypothesis of the RVoG model is not valid. Hence, rank{Υ} represents a binary test about the validity of the RVoG model assumption. In the case of real PolInSAR data, this test is not able to determine how close is the real behavior of data to CL, and in this case, it is also not able to determine which effects explain this mismatch, i.e., the speckle noise or just the invalidity of the RVoG. One option to soften this test may be to use the philosophy of the Anisotropy parameter employed in SAR polarimetry [10], considering, however, the SVD decomposition of matrix Υ. If we consider the diagonal elements of the Σ matrix, i.e., the singular values of Υ in decreasing order σ1 ≥ σ2 ≥ σ3 ≥ 0, the following RVoG test may be defined: ΛA RVoG =

σ2 − σ3 . σ2 + σ3

(34)

If ΛA RVoG = 1, then σ3 = 0, or equivalently rank{Υ} = 2, and the RVoG hypothesis may be considered valid. Otherwise, 1 > ΛA RVoG ≥ 0, in such a way that the hypothesis of the RVoG becomes more valid the closer it is to 1. The previous section considered the AT of a generic matrix, which has been considered for the transform of the contraction  12 . Therefore, when applied to a PolInSAR coherency matrix Ω matrix T matrix [see (4) and (5)], the AT matrix, which is denoted by Tτ , is defined as   1/2    τ  1/2 I Ω 12 T11 0 0 T11 τ   H T = τ 1/2 1/2  I Ω 0 T22 0 T22 12 (35)  = aH1 + jbH2 + cI3 , and H1 and H2 are the Herwhere Ω 12  12 , respectively [see (21)]. mitian and non-Hermitian parts of Ω  12 = In the same way, the IAT is defined as in (5), where Ω  τ  τ  τ τ a H1 + jb H2 + c I3 , and H1 and H2 are the Hermitian and  τ , respectively. non-Hermitian parts of Ω 12 τ

IV. M AXIMUM L IKELIHOOD A NALYSIS Invertible ATs allow to identify the conditions assumed by  12 to be  12 . As shown, CL imposes Ω the RVoG model on Ω affine equivalent with a Hermitian matrix. This result may be now embedded into T to determine which is the internal structure of this matrix assumed by the model, according to the list presented in Section II-B. First, it is not necessary to impose IPS as it is included into CL. Second, we need to impose PS. Considering (5), under the hypothesis of the RVoG, T presents the following structure: TRVoG  =

1/2 T11

T11 1/2 (a Hτ1 + c I3 ) T11 

1/2 T11

(a Hτ1 + T11

1/2 c I3 ) T11



(36)

where Hτ1 is a Hermitian matrix and a and c correspond to the  12 = a Hτ + c I3 . On the contrary, if coefficients of the AT Ω 1 the RVoG model does not apply, T presents a general structure, [see (4)]. Given (36), and under the assumption of data to be distributed according to (7), it is possible to consider the MLE of TRVoG , i.e., the MLE of the coherency matrix under the RVoG model hypothesis. In addition, one may also consider the GLRT to test the hypothesis of the RVoG model on real PolInSAR data. In order to derive the GLRT, the null model is established by (36), whereas the alternative model is given by a generic coherency matrix, i.e., H0 : T = TRVoG H1 : T.

(37) (38)

Hence, the GLRT is defined as ΛRVoG =

 RVoG , H0 ) H0 pZ (Z; T ≷γ  H1 ) H 1 pZ (Z; T,

(39)

 i is the MLE of Ti , which maximizes pZ (Z; T  i , Hi ). where T  In the case of H1 , T is the MLE of T, which corresponds to the sample coherency matrix (6), and the maximum of the  H1 ) = pZ (Z; Z). For H0 , T  RVoG is denominator is pZ (Z; T,  the MLE of TRVoG . The analysis of pZ (Z; TRVoG ) to find the MLE of TRVoG , maximizing it, represents a complex process as it would be necessary to derive it with respect to T11 , Hτ1 , a , and c . The solution is to use ATs. In Section III-C, we defined the AT minimizing the norm of Hτ2 that represents the shift from the coherence region to the real axis in the complex plane. Since the transform is invertible, it is proposed to consider the GLRT problem in this transformed domain instead of considering it in the original domain, i.e., (39). Given T, and considering the AT coefficients obtained by (32), it is possible to obtain the transformed coherency matrix Tτ (35). According to [39], if k6 is distributed according to (3), the vector kτ6 = Tτ 1/2 T−1/2 k6 is also distributed according to a zero-mean multidimensional Gaussian pdf with a coherency matrix equal to Tτ . The most relevant property of the transformed coherency matrix, in the PolInSAR context, is that to determine the validity of the RVoG scattering model, the as τ is Hermitian sumption is reduced to test whether the matrix Ω 12  τ is Hermitian, its eigenvalues will or not. In summary, if Ω 12 be real, and the coherence region will be a line. If not, the eigenvalues will be complex, and the coherence region will be also complex. In the transformed domain, under the hypothesis of the RVoG, Tτ presents the following structure:    1/2 1/2  T11 T11 Hτ1 T11 T11 Ωτ12 τ TRVoG = = 1/2 1/2 Ωτ12 T11 T11 Hτ1 T11 T11 (40) where Hτ1 and Ωτ12 are Hermitian. Hence, the hypothesis of the RVoG model may be tested through the following hypotheses: H0 : Tτ = TτRVoG H1 : T τ .

(41) (42)

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and the GLRT can be defined accordingly   τ pZ τ Z τ ; T , H 0 H0 RVoG ΛτRVoG = ≷ γ. τ τ  pZτ (Z ; T , H1 ) H1

TABLE I F LIGHT G EOMETRY FOR S IMULATED P OL I N SAR DATA

(43)

As demonstrated in Appendix B, the MLE of a generic coherency matrix in the rotated domain Tτ is simply the rotated sample covariance matrix Zτ , whereas the MLE of the coherency matrix under the hypothesis of the RVoG, i.e., TτRVoG , corresponds to the following matrix:   1 Z11 + Z22 Zτ12 + Zτ12 H τ ZRVoG = . (44) H Z11 + Z22 2 Zτ12 + Zτ12 Finally, the GLRT takes the compact expression  n H0 |Zτ | τ ΛRVoG = ≷ γ. |ZτRVoG | H1

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(45)

It is now interesting to perform an analysis of the previous two equations. If we consider the MLE in the transformed domain under the hypothesis of the RVoG, certain similarities with previous results are observed. Regarding the diagonal elements of ZτRVoG , the MLE corresponds to the estimator derived by [22] and [40] under the PS hypothesis. The interpretation of the MLE of the off-diagonal elements results to simply considering the concept of AT. Theorem 1 states that the coherence region of a matrix is a line if it is affine equivalent to a Hermitian matrix. In the transformed domain, which is obtained through an AT, the previous theorem simplifies to test if the matrix is Hermitian. Thus, what the MLE takes is the Hermitian part of the PolInSAR rotated matrix Zτ12 . Hence, this matrix will have real eigenvalues; therefore, the coherence region will be a line. Equation (45) is interesting as it allows to test the hypothesis of the RVoG on real data. If we consider the block structure of the matrices Zτ and ZτRVoG , then   ⎞n n ⎛   τ Z  τ  I − Z  12 21 |Z11 Z22 | ⎝ ⎠ ΛτRVoG =      Z11 +Z22 2 τ τ − Z Z I 2 12,RVoG 21,RVoG  (46)  τ = Z−1/2 Zτ Z−1/2 and where Z 12 12 22 11  − 12 Z11 + Z22 τ  Z12,RVoG = 2  τ  − 12 Z12 + Zτ12 H Z11 + Z22 × . (47) 2 2 The first term of the GLRT corresponds to the test of the PS hypothesis [22], [40], i.e., n  |Z11 Z22 | τ . (48) ΛPS =    Z11 +Z22 2 2

The second multiplicative term corresponds to the test of the CL property that consists of testing the Hermitian property, i.e.,   ⎞n ⎛  τ Z  τ  I − Z 12 21 ⎠ . (49) ΛτCL = ⎝   τ τ   I − Z12,RVoG Z21,RVoG 

In the following, the GRLT shall be considered in the form −2 ln(Λ) due to its statistical properties. Equation (44) corresponds to the estimation of the covariance matrix under the RVoG hypothesis in the transformed domain, where the contraction matrix has been shifted to the real axis and its coherence region corresponds to a line. The final step to retrieve the estimated covariance matrix in the original domain under the RVoG assumption, i.e., ZRVoG , consists of applying τ the IAT to the contraction matrix Z 12,RVoG . V. A NALYSIS AND R ESULTS In the following, an evaluation of the RVoG scattering model tests, as well as the analysis of its estimation in the ML context, is presented. An accurate evaluation based exclusively on experimental data presents severe problems since it would be practically impossible to uncouple the filtering or the estimation effects from the coherent scattering model. Hence, a quantitative evaluation is first presented in terms of simulated data, which is then extended to real data. A. Simulated PolInSAR Data The simulation of PolInSAR data may be considered using different approaches, under the limitation to be coherent to preserve phase information. The simulation considered here consists of a stochastic simulation [41] based on the multidimensional zero-mean complex Gaussian pdf. Thus, any deviation of the estimated information from the actual values can be attributed to the estimation process. In order to construct T, it is also necessary to specify the acquisition geometry. We use parameters of the L-band DLR E-SAR system with two parallel horizontal tracks (see Table I). Under this geometry, a particular scenario of a forest is assumed according to the RVoG model. The volume scattering contribution will consider different forest heights hv . The ground contribution, located at a reference height of φ1 rad, considers a flat, rough, and loamy terrain with 2.2% water content simulated according to the X-Bragg rough surface scattering model [42]. Both scattering contributions are related with a ground-to-volume ratio of −5 dB. Considering the previous framework, four scenarios are analyzed (see Table II). The first two scenarios simulate data according to the RVoG model, whereas the other two do not follow it. In the case of the third and fourth scenarios, in order to  12  12 does not follow the RVoG model, a matrix Ω assure that Ω following it but contaminated by speckle noise is considered. The same matrix is considered in both scenarios. In all the cases, T is estimated by a multilook filter, which dimensions range from 3 × 3 to 21 × 21 pixels.

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TABLE II S IMULATED F OREST S CENARIOS

Fig. 2 shown the different tests as a function of the multilook size, where we have considered −2 ln(Λ). For the first two scenarios, −2 ln(ΛτRVoG ) presents a low value, which is almost constant with the number of looks, indicating the validity of the RVoG model assumption. For the third and fourth scenarios, −2 ln(ΛτRVoG ) presents larger values, indicating that the RVoG model hypothesis cannot be considered true. In these two cases, the value of the GLRT increases with the number of looks since the larger the multilook filter, the more reliable the estimation of T and −2 ln(ΛτRVoG ). In all the cases, it is also important to study the value of −2 ln(ΛA RVoG ). As it was noted, this test considers speckle noise only through the coherence region shape, without taking into account its pdf. In addition, one may observe that it presents a lower dynamic range. It is clear that speckle has an important impact on the coherence region shape, even when a large filter is employed to remove it. For forest areas, assuming the RVoG model, speckle alter the shape of the coherence region in such a way that it is not a line. If this result is extended to the other type of scatterers with arbitrary coherence region shapes, it would be possible to conclude that the use of the shape itself to characterize them is affected by speckle; therefore, it should be carefully removed to have a reliable interpretation of the shape. Fig. 2 considers the values of −2 ln(ΛτPS ) and −2 ln(ΛτCL ) separately to observe their influence on −2 ln(ΛτRVoG ), for the four simulated scenarios. Whereas in the case of data simulated according to the RVoG model, both contribute in a similar way. When this is not the case, i.e., the third and the fourth scenarios, −2 ln(ΛτPS ) and −2 ln(ΛτCL ) may contribute differently. In the case that PS and CL are not valid and the RVoG hypothesis is also therefore not valid, for instance, in the particular case of the fourth scenario, −2 ln(ΛτPS ) seems to dominate −2 ln(ΛτRVoG ) over −2 ln(ΛτCL ). In conclusion, in order to decide about the validity of the RVoG hypothesis, the decision should not be based exclusively on −2 ln(ΛτRVoG ), and −2 ln(ΛτPS ), and −2 ln(ΛτCL ) should be also considered. Concerning the numerical values of the tests, one could establish at this point a threshold to decide about the validity of the RVoG model assumption. Nevertheless, a closer analysis of Fig. 2 shows that the test values, particularly for the first and second scenarios, depend on the forest structure itself. In other words, the numerical values depend on the underlying scattering mechanism represented by T, making it difficult to establish a universal threshold. Thus, the solution would be to determine this threshold depending on the data being analyzed. Section IV considered also the ML estimation of the coherency matrix under the RVoG assumption TRVoG , i.e., ZRVoG . In the case of the first two scenarios, Fig. 3 shows the  12 , coherence region for two random pixels obtained from Z together with the true coherence regions, i.e., a line in the

τ Fig. 2. RVoG tests. (−) − 2 ln(ΛA RVoG ), (−−) − 2 ln(ΛRVoG ), (· · ·) − 2 ln(ΛτCL ), (−.) − 2 ln(ΛτPS ). In the last figure, a logarithmic scale is employed to improve visibility. (a) First scenario. (b) Second scenario. (c) Third scenario. (d) Fourth scenario.

complex plane (× − ×). As observed, the coherence regions of these pixels do not follow a line due to speckle. In order to be able to determine −2 ln(ΛτRVoG ), we derive the AT as indicated  τ . Fig. 3 presents also the in Section III-C, leading to matrix Z 12 coherence region of this matrix where it can be observed that the effect of the AT is to shift the coherence region to the real axis. At this point, the coherency matrix is estimated according to (44), which the coherence region corresponds to a segment of the real axis (♦ − ♦). Finally, data are inversely transformed to estimate the coherency matrix at the original position. Fig. 3 shows the corresponding coherence regions after the estimation of the coherency matrix under the RVoG model assumption ( − ). As one may observe, the estimated data under the RVoG hypothesis present a linear behavior of coherence as a function of polarimetry, which is very close to that of the original coherency matrix employed to simulate data. First of all, as one may observe, the coherency matrix T is estimated in such a way that the coherence regions correspond to lines as it would be expected in the case of the RVoG model.

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TABLE III R EAL P OL I N SAR DATASETS

Fig. 3. Original and processed coherence regions. (× − ×) original CR. (♦ − ♦) CR obtained from (44). ( − ) CR obtained after the IAT of (44). 12 and Z τ12 . (a) 9 × 9 Dots correspond to the CR from simulated data for Z multilook. (b) 15 × 15 multilook.

Nevertheless, it is important to highlight that the estimation does not estimate the complete line, but the segment corresponding to the actual data. It is also important to remark the accuracy of the estimation as the estimated segments almost match the actual segments. A final aspect to remark is related with the coherence optimization process. As ATs preserve collinearity and ratios of vectors along a line, one may see that the extreme values of the  12 are affine related with the coherence region obtained from Ω τ  extreme values of Ω12 , which correspond to the eigenvalues of this matrix. Consequently, the optimum coherence values  12 may be obtained, considering the IAT of the real of Ω τ . eigenvalues of Ω 12 B. Experimental PolInSAR Data Here, we consider real PolInSAR data with four data sets acquired at two frequencies (P-band and L-band), at different

interferometric configurations, and over tropical and boreal forests (see Table III). These data sets were acquired by the German E-SAR system, operated by DLR, during the INDREX-II and the BioSAR campaigns. Fig. 4 details the RGB Pauli images of the master data sets. The INDREX-II campaign was conducted in 2004 on the Kalimantan island of Indonesia [8], where the area under study corresponds to a relatively flat surface covered by a tropical peat swamp forest type where forest height ranges from 15 to 30 m. In this case, almost the same area is considered at both frequencies. The BioSAR campaign was conducted in 2007 on the region of Remningtorp, which is located in southern Sweden. This area presents also a fairly flat topography where the dominant tree species are Norway spruce, Scots pine, and birch. Nevertheless, in this case, different areas are imaged at the two frequencies. To assess the validity of the RVoG model assumption, the first step consists of the estimation of T. An accurate estimation of −2 ln(ΛτCL ) and −2 ln(ΛτPS ) is obtained by a 15 × 15-pixel multilook. Based on the estimated T, the different tests have been derived. Fig. 4 depicts the full test −2 ln(ΛτRVoG ), whereas −2 ln(ΛτCL ) and −2 ln(ΛτPS ) are presented in Fig. 5. In all the cases, black indicates the validity of the given hypothesis, whereas white corresponds to the nonvalidity. As indicated, the analysis of −2 ln(ΛτRVoG ) needs to take into account −2 ln(ΛτCL ) and −2 ln(ΛτPS ), separately. A global overview of Figs. 4 and 5 shows a dependence of the RVoG model hypothesis on the parameters of the imaging system, as well as on the forest characteristics. First of all, particularly for tropical forest, an incidence angle effect is observed, where the radar incidence angle varies from near to far range from 25◦ to 55◦ . A steeper incidence provides a stronger backscattering from the ground and a weaker one from the canopy. On the contrary, higher incidences make the sensor to be more sensitive to the volume scattering. This behavior confirms, to a certain extent, the observation with simulated data, indicating a dependence of the numerical values of the tests on the total underlying scattering mechanism. Regarding the dependence of the RVoG model hypothesis with respect to frequency, a general observation shows that, in the case of tropical forests, minimum differences are observed between P- and L-band; these are more pronounced in the case of boreal forests. In the case of −2 ln(ΛτCL ), the highest values of the test, i.e., nonacceptance of the CL hypothesis, appear at the boreal forest L-band data set. When introducing the effect of −2 ln(ΛτPS ) to obtain −2 ln(ΛτRVoG ), for the two forest sites, the hypothesis of the RVoG seems to be more valid at P-band. Considering now the forest type, the RVoG model hypothesis appears to be more plausible in the case of the homogeneous tropical forest where, as indicated, minimum differences are

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Fig. 4. (a), (b), (c), and (d) Pauli RGB compositions (R = |Shh − Svv |, G = 2|Shv |, B = |Shh + Svv |). (e), (f), (g), and (h) Tests −2 ln(ΛτRVoG ) for the RVoG scattering model. (a) 1402–1405. (b) 1302–1305. (c) 0411–0407. (d) 0401–0402. (e) 1402–1405. (f) 1302–1305. (g) 0411–0407. (h) 0401–0402.

observed between both frequencies. In the case of the more heterogeneous boreal forest, the RVoG model hypothesis is less valid if compared with the tropical forest. For the boreal forest, the most important observation is that, in the case of P-band, −2 ln(ΛτCL ) would suggest the acceptance of the CL hypothesis and, finally, the acceptance of the RVoG hypothesis; however, this is not the case at L-band. The quantitative evaluation of the tests must be carefully considered, since as indicated, the numerical values depend on the underlying scattering mechanism. In addition, the previous analysis must be contemplated in relative terms, i.e., it does not demonstrate that, at P-band, the RVoG hypothesis is completely valid. Even in the case of the tropical forest imaged at P-band, some values would suggest that the RVoG model hypothesis could not be considered as valid. In all the cases, there are two main reasons that could explain the low validity of the RVoG

model assumption. First is the statistical distribution of the data. In this paper, the complex Gaussian pdf has been considered, which does not take into account signal variability associated to texture. Second is the validity of the RVoG itself. The presence of orientation effects in the forest scattering would also explain the low values of the RVoG model hypothesis. This could be the main reason explaining the results for the boreal forest results at L-band, but it is also plausible to consider it for the tropical forest results at P-band. Finally, the model does not consider temporal decorrelation, which could explain the subtle diagonal strips that are noticeable in the case of the tropical forest imaged at L-band. Fig. 6 shows −2 ln(ΛA RVoG ). This test only analyzes the shape of the coherence region, without considering speckle noise. Since data have suffered a 15 × 15 multilook, a correct interpretation of −2 ln(ΛA RVoG ) is possible. The comparison

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Fig. 5. (a), (b), (c), and (d) Tests −2 ln(ΛτCL ). (e), (f), (g), and (h) Tests −2 ln(ΛτPS ). (a) 1402–1405. (b) 1302–1305. (c) 0411–0407. (d) 0401–0402. (e) 1402–1405. (f) 1302–1305. (g) 0411–0407. (h) 0401–0402.

of Fig. 6 with Fig. 5 shows that the validity of the RVoG τ model predicted by −2 ln(ΛA RVoG ) is similar to −2 ln(ΛRVoG ). A However, the noisier aspect of −2 ln(ΛRVoG ) is due to the fact that this test does not take into account speckle. A closer analysis of Figs. 4 and 5, particularly over tropical forests, allows to identify particular behaviors. In this case, one may easily identify a river on the bottom part, where the hypothesis of the RVoG is clearly nonvalid since the signal in this area is dominated by noise. Nevertheless, over this river, one may identify the areas with the lowest value of the GRLT, i.e., the maximum validity of the RVoG. As observed in the RGB images, these correspond basically to bare areas. In this case, the interferometric coherence is close to 1 and its dependence on polarization presents a low variability. Two examples are shown in Fig. 7, where the coherence regions that are near the unit circle correspond to pixels of this area. This

figure also shows the original coherence regions from random pixels of the four data sets along with the coherence regions estimated with the procedure proposed in this paper ( − ). In all the cases, it can be observed that the estimated covariance matrices present a linear coherence region. Comparing both plots in Fig. 7, one can observe that, in the case of the tropical forest, the coherence regions are narrower than in the case of the boreal forest. This is explained by the fact, as observed in Fig. 4, that the tests indicate that the RVoG is more valid in the case of tropical forests. VI. D ISCUSSIONS AND C ONCLUSION This paper has considered the analysis of the validity of the RVoG coherent scattering model hypothesis for forest scattering on PolInSAR data. The MLE of the coherency matrix, under

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Fig. 6. Test −2 ln(ΛA RVoG ). (a) 1402–1405. (b) 1302–1305. (c) 0411–0407. (d) 0401–0402.

the hypothesis of the RVoG, has been also derived. Both the assessment and the estimation have been considered under the assumption of data distributed according the zero-mean complex Gaussian pdf. The RVoG coherent scattering model assumes certain properties on the PolInSAR data, which are reflected on the coherency matrix structure. The first condition refers to PS, i.e., T11 =  12 T22 . The second condition is the IPS, which assumes Ω to be a normal matrix. Finally, the RVoG assumes also CL, which implies a linear behavior of the interferometric complex coherence with respect to polarization. In order to determine the validity of the RVoG model assumption, these conditions need to be tested on the data. While PS has been already addressed,  12 is necessary CL is not. As demonstrated, the normality of Ω but not a sufficient condition to assure CL. CL is valid if matrix  12 is affine equivalent to a Hermitian matrix, which imposes Ω  12 to be normal. Hence, in order to determine the validity of Ω the RVoG hypothesis, it is necessary to test the PS and the affine  12 with a Hermitian matrix. equivalence of Ω The use of invertible ATs of the coherence region allows the test of the RVoG hypothesis. In the absence of speckle noise and considering that CL applies, the effect of the AT, as considered here, is to shift the coherence region, i.e., a line, to the real axis leading to a Hermitian matrix. In the case that the coherence region is not a line, we have obtained the AT coefficients that allow shifting the coherence region to the real axis, while minimizing the non-Hermitian part of the resulting matrix. In this case, despite the fact that speckle noise has not been directly considered, we introduced the concept of the trace matrix. This matrix depends on the shape of the coherence region in such a way that its rank depends on the validity of the RVoG assumption. This observation has introduced a first validity test for the RVoG model hypothesis. In order to consider specifically the effect of speckle noise, the AT matrix has been considered in a ML context under the complex Gaussian pdf assumption. It has been possible

to derive both the MLE of the coherency matrix under the RVoG assumption and the GLRT. As it has been demonstrated, the main effect of speckle noise in PolInSAR data, under the assumption of the RVoG, is to break the affine equivalence of  12 with a Hermitian matrix. The GLRT analyzes this link to Ω determine the validity of the RVoG assumption, whereas the MLE restores it. Regarding the estimation of the covariance matrix under the RVoG assumption, it has been shown that the MLE of the polarimetric matrices corresponds to the average of T11 and T22 , whereas the estimation of the polarimetric and interferometric matrix corresponds to the Hermitian part of the transformed  τ . It is worth noting that the proposed estimation matrix Ω 12 procedure is not based on the adjustment of a line, and it does not need the direct calculation of the coherence region. In addition, this procedure is able to estimate a matrix which coherence region is a line segment according to the RVoG. That is, it does not only estimates a line but also the segment, which limits correspond to the extreme coherences. These limits can  12 be obtained both in the original domain of the estimated Ω matrix and in the transformed domain where they correspond directly to the real eigenvalues of the Hermitian transformed matrix. As a consequence of the ML analysis, it has been possible to derive a GLRT of the RVoG hypothesis. This test is composed of two tests. One is a test for the PS and the other is a test for the CL condition. As shown, the correct analysis of the RVoG needs to consider these two components separately. This test, as well as the test based on the trace matrix, has been derived to test the validity of the RVoG model hypothesis. Hence, the tests could be also employed to check the hypothesis of random versus oriented volume. The test and estimation procedures have been validated in terms of simulated and real PolInSAR data sets. The use of simulated data has shown that the resulting T matrix presents a linear coherence region close to the actual lines. Nevertheless,

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Finally, temporal decorrelation effects could also explain the nonvalidity of the RVoG model assumption. A PPENDIX A A FFINE T RANSFORM  12 , under the hypothesis of In the absence of speckle noise, Ω the RVoG model, is affine equivalent with an Hermitian matrix [see (27)]. The coefficients of the transform are (aH1 + jbH2 + cI3 ) − (aH1 + jbH2 + cI3 )H = (a − aH )H1 − j(b + bH )H2 + (c − cH )I3 = 0 I{a}H1 + R{b}H2 + I{c}I3 = 0.

(50)

Considering Theorem 4, (50) can be considered in terms of coherence or the numerical range, i.e., I {τ (ρ(w))} ! 

τ =I W Ω 12

"

= I{a}vH H1 v − jR{b}vH H2 v + I{c} = 0.

(51)

 12 )} = vH H1 v and I{ρ(w)} = Since R{ρ(w)} = R{W (Ω H  I{W (Ω12 )} = v H2 v(see (25) and (26), respectively) and since I{ρ(w)} = mR{ρ(w)} + k as coherence describes a line in the complex plane, we have I {τ (ρ(w))} 

Fig. 7. Coherence regions of the processed data sets. (♦ − ♦) CR obtained from (44). ( − ) CR obtained after the IAT of (44). Dots correspond to the 12 and Z τ12 . (a) INDREX-II data sets. (b) BioSAR CR from simulated data for Z data sets.

the numerical values of the tests depends on the underlying forest structure and imaging geometry, making it difficult to establish a universal threshold to determine the validity of the RVoG model assumption. In the case of real PolInSAR data, the tests have been considered on different scenarios. The results presented previously suggest that the validity of the RVoG model assumption is higher at P-band than in Lband and is higher in the case of homogeneous rather than heterogeneous forests. Nevertheless, the previous statement needs to be carefully considered since, even in the case of the tropical forest imaged at P-band, in some forested areas, the RVoG model assumption could not be valid. Physically, this could be explained by the presence of orientation effects, even at P-band. Another potential explanation could be the presence of data texture not considered within the complex Gaussian pdf.

= (I{a} + mR{b}) R {ρ(w)} + (kR{b} + I{c}) = 0    I{a} I{c} + m R {ρ(w)} + k + = 0. (52) R{b} R{b}

From the first equation, it comes out that the affine equivalence between matrices is independent from R{a}, I{b}, and R{c}. The second equation indicates that the equivalence is indepen 12 dent of R{b}. Therefore, the parameters of the AT relating Ω with a Hermitian matrix are those that fulfill (I{a} + m) R {ρ(w)} + (k + I{c}) = 0.

(53)

Hence, the AT is determined by two real parameters. In order to have an invertible transform, ab = 0 and ab−1 must not be purely imaginary; hence, R{a}, R{b}, I{b}, and R{c} must be fixed accordingly.  12 is not affine equivalent with In the presence of speckle, Ω a Hermitian matrix, but one may consider to find the AT linking it with a matrix presenting a minimum non-Hermitian part [see (30)]. To find the coefficients of the AT, the following complex differential is considered: #  % 2 $ & A − AH A − AH A − AH d = tr 2 d tr . 2j 2j 2j (54) The complex differential is employed to find the derivatives with respect to a, b, and c, which are made equal to zero to

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find the minimum of (30). Then [43]

from where the following derivatives result:

 " ∂ !  τ  τ H H1 = 0 tr Ω12 − Ω 12 ∂a

∂ ln (|T|) = (TT )−1 ∂T ∂ tr(T−1 Z) = − (TT )−1 ZT (TT )−1 . ∂T

I{a}tr(H1 H1 ) + R{b}tr(H1 H2 ) + I{c}tr(H1 ) = 0 (55)  " ∂ !  τ  τ H H2 = 0 tr Ω12 − Ω 12 ∂b

max pZ (Z; T, H0 ) =

(56)

I{a}tr(H1 ) + R{b}tr(H2 ) + 3I{c} = 0.

(57)

These equations can be arranged in a matrix [see (32)]. As in the case of absence of speckle noise, the solution does no depend on R{a}, I{b}, and R{c}. Nevertheless, in this case, the AT is defined by three real parameters due to speckle noise. Similarly, in order to have an invertible transform, R{a}, I{b}, and R{c} need to be fixed accordingly. It is worth to observe the behavior of the previous three equations when considering the CL restriction. Under this assumption, the complex coherence describes a line, or equivalently, H1 and H2 are linearly dependent [see (50)], i.e., H2 = −

I{a}H1 + I{c}I3 . I{b}

(58)

Thus, introducing (58) into (55)–(57), one may easily demonstrate that (56) is a linear combination of (55) and (57), and the solution does not depend on b as expected.

(64)

nmn |Zτ |n−m exp(−6n).  m (n) |Zτ |n Γ

max pZτ (Zτ ; Tτ , H0 ) =

(65)

When analyzing the null model H0 in the AT domain, it results into determining the MLE of the structured coherency matrix TτRVoG (40) and the corresponding maximum of (59). At this point, we may also consider a similar block partition of the sample covariance matrix, i.e.,   Z11 Zτ12 Zτ = (66) Zτ12 H Z22  τ Z . Considering the block partitions where Zτ12 = Z11 Z 12 22 τ τ of TRVoG and Z , the term tr(Tτ −1 Zτ ) of the log-likelihood function is   tr TτRVoG −1 Zτ     τ −1 τ Z11 − Zτ12 T−1 = tr T11 − Ωτ12 T−1 11 Ω12 11 Ω12 1/2

1/2

    τ −1 τ H −1 τ +tr T11 − Ωτ12 T−1 Ω − Z T Ω Z 22 12 11 12 11 12

The analysis of the likelihood function of the complex Wishart pdf may be considered in terms of its log-likelihood function. Considering (7), its log-likelihood function is   nmn |Z|n−m ln pZ (Z; T) = −ln −n ln (|T|) − ntr(T−1 Z)  Γm (n) (59) where tr(·) indicates the matrix trace. When analyzing the alternative model H1 , in the original domain, it results into determining the MLE of a generic coherency matrix T and the corresponding maximum of (59). In order to derive (59) with respect to T, the following differential forms are necessary [43]:

d tr(T−1 Z) = tr(−T−1 ZT−1 dT)

nmn |Z|n−m exp(−6n).  m (n) |Z|n Γ

Now, if the alternative model H1 is considered in the AT domain, the analysis is parallel to the one shown previously. The MLE of a transformed generic coherency matrix Tτ is the transformed sample covariance matrix Zτ , and the maximum of the log-likelihood function is

A PPENDIX B L IKELIHOOD A NALYSIS

d ln (|T|) = tr(T−1 dT)

(63)

If (59) is derived with respect to T and is made equal to zero, the MLE of the coherency matrix T corresponds to the sample covariance matrix Z, and the corresponding maximum of the likelihood function is

I{a}tr(H1 H2 ) + R{b}tr(H2 H2 ) + I{c}tr(H2 ) = 0 " ∂ !  τ  τH =0 tr Ω12 − Ω 12 ∂c

(62)

(60) (61)

= tr



τ T11 − Ωτ12 T−1 11 Ω12

−1

    τ × (Z11 + Z22 ) − Zτ12 + Zτ12 H T−1 11 Ω12  = tr



τ T11 − Ωτ12 T−1 11 Ω12

−1



Z11 + Z22 Zτ + Zτ12 H −1 τ − 12 T11 Ω12 × 2 2   τ −1 + tr T11 − Ωτ12 T−1 11 Ω12  ×

Z11 + Z22 Zτ + Zτ12 H −1 − 12 T11 Ω12 2 2

  = tr TτRVoG −1 ZτRVoG





(67)

LÓPEZ-MARTÍNEZ AND ALONSO-GONZÁLEZ: ASSESSMENT AND ESTIMATION OF RVOG MODEL IN POLINSAR

where



ZτRVoG =

1 Z11 + Z22 H 2 Zτ12 + Zτ12

Zτ12

Zτ12 H

+ Z11 + Z22

 .

(68)

If we now consider (44) and (67) into (59), it is already possible to prove that ZτRVoG corresponds to the MLE of TτRVoG . In a similar way, the maximum of the likelihood function under the hypothesis of the null model, i.e., the RVoG scattering model, is max pZτ (Zτ ; Tτ , H1 ) =

nmn |Zτ |n−m exp(−6n). (69)  m (n) |Zτ |n Γ RVoG

ACKNOWLEDGMENT The authors would like to thank the European Space Agency for the data that they have provided. R EFERENCES [1] “Global forest resources assessment 2005,” Food Agric. Org. United Nations (FAO), Rome, Italy, FAO Forestry Paper 147, 2005. [2] R. Treuhaft, S. Madsen, M. Moghaddam, and J. van Zyl, “Vegetation characteristics and underlying topography from interferometric radar,” Radio Sci., vol. 31, no. 6, pp. 1449–1485, Nov./Dec. 1996. [3] R. N. Treuhaft and S. R. Cloude, “The structure of oriented vegetation from polarimetric interferometry,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 5, pp. 2620–2624, Sep. 1999. [4] R. N. Treuhaft and P. R. Siqueira, “Vertical structure of vegetated land surfaces from interferometric and polarimetric radar,” Radio Sci., vol. 35, no. 1, pp. 141–177, Jan./Feb. 2000. [5] S. R. Cloude and K. P. Papathanassiou, “Polarimetric SAR interferometry,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 5, pp. 1551–1565, Sep. 1998. [6] K. P. Papathanassiou, S. R. Cloude, A. Reigber, and W. M. Boerner, “Multi-baseline polarimetric SAR interferometry for vegetation parameters estimation,” in Proc. IEEE IGARSS, 2000, pp. 2762–2764. [7] S. Cloude and K. Papathanassiou, “Three-stage inversion process for polarimetric SAR interferometry,” Proc. Inst. Elect. Eng.—Radar, Sonar Navig., vol. 150, no. 3, pp. 125–134, Jun. 2003. [8] I. Hajnsek, F. Kugler, S.-K. Lee, and K. P. Papathanassiou, “Tropicalforest-parameter estimation by means of Pol-InSAR: The INDREX-II campaign,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 2, pp. 481– 493, Feb. 2009. [9] J. Praks, F. Kugler, K. P. Papathanassiou, I. Hajnsek, and M. Hallikainen, “Height estimation of boreal forest: Interferometric model-based inversion at L- and X-band versus HUTSCAT profiling scatterometer,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 3, pp. 466–470, Jul. 2007. [10] S. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens., vol. 34, no. 2, pp. 498–518, Mar. 1996. [11] N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Math. Statist., vol. 34, no. 1, pp. 152–177, Mar. 1963. [12] E. Luneburg and S. Cloude, “Contractions, hadamard products and their application to polarimetric radar interferometry,” in Proc. IEEE IGARSS, 1999, vol. 4, pp. 2224–2226. [13] R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A, vol. 499, no. 1937, pp. 567–589, Jun. 1995. [14] J.-S. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 5, pp. 1017–1028, Sep. 1994. [15] R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” Inverse Problems, vol. 14, no. 4, pp. R1–R54, Aug. 1998. [16] E. Colin, C. Titin-Schnaider, and W. Tabbara, “An interferometric coherence optimization method in radar polarimetry for high-resolution imagery,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 1, pp. 167–175, Jan. 2006. [17] M. Neumann, L. Ferro-Famil, and A. Reigber, “Multibaseline polarimetric SAR interferometry coherence optimization,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 1, pp. 93–97, Jan. 2008.

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Carlos López-Martínez (S’97–M’04–SM’11) received the M.Sc. degree in electrical engineering and the Ph.D. degree from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1999 and 2003, respectively. From October 2000 to March 2002, he was with the Department of Frequency and Radar Systems, Microwaves and Radar Institute, German Aerospace Center (DLR), Oberpfaffenhofen, Germany. From June 2003 to December 2005, he was a member of the Synthetic Aperture Radar (SAR), Polarimetry, Holography, Interferometry, and Radargrammetry (SAPHIR) Team of the Image and Remote Sensing Group with the Institute of Electronics and Telecommunications of Rennes, Rennes, France. In January 2006, he joined the UPC as a Ramón-y-Cajal Researcher. He is currently an Associate Professor of remote sensing and microwave technology with UPC. His research interests include SAR and multidimensional SAR, radar polarimetry, physical parameter inversion, digital signal processing, estimation theory, and harmonic analysis. Dr. López-Martínez is an Associate Editor for the IEEE J OURNAL OF S ELECTED T OPICS IN A PPLIED E ARTH O BSERVATIONS AND R EMOTE S ENSING , and he served as a Guest Editor for the European Association for Signal Processing Journal on Advances in Signal Processing. He has organized different invited sessions in international conferences on radar and SAR polarimetry. He has presented advanced courses and seminars on radar polarimetry to a wide range of organizations and events. He was the recipient of the Student Prize Paper Award at the 2002 European Conference on Synthetic Aperture Radar (EUSAR) and a corecipient of theFirst Place Student Paper Award at the 2012 EUSAR Conference.

Alberto Alonso-González (S’11) was born in León, Spain, in 1984. He received the B.Sc. degree in computer science and the M.Sc. degree in telecommunication engineering from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree in telecommunication engineering at the UPC. In 2009, he joined the Department of Signal Theory and Communications, UPC. His Ph.D. thesis is focused on multidimensional synthetic aperture radar (SAR) data modeling and processing. His research interests include multidimensional SAR, SAR interferometry and polarimetry, digital signal and image processing, and data segmentation and simplification techniques. Dr. Alonso-González received the First Place Student Paper Award at the 2012 European Conference on Synthetic Aperture Radar (EUSAR) Conference.