Formulation and Validation of a Multidimensional SAR Data Speckle Noise Model Lopez-Martinez, Carlos1, Fabregas, Xavier2, Pottier, Eric1 1
Institut d'Electronique et de Télécommunications de Rennes 2Signal Theory and Communications Department Equipe Imagerie Radar - Teledetection Polarimetrie Technical University of Catalonia 08034 Barcelona Spain Universite de Rennes 1 35042 Rennes France Email:
[email protected] Telephone/Fax: +33 (0)2 2323 6822/6963 Email:{carlos.lopez,eric.pottier}@univ-rennes1.fr
Abstract The availability of multidimensional Synthetic Aperture Radar (SAR) imagery makes possible the physical characterization of scatterers through the inversion of electromagnetic scattering models. Natural scatterers need to be characterized stochastically as a consequence of the coherent nature of SAR systems, leading to speckle. At the resolution at which SAR systems operate, speckle turns out to be a noise source. Consequently, a complete characterization of this noise source, specially for correlated multidimensional SAR imagery is crucial for a correct information interpretation from a quantitative point of view. In this direction, this paper presents a novel multidimensional speckle noise model, in a matrix-based formulation, as well as, its validation by means of real interferometric and polarimetric SAR data.
1
Introduction
The process to determine the physical properties of a given scatterer, in the microwave region of the spectra, is only possible through a multiparameter characterization. Consequently, a multidimensional imaging system is necessary. In this scenario, mutichannel SAR systems have raised as a powerful technique to perform such a task. Additionally, these systems are capable to cover large areas, independently of the weather conditions in a day-night basis and with a high spatial resolution. The different configurations a multichannel SAR system can adopt depend on which parameter of the system is varied between each channel data or survey. A change on the spatial position of the platform makes possible to be sensible to the vertical structure of the imaged scene, hence, making possible single or multiple baseline SAR interferometry (InSAR) [1]. Another possibility is to vary, both, the polarization with which the SAR transmits the wave and the polarization with which the same system records the echo. Hence, data are sensible to the polarimetric properties of the scatterer. This configuration is called SAR Polarimetry (PolSAR) [2]. Other configurations are based, for instance, on a variation of the working frequency. The increase of information is not only due to the fact that more data channels are available, but also, to the fact that these channels present a degree of correlation between them. As mentioned, one of the main features of a SAR system is the capacity to provide high spa-
tial resolution data thanks to a coherent processing of the returned echoes. This coherent processing is also the origin of one of the most important problems of SAR data, i.e., speckle [3]. Despite being a true electromagnetic measurement, the spatial resolution at which SAR system operate, makes necessary to consider speckle as a disturbance, i.e., as a noise source. Speckle noise, for one-dimensional SAR systems can be considered as a solved problem, since, for homogeneous areas, this noise is characterized by a multiplicative nature [4]. On the contrary, since multidimensional SAR data are normally correlated, an extension of the multiplicative model is not possible. Recently, the authors have proposed, and validated, a novel speckle noise model for multidimensional SAR imagery on the basis of the Hermitian product of two SAR images [5]. In what it follows, the authors present the extension of this noise model, through a matrix formulation, making possible to consider arbitrary multidimensional SAR data. Additionally, a complete validation in different frequency bands by means of real data is reported.
2
Multidimensional Speckle Noise Model
There exist different ways in which multidimensional SAR data can be analyzed for distributted scatterers: the covariance matrix C, the coherency matrix T and the Müeller matrix M. Every one of the previous alternatives presents some characteristics which make each of them suitable for a different type of analysis. For an stochastic analsyis of the data, the formulation
based on the covariance matrix C is the most recommendable, since all the elements of this matrix have the same nature, i.e., they consist of the Hermitian product of two SAR images.
2.1
Hermitian Product Noise Model
A multidimensional SAR system measures the target vector k = [ S1 ,…, S m ] . T
(1)
Under the Gaussian scattering hypothesis for homogeneous data, (1) is characterized by a zero-mean, complex, multidimensional Gaussian distribution k ∼ N ( 0, C ) [4]. Hence, for distributed scatterers it is not possible to extract useful information from (1) due to the fact that data mean equals zero. For instance, this fact can be observed in PolSAR data, since the target vector k is not capable to completely characterize distributed scatterers. Thus, the stochastic behavior of (1) must be consider by analyzing higher moments, i.e, the covariance, the coherency or the Müeller matrices. Under the Gaussian scattering assumption, (1) is completely characterized by the covariance matrix
{
C = E kk
H
}
{
}
E S1S1∗ = ∗ E S m S1
{
}
{
}
E S1S m∗ ∗ E Sm Sm
{
}
(2)
where * indicates the complex conjugate, H is the transpose complex conjugate and E{x} is the ensemble average. Under the hypothesis that the processes involved in (2), i.e., Sp, Sq and Sp S*q are homogeneous and ergodic, the ensemble average can be substituted by an spatial average. Thus, the matrix C can be estimated by means of the sample coherence estimator Zn =
1 n 1 n ∑ k ik iH = n ∑ i =1 Zi n i =1
(3)
where n refers to the number of averaged samples, and every Zi is the one-look covariance matrix of every pixel. Since the k-vector is a random variable, the matrix Zn is also a random variable characterized by the Wishart distribution W ( C, n ) pZn ( Z n ) =
n mn Z n
n−m
C Γm ( n ) n
(
etr −nC−1Z n
)
(4)
where etr(X) refers to the exponential of the trace, and the multidimensional gamma distribution is Γm ( n ) = π
m ( m −1) 2
∏
m i =1
Γ ( n − i + 1).
(5)
As one can observe in (3), every element of Zi consists of the Hermitian product of two elements of (1). Hence, for every of these elements it is possible to write
(
)
S p S q∗ = S p S q∗ exp j (φ p − φq ) = z exp ( jφ ) .
(6)
When (6) refers to a diagonal element of (3), speckle is characterized by a multiplicative behavior. Nevertheless, this model can not be extended to the offdiagonal elements of (3), as the multiplicative noise model alone is not capable to explain data variance for those matrix entries characterized by a low coherence level. In [5], through an extensive statistical study of the different properties of (6), the authors proposed a model for the Hermitian product of two complex SAR images S p S q∗ = ψ N c zn nm e jφx + ψ ( ρ − N c zn ) e jφx + ψ ( nar + jnai ) . Multiplicative term
(7)
Additive term
where the average power is
( { } E { S })
ψ = E Sp
2
2
(8)
12
q
and zn is the normalized amplitude of Sp S*q. The phase φx refers to the average phase of Sp S*q. Finally, |ρ| is the amplitude of the correlation coefficient of the Hermitian product Sp S*q and Nc takes the value, for n=1, Nc =
π 4
1 1
2
ρ 2 F1 , ;2; ρ . 2 2
(9)
In (7), the first noise term is given by nm with an expectation and a variance equal to one. Hence, this noise term is characterized by a multiplicative nature. On the contrary, the two additive noise terms, nar and nai, have a mean equal to zero, and the following variance var {nai } = var {nar }
(1 2 ) (1 − ρ )
1.32
.
(10)
Consequently, the additive noise terms nar and nai are not homogeneous as they depend on the level of coherence between Sp and Sq. Basically, for the Hermitian product (7), noise is characterized by a multiplicative nature for high coherences, whereas it turns to an additive nature for low coherences. Additionally, the final nature of speckle noise for the Hermitian product of a pair of SAR images depends also on the value of the average phase φx, since one can observe in (7), it also modulates the multiplicative noise term. Therefore, the final nature of speckle will vary between the real and imaginary parts of (7). As a result, it is possible to observe situations in which the additive noise nature dominates, even for high coherences. Finally, the model given in (7) reduces to the well known multiplicative speckle noise model when applied to the diagonal elements of (3). As a brief conclusion of this section, one can observe that the nature of speckle noise is not simple for the case of the Hermitian product of correlated SAR images. In the following, the model given in (7) is
taken as a basic building block to construct a noise model for the complete sample covariance matrix (3).
2.2
Multidimensional Speckle Noise Model: Matrix Formulation
In order to derive a matrix formulation under which the noise model (7) can be applied to obtain a multidimensional noise model for the sample covariance matrix, one needs to consider the nature of the two noise mechanisms for every element of the sample covariance matrix. As it will be shown, a formulation based on m by m matrices of the multidimensional speckle noise model is not possible when the objective is to separate the useful signal components from the noise components.
2.2.1
Multidimensional Multiplicative Speckle Noise Component
n1mm n1m1
(11)
and the next m by m matrix that contains the useful signal component ψ 11 Cm = m1 m1 m1 jφxm1 ψ N c zn e
ψ 1m N c1m zn1me jφ 1m x
ψ mm
,
(12)
the multiplicative term of (7) can be considered in a multidimensional formulation as diag ( vec ( N m ) ) vec ( Cm )
(13)
where vec(X) represents a mm by 1 vector formed by stacking the columns of X under each other, T represents the transpose of a vector, and diag(x) forms a diagonal matrix with the elements of the vector x.
2.2.2
Multidimensional Additive Speckle Noise Component
The additive nature of the additive term of (7) would make possible to establish a formulation based on m by m matrices. Nevertheless, the formulation of the multiplicative speckle components imposes in this case, also, a vector-based formulation in order to obtain a multidimensional noise model. For the additive
(
)
12 0 ψ 12 ρ 12 − N c12 zn12 e jφx 21 ψ 21 ρ 21 − N 21 z 21 e jφx 0 c n Ca = ψ m1 ρ m1 − N m1 z m1 e jφxm1 c n
(
)
(
)
(
)
ψ 1m ρ 1m − N c1m zn1m e jφ
0
1m x
(14) and the additive speckle noise components are considered as follows 0 ψ 21 nar21 + jnai21 Νa = m1 m1 m1 ψ nar + jnai
(
By considering the noise model given by (7), one can observe that the multiplicative component nm appears in the diagonal, as well as, the off-diagonal elements of the sample covariance matrix. Consequently, the first addend of (7) can not be expressed as the product of two different m by m matrices, considering separately useful signal from noise. If one wants to maintain this two components separately, it is necessary to consider the first addend of (7), in a multidimensional scheme, by a vector-based formulation. Thus, if one considers the following m by m matrix containing the multiplicative noise components n11 m Νm = nmm1
speckle noise component, one has to consider that it appears only over the off-diagonal elements of the sample covariance matrix. Hence, if the useful signal component is considered to be contained in the following singular m by m matrix
(
12 ψ 12 ( n12 ar + jnai )
)
ψ 1m ( n1arm + jn1aim )
0
)
0
(15)
the additive terms of (7), in a multidimensional formulation can be considered as vec ( Ca ) +vec ( N a ) .
2.2.3
(16)
Multidimensional Noise Model
The final multidimensional noise model for the sample covariance matrix is obtained by considering (7) for each element of (3), under the formulation presented in (13) and (16). Consequently vec ( Z n ) = diag ( vec ( N m ) ) vec ( Cm ) + vec ( Ca ) +vec ( N a ) (17)
is a speckle noise model able to represent separately the noise components from the useful signal component, which equals vec ( C ) =vec ( Cm ) + vec ( Ca ) ⇒ C = Cm + Ca .
(18)
This multidimensional speckle noise model is valid to consider any type of multidimensional SAR configuration, as it takes into account the correlation structure of all the SAR data channels. For instance, it is valid to consider interferometric, polarimetric and polarimetric-interferometric SAR data.
2.3
Multifrequency Model
An interesting application of the multidimensional speckle noise model presented in the previous section is the possibility to consider multifrequency SAR imagery. In this sense, the speckle noise model presented in (17) considers the possible correlations between the different frequency bands. In this direction, the model presented by (7) can be rewritten as follows S p S q∗ = ψ N c zn nm e
j Φ ( ∆f )
+ψ ( R ( ∆f ) − N c zn ) e
j Φ ( ∆f )
(19) + ψ ( nar + jnai ) .
where R ( ∆f ) represents the normalized frequency correlation function and Φ ( ∆f ) denotes the phase of
the frequency correlation function [6]. It has to be mentioned that (19) is valid only if the Gaussian scattering assumption applies to the multifrequency data.
3
Quantitative Noise Model Validation
5
In the last part of this paper, a complete multifrequency validation process of the speckle noise model given by (7), or (17), is presented. In order to perform this analysis, real multidimensional data has been employed. Table 1 summarizes the main characteristics of these data. Freq. Band Sensor
P-.45GHz
L-1.26GHz
C-5.31GHz
X-9.6GHz
AIRSAR
AIRSAR
AIRSAR
E-SAR
Rg. Resol.
7.5 m
3.75 m
1.875 m
2.2 m
Az. Resol.
1m
1m
1m
Her. Prod. ρ
S
hh
0.74e
S
∗ vv
j 0.18
S
hh
0.67 e
S
∗ vv
j 0.02
S
hh
0.46e
S
1.8 m
∗
S
vv
j 0.02
v v1
0.87 e
S
∗ v v1
− j 0.65
Table 1: Multidimensional SAR data features.
As it is presented in Table 1, for each one of the datasets, an homogeneous area, of a given Hermitian product of the data, has been considered. The correlation coefficient of each area is presented in the last row of Table Table 1. The P-, L-, and C- bands correspond to polarimetric data, whereas the X-band correspond to interferometric data. The validation consists of the analysis of a least squares regression test between the theoretical values of the mean and the standard deviation of the multiplicative and additive terms of (7) [5] and the values derived from real data. This analysis has four parameters. First, the coefficients of zero and first order, a0 and a1 respectively, which characterize the regression line. A perfect mach is obtained when a0=0 and a1=1. The second set of parameters is the one given by s and r. The former is the estimate standard error and the later is the correlation coefficient. A complete match between theoretical and real values are obtained for s=0 and r=1. The Tables 1, 2, 3 and 4 contain the results of the least squares regression test for every dataset in Table 1. In each of the four tables, Mn refers to mean, Sdv to standard deviation, Mult to the multiplicative term and Add to the additive term. From the results presented in the four tables, one can observe that the speckle noise model presented in this paper can be considered as valid for a wide range of frequencies on the microwave region of the spectra. Hence, this set of results assures the usefulness of the multifrequency model given in Section 2.3.
4
the entries of the sample covariance matrix. Finally, this model has been validated in P-, L-, C- and Xbands, allowing to extend the multidimensional speckle noise model validity to a multifrequency configuration of a SAR system.
Conclusions
This paper presents the extension of the speckle noise model of the Hermitian product of two SAR images to a set of SAR images. As it is shown, a direct matrix formulation is not possible as a consequence of the multiplicative speckle noise component affecting all
Acknowledgements
This work has been funded by the EU RTN Network AMPER, contract number: HPRN-CT-2002-00205.
6
References
[1] Bamler, R., Hartl, P.: Synthetic aperture radar interferometry. Inverse Problems, 14, (1998), pp.: R1-R54. [2] Ulaby, F.T., Elachi. C.: Radar Polarimetry for Geoscience Applications. Arthech House. Norwood, MA. USA 1990. [3] Goodman, J.W.: Some fundamental properties of speckle. J. Opt. Soc. Am., vol 66, no. 11, (1976) pp.: 1145-1149. [4] Lee, J.S.: Speckle Analysis and smoothing of SAR images. Com. Garph. and Im. Proc., vol 17, (1991), pp.: 24-32. [5] Lopez-Martinez, C., Fàbregas, X.: Polarimetric SAR Speckle Noise Model. IEEE Trans. Geos. Rem. Sensing, vol. 41, no. 10 (2003), pp.22322242. [6] Sarabandi, K.: ∆k-radar equivalent of interferometric SAR’s. IEEE Trans. Geos. Rem. Sensing, vol. 35, no. 5, (1997), pp.1267-1276. a0
a1
s
r
Mn. Mult. 0.004 0.987 0.012 0.990 Sdv. Mult. -0.026 0.997 0.078 0.774 Mn. Add. 0.008 0.960 0.012 0.675 Sdv. Add. 0.022 0.929 0.048 0.820 Table 2: P-band least squares regression test. a0
a1
s
r
Mn. Mult. 0.003 0.985 0.012 0.990 Sdv. Mult. 0.033 0.865 0.067 0.813 Mn. Add. -0.021 1.153 0.013 0.682 Sdv. Add. -0.067 1.104 0.052 0.854 Table 3: L-band least squares regression test. a0
a1
s
r
Mn. Mult. 0.001 0.992 0.012 0.985 Sdv. Mult. 0.005 0.930 0.036 0.865 Mn. Add. -0.002 1.034 0.012 0.939 Sdv. Add. -0.128 1.207 0.075 0.740 Table 4: C-band least squares regression test. a0
a1
s
r
Mn. Mult. 0.006 0.986 0.007 0.997 Sdv. Mult. -0.057 1.164 0.156 0.664 Mn. Add. 0.006 0.950 0.007 0.960 Sdv. Add. -0.041 1.120 0.057 0.860 Table 5: X-band least squares regression test.
