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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 1, JANUARY 2014
SNR and Noise Variance Estimation in Polarimetric SAR Data Michelangelo Villano, Student Member, IEEE
Abstract—Thermal noise affects polarimetric measurements; but a first order noise correction can be applied if the noise variance (or power) is known. This letter deals with the estimation of the noise variance and the signal-to-noise ratio (SNR) of the cross-polarized channels in polarimetric synthetic aperture radar (SAR) data. Cram´er–Rao lower bounds (CRLB) and maximum likelihood (ML) estimators are derived. The ML noise variance estimator is unbiased and efficient, while the ML SNR estimator is biased, but an unbiased SNR estimator can be derived from the biased one. It is also shown that commonly used noise variance and SNR estimators are biased. The results are finally validated using TerraSAR-X fully polarimetric data. Index Terms—Estimation, noise, polarimetry, signal-to-noise ratio, synthetic aperture radar.
I. Introduction
F
ULLY polarimetric synthetic aperture radar (SAR) systems allow the retrieval of all the elements of the scattering (or Sinclair) matrix, from which several other measurements, such as the covariance matrix and the Stokes matrix, can be derived [1]. As pointed out in [2], the scattering matrix terms are affected by thermal noise and so are the polarimetric measurements derived from the scattering matrix. Thermal noise cannot be removed from the measurements; however a first order correction can be applied to averaged covariance matrix or Stokes matrix values if the noise variance (or power) is known [1], [2]. If such first order noise corrections are not applied, several measures commonly derived from polarimetric SAR data may give erroneous results, hence the importance of an unbiased and accurate estimation of the noise variance [2]. A crude estimate of an upper bound to the noise variance can be obtained from the backscatter of one of the crosspolarized channels of a target demonstrating known Bragg scattering behavior (e.g., a water surface) [2]. For a monostatic system, a reciprocal propagation medium and reciprocal scatterers, the property for which the data of the two crosspolarized channels are equal, but for the thermal noise, which adds in the receivers, can be exploited, as suggested in [3], where a noise variance estimator is also derived. The latter estimator, used for the first-order noise correction in [4]–[7], is however negatively biased, as shown in the following.
Manuscript received November 28, 2012; revised February 22, 2013 and March 20, 2013; accepted March 26, 2013. Date of publication June 7, 2013; date of current version November 8, 2013. The author is with the Microwaves and Radar Institute, German Aerospace Center (DLR), Wessling 82234, Germany (e-mail:
[email protected]). 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/LGRS.2013.2255860
Furthermore, the estimation of the signal-to-noise ratio (SNR) of the different polarimetric channels is also of importance; as different applications have different requirements in terms of SNR, the estimated SNR lets us understand whether or not a dataset or part of it is suitable for a given application [1]. In particular, due to the low backscatter, the SNR of the cross-polarized channels can be particularly critical. II. Statement of the Problem Let us assume that single-look fully polarimetric SAR data, acquired by a monostatic system, are available, that channel imbalance and cross-talk calibration has been performed, and that data have not been symmetrized. Let us denote as u1 [i], i = 0 . . . N – 1, and u2 [i], i = 0 . . . N – 1, the complex amplitudes of the two cross-polarized channels for N independent resolution cells, over which a distributed target extends. Following the property, for which the data of two cross-polarized channels are equal, but for an additive term due to the thermal noise, each of the two sequences uk [i], k = 1, 2, can be written as the sum of a common sequence s[i], representing the useful signal, and a sequence wk [i], k = 1, 2, representing the additive thermal noise contribution uk [i] = s [i] + wk [i] , i = 0 . . . N − 1, k = 1, 2.
(1)
It is assumed that s[i], i = 0 . . . N – 1, are N independent realizations of a circularly symmetric Gaussian random variable with mean zero and variance A2 . This is, in fact, the behavior of a distributed target, whose radar cross-section (RCS) is equal to A2 , in case the speckle is fully developed [8]. It is also assumed that wk [i], k = 1, 2, i = 0 . . . N – 1, are 2N independent realizations of a circularly symmetric Gaussian random variable with mean zero and variance σ 2 . Moreover, it is assumed that s[i], i = 0 . . . N – 1, and wk [j], k = 1, 2, j = 0 . . . N – 1, are uncorrelated. The SNR of the cross-polarized channels SNR is defined as A2 SNR = 2 . (2) σ We would like to estimate the SNR and the noise variance σ 2 from uk [i], i = 0 . . . N – 1, k = 1, 2, under the stated assumptions. III. Cram e´ r−Rao Lower Bound The Cram´er–Rao lower bound (CRLB) is the minimum variance achievable by any unbiased estimator [9]. If an estimator is unbiased and its variance is equal to the CRLB, it is the minimum variance unbiased (MVU) estimator [9].
c 2013 IEEE 1545–598X
VILLANO: SNR AND NOISE VARIANCE ESTIMATION IN POLARIMETRIC SAR DATA
The CRLBs for the joint estimation of SNR and σ 2 are given by the diagonal elements of the inverse of the 2 × 2 Fischer information matrix J(SNR, σ 2 ) [9], defined as ⎡
⎤ ∂2 ln px (x) ∂2 ln px (x) −E ⎢ −E ⎥ ⎢ ∂SNR∂(σ 2 ) ⎥ ∂SNR2 2 ⎢ ⎥
J SNR, σ = ⎢ ∂2 ln px (x) ∂2 ln px (x) ⎥ ⎣ ⎦ −E −E ∂(σ 2 )SNR ∂(σ 2 )2 (3) where E{·} denotes the expected value and px (x) is the joint probability density function (pdf) of the vector x, defined as
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and the natural logarithm of px (x) is given by ln px (x) = −2N ln (π) − 2N ln σ 2 − N ln (2SNR + 1) 2N−1 N−1 ∗ 1 SNR + 1 SNR 2 |xi | − − 2 2Re xi xi+N . σ 2SNR + 1 i=0 2SNR + 1 i=0 (11) The 2 × 2 Fischer information matrix J(SNR, σ 2 ) can be therefore obtained substituting (11) in (3) (2SNR + 1)2 2N σ 2 (2SNR + 1) 4N 2 J SNR, σ = 2N σ 2 (2SNR + 1) 2N σ 4 (12)
x= [u1 [0] u1 [1] · · · u1 [N−1] u2 [0] u2 [1] · · · u2 [N − 1]]T . (4) Under the stated assumptions px (x) is given by exp −xH C−1 x x px (x) = (5) π2N det(Cx )
and its inverse J−1 (SNR, σ 2 ) is given by 2 2 (2N) (2SNR (2N) (2SNR+1) −σ + 1) . J−1 SNR, σ 2 = −σ 2 (2SNR+1) (2N) σ4 N (13)
where Cx is the 2N × 2N covariance matrix of x, the apices H and −1 indicate the Hermitian transpose and matrix inversion operators, and det( · ) indicates the matrix determinant. The elements of Cx are given by ∗ Cxi,i = E s [i] +w1+floor(i/N) [i] s [i] + w1+floor(i/N) [i]
The CRLBs for the joint estimation of SNR and σ 2 are therefore given by
Cx i,i+N
= A2+ σ 2 = σ 2 (SNR + 1) , i = 0 . .. 2N − 1 = E (s [i] + w1 [i]) (s [i] + w2 [i])∗
Cx i+N,i
= A2 = σ 2 SNR, i = 0..N − 1 = E (s [i] + w2 [i]) (s [i] + w1 [i])∗
Cx i,j
var{σˆ 2 } ≥
= A2 = σ 2 SNR, i = 0 . . . N − 1 = 0, for all the other elements of the matrix.
(6)
Cx can be therefore rewritten as a block matrix 2 σ (SNR + 1) IN σ 2 SNR IN Cx = σ 2 SNR IN σ 2 (SNR + 1) IN
(7)
where IN is the identity matrix of size N. The determinant and the inverse of the block matrix Cx are given by [10], [11] det (Cx ) = σ C−1 x
4N
(2SNR + 1)
1 = (2SNR + 1) σ 2
N
(8)
(SNR + 1) IN −SNR IN
−SNR IN (SNR + 1) IN
(9)
respectively. The quantity xH C−1 x x in (5) can be expanded as xH C−1 x x =
2N−1 2N−1 i=0
=
i=0
1 = 2 σ
−1 xi∗ Ci,j xj
−1 |xi |2 + Ci,i
N−1
∗ −1 ∗ Ci,i+N xi xi+N + xi+N xi
i=0 2N−1 SNR + 1 SNR |xi |2 − 2SNR + 1 i=0 2SNR + 1 N−1 2Re xi∗ xi+N i=0
(2SNR + 1)2 2N
σ4 N
(14) (15)
respectively, where ∧ denotes the estimate of a quantity. In case the noise variance σ 2 is known (e.g., accurate physical measurements are available or a very accurate estimate has been carried out using the entire dataset, in case σ 2 is stationary over the entire dataset), the CRLB for the SNR estimation is given by the inverse (or reciprocal) of the singleelement Fischer information matrix J(SNR), given by J (SNR) = −E
∂2 ln px (x) ∂SNR2
=
4N . (2SNR + 1)2
(16)
It therefore holds ˆ var{SNR} ≥
(2SNR + 1)2 . 4N
(17)
By comparison with (14), it can be noticed that the CRLB is by a factor of two better, if the noise variance σ 2 is known.
IV. Maximum Likelihood Estimation
j=0
2N−1
ˆ var{SNR} ≥
(10)
The maximum likelihood (ML) estimates of SNR and σ 2 are the values of SNR and σ 2 for which the pdf of the observation vector in (5) is maximum [9]. In order to derive a closed-form expression for the ML estimates, the expression in (5) has to be maximized with respect to each of the two variables. As the natural logarithm is a strictly monotonic function, this is equivalent to maximize the natural logarithm of the pdf in (5), which is given in (11). In particular, the first-order partial derivatives (with respect to SNR and σ 2 ) of the expression in (11) have to be set equal to zero ∂ ln px (x) ∂SNR = 0 (18) ∂ ln px (x) ∂ σ 2 = 0.
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 1, JANUARY 2014
By solving for SNR and σ 2 , one obtains the ML SNR and noise variance estimates, which are given by N−1
2
i=0 N−1
ˆ ML = SNR
Re u∗1 [i] u2 [i] (19)
|u1 [i] − u2 [i] |
2
to zero the first-order partial derivative with respect to SNR of the expression in (11) ∂ ln px (x) = 0. (26) ∂SNR The ML estimate of SNR, which is also function of the noise variance σ 2 , is therefore given by N−1
i=0 N−1 i=0
2 σˆ ML =
(20)
2N
respectively. On the basis of the assumptions of Section II, the numerator of the expression on the right side of (20) is a gamma distributed random variable with shape factor N and scale factor 2σ 2 . It follows that [12]
E
N−1 i=0
2 E σˆ ML =
var
2 σˆ ML
N−1 i=0
=
N 2σ
=
(2N)
|u1 [i] − u2 [i] |2
(21)
(2N)2
2 2 2
2Nσ 2 = σ2 2N
=
2N var
=
|u1 [i] − u2 [i] |2
σ4 N
(22)
where var{ · } indicates the variance. The ML noise variance estimator is therefore unbiased and efficient, as its variance is equal to the CRLB given in (15). As far as the ML SNR estimator is concerned, as shown in Appendix, it is biased, as it holds ˆ ML = SNR + 2SNR + 1 . E SNR (23) 2 (N − 1) However, from the biased ML SNR estimator of (19) an unbiased SNR estimator can be defined as follows: N −1 ˆ ˆ ML + 1 . SNRunb = SNR (24) N 2N As shown in Appendix, the variance of the unbiased estimator of (24) is given by
ˆ unb = var SNR
2N − 1 2N − 4
(2SNR + 1)2 . 2N
(25)
By comparison with the CRLB given in (14), this estimator is not efficient, but it becomes efficient, as N get large. In case the noise variance σ 2 is known, the ML estimate of the SNR of the cross-polarized channels is obtained by setting ⎛ 2 σˆ EB =
1⎜1 ⎝ 2 N
N−1 i=0
|u1 [i]|2 +
1 N
N−1 i=0
1 − . (27) 4Nσ 2 2 On the basis of the assumptions of Section II, the numerator of the first term on the right-hand side of (27) is a gamma distributed random variable with shape factor N and scale factor σ 2 (2SNR + 1). It follows that [12] N−1
2 |u1 [i] + u2 [i]| E 2 i=0 ˆ ML σ −1 2 = E SNR 2 4Nσ 2Nσ 2 (2SNR + 1) 1 = − = SNR (28) 4Nσ 2 2 N−1 2 2 var i=0 |u1 [i] + u2 [i]| ˆ ML σ var SNR = (4Nσ 2 )2 4 4Nσ (2SNR + 1)2 (2SNR+1)2 = = . (29) 16N 2 σ 4 4N In case the noise variance is known, the ML SNR estimator is therefore unbiased and efficient, as its variance is equal to the CRLB given in (17). ˆ ML (σ 2 ) = SNR
|u1 [i] − u2 [i] |2
|u1 [i] + u2 [i] |2
i=0
V. Other Estimators As mentioned in Section I, a noise variance estimator has been proposed in [3], where the noise variance is estimated as the smallest eigenvalue of the estimated covariance matrix of the two-element vector [u1 u2 ]. The expression of the noise variance estimate, from now on referred to as eigenvalue-based (EB), is given by (30), as shown at the bottom of the page [11]. The pdf of the EB noise variance estimate in (30) is provided in [13, eq. 39]. Having defined for an estimator aˆ the relative bias θrel and the relative accuracy σrel as E {ˆa} − a θrel = (31) a √ var {ˆa} (32) σrel = a respectively, where a is the true value of the parameter to be estimated, the relative bias and accuracy of the EB noise variance estimator can be obtained by numerical integration and are displayed in Fig. 1 for different values of N. CRLBs are indicated using dashed lines.
⎞ $ N−1 $2 " N−1 #2 N−1 $ $ 1 1 $1 $ ⎟ |u2 [i]|2 −! |u1 [i]|2 − |u2 [i]|2 + 4 $ u∗1 [i]u2 [i]$ ⎠ $N $ N i=0 N i=0 i=0
(30)
VILLANO: SNR AND NOISE VARIANCE ESTIMATION IN POLARIMETRIC SAR DATA
Fig. 1. (a) Relative estimation bias and (b) accuracy for the EB noise variance estimator for different values of N. Dashed lines correspond to the CRLB.
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Fig. 3. (a) Histograms of ML and EB noise variance estimates in decibel. (b) Histograms of unbiased (24) and CB SNR estimates in decibel.
Data have been calibrated and a 1024 × 1024 pixel patch, corresponding to a rather homogeneous area has been extracted. The noise variance estimators given by (20) and (30), as well as the SNR estimators given in (24) and (33) have been applied to the data, using an 11 × 11 pixel window for spatial averaging. This corresponds to N = 35 independent resolution cells (the range/azimuth pixel spacing is smaller than the range/azimuth resolution). The histograms of the noise Fig. 2. (a) Relative estimation bias and (b) accuracy for the CB SNR variance estimates and the SNR estimates are provided in estimator for different values of N. Dashed lines correspond to the CRLB. decibel in Fig. 3(a) and (b), respectively. As apparent from Fig. 3(a), the noise variance EB estimator As apparent, the EB noise variance estimator is negatively gives smaller estimates than the ML one and this is consistent biased and the relative bias is a function of SNR. As the ML with the fact that the EB estimator is negatively biased and noise variance estimator, given in (20), is unbiased, the ML the ML estimator is unbiased. For a quantitative analysis, the estimator should be preferred to the EB estimator. As apparent difference of the mean values of the EB and ML noise variance from Fig. 1(b) the relative accuracy of the EB noise variance estimates, divided by the mean value of the ML estimate, as estimator is better than the CRLB. This is not in contradiction resulting from the histograms, is equal to −0.053. This is in with the CRLB definition, as the CRLB represents a bound good agreement with the relative bias given in Fig. 1(a) for N = 35 for values of SNR comprised between 1 (0 dB) and 5 only for unbiased estimators [9]. The SNR of the cross-polarized channels can be estimated (7 dB), where the range of values of SNR are deducted from from the estimated coherence magnitude between the two the histograms of Fig. 3(b). As apparent from Fig. 3(b), instead, the SNR CB estimator cross-polarized channels [1]. The resulting SNR estimate, from now on referred to as coherence-based (CB) SNR estimate, is gives larger estimates than the unbiased one of (24) and this is consistent with the fact that the CB estimator is positively defined as biased and the estimator of (24) is unbiased. For a quantitative $N−1 $( N−1 N−1 analysis, the difference of the mean values of the CB and $ $ $ $ unbiased SNR estimates, divided by the mean value of the |u1 [i]|2 |u2 [i]|2 u1 [i]u∗2 [i]$ ! $ $ $ unbiased one, as resulting from the histograms, is equal to i=0 k=0 k=0 ˆ CB = SNR . 0.098. This is in good agreement with the values of the curve $N−1 $( N−1 $ $ N−1 N = 35 in Fig. 2(a) for values of SNR comprised between 1 $ $ |u1 [i]|2 |u2 [i]|2 1−$ u1 [i]u∗2 [i]$ ! (0 dB) and 5 (7 dB). i=0 $ $ k=0
k=0
(33) Fig. 2 displays the relative bias and accuracy for the CB SNR estimator, as obtained by Monte-Carlo simulation. Gaussian distributed random variables u1 and u2 have been generated, correlated as described in Section II. The quantity given in (33) has been then computed. As apparent, unlike the SNR estimator given in (24), the CB SNR estimator is positively biased. As the SNR estimator, given in (24), is unbiased, the latter should be preferred to the CB estimator. As apparent from Fig. 2(b), except for very low SNR values, the relative accuracy of the CB SNR estimator is worse than the CRLB.
VII. Conclusion This letter proposed an unbiased and efficient noise variance estimator, which should be preferred to the biased EB one for first-order noise correction of polarimetric SAR data. Furthermore, an unbiased estimator for the SNR of the crosspolarized channels was proposed, which becomes efficient as N gets large and which should be preferred to the biased CB one. Evidence of the biases was shown using TerraSAR-X fully polarimetric data. Appendix Let us consider the following estimator: N−1
VI. Validation of the Results with Real Data The results are validated by using TerraSAR-X fully polarimetric data, acquired over Austfonna, Svalbard, Norway, during the dual-receive antenna (DRA) campaign.
ˆ = Z
i=0 N−1 i=0
|u1 [i] + u2 [i] |2 . |u1 [i] − u2 [i] |2
(A1)
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 1, JANUARY 2014
N −1 2N
ˆ −1 Z ⎞ ⎛ 2 N−1 2 |u1 [i] + u2 [i] | ⎟ ⎜ ⎟ N −1 ⎜ ⎟ ⎜ i=0 = ⎟− ⎜ N−1 ⎟ ⎜ 2N ⎝ 2⎠ |u [i] + u [i] |
ˆ unb = SNR
1
(N − 1)
2
"N−1i=0
|u1 [i] + u2 [i] | − 2
i=0
=
2N " (N − 1) 4
N−1
N −1 N
|u1 [i] − u2 [i] |
Re u∗1 [i] u2 [i]
−
2
−
N−1
|u1 [i] − u2 [i] |2
i=0
|u1 [i] − u2 [i] | N−1
(A5)
2
|u1 [i] − u2 [i] |2
i=0
N−1
|u1 [i] − u2 [i] |2
i=0
ˆ ML + SNR
1 2N
On the basis of the assumptions of Section II, and considering that (u1 [i] + u2 [i]) and (u1 [i]−u2 [i]) are uncorrelated Gaussian random variables, the estimator of (A.1) has a beta pdf and its expected value is given by [12] ˆ = E Z
#
i=0 #
2N =
N−1
i=0 N−1
i=0
=
1 2
N (2SNR − 1) N −1
(A2)
An unbiased SNR estimator can be therefore obtained from the one given in (A.1), considering (A.2), as ˆ unb = N − 1 Z ˆ −1 SNR (A3) 2N 2 The variance of this unbiased estimation is given by [12] 2 ˆ unb = N − 1 var Z ˆ var SNR 2N " # (2SNR + 1)2 2N − 1 = (A4) 2N − 4 2N It can be shown that the unbiased estimator of (A3) coincides with the estimator of (26), i.e., the estimator obtained from the ML biased one As the estimator of (26) and (A3) coincide, they also have the same expected value and variance. The estimator of (26) is therefore unbiased and its variance is given by (27) or (A4). This reasoning also demonstrates that the expected value of the ML SNR estimator is given by (25).
Acknowledgment The author would like to thank his colleagues F. De Zan, M. Martone, and K. P. Papathanassiou for many constructive discussions, and the anonymous reviewers for their helpful comments to the manuscript.
References [1] S. R. Cloude, Polarization: Application in Remote Sensing. New York, NY, USA: Oxford University Press, 2010. [2] A. Freeman, “The effect of noise on polarimetric SAR data,” in Proc. Int. Geosci. Remote Sens. Symp., Tokyo, Japan, 1993, pp. 799–802. [3] I. Hajnsek, K. P. Papathanassiou, and S. R. Cloude, “Removal of additive noise in polarimetric eigenvalue processing,” in Proc. Int. Geosci. Remote Sens. Symp., Sydney, Australia, 2001, pp. 2778–2780. [4] I. Hajnsek, E. Pottier, and S. R. Cloude, “Inversion of surface parameters from polarimetric SAR,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 4, pp. 727–743, Apr. 2003. [5] M. Migliaccio, A. Gambardella, F. Nunziata, M. Shimada, and O. Isoguchi, “The PALSAR polarimetric mode for sea oil slick observation,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 12, pp. 4032–4041, Dec. 2009. [6] J. J. Sharma, I. Hajnsek, K. P. Papathanassiou, and A. Moreira, “Polarimetric decomposition over glacier ice using long-wavelength airborne PolSAR,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 1, pp. 519–535, Jan. 2011. [7] J. M. Lopez-Sanchez, S. R. Cloude, and J. D. Ballester-Berman, “Rice phenology monitoring by means of SAR polarimetry at X-band,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 7, pp. 2695–743, Jul. 2012. [8] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images. Raleigh, NC, USA: SciTech Publishing, 2004. [9] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Upper Saddle River, NJ, USA: Prentice-Hall, 1993. [10] J. R. Silvester, “Determinants of block matrices,” Math. Gazette, vol. 84, no. 501, pp. 460–467, Nov. 2000. [11] F. Ayres, “The inverse of a matrix,” in Theory and Problems of Matrices. Shaum’s Outline Series. New York, NY, USA: McGraw-Hill, 1962, pp. 55–63. [12] K. Krishnamoorthy, Handbook of Statistical Distributions with Applications. Boca Raton, FL, USA: Chapman & Hall/CRC, 2006. [13] I. C. Sikaneta and J.-Y. Chouinard, “Eigendecomposition of the multichannel covariance matrix with applications to SAR-GMTI,” Signal Process., vol. 84, no. 9, pp. 1501–1535, Sep. 2004.
