Thursday ACFrery DistancesCCC

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References

Contrast Measures based on the

Complex Correlation Coefficient

for PolSAR Imagery

Alejandro C. Frery Renato J. Cintra Abraão Nascimento

LaCCAN

Laboratório de Computação Científica e Análise Numérica

Universidade Federal de Alagoas, Brazil

APSAR 2013 September 26

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Motivation & Model

Summary

1 Motivation and Model

2 Information theory

Distances between two CCCs as PolSAR attributes New contrast method for two PolSAR regions based on CCCs

3 Application to actual data

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General Discussion: Literature Review

1 Existent Theoretical Tools: Lee et al. (1994) proposed

a reparametrization for the covariance matrix in terms of the complex correlation coefficient. That work provides statistical models which describe the multilook phase difference, the magnitude of the complex product, and both intensity and amplitude ratios between two

components of the scattering matrix. The resulting density functions have closed forms which depend on the complex correlation coefficient (CCC) and on the number of looks.

2 Statistical Analysis of a PolSAR Attribute: Many

authors have utilized the complex correlation coefficient as an important quantity for analyzing PolSAR images. For instance, Ainsworth et al. (2008) presented evidence that the complex correlation coefficient between channels can be utilized to identify man-made targets.

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General Discussion: Literature Review

1 Existent Theoretical Tools: Lee et al. (1994) proposed

a reparametrization for the covariance matrix in terms of the complex correlation coefficient. That work provides statistical models which describe the multilook phase difference, the magnitude of the complex product, and both intensity and amplitude ratios between two

components of the scattering matrix. The resulting density functions have closed forms which depend on the complex correlation coefficient (CCC) and on the number of looks.

2 Statistical Analysis of a PolSAR Attribute: Many

authors have utilized the complex correlation coefficient as an important quantity for analyzing PolSAR images. For instance, Ainsworth et al. (2008) presented evidence that the complex correlation coefficient between channels can be utilized to identify man-made targets.

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General Discussion: Goals

Proposed Contrast Measures between CCCs: In summary, our contributions are two-fold:

1 Based on the parametrization proposed by Lee et al.

(1994), we derive four contrast measures which depend on the complex correlation coefficient. These measures were obtained considering four distances from the h-φ class of distances proposed by Salicrú et al. (1994): the

Kullback-Leibler, Rényi (of order β), Bhattacharyya, and Hellinger distances between reparametrized scaled complex Wishart distributions.

2 We study the asymptotic properties of these measures, and

we propose new confidence regions for these distances which allow comparing two PolSAR regions.

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General Discussion: Goals

Proposed Contrast Measures between CCCs: In summary, our contributions are two-fold:

1 Based on the parametrization proposed by Lee et al.

(1994), we derive four contrast measures which depend on the complex correlation coefficient. These measures were obtained considering four distances from the h-φ class of distances proposed by Salicrú et al. (1994): the

Kullback-Leibler, Rényi (of order β), Bhattacharyya, and Hellinger distances between reparametrized scaled complex Wishart distributions.

2 We study the asymptotic properties of these measures, and

we propose new confidence regions for these distances which allow comparing two PolSAR regions.

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Lee model based on the correlation coefficient

Lee et al. (1994) presented a reparametrization of a particular case of the complex Wishart distribution based on a

two-element scattering vector y(k)₌_y(k) 1 , y

(k) 2

>

at the kth look, for which the sample covariance matrix is written as

Z = 1 L L X k=1 y(k)y(k)∗= z11 αei∆ αe−i∆ _z 22 ,

where> is the transposition operator,∗ represents the Hermitian operator, α and ∆ are the sample multilook magnitude and phase, respectively, and

zii = L−1P L k=1y (k) i y (k)∗

i , for i = 1, 2. The expected value of Z,

E(Z), is Σ = σ11 √ σ11σ22|ρc|eiδ √ σ11σ22|ρc|e−iδ σ22 , (1)

where σii = E(zii), δ is the population multilook phase, and ρc

is the complex correlation coefficient between z11 and z22.

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Lee model based on the correlation coefficient

In order to study the preservation of polarimetric properties along the process of filtering PolSAR imagery, Lee et al. (1999) studied the correlation coefficient between polarization channels SHHand SVV. It is given by

ρc=

E(SHHSVV∗ )

pE(|SHH|2) E(|SVV|2)

. (2)

The normalized quantities B1= z11 σ11 , B2= z22 σ22 , η =√ α σ11σ22 ,

and ∆ obey the distribution characterized by the following joint probability density function:

f (B1, B2, η, ∆; ρc, L) = η(B1B2− η2)L−2 π(1 − |ρc|2)LΓ(L)Γ(L − 1) × expn−B1+ B2− 2η|ρc| cos(∆ − δ) 1 − |ρc|2 o . (3)

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Information theory

Summary

4 Conclusion

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General Discussion

Statistical Information Theory

The information provided by the event ξ can be computed as − log Pr(ξi).

The entropy is the expected value of the information; it measures the uncertainty, how unpredictable a process is. In particular, the Shannon entropy of an absolutely continuous random variable X with density f is

H(X) = E(− log f (X)) = − Z

f log f, i.e., the expected value of the information, where E(·) is expectation operator.

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The h-φ class of distances

Definition

Let X and Y be random matrices with densities fX(Z0; θ1)

and fY(Z0; θ2), respectively, with support A. The h-φ class of

divergences is given by Dφh(X, Y ) = h Z A φfX(Z 0_{; θ} 1) fY(Z0; θ2) fY(Z0; θ2)dZ0 , where h : (0, ∞) → [0, ∞) is a strictly increasing function with h(0) = 0 and φ : (0, ∞) → [0, ∞) is a convex function such that 0 φ(0/0) = 0, and 0 φ(x/0) = limx→ ∞φ(x)/x.

Symmetrization

Divergences can be turned into distances dh_φ(X, Y ) = 1 2D h φ(X, Y ) + D h φ(Y , X).

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Distances obtained by the symmetrization of

divergences

They satisfy the following properties:

1 Non-negativity dh_φ(X, Y ) ≥ 0 2 Symmetry dh_φ(X, Y ) = dh_φ(Y , X) 3 Definiteness dh_φ(X, Y ) = 0 ⇔ X = Y

Nascimento et al. (2010) applied these distances to the analysis of contrast between regions of intensity SAR data, while Frery et al. (in press-) derived analytic expressions of these distances under the multilook complex Wishart model.

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Table 1 : h-φ distances and related functions

Distances Functions h and φ

Kullback-Leibler : dKL(X, Y ) h(y) = y/2 and

1

2R (fX− fY) logffX_Y φ(x) = (x − 1) log x

Rényi (with order β) : dβ_R(X, Y ) h(y) = _β−11 log((β − 1)y + 1),

1 β−1log R fβ Xf 1−β Y +R f 1−β X f β Y 2 φ(x) =x1−β+x_2(β−1)β−β(x−1)−2, 0 ≤ y < (1 − β)−1and 0 < β < 1 Hellinger : dH(X, Y ) h(y) = y/2

1 −R √fXfY

φ(x) = √x − 12

and 0 ≤ y < 2 Bhattacharyya : dB(X, Y ) h(y) = − log(1 − y)

− logR √fXfY

φ(x) = −√x +x+1₂ and 0 ≤ y < 1

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Hypothesis tests based on distance measures

Let bθ1= (bθ11, . . . , bθ1M) and bθ2= (bθ21, . . . , bθ2M) be ML

estimators of θ1 and θ2 based on random samples of sizes N1

and N2, respectively.

Under the regularly conditions discussed by Salicrú et al. (1994, p. 380), the following lemma can be verified:

Lemma

If N1 N1+N2 −−−−−−−→_N1,N2→∞ λ ∈ (0, 1) and θ1= θ2, then S_φh( bθ1, bθ2) = 2N1N2 N1+ N2 dh φ( bθ1, bθ2) h0_(0)φ00₍₁₎ D −−−−−−−→ N1,N2→∞ χ2_M.

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Hypothesis tests based on divergence measures

Based on the previous lemma, tests for the null hypothesis H0: θ1= θ2 can be derived as in the following proposition.

Proposition

Let N1 and N2 be sufficiently large and Sφh( bθ1, bθ2) = s, then the

null hypothesis H0: θ1= θ2 is rejected at level α whenever

Pr(χ2

M > s) ≤ α.

In the next slides, we will apdot SM for denoting the derived

stochastic statitics, for M ∈ {KL, H, B, R}.

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Information theory Distances between CCCs

Analytical Contrast Measures

Let θi= [ρi, L]>, for i = 1, 2. We derived closed form

expressions for the distances presented in the last table:

dKL(θ1, θ2) = L (1 − |ρ1||ρ2|)(2 − |ρ1|2_{− |ρ2|}2₎ (1 − |ρ1|2_{)(1 − |ρ}_2|2₎ − 2 . (4) dβ_R(θ1, θ2) = log 2 1 − β+ 1 β − 1log (" (1 − |ρ1|2₎1−β_{(1 − |ρ}_2|2₎β 1 − {|ρ1| − β(|ρ1| − |ρ2|)}2 #L + " (1 − |ρ2|2₎1−β_{(1 − |ρ}_1|2₎β 1 − {|ρ2| − β(|ρ2| − |ρ1|)}2 #L) . (5) dB(θ1, θ2) =L log(1 − |ρ1|2_{) + log(1 − |ρ2|}2₎ 2 − 2 log 2 + log _{4 − (|ρ}_{1| + |ρ2|)}2 (1 − |ρ1|2)(1 − |ρ2|2) . (6) dH(θ1, θ2) = 1 − " 4p(1 − |ρ1| 2_{)(1 − |ρ}_2|2₎ 4 − (|ρ1| + |ρ2|)2 #L . (7)

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Information theory Distances between CCCs

An implication

Statistics derived from these distances can be used to measure the influence of the number of looks on the complex correlation coefficient in Z. The plots show the resulting statistics for |ρ1|2= 0.5 and L1= L2= L ∈ {2, 5} (Figures 1(a) and 1(b),

respectively), while |ρ2| varies on [0, 1].

(a) L = 2 (b) L = 5

Figure 1 : Sensitivity of statistics in (4)-(7) for L1= L2= L = {2, 5}, and |ρ1|2= 0.5.

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Information theory New contrast measure

New contrast method for two PolSAR regions

based on CCCs

From the previous lemma, four new hypothesis tests are derived for checking whether two samples, of sizes N1 and N2, come

from regions with statistically similar correlation coefficients between channels. As a consequence, the resulting statistics can be also used as confidence regions. For simplicity, we present only the results relative to the Kullback-Leibler and Hellinger distances.

The statistics based on Kullback-Leibler and Hellinger

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New contrast method for two PolSAR regions

based on CCCs

Given a PolSAR image, let L be the number of looks known and constant on this image,

RKL(ρ1, ρ2| L) = n (ρ1, ρ2) ∈ C × C : (1 − |ρ1||ρ2|)(2 − |ρ1|2− |ρ2|2) (1 − |ρ1|2)(1 − |ρ2|2) | {z } ξ1 ≤(N1+ N2) 2 L N1N2 χ2₁(η) + 2 | {z } tKL o (8) and RH(ρ1, ρ2| L) = n (ρ1, ρ2) ∈ C × C : p(1 − |ρ1|2)(1 − |ρ2|2) 4 − (|ρ1| + |ρ2|)2 | {z } ξ2 ≥1 4 h 1 −(N1+ N2) 2 L N1N2 χ2₁(η)i 1/L | {z } tH o , (9)

where η is the specified nominal level.

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New contrast method for two PolSAR regions

based on CCCs

Given two polarimetric data samples, their correlation coefficients can be estimated according to Eq. (2). Quantities b

ξ1, bξ2, tKL, and tH are evaluated by means of (8) and (9),

replacing ρ1 and ρ2by their estimates (or sample counterparts)

b

ρ1 andρ_b2. We then propose the following decision rules:

• Kullback-Leibler criterion: If bξ1≤ tKL, then we have

statistical evidence that the samples come from populations with similar correlation between HH and VV channels. • Hellinger criterion: If bξ2≥ tH, the samples present

equivalent correlation between channels.

Thus, regions RKL(ρ1, ρ2| L) and RH(ρ1, ρ2| L) along with (2)

are methods for comparing two regions based on their estimates for the correlation coefficient.

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Application to actual data

Summary

4 Conclusion

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Application to actual data

In order to illustrate this methodology, consider the samples highlighted in Fig. 2 (which is extracted from an E-SAR image from surroundings of Weßling, Germany).

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Application to actual data

Table 2 lists the quantities observed, along with the decisions they led to. Notice that both rules discriminate well.

Table 2 : Results based on confidence regions

Regions ξb1 tKL Decision ξb2 tH Decision

D1-D2 2.0220 2.0016 Distinct 0.2486 0.2499 Distinct D1-D3 2.0068 2.0012 Distinct 0.2495 0.2499 Distinct D1-D4 2.0734 2.0014 Distinct 0.2455 0.2499 Distinct D2-D3 2.0534 2.0018 Distinct 0.2467 0.2499 Distinct D2-D4 2.0149 2.0020 Distinct 0.2490 0.2498 Distinct D3-D4 2.1254 2.0016 Distinct 0.2424 0.2499 Distinct

Whereas subsamples from the same area did not provide enough evidence for rejecting the null hypothesis.

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Distances between Complex Correlation Coefficients on PolSAR Images A. C. Frery Motivation & Model Information theory Distances between CCCs New contrast measure Application to actual data Conclusion References Conclusion

Summary

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Conclusion

In this paper, we have derived four contrast measures in terms of the correlation coefficient and of the number of looks. Using asymptotic results for these measures, two methodologies based on the Kullback-Leibler and Hellinger distances were proposed as new discrimination techniques between PolSAR image regions. These methods were applied to actual data and the obtained results present evidence in favor of both the proposals.

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References I

Ainsworth, T. L., Schuler, D. L. & Lee, J. S. (2008), ‘Polarimetric SAR characterization of man-made structures in urban areas using normalized circular-pol correlation coefficients’, Remote Sensing of Environment 112(6), 2876–2885.

Frery, A. C., Nascimento, A. D. C. & Cintra, R. J. (in press-), ‘Analytic expressions for stochastic distances between relaxed complex Wishart distributions’, IEEE Transactions on Geoscience and Remote Sensing. Lee, J. S., Grunes, M. R. & de Grandi, G. (1999), ‘Polarimetric SAR

speckle filtering and its implication for classification’, IEEE Transactions on Geoscience and Remote Sensing 37(5), 2363–2373. Lee, J. S., Hoppel, K. W., Mango, S. A. & Miller, A. R. (1994), ‘Intensity

and phase statistics of multilook polarimetric and interferometric SAR imagery’, IEEE Transactions on Geoscience and Remote Sensing 32(5), 1017–1028.

Nascimento, A. D. C., Cintra, R. J. & Frery, A. C. (2010), ‘Hypothesis testing in speckled data with stochastic distances’, IEEE Transactions on Geoscience and Remote Sensing 48(1), 373–385.

Salicrú, M., Menéndez, M. L., Pardo, L. & Morales, D. (1994), ‘On the applications of divergence type measures in testing statistical hypothesis’, Journal of Multivariate Analysis 51, 372–391.

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Contact:

Alejandro C. Frery acfrery@gmail.com