Detecting Changes in Fully Polarimetric SAR Imagery with Statistical Information Theory
Abraão D. C. Nascimento Alejandro C. Frery Renato J. Cintra
cs.IT math.IT







Published in the IEEE TGRS
Nascimento et al.: Nascimento



Detecting Changes in Fully Polarimetric SAR Imagery with Statistical Information Theory
    Abraão D. C. Nascimento,
        Alejandro C. Frery, Senior Member, 
        and
        Renato J. Cintra, Senior Member
				
    May 3, 2018
=================================================================================================================================


Images obtained from coherent illumination processes are contaminated with speckle. 

A prominent example of such imagery systems is the polarimetric synthetic aperture radar (PolSAR).

For such remote sensing tool the speckle interference pattern appears in the form of a positive definite Hermitian matrix, 
which requires specialized models 
and makes change detection a hard task.

The scaled complex Wishart distribution is a widely used model for PolSAR images.

Such distribution is defined by two parameters: the number of looks and  the complex covariance matrix.

The last parameter contains all the necessary information to characterize the backscattered data and, thus, identifying changes in a sequence of images can be formulated as a problem of verifying whether the complex covariance matrices differ at two or more takes.

This paper proposes a comparison between a classical change detection method based on the likelihood ratio
and three statistical methods
that depend on information-theoretic measures: the Kullback-Leibler distance and two entropies.

The performance of these four tests was quantified in terms of their sample test powers and sizes using simulated data.

The tests are then applied to actual PolSAR data.

The results provide evidence that tests based on entropies may outperform those based on the Kullback-Leibler distance and likelihood ratio statistics.


contrast, information theory, Wishart, hypothesis test, change detection.




§ INTRODUCTION


Synthetic aperture radar (SAR) has been widely used as an important system for information extraction in remote sensing applications.

Such microwave active sensors have as main advantages the following features:
(i) their operation is not determined by day time, neither weather conditions
and
(ii) they are capable of providing high spatial image resolution.

In recent years, the interest in understanding such type of imagery in a multidimensional and multilook perspective has increased. 

Such systems are called “polarimetric SAR” (PolSAR).

In this case, obtaining of PolSAR data obeys the following dynamic: a scene is mapped with polarized pulses which are backscattered by the scene and captured by a sensor to form an image. 

As a result, PolSAR measurements record the amplitude and phase of backscattered signals for possible combinations of linear reception and transmission polarizations: HH, HV, VH, and VV (H for horizontal and V for vertical polarization).

However, since the acquired images stem from a coherent illumination process, they are affected by a signal-dependent granular noise called “speckle” <cit.>. 

Such noise has a multiplicative nature and its intensity does not follow the Gaussian law. 

Thus, analyzing PolSAR images requires tailored image processing based on the statistical properties of speckled data.

PolSAR theory prescribes that the returned (backscattered) signal of distributed targets is adequately represented by its complex covariance matrix.

Under the assumption that the complex scattering coefficients are jointly circular Gaussian, the Wishart distribution is the statistical model for multilook PolSAR data. 

This paper adopts the assumption that a PolSAR image is well described by such distribution.

Change detection methods aim at identifying differences in the scene configuration at distinct observation instants.
Such procedures have achieved a prominent position in recent decades <cit.>.
Indeed, literature reports several approaches for change detection problems, among them:



  * image ratioing <cit.>,

  * multitemporal coherence analysis <cit.>,

  * spatiotemporal contextual classification <cit.>,

  *  Hotelling-Lawley and likelihood ratio tests <cit.> and robust tests <cit.>,

  * combination of image ratioing and the generalized minimum-error method <cit.>, 

  * detection algorithms based on Lagrange optimization <cit.>,


  *  information-theoretic measures for change detection <cit.>
and

  * change detection with post-classification <cit.>. 


This paper advances points (<ref>) and (<ref>) above.

The change detection process is theoretically rooted in the hypothesis test theory and the proposal of statistical similarity measures <cit.>. 

In particular, hypothesis tests based on the complex covariance matrix have been sought for PolSAR data analysis. 

Many statistical approaches have been developed in order to reach this goal.

Conradsen et al. <cit.> proposed a methodology based on the likelihood ratio test defined by two random samples from the complex Wishart distribution. 

Subsequently, this technique was applied to edge detection in PolSAR images by Schou et al. <cit.>.

Recently, 
Conradsen et al. <cit.> 
extended likelihood-based detection for 
PolSAR time series.

Kersten and Ainsworth <cit.> compared three test statistics (the contrast ratio, ellipticity, and Bartlett tests).

It was found that the method based on the contrast ratio is more robust to variations in the covariance estimates on actual data.

In a complementary study, Molinier and Rauste <cit.> compared six polarimetric change detection methods.

As a conclusion, the methods directly derived from the Wishart distribution outperformed other approaches
as they provide explicit thresholds.

Recently, Akbari et al. <cit.> proposed a change detector involving the Hotelling-Lawley trace (HLT) which, asymptotically, follows the Fisher-Snedecor distribution.

The authors provided evidence that 
the HLT test may outperform the Bartlett test in some scenarios.

Several works have employed information-theoretic tools as a pre-processing step for change detection in PolSAR images.

They can be categorized into two approaches: 
one is based only on discrimination measures,
whereas the other considers the asymptotic distribution of such tools.

In the first category, Inglada and Mercier <cit.> proposed a new similarity measure for automatic change detection in multitemporal SAR images.

Such measure was derived considering the symmetrized Kullback-Leibler (KL) divergence (or distance) between the Edgeworth series expansions for two distinct elements of the 𝒦 distribution from the Pearson System <cit.> for intensity SAR data.

In <cit.>, the KL measure is improved by means of copula-based quantile regression to generate local change measures.

Further, Erten et al. <cit.> proposed a new method based on mutual information for quantifying the coherent similarity between temporal multichannel PolSAR images.
Atto et al. <cit.> used the KL divergence for spatio-temporal change detection in image time series.

In the second category, Nascimento et al. <cit.> derived hypothesis tests based on several distance measures between 𝒢^0 distributions <cit.>. 

In terms of the nature of the image data, these results were extended in <cit.> and applied to boundary detection <cit.> and filtering <cit.> in PolSAR images.  

All these references derived new proposals using contrast measures
designed from the scaled complex Wishart law.

Recently,
Akbari et al. <cit.>
introduced
a change detector with
the HLT statistics as
the contrast measure based on the relaxed scaled Wishart likelihood.

This paper proposes three new change detection methodologies for fully polarimetric data.

Additionally, a new expression for the likelihood ratio statistics obtained from the scaled Wishart distribution is achieved, 
and its relationship with the individual distributions of the intensity channels is discussed.    

Using Monte Carlo simulation, we quantify the performance of four parametric methodologies for detecting the change: two considering Shannon and Rényi entropies, one stemming from the Kullback-Leibler distance, and one based on the classic likelihood ratio statistics.

The methods are compared by their empirical test size and power.

Finally, two experiments with actual PolSAR data are performed. 

Results provide evidence that the methods based on entropies
are superior.

This paper is organized as follows.

Section <ref> provides the background of the statistical modeling.

A brief survey on parametric methodologies for hypothesis testing on complex covariance matrices is provided in Section <ref>.

In Section <ref>, we present a comparative study of change detection methods by means of Monte Carlo simulation.

Additionally, we perform two experiments with actual PolSAR data. 

Section <ref> summarizes the main results.




§ STATISTICAL MODELING FOR POLSAR DATA


PolSAR systems represent each resolution cell by p polarization elements comprising a complex random vector:


    y=[S_1  S_2 ⋯  S_p]^⊤
    ,


where the superscript ^⊤ is the vector transposition.

In single-look PolSAR image processing,
y is admitted to obey the multivariate complex circular Gaussian distribution with zero mean <cit.> whose probability density function (pdf) is:

    f_y(ẏ;Σ)=1/π^p|Σ|exp(-ẏ^*Σ^-1ẏ),

where ẏ is an outcome of y,  |·| is the matrix determinant, the superscript ^* denotes the complex conjugate transpose of a vector, Σ is the covariance matrix of  y such that Σ=E{yy^*}, and E{·} is the statistical expectation operator.

This distribution is denoted by y∼𝒩^C( 0,Σ).

Besides being Hermitian and positive definite, Σ contains all the necessary information to characterize the backscattered data <cit.>.

In order to improve the signal-to-noise ratio, L independent and identically distributed samples are usually averaged in order to form the L-looks covariance matrix <cit.>:	

    Z=1/L∑_i=1^L y_iy_i^*,

where y_i, i=1,2,…,L are realizations of (<ref>).

Under the aforementioned hypotheses, Z follows a scaled complex Wishart distribution. 

Having Σ and L as parameters, such law is characterized by the following pdf:

    f_Z(Ż;Σ,L) = L^pL|Ż|^L-p/|Σ|^L Γ_p(L)exp[
    -Ltr(Σ^-1Ż)],

where Γ_p(L)=π^p(p-1)/2∏_i=0^p-1Γ(L-i), L≥ p, Γ(·) is the gamma function, and tr(·) is the trace operator.

We denote it by Z∼𝒲(Σ,L).
This distribution satisfies E{Z}=Σ, which is a Hermitian positive definite matrix <cit.>.

In practice, L is treated as a parameter and must be estimated.

The resulting distribution is the relaxed Wishart distribution, and it is denoted by 𝒲_ℛ(Σ,L) <cit.>. 

Due to its optimal asymptotic properties, 
we employ the maximum likelihood (ML) approach to estimate the parameters Σ and the equivalent number of looks L.

Let Z={Z_1,Z_2,…,Z_N} be a random sample of size N obtained from Z∼𝒲_ℛ(Σ,L). 

Setting ℓ_k(θ)=log f_Z(Z_k;Σ,L) for θ=[vec(Σ)^⊤, L]^⊤ as the log-likelihood of the kth random matrix, Z_k, from Z, solving N^-1∑_k=1^N ∇ℓ_k(θ)=0, we have that
Σ=N^-1∑_k=1^N Z_k,
and

    plogL+1/N∑_k=1^N log|Z_k|-log|Σ|-ψ_p^(0)(L)=0,

where vec(·) is the vectorization operator, 
ψ_p^(0)(·) is the zero order term of the vth-order multivariate polygamma function:

    ψ_p^(v)(L)=∑_i=0^p-1ψ^(v)(L-i),

and ψ^(v)(·) is the ordinary polygamma function expressed by

    ψ^(v)(L)=∂^v+1logΓ(L)/∂ L^v+1,

for v≥ 0; note that ψ^(0) is the digamma function <cit.>.

Thus, the ML estimator of Σ is the sample mean, while L is obtained
by solving the system shown in (<ref>). 
We used the Newton-Raphson iterative method <cit.> to solve it.

The work by Anfinsen et al. <cit.> is an important reference on how to efficiently estimate L.


Fig. <ref> shows an area from the AIRSAR image of Flevoland, the Netherlands, obtained on August 1989 <cit.> with four nominal looks.
We delimited three regions of interest.



Table <ref> lists the ML parameter estimates as well as the sample sizes.
Each sample is taken from a single class without evidence of texture.
Notice that the estimates for the equivalent number of looks are very close, although lower than the nominal value.
We also show the determinant of the estimated covariance matrix.
This quantity, called geometric intensity in <cit.>, is the generalized variance in multivariate analysis; 
it can be used as a measure of mean backscatter <cit.>.
According to it, region B_2 presents the highest return, followed by B_1 and by B_3; this is in agreement with what is observed in channel HH, cf. Fig. <ref>.




Fig. <ref> depicts the empirical densities of data from the agricultural regions along with the fitted marginal densities.
The scaled Wishart density collapses to the Gamma density:

    f_Z_i(z_i;θ_i,L)=L^L z_i^L-1/Γ(L) θ_i^Lexp[-L θ_i^-1z_i],

where i∈{HH,HV,VV}, θ_k is the element (k,k) of Σ, and Z_k is the (k,k)-th entry of Z.
In practice, θ_i represents the mean polarization channel i∈{1 (HH), 2 (HV), 3 (VV)}.
Figs. <ref>-<ref> show the data and the densities for the estimated number of looks
𝒲_ℛ(Σ,L) (black curve) and the fixed value
𝒲(Σ,4) (gray curve).
These densities are remarkably close, and also to the histograms, so the Gamma assumption is reasonable.



According to Akbari et al. <cit.>, if {Z_i;i=1,2,…,n} is a random sample drawn from Z∼𝒲(L,Σ) and Σ represents the maximum likelihood estimator of Σ, 
then tr(Σ^-1Z_i) follows a Gamma distribution for i=1,2,…,n.

Fig. <ref> displays fitted and empirical densities of such transformed data for the three selected regions.

These results indicate that data may follow a scaled complex Wishart model.

Additionally, the Kolmogorov-Smirnov statistic p-values for checking the adequacy of the Gamma model to the transformed data
are 0.1377, 0.4923, and  0.3911 for regions B_1, B_2, and B_3, respectively.



We used likelihood ratio tests for two and three samples in order to quantify the similarity among these samples, 

The results presented in Table <ref> point out that B_1 is different from B_2 and B_3, but these last two are similar.

Although a visual inspection of areas B_2 and B_3 (Fig. <ref>) suggests regions of different nature, their observations projected via tr(Σ^-1Z_i) 
are statistically similar.







These samples are used to validate our proposed methods in Section <ref>.




§ HYPOTHESIS TESTS IN POLSAR DATA: A SURVEY


This section provides a survey concerning three hypothesis tests which have been studied in the PolSAR data literature.

We assume that PolSAR data follow a scaled complex Wishart distribution.

Change detection is often formulated as a statistical test for
ℋ_0: Σ_1=Σ_2
assuming L known.

The two main approaches in the literature are: 
(i) likelihood ratio <cit.>
and 
(ii) stochastic distances <cit.>.

In this paper, the former proposal is extended to the context of scaled complex Wishart distributions, since the original approach used the nonscaled Wishart law.

Moreover, this paper also introduces an alternative way for validating ℋ_0 by means of entropy measures <cit.>. 

Subsequently, these methodologies are introduced and discussed. 

In order to obtain more general results,
we will provide expressions for testing
ℋ_0: 
(Σ_1,L_1)
=
(Σ_2,L_2).




 §.§ Likelihood Ratio Statistics


The log-likelihood ratio (LR) statistic has great importance in inference on parametric models. 

Let S_LR be the LR statistic for assessing the simple null hypothesis ℋ_0.

As discussed in <cit.>, such statistic based on ℋ_0 has an asymptotic distribution χ^2_q,
where q is the difference between the dimensions of the parameter spaces under the alternative and the null hypotheses.
We denote such spaces by Θ_1 and Θ_0,
respectively.

Let {X_1,X_2,…,X_N_1} and  {Y_1, Y_2, …, Y_N_2} be  two random samples from 𝒲_ℛ(Σ_1,L_1) and 𝒲_ℛ(Σ_2,L_2) of sizes N_1 and N_2, respectively.

The LR statistic is given by   

    S_LR=-2logλ_𝒲_ℛ(Σ,L),

where 
λ_𝒲_ℛ(Σ,L)
=
sup_θ∈Θ_0ℓ(θ)/sup_θ∈Θℓ(θ), 
Θ=Θ_0∪Θ_1,
and
Θ_0∩Θ_1=∅.

Thus,
we have that


    log    λ_𝒲_ℛ(Σ,L)
    = 
    𝒜(p) + log|Σ_1|^N_1 L_1|Σ_2|^N_2 L_2/ |Σ_c|^(N_1+N_2) L_c
       +(L_c-L_1)∑_i=1^N_1log|X_i| + (L_c-L_2)∑_i=1^N_2log|Y_i|
       +∑_i=1^N_1tr[(L_1Σ_1^-1-L_cΣ_c^-1)X_i]
       +∑_i=1^N_2tr[(L_2Σ_2^-1-L_cΣ_c^-1)Y_i],

and

    𝒜(p)=plogL_c^L_c(N_1+N_2)/L_1^L_1N_1L_2^L_2N_2+logΓ_p(L_1)^N_1Γ_p(L_2)^N_2/Γ_p(L_c)^N_1+N_2,

where L_c and Σ_c represent the number of looks and covariance matrix under the null hypothesis, respectively.
Akbari et al. <cit.>
discuss the two-sample LR test
under the 𝒲_ℛ model.


Sections III-B and -C discuss tests for ℋ_0 based on
information-theoretic measures.



 §.§ The Kullback-Leibler distance


The Kullback-Leibler divergence (D_KL) is one of oldest discrepancy measures between stochastic models; it has a central role in Information Theory <cit.>.

This quantity was firstly understood as a measure of the error in choosing a model when another is the true one.

It has been used in image processing for 
segmentation <cit.>, 
classification <cit.>, 
boundary detection <cit.>, 
and change detection <cit.>.

Moreover, D_KL has a close relationship with the Neyman-Pearson lemma <cit.>, 
and its symmetrization has been suggested as a correction form 
for another important goodness-of-fit measure for comparing statistical models:
the Akaike information criterion <cit.>.

Let X and Y be two random matrices defined over the common support 𝒳 of positive definite complex matrices of size p× p.

The Kullback-Leibler distance is defined by 

    d_KL(X,Y)   =1/2 [D_KL(X,Y)+D_KL(Y,X)]
       = 1/2[ ∫_𝒳 f_Xlogf_X/f_YdŻ + ∫_𝒳 f_Ylogf_Y/f_XdŻ] 
       =1/2∫_𝒳(f_X-f_Y)logf_X/f_YdŻ,

with differential element dŻ given by

    dŻ=∏_i=1^pdz_ii∏^p_i,j=1_i<jd{z_ij}d{z_ij},
 
where z_ij is the (i,j)-th entry of matrix Ż; and  and  denote the real and imaginary part operators, respectively <cit.>.

When distances are taken between particular cases of the same distribution, only the parameters are relevant.
In this case, the parameters θ_1 and θ_2 replace the random variables X and Y.

Salicrú et al. <cit.> proposed a hypothesis test based on d_KL.
Let θ_1=(θ_11,θ_12,…,θ_1M)^⊤ and θ_2=(θ_21,θ_22,…,θ_2M)^⊤ be the ML estimators for θ_1 and θ_2 based on random samples of size N_1 and N_2, respectively.
Under the regularity conditions discussed in <cit.>, the following lemma holds.

If N_1/N_1+N_2λ∈(0,1) and θ_1=θ_2, then

    S_KL(θ_1,θ_2)=2 N_1 N_
    Y/N_1+N_2d_KL(θ_1,θ_2)/ h'(0) ϕ”(1)χ_M^2,

where “” denotes convergence in distribution.


Proposition <ref> is a test for the null hypothesis θ_1=θ_2 based on Lemma <ref>.


Let S_KL(θ_1,θ_2)=s and θ_1 and θ_2 be ML estimates obtained from two sufficiently large random samples of sizes N_1 and N_2, respectively; then the null hypothesis θ_1=θ_2 can be re­jec­ted at le­vel α if ( χ^2_M>s)≤α. 



Frery et al. <cit.> presented closed expressions for d_KL when
the random matrices X and Y follow
the Wishart distribution:

    d_KL   (θ_1,θ_2)=L_1-L_2/2{log|Σ_1|/|Σ_2|-plogL_1/L_2
       +ψ_p^(0)(L_1)-ψ_p^(0)(L_2)}-p(L_1+L_2)/2
    
       + tr(L_2Σ_2^-1Σ_1+L_1Σ_1^-1Σ_2)/2,

from which the S_KL test statistic follows.



 §.§ Shannon and Rényi Entropies


The Shannon entropy has achieved a prominent position in PolSAR imagery.

Morio et al. <cit.> applied it for extracting features from polarimetric targets, assuming the circular Gaussian distribution.

The Shannon entropy has also been used for classifying PolSAR textures <cit.>.

In the subsequent discussion, we present a comprehensive examination of hypothesis tests based on Shannon and Rényi entropies. 

Let f_Z(Z;θ) be a pdf with parameter vector θ.

The Shannon and Rényi (with order β) entropies are defined, respectively, as:

    H_S(θ)=   -∫_𝒳 f_Z(Ż;Σ,L)log f_Z(Ż;Σ,L)  dŻ
    
    =   E{-log f_Z(Z)}
    and+13ex   
    
    H_R^β(θ)=   (1-β)^-1log∫_𝒳 f_Z^β(Ż;Σ,L)
     dŻ
    
    =   (1-β)^-1logE{f_Z^β-1(Z)}.


Pardo et al. <cit.> derived an important result which paves the way for asymptotic statistical inference methods based on entropies.

Let θ=[θ_1 θ_2 ⋯ θ_M]^⊤ be the ML estimate of the parameter vector θ=[θ_1 θ_2 ⋯ θ_M]^⊤ based on an N-point random sample from a model Z having pdf f(Ż;θ). 
Then

    √(N)[H_ℳ(θ)-H_ℳ(θ)] 𝒩(0,σ_ℳ^2(θ)),

where ℳ∈{S,R}, 𝒩(μ,σ^2) is the Gaussian distribution with mean μ and variance σ^2, 

    σ_H^2(θ)=δ^⊤𝒦(θ)^-1δ,

𝒦(θ)=E{-∂^2 log f_Z(Z;θ)/∂θ^2} is the Fisher information matrix, and δ=[δ_1 δ_2 ⋯ δ_M]^⊤ such that 
δ_i=∂ H_ℳ(θ)/∂θ_i for i=1,2,…,M.


Now we introduce a methodology for hypothesis tests and confidence intervals based on entropies.
We aim at testing the following hypotheses:

    ℋ_0    
    H_ℳ(θ_1)
    =
    H_ℳ(θ_2)=v
    ,
    
    ℋ_1    
    H_ℳ(θ_1)
    ≠
    H_ℳ(θ_2)
    ,

where ℳ∈{S,R}.
In other words, is there any statistical evidence for rejecting the assumption that two PolSAR samples come from the same model? 

Let θ_i be the ML estimate for θ_i based on a random sample of size N_i from Z_i for i=1,2,…,r and r≥ 2.  
From Lemma <ref>,
we have that 

    ∑_i=1^rN_i(H_ℳ(θ_i)-v)^2/σ_ℳ^2(θ_i)χ^2_r-1,

where

    v=[∑_i=1^rN_i/σ_ℳ^2(θ_i)]^-1∑_i=1^r N_i H_ℳ(θ_i)/σ_ℳ^2(θ_i).

Then we obtain the following test statistic:

    S_ℳ(θ_1,θ_2,…,θ_r)=∑_i=1^rN_i(H_ℳ(θ_i)-v)^2/σ_ℳ^2(θ_i);
 
the expressions for 
H_ℳ(θ_i)
and
σ_ℳ^2(θ_i)
are presented the Appendix.
We are now in position to state the following result.


Let N_i, i=1,2,…,r, be sufficiently large.
If S_ϕ^h(θ_1,θ_2,…,θ_r)=s, then the null hypothesis ℋ_0 can be rejected at a level α if ( χ^2_r-1>s)≤α.


Whereas tests based on stochastic distances, such as d_KL, allow contrasting only two samples, those based on entropies permit assessing r samples at once; cf. (<ref>).

For issues involving more than two populations (r> 2 in (<ref>)), this is a major advantage of the latter over the former.
In the case of comparing two samples of the same size, i.e. r=2 and N_1=N_2=N, (<ref>) reduces to

    S_ℳ(θ_1,θ_2)
     = 
    N 
    [H_ℳ(θ_1) - H_ℳ(θ_2)]^2
    /σ_ℳ^2(θ_1) + σ_ℳ^2(θ_2)
    .






§ PERFORMANCE ANALYSIS


In this section we assess the performance of the methodologies proposed with three experiments involving simulated (under the scaled Wishart complex law) and actual PolSAR data. 

Firstly, we use Monte Carlo experiments to measure 
(i) test size (false alarm rate)
and
(ii) test power (1-false negative rate).


For the test size,
we check whether two samples from X∼ W(B_1,4)
are from the same distribution, i.e., in a scenario where there was no change and there might be false positives.
We assess the test power
checking if
two samples from
X∼ W(B_1,4)
and
X∼ W(B_1· (1+k),4),
for k=0.2,0.3,0.4,
are correctly identified as a situation where there was a change.

We then perform two experiments with actual PolSAR data. 






 §.§ Simulated Data


We compare the following hypothesis tests:



  * Likelihood ratio S_LR;

  * Kullback-Leibler distance S_KL;

  * Statistics based on Shannon S_S and Rényi S_R^β entropies.


We fixed β=0.1, since this value was found in Ref. <cit.> to provide good discrimination in hard-to-deal-with situations.

We assume that the number of looks is known, as in <cit.>.

Therefore, we are able to compare
information-theoretic measures with the methodology proposed by Conradsen et al. <cit.>.

The samples are generated according to Algorithm <ref>.




The parameters used for assessing the null hypothesis ℋ_0Σ_1=Σ_2 are
L_1=L_2=4, and (<ref>), the sample covariance matrix of area B_1, Fig. <ref>.

As we are interested in the behavior of the tests with small sample sizes, we computed the size of the hypothesis at α∈{1, 5, 10} for N_1=N_2=N ∈{10,11,…,50}.

Let T be the number of Monte Carlo replications and C the number of occurrences under ℋ_0 (i.e., pairs of samples are taken from the same model) on which the null hypothesis is rejected at the nominal level α. 
The empirical test size (ETS) or false positive rate is defined by 
α_ETS=C/T.
We used 
T=5500, 
as suggested in <cit.>, and α_ETS did not suffer expressive changes for larger values.



Table <ref> shows: 
(i) the empirical test size at nominal levels 1, 5, 10, and
(ii) the mean test statistic (S_∙)
of the four statistics. 



In average, all test statistics behave as expected when the sample sizes increase: S̅_S and S̅_R^0.1 tend to one, while S̅_LR and S̅_KL tend to nine.
Recall that the asymptotic distribution of the two former is χ^2_9, while the two latter are χ^2_1.

The S_LR and S_S tests exhibit the closest empirical sizes to the nominal levels, 
as confirmed by Fig. <ref>.

The ETS associated with S_R^0.1 and S_KL are biased, however the bias reduces as the sample size increases.

We conclude that these two statistics require larger sample sizes to achieve the expected asymptotic behavior.

In general terms, Table <ref> suggests this inequality:

    ETS_S_KL≥ETS_S_LR≥ETS_S_S≥ETS_S_R^0.1.


The size of tests (False Positive rates) based on the Shannon entropy and likelihood ratio are the closest to the nominal level.














We also studied the test power.
We wish to reject the hypothesis ℋ_0 given two samples drawn from X∼𝒲(B_1,4) and Y∼𝒲(B_1· (1+k),4) where k=0.2,0.3,0.4; i.e, 
under ℋ_1.
The rate
η=(T-C^*)/T,
where C^* is the number of rejections of ℋ_0 under
ℋ_1,
estimates the Type II error or false negative <cit.>, and we aim at quantifying the test power 1-η.

Fig. <ref> presents the estimated power for several samples sizes. 
The test based on Shannon entropy performs best.


In this case, we obtain the inequality:

    (1-η)_S_S≥
    (1-η)_S_R^0.1≥
    (1-η)_S_KL≥
    (1-η)_S_LR.


The relation between discriminatory powers within groups
{S_KL, S_LR} and {S_S, S_R^β}
has been discussed in the statistical literature.

This fact can be explained twofold, namely
(i) the relationship between the Neyman and Pearson lemma and the Kullback-Leibler distance <cit.>, and 
(ii) the fact that 
lim_β→ 1 S_R^β = S_S <cit.>.

The best test statistics should have both empirical size near to the nominal level, and the highest estimated power.
Thus, based on this evidence and on the estimated size, we suggest S_S as the best discriminator on scenarios which follow the scaled complex Wishart distribution.


































 §.§ Experiments with Data from Sensors


In this section
we apply the proposed test statistics
to two studies:
(i) for assessing ℋ_0:Σ_1=Σ_2 to the data presented in Fig. <ref> (single date)
and
(ii) for detecting changes on two PolSAR images captured at different instants, as displayed in Fig. <ref> (multitemporal data).

ENL is assumed constant.



  §.§.§ Single date experiment


Our first experiment aims at assessing α_ETS, the empirical test size (Type I error or Probability of False Alarm), using pairs of disjoint samples from the same target.


This experiment is outlined in Algorithm <ref>.
We used samples of size N∈{3× 3, 4× 4,…,23× 23}.



Fig. <ref> shows the observed α_ETS.
Inequality (<ref>) is also verified on actual data.

For α=1,  
11.67≤α_ETS(S_LR)<α_ETS(S_KL)≤18.93
and
1.527≤α_ETS(S_S)<α_ETS(S_R^0.1)≤6.909;
i.e., all tests overestimate α,
but S_S and S_R^0.1 presented better results than S_LR and S_KL.

For B_2 and B_3, 
α_ETS(S_R^0.1)≤1.49,
overcoming
2.091≤α_ETS(S_S)≤5.055, 
6.745≤α_ETS(S_LR)≤10.618
and  
7.164≤α_ETS(S_KL)≤14.109.


PolSAR regions
are Wishart, our explanation for the better performance of S_R^0.1 is deviations from this hypothesis.




These results present evidence that the test statistics based on S_R^0.1 outperforms the other ones.
This test presented good results even for small samples.
Thus, this measure is suggested as a relevant change detection tool for PolSAR imagery.



  §.§.§ Multitemporal data


Fig. <ref> presents the study areas for this experiment: 
surroundings of the city of Los Angeles, CA, USA.

These pictures refer to a dense urban area whose changes are caused by the urbanization process.

Ratha et al. <cit.> employed these data in the proposal of change detectors for single look polarimetric data using a geodesic distance.

Here we apply the four multilook PolSAR data detectors discussed in Section <ref>.

Fig. <ref> shows the Pauli decomposition of 
two UAVSAR images obtained by JPL's UAVSAR (Uninhabited Aerial Vehicle Synthetic Aperture Radar) sensor at two different instants (23 April 2009, and 3 May 2015).







Using windows of size 3×3 on both dates, we computed the S_LR, S_KL, S_S, and S_R^0.1 test statistics and, from them, p-value maps; cf. Figs. <ref> and <ref>.
Probability values higher than 0.01 are drawn in black, as they provide no evidence of change.
Values below 0.01 range vary from red to dark blue (from strong to weak evidence of change).







It is noticeable
that S_R^0.1 and 
S_S are similar,
cf. Figs. <ref>, <ref> and <ref>, <ref>,
while S_LR and S_KL look alike, see Figs. <ref>, <ref>, <ref>, and <ref>, but somewhat different from the previous pair.

Fig. <ref> shows the relationship between S_R^0.1 and S_S for the second scene, along with the identity function for reference.

The p-values associated to the 
Shannon statistic are smaller than that those related to the Rényi statistic, so the former tends to reject more than S_R^0.1, as discussed in the simulation experiments.



Finally,
Figs. <ref>, <ref>, <ref>, <ref>
and <ref>, <ref>, <ref>, <ref>
show 
binary images resulting from thresholding 
the S_R^0.1, S_S, S_LR, and 
S_KL statistics for the first and second scenes: p-values larger than 10^-4 are shown in white, otherwise in black.

The results, again, favor entropy-based detectors.

To confirm the qualitative discussion, we quantify the performance of detectors with respect to reference maps in Figs <ref>-<ref> in terms of five measures:



  * False positive (FP):  Number of pixels indicated as change by Reference map (RM), but classified as no change; 

  * False negative (FN): Number of pixels indicated as no change by RM, but classified as change;

  * False alarm rate (FA): (FP+FN)/N, where N is the number of unchanged pixels according to the detector;

  * Detection rate (DR): TP/CG, where TP is the number of pixels indicated as change by both RM and the detector, and CG is the number of changed pixels according to the detector; and 

  * Kappa coefficient: κ=(A-B)/(1-B), where 
A=1-p_FP-p_FN and
B=(p_TP+p_FP)(p_TP+p_FN)+(p_TN+p_FP)(p_TN+p_FN),
where p_𝒞 is the proportion of pixels under the condition 𝒞
relative to the total number of pixels
and TN is the number of pixels indicated as no change by both RM and the detector.

The reference maps were prepared by specialists with Bing and Google Earth imagery; cf. Ref. <cit.>.

Table <ref> shows the results.

S_R obtained the best performance, followed by S_S, for both data sets with respect to κ and DR.

These detectors presented lower FN and FA than S_LR and S_KL.

S_KL and S_LR performed better than entropy-based detectors with respect to FP.

The values of FP were smaller than 5 in all cases, so this is not an issue for any detector. 







§ CONCLUSIONS


We quantified and compared the performance of four change detection methods for fully polarimetric SAR data. 
These methods are based on the likelihood-ratio statistic, on the Kullback-Leibler distance, and on the Rényi and Shannon entropies. 
We used empirical test powers and sizes as comparison criteria.

Firstly, the performance of the methods was quantified through a Monte Carlo study using scenarios modeled by the scaled complex Wishart law.
The empirical test sizes showed evidence that the detectors based on the likelihood ratio and Shannon entropy statistics presented the best performance.
In particular, the one based on the entropy is the best for small samples and statistically similar to the S_LR.

Additionally, the tests based on the Kullback-Leibler and on the likelihood ratio statistics tend to overestimate the nominal level, while those which employ entropies underestimate it.

Regarding the empirical test power, the test based on the Shannon entropy presented, in a consistent fashion, the best results.

Computational costs are quite different. 
The test statistic based on the likelihood ratio S_LR requires evaluating (<ref>), 
while S_KL depends only on the Kullback-Leibler distance (<ref>).
The latter is less demanding than the former by an order of magnitude.

Thus, on those situations in which S_KL and S_LR are competitive (for moderate and large sample sizes), the Kullback-Leibler test is more attractive because it has the lowest computational cost.

Secondly, and since estimated test sizes were quite competitive, two experiments with actual data were performed.

For the single date experiment, in all the situations considered, the test based on Rényi entropy with order β=0.1 presented the best results.

The multitemporal data experiments suggests that change detectors equipped by entropies provide better performance than those based on the Kullback-Leibler distance and those based on the likelihood ratio statistic.

Finally, the diversity of tests statistics stemming from Information Theory opens the venue for investigation of composite decision rules, as in Ref. <cit.>.

Future works  will aim to adapt developments made in this paper to more general distributions as, for instance, the 𝒢_Pol and its particular cases (𝒦_Pol, 𝒢^0_Pol, and 𝒢^H_Pol), the Kummer-𝒰, and ℳ laws; cf. <cit.>, and <cit.>.





§ 
DERIVATION OUTLINE OF EXPRESSIONS FOR SHANNON AND RÉNYI STATISTICS




Applying (<ref>) in (<ref>) and (<ref>), 
we obtain the following entropies <cit.>:

    H_S   (θ)=p(p-1)/2logπ- p^2 log L +p log |Σ| + p L
       +(p-L)ψ_p^(0)(L) +∑_k=0^p-1logΓ(L-k),  and
    
    H_R^β   (θ)= p(p-1)/2logπ- p^2 log L +p log |Σ|
       -pqlogβ/1-β+∑_i=0^p-1[logΓ(q-i)-βlogΓ(L-i)]/1-β,

where q=L+(1-β)(p-L).

Under the scaled complex Wishart law, Frery et al. <cit.> derived the following variances:


  * Shannon:


    σ_S^2=   [(p-L)ψ_p^(1)(L)+p-p^2/L]^2/ψ_p^(1)(L) - p/L
       +p^2/Lvec(Σ^-1)^* ( Σ⊗Σ)  vec(Σ^-1).



  * Rényi entropy:


    σ_R,β^2=   {β/1-β[ψ_p^(0)(q)-ψ_p^(0)(L)]-pβln(β)/1-β-p^2/L}^2/ψ_p^(1)(L) - p/L
       +p^2/Lvec(Σ^-1)^* (Σ⊗Σ)vec(Σ^-1).


		

IEEEtran


[
    < g r a p h i c s >
]Abraão D. C. Nascimento
holds B.Sc.  M.Sc.  and D.Sc. degrees in Statistics from Universidade Federal de Pernambuco (UFPE), Brazil, in 2005, 2007, and 2012, respectively.
In 2014, he joined the Department of Statistics at UFPE as Adjoint Professor.
His research interests are statistical information theory, inference on random matrices (with emphasis for applications on PolSAR imagery), statistical theory of shape, spatio-temporal processes, survival analysis, and asymptotic theory.



[
    < g r a p h i c s >
]Alejandro C. Frery (S'92–SM'03)
received the B.Sc. degree in Electronic and Electrical Engineering from the Universidad de Mendoza, Mendoza, Argentina.
His M.Sc. degree was in Applied Mathematics (Statistics) from the Instituto de Matemática Pura e Aplicada (IMPA, Rio de Janeiro) and his Ph.D. degree was in Applied Computing from the Instituto Nacional de Pesquisas Espaciais (INPE, São José dos Campos, Brazil).
He is currently the leader of LaCCAN – Laboratório de Computação Científica e Análise Numérica, Universidade Federal de Alagoas, Maceió, Brazil.
His research interests are statistical computing and stochastic modelling.


[
    < g r a p h i c s >
]Renato J. Cintra (S'00, M'05, SM'10)
received the B.Sc., M.Sc., and D.Sc. degrees in electrical engineering from the Universidade Federal de Pernambuco (UFPE), Recife, Brazil, in 1999, 2001, and 2005, respectively. He joined the Department of Statistics, UFPE, in 2005. In 2014–2015, he was visiting professor at the Département Informatique, INSA, Lyon, France. During 2017–2018, he is a visiting professor at the University of Calgary, Canada. He is an associate editor for IEEE Geoscience and Remote Sensing Letters; Springer Circuits, Systems, and Signal Processing; IET Circuits, Devices & Systems; and Journal of Communication and Information Systems. His long-term topics of research include: approximation theory for discrete transforms, theory and methods for digital signal processing, and
statistical methods.



