Application of Bootstrap Techniques for the Estimation of Target Decomposition Parameters in RADAR Polarimetry

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1 Application of Bootstrap Techniques for the Estimation of Target Decomposition Parameters in RADAR Polarimetry Samuel Foucher Research & Development Dept Computer Research Institute of Montreal Montreal, Canada Grégory Farage, Goze B. Bénié Geography Dept. University of Sherbrooke Sherbrooke, Canada Abstract The precise estimation of the eigenvalues of PolSAR reponses is essential in the derivation of Target Decomposition parameters such as the Cloude-Pottier parameters (Entropy, Anisotropy and average angle Alpha). However, sample eigenvalues are strongly biased for small sample sizes leading to underestimated Entropy and overestimated Anisotropy values. In this paper, we investigate the use of a particular bootstrap technique for the correction of the bias. Bootstrap techniques are attractive because they can deal with very small sample sizes under minimal assumptions on the signal distribution. Here, we are using the jackknife bias correction technique which has been successfully applied to various signal processing problems. Monte-Carlo simulations reveal that the jackknife bias correction directly applied on the Cloude-Pottier parameters lead to better bias reduction. Keywords-bootstrap; jackknife; polarimetry; I. INTRODUCTION Target Decomposition (TD) parameters such as the Cloude- Pottier Parameters [3] and the Freeman-Durden parameters [4] are essential to the understanding of the scattering mechanisms of extended targets. Their computation is dependent upon the correct estimation of the sample covariance eigenvalues and eigenvectors. However, these parameter estimates are known to be sensitive to the number of looks involved in the estimation (i.e. the number of independent samples). For instance, for a small number of looks, the sample entropy has been shown to underestimate the true value [1]. Previous approaches for bias-correction in sample eigenvalues include an asymptotic analysis by Lopèz-Martinez et al. [2] based on a derivation of the sample eigenvalue probability density function (pdf). The resulting estimator (AQ- MLE) has been shown to significantly reduce the bias for the Anisotropy and the Entropy but not for alpha. However, an optimal bias correction on TD parameters would require an analytical derivation of the pdfs of the Cloude-Pottier parameters which remains an ongoing problem. Bootstrap techniques have been widely applied on statistical and signal processing problems [6][7]. For instance, Brich et al. [7] have applied bootstrap techniques for source detection in array processing. Bootstrap techniques are very attractive for two main reasons: 1) they are valid for a large class of Gaussian and non-gaussian signals; and 2) the finite sample, and not the asymptotic, distributions are estimated removing the requirement for large sample sizes. The bootstrap paradigm is based on a resampling technique. Reseampling can be performed with or without replacement [6][7][8]. The latter is referred to as jackknife. Among the different bootstrap techniques, the jackknife technique is computationally more efficient. We propose to apply the jackknife technique for the bias correction on eigenvalues of the covariance/coherency matrix and for the direct bias correction of the Cloude-Pottier parameters ( H / A/ α ). The paper is organized as follows. In section II, some background on bootstrap techniques is provided. In Section III, we perform Monte-Carlo simulations are performed on typical targets in order to assess the bias on the TD parameters. II. BACKGROUND ON BOOTSTRAP TECHNIQUES A. The Bootstrap Principle Let X = { X1, X2,..., X N } be a sample of N i.i.d. random variables drawn from a completely unspecified distribution F. Let θ be a statistic of interest of F such as the mean. Only an estimator ˆθ of θ is available to us such as the sample mean. If we want to infer θ from ˆθ, we need to find the distribution of ˆθ. The bootstrap is composed of the following steps: 1. Let x = { x1, x2,..., x N } be the observed realization of X (i.e. a particular experiment) from which we derive ˆθ. 2. Construct the sample probability distribution ˆF, putting mass 1/ N at each observation x i. 3. with fixed ˆF, draw a random sample x * (called a * * * bootstrap resample) of size N: X = x, X Fˆ i i i 4. Approximate the distribution of ˆθ by the distribution * * of ˆθ derived from the resample x. This work has been supported in part by the NSERC of Canada (Discovery Grant) and the MDEIE of the Gouvernement du Québec.

2 In short, the bootstrap procedure is observing the variability of ˆθ in our artificial resamples in order to assess the natural variability of θ without the requirement of a large N value [6][8]. The bootstrap does not work in all of the situations because the resamples are not generated by F but by ˆF that may have not taken into account some important features of F despite being a consistent estimator. In such a case, the jackknife has been shown to be more appropriate [8][7]. The jackknife is based on a resampling without replacement or subsampling and is less computer intensive when the subsamples are N-1 in size. A. Eigenvalue Estimation We wanted to estimate the bias reduction on sample eigenvalues of the coherency matrix. We fixed the true eigenvalues to {4,3,2}. The result of the Monte-Carlo simulation is provided in Fig. 2. We observed that the bias reduction from the AQ-MLE [2] and the jackknife are similar (a slightly better bias reduction was obtained for the third eigenvalue with the jackknife). III. APPLICATION IN POLARIMETRY A. General Approach Suppose we are given a sample of N target vectors x= { k,..., 1 k N }. The jackknife method is based on the sample delete-one-observation (i.e. a subsample) at a time [6][7], {,,...,,,..., } x = k k k k k (1) () i 1 2 i i+ 1 N from which we can derive the subsample coherency matrix given by N () i 1 H T = kjk j (2) N 1 j= 1, j i Suppose that s( x ) is a statistic of interest (e.g. the coherency matrix, sample eigenvalues, etc.). We want to estimate the bias on the estimate based on the entire sample Θˆ = s( x ). For each ith jackknife sample, we calculate the ith jackknife estimate, i.e., ˆ () i = s( () i ) Θ x (3) The jackknife estimate of the bias on ˆΘ is then N ˆ 1 ( 1) ˆ ˆ Θ = N Θ Θ (4) N i = 1 () i ( ) Bias JCK IV. MONTE-CARLO SIMULATION We followed an approach similar to Lee et al. [1]. We consider three types of targets taken from an AIRSAR image: grass, forest and urban. All the Monte-Carlo experiments are conducted with at least 20,000 runs using Matlab. Simulations are performed for sample sizes N ranging from 2 to 16 and then for 25,36,49,64 and 81. The eigendecompositions are performed using the SVD algorithm. Figure 2. Eigenvalue sample means (top) and standard deviation on the first eigenvalue (bottom). B. H / A/ α Bias Estimation We performed the bias correction using three different approaches: 1) the AQ-MLE correction [2]; 2) the jackknife bias correction on the sample eigenvalues (JACK-EIG); and 3) the jackknife bias correction on the sample parameters directly (JACK-PAR). The grass area (dominant surface scattering) has the following parameter values H / A/ α =0.17/0.1/21.6. The forest area (dominant volume scattering) has H / A/ α = 0.96/0.18/51.9. The urban area (dominant double bond) has H / A/ α =0.69/0.78/56.1. Figure 1. Areas in the AIRSAR image where sample coherency matrices were estimated for use in the Monte-Carlo simulation.

3 Figure 3. Monte-Carlo Entropy means for grass, forest and urban areas 1) Entropy: The result for the entropy for the 3 different target types is provided in Fig. 3. The reduction of the bias is substantial especially with the JACK-PAR procedure where the bias is correctly estimated for as low as 4-5 looks. Figure 4. Monte-Carlo anisotropy mean. 2) Anisotropy: The bias reduction on the Anisotropy (Fig. 4) provided by the JACK-EIG is superior to the AQ-MLE as a results of a better bias reduction on the third eigenvalue (see Fig. 2). However, the bias reduction for the Urban area appears to be less stable as compared to the AQ-MLE particularly in respect to the JACK-PAR.

4 Figure 5. Monte-Carlo angle alpha mean. 3) Average alpha angle: The result for the average angle alpha for the 3 targets is provided in Fig. 5. We observe that bias correction of the eigenvalues (AQ-MLE and JACK-EIG) does not lead to an accurate correction for the average angle because of the effect of the eigenvectors. However, the direct estimation of the bias on α (JACK-PAR) appears to have reduced bias especially for the grass target. Figure 6. Monte-Carlo standard deviations obseved for the grass target. 4) Variance: A second important quality for any estimator is its variance. We observe that the standard deviation is increasing when bias correction techniques are applied especially for the Anisotropy. The JACK-PAR leads to greater estimator variances.

5 V. RESULTS ON IMAGES In Fig. 7, we exhibit the Entropy images calculated on an AIRSAR/JPL image based on a 3x3 boxcar filter. We can appreciate in Fig. 7.c the effect of the jackknife bias correction where values for the Entropy are corrected upwardly, in particular, on forest areas. a) b) compared to a bias correction only on the eigenvalues (JACK- EIG): The bias correction on the Entropy parameter produces very good results especially for small number of looks. Correction on the average angle alpha parameter appears to surpass the AQ-MLE. The performance of the jackknife bias correction does not seem to be dependent upon the type of target. There is a fairly large body of literature covering the subject of bootstrapping and we have not yet begun to explore its potential for PolSAR applications. The preliminary results observed here are encouraging and lead to the possibility of formulating a simple bias correction technique for small number of looks. Future work will focus on the bootstrapping of eigenvectors, which remains a difficult problem, as well as the evaluation of resampling on other TD parameters (e.g. the Freeman-Durden parameters [4]). Additionally, the effect of correlation between samples has not been investigated. ACKNOWLEDGMENT This work has been supported in part by the NSERC of Canada (Discovery Grant) and the MDEIE of the Gouvernement du Québec. The authors wish to thank Lisa Hollinger and Elaine Rosenberg for their linguistic expertise. c) Figure 7. AIRSAR image of the San Francisco Bay: a) color composite; b) Entropy estimated using a 3x3 boxcar filter; c) Same as b) but with a jackknife bias correction. VI. CONCLUSION This paper has explored the use of a particular bootstrap technique, known as the jackknife. Two approaches were evaluated: 1) the bias correction of the eigenvalues; and 2) the bias correction of the H / A/ α parameters. Based on the results of Monte-Carlo experiments, we were able to make the following observations: The jackknife technique is less computer intensive and easier to implement than a resampling with replacement. The bootstrap on eigenvalues (JACK-EIG) gives a similar bias correction to the AQ-MLE. The use of a jackknife bias correction directly on the TD parameters (JACK-PAR) produces superior results as REFERENCES [1] J.-S. Lee, T. L. Ainsworth, and C. Lopez-Martinez, Monte Carlo Evaluation of Multi-Look Effect on Entropy/Alpha/ Anisotropy Parameters of Polarimetric Target Decomposition, IGARSS'2006. [2] C. Lopez-Martinez, E. Pottier, and S. R. Cloude, Statistical Assessment of Eigenvector-based Target decomposition Theorems in Radar Polarimetry IEEE Trans. Geosci. Remote Sens., vol. 40, no. 4, Apr. 2003, pp [3] S. R. Cloude, and E. Pottier, A Review of Target Decomposition Theorems in Radar Polarimetry, IEEE Trans. Geosci. Remote Sens., vol. 34, no. 2, pp , March [4] A. Freeman and S. L. Durden, A Three Component Scattering Model for Polarimetric SAR Data, IEEE Trans. Geosci. Remote Sens., vol. 33, no. 3, pp , May [5] R. Touzi, Target Scattering Decomposition in Terms of Roll-Invariant Target Parameters, IEEE Trans. Geosci. Remote Sens., vol. 45, no. 1, Jan. 2007, pp [6] A. Zoubir and B. Boashash, The Bootstrap and its Application in Signal Processing, IEEE Signal Processing Mag., pp , January [7] R. F. Brcich, A. M. Zoubir, and P. Pelin, Detection of Sources using Bootstrap Techniques, IEEE Trans. Signal Process., vol. 50, no 2, Date: Feb 2002, pp [8] D. N Politis, Computer-Intensive Methods in Statistical Analysis, IEEE Signal Processing Magazine, 1998, pp