selected. In other words, if both the SIRV model and the Gaussian model arefitting the data, the latter will have better estimation performances with a finite number of samples. Based on the results presented in this chapter, we provide the methodological framework to asses multivariate SAR data conformity, based on the following tests:
1. Zero-mean test, 2. Circularity test, 3. Sphericity test,
4. Spherical symmetry test, 5. non-Gaussianity test.
The algorithmic representation of proposed methodological framework is provided in Fig.
II.5.
By successively performing the proposed tests for a specific multivariate SAR dataset, it is possible to asymptotically evaluate the pertinence of various model-based statistical processing schemes (filtering, segmentation or detection).
In terms of theoretical performance analysis, the adjusted generalized LRT is asymptot-ically uniformly most powerful according to the Neyman-Pearson lemma. This "optimality"
holds provided the estimators plugged into the LRT (or the Wald test) are consistent and unbiased, which is the case for our study.
Special care must be taken when applying the tests with multivariate SAR data. In theory, the proposed conformity testing holds as long as the observed number of samples (estimation neighborhood) is large enough with respect to the dimension of the target vector, especially for the spherical symmetry test (based on the specific structure of the quadricovariance).
Finally, it is important to stress that no predefined analytical form was imposed on the texture probability function when establishing the conformity tests. Therefore, they can be di-rectly applied for a wide class of stochastic processes currently used for describing multivariate high-resolution SAR data.
II.6 Conclusions
In this chapter we have presented a new methodological framework to asymptotically asses the conformity of multivariate high-resolution SAR data. The proposed approach consist of applying successively three statistical hypotheses tests for verifying three important statistical properties: circularity, sphericity and spherical symmetry, briefly summarized in Fig. II.6.
The latter is asymptotically equivalent, under certain hypotheses, to the conformity of the
50 Chapter II. Statistical assessment of high-resolution POLSAR images
Figure II.5: Methodological framework
II.6. Conclusions 51 experimental data with respect to the SIRV product model. In addition, the zero-mean and the non-Gausiannity [44] tests can be used to decide which model is better suited to asymptotically fit the experimental data.
The proposed framework, aside from the fact that it is gathering the most notable advances in thefield of signal processing, is introducing the extension of both the sphericity and the spherical symmetry tests with zero-mean complex circular SIRV assumption.
The effectiveness of the proposed detection schemes was illustrated by synthetic and very high resolution ONERA RAMSES POLSAR data. Additionally, the analysis has been strengthened by high-resolution TerraSAR-X multi-pass InSAR. The conclusions driven from the analysis of the obtained results are important with respect to the two tested real datasets:
circularity is important for very-high resolution POLSAR data, while non-sphericity can be an important issue for high-resolution multi-pass InSAR.
It has been illustrated that in strong heterogeneous clutter, such as the urban environment, the SIRV model can fail. The bottom line is that characterization of urban regions is much more complex (and difficult) - since a more complex model, with more parameters, may be required. In the light of the results shown in this chapter, SIRV models may be less appropriate for urban areas characterization. However, alternative explanations are possible.
As an example, the root of this inappropriateness might as well be the assumed ergodicity / stationarity (in spatial sense) for the backscattered signal and, also, even in the hypothesis of randomness: targets exhibit a deterministic behavior.
First, the very use of a sliding analysis window for estimating the stochastic parameters of the scattered signal may be questioned, as it implicitly assume that the considered signal is ergodic / stationary in spatial sense (homogeneous). While this hypothesis holds for dis-tributed and uniform targets, where the physical parameters (and, thus, the electromagnetic scattering behavior) differs very little from one resolution cell to another, in urban areas the physical structure (and, as such, its electromagnetic behavior) may change considerably from one resolution cell to the next. This makes the hypothesis of ergodicity / stationarity less applicable.
Second, one should note that even the randomness of the radar echo is not given, but assumed. This is mainly a way to deal with the inherent complexity of the signal. Anyway, for identical measuring conditions, the recorded radar data is perfectly identical. Even if small differences in measuring conditions lead to strong discrepancies in the recorded data, this is not an evidence for randomness, as such behaviour can be fully explain under a deterministic paradigm - the chaotic models. Various parameters, such as meteorological conditions and, even more important, the changes that the target suffers in time (between two succeeding acquisitions, for example), account for the observed randomness of the recorded data. How-ever, these changes of the target are more significant for green targets (such as forests and agriculturalfields), where humidity and wind modify both their physical structure and their electromagnetic behavior. On the other hand, those changes are less significant for urban targets and, as such, randomness is less likely for the latter.
52 Chapter II. Statistical assessment of high-resolution POLSAR images
Figure II.6: Brief summary of the analysed statistical properties.
In perspective, applying chaotic (or pseudo-chaotic) [90, 91, 92] models to POLSAR / InSAR data from urban areas can be a possible solution. These models should be able to take into account the deterministic features of those areas (presence of dihedral angles, straight edges, cavities, etc.), while still leaving room for some unpredictability (orientation of those elements). Using chaotic models in POLSAR and multi-pass InSAR data will make the object of our future work.
This work has many interesting perspectives. We believe that it contributes toward the de-scription and the analysis of heterogeneous clutter over scenes exhibiting complex polarimetric signatures. Firstly, the exact texture normalization condition for the PWF-SCM estimator has been derived under the SIRV clutter hypothesis. A novel estimation / detection strategy has been proposed which can be used with conventional boxcar neighborhoods directly. Finally, the proposed estimation scheme can be extended to other multidimensional SAR techniques using the covariance matrix descriptor, such as the following: repeat-pass interferometry, po-larimetric interferometry, or multifrequency polarimetry.
Chapter III
Polarimetric decomposition by means of BSS
III.1 Introduction . . . 54 III.2 PCA and ICA . . . 55 III.3 Method . . . 56 III.3.1 Estimation of the independent components . . . 56 III.3.2 Roll-Invariance . . . 59 III.4 Performance analysis . . . 62 III.4.1 Data selection . . . 62 III.4.2 Synthetic data set . . . 63 III.4.3 Data set I: Urban area . . . 64 III.4.4 Data set II: Mountainous region . . . 68 III.5 Conclusion . . . 73
This chapter represents the focal point of the thesis. Here, we generalize Incoherent Target Decomposition concept to the level of BSS, by replacing conventional eigenvector decompo-sition with ICA algorithm. The proposed decompodecompo-sition exploits the higher order statistical information, emerging from POLSAR clutter heterogeneity [1]. The result is a set of mutually independent, non-orthogonal target vectors, characterizing dominant scatterers over a station-ary set of observed target vectors. The most dominant component appears to be quite similar to the ones obtained by means of conventional methods, but the second ones carries a new information.
Firstly, relying on Section I.3, we discuss the role of BSS in ICTD. The description of the method, comprising the elaborated comparison of applied ICA algorithms, is provided there-after. The roll-invariance properties are as well discussed at this point. Finally, we present the application of the decomposition on two real data-sets, followed by corresponding discussion.
The last contains an application on a synthetic data set as well, used to demonstrate the ca-pability of retrieving non-orthogonal mechanisms. The polarization basis invariance analysis is demonstrated using one of the real data sets.
53
54 Chapter III. Polarimetric decomposition by means of BSS
III.1 Introduction
As already stated in Section I.3, under certain constraints, the eigenvector decomposition of the scattering coherence matrix, provides the same results as the Principal Component Analysis (PCA) of the corresponding representative target vector [93]. Thus, the conventional approach in POLSAR images incoherent target decomposition, elaborated in Section I.1.2.2, results in deriving uncorrelated components. This is adequate if we consider the conventional statistical model assuming Gaussian homogeneous clutter [94]. However, as being elaborated in the previous two chapters, the high and very high POLSAR data can be characterised by Non-Gaussian heterogeneous clutter [30]. In this case decorrelation cannot be considered as the most meticulous way for separating the scattering sources present in the scene. It appears that more advantageous solutions, capable of deriving independent components, are needed.
Applying the Independent Component Analysis (ICA) seems to be one of such solutions.
The research presented in this chapter would be our effort to answer the second question posed in the preface of this thesis. Namely, by accepting the altered statistical models we are accessing in a different manner to the target decompositions, trying to exploit the information contained in the higher statistical orders. This information allows different characterization of POLSAR data, which could prove to be advantageous with respect to the conventional one.
The ICA methods have been already successfully employed on SAR data: in speckle re-duction, feature extraction and data fusion [95, 96]. The application on polarimetric data was, however, either restricted to the analysis of two-components polarimetric target vector [97], either rather related to the POL-InSAR data analysis [98].
The main idea of this chapter is to propose a generalization of the polarimetric decom-positions to the level of blind source separation techniques by introducing the ICA method instead of the eigenvector decomposition. Essentially, our motivation is the possibility to exploit higher order statistics of the non-Gaussian target vector in order to recover a set of in-dependent dominant scatterers. The recovered linearly inin-dependent scattering target vectors are not necessarily mutually orthogonal, which is demonstrated using a synthetic data set.
Atfirst, we apply the statistical classification algorithm (for example [43]) in order to obtain stationary sets of polarimetric observations - scattering matrices projected onto the Pauli ba-sis. Then, the target vectors of the single scatterers are estimated by applying comparatively the representative ICA algorithms, introduced in Section I.3.2, on each of the sets derived in the previous step. They are parametrised using the TSVM, allowing the adapted Poincaré sphere representation with direct physical interpretation [99]. The share of the component in the total backscattering is computed by the squared `2 norm of the single scatterer target vector.
The proposed method, based on the particular version of the FastICA algorithm [100] is invariant both under the rotation of the line of sight and under the change of polarization basis.
The latter is demonstrated using the projection of the observations onto the circular basis, coupled with the Circular Polarization Scattering Vector (CPSV) model [101] and furthermore, by additionally employing↵−β−γ−δ model in Pauli basis [15].
III.2. PCA and ICA 55