III.4 Performance analysis
III.4.3 Data set I: Urban area
The results presented in this section are obtained by applying the proposed ICTD on the RAMSES POLSAR X-band image acquired over Brétigny, France. Fig. III.1 illustrates the Pauli RGB coded image and shows the classiﬁcation map used to deﬁne the observation data sets for the ICA algorithm.
III.4.3.1 The criterion selection
In theﬁrst place, the goal is to compare the ICA methods and choose the appropriate in the context of ICDT.
The ﬁrst point of comparison between the proposed methods in complex independent components derivation is the possibility of identifying the class of trihedral reﬂectors present in the scene (Class 8 in Fig. III.1). The mask derived from the classiﬁcation map allowed us to select the observation data set containing only target vectors from the regions in the image where the reﬂectors were placed. Further, one mixing matrix is estimated using each of the three criteria. In each case, theﬁrst and the second dominant components are presented on the symmetric scattering target adapted Poincaré sphere (Fig. III.6) . The third component parameters are provided in the Table V.1 but, due to the values of helicity and symmetric scattering type phase, the illustration using a sphere is not possible for each of the applied methods.
All the methods are able to identify the class corresponding to the trihedral reﬂectors placed in the scene. A curious fact is that the second dominant component in each case appears to be symmetric as well. Concerning the FastICA,kurtosis criterion results however in both ﬁrst and second components almost matching trihedral. This indicates apparent
"splitting" of the trihedral on the two dominant components, which cannot be granted as a good estimation. On the other side, in case of thelogarithm and the square root criteria, the second component, although symmetric, rather represents weaker dipole backscattering. In
III.4. Performance analysis 65
(a) (b) (c)
(d) (e) (f) (g)
Figure III.6: RAMSES POLSAR X-band, Brétigny, France: Adapted Poincaré sphere repre-sentation of the trihedral class (Class 8) single scatteres (ﬁrst component,second component, third component,trihedral) using (a) PCA, (b) ICA - C2 criterion, (c) ICA - C3 criterion, (d) ICA - FOBI, (e) ICA - JADE, (f) ICA - C1 criterion, (g) ICA - SOBI.
Table III.1: RAMSES X-band POLSAR data over Brétigny, France: roll-invariant parameters of the single scatterers in the trihedral class (Class 8). Trihedral expected values are: ⌧m = 0◦,↵s= 0◦,Φ↵s = [−90,+90].
Comp. PCA ICA-C2 ICA-C3 FOBI JADE ICA-C1 SOBI
1st -0.23 -0.28 -0.28 -0.32 -0.33 -0.33 0.32
2nd -37.15 -0.24 -0.36 0.50 0.43 -0.42 -1.18
3rd 36.15 19.84 5.77 -1.80 -1.67 7.11 1.45
1st 0.50 0.53 0.53 0.90 1.14 1.49 3.09
2nd 89.21 39.91 41.20 13.54 10.25 7.34 2.23
3rd 87.90 58.49 54.97 36.19 34.43 24.82 21.84
1st -51.25 -27.42 -27.70 5.93 7.01 7.54 -77.67
2nd -18.64 2.56 -3.33 12.49 11.09 -9.60 -80.66
3rd 68.86 77.92 -68.60 34.38 60.93 -83.22 -34.70
66 Chapter III. Polarimetric decomposition by means of BSS case of tensorial methods the third component appears to be symmetric, as well. However, with respect to established criterion, FOBI and JADE are placed behind the FastICA logarithm andsquare root criteria, although the trihedral "splitting" appears to be less conspicuous than in the case ofkurtosis. SOBI completely fails to separate two dominant components.
The second point of comparison is the entropy estimation  (Fig. III.3). Having PCA based classic decomposition as a reference, we have compared the overall estimation of entropy (all classes), paying particular attention to the trihedral class. The entropy estimation scheme appears to be by far the best with the criteria (C2 and C3). Actually, the gradation of methods corresponds exactly to the one obtained at the ﬁrst point of comparison. This is however, not surprising, given that the inability to concentrate the energy of trihedral in the ﬁrst component implies the "splitting" of the same and thus the inevitable increase of entropy.
The overall performance of the analysed ICA criteria in the frame of ICTD, seems to depend directly on the growth rate of the employed nonlinear function. The ICA based on slowly growing nonlinear functions (logarithm and square root) are more eﬃcient in both identifying trihedral as the most dominant backscattering mechanism and, although it is an implication, in estimating entropy. The poor performances of selected tensorial decompositions rise from the fact elaborated in section I.3.2 - they depend too much on the particularities of the data 2nd and 4rd order structures. On the other side, FastICA is far more adaptive.
After choosing the second criterion (C2) of the NC FastICA as the most appropriate one, we have compared the ICA based ICDT with the PCA classic counterpart. The estimated ﬁrst dominant component is nearly equivalent in both cases (Fig. III.8). It was this fact which inclined us toward the comparison of the estimated entropy as one of the criteria for selecting the appropriate non-linearity.
The second component, however, appears to be signiﬁcantly diﬀerent (Fig. III.8). This is both due to the constraint of mutual orthogonality present in the conventional approach and due to the useful information contained in the higher order statistical moments. The same class used in comparing the diﬀerent criteria (Class 8) happens to be favourable for demonstrating the utility of the second dominant component. Namely, dipole as the second strongest single scatterer indicates the capability of recognizing the trihedral’s edge diﬀraction, eventually.
III.4.3.2 Polarisation basis invariance
The same dataset was used to demonstrate the invariance with respect to more complex uniform transform - the change of the polarization basis. The observed scattering matrices are projected on the circular polarization basis and the obtained components parametrized using Circular Polarization Scattering Vector (CPSV) [111, 112]:
III.4. Performance analysis 67
Figure III.7: RAMSES POLSAR X-band, Brétigny, France: comparison between the TSVM parameters obtained by means of PCA (ﬁrst most dominant component (i) and second most dominant component (iii) and by means of ICA (ﬁrst most dominant component (ii) and second most dominant component (iv)): (a) ⌧m, (b)↵s, (c)Φ↵s.
68 Chapter III. Polarimetric decomposition by means of BSS
SPANexpjΦ 2 4
sin↵ccosβcexpj(−43⌥c−2 ) cos↵cexpj83⌥
−sin↵csinβcexpj(−43⌥c+ 2 ) 3
Among four parameters invariant to the rotation around the LOS ( ) and to the tar-get absolute phase (Φ): energy (SPAN), angle ⌥c, angle ↵c and helicity deﬁned as Helc = sin2↵c⇥
, we have compared the last three with their counterparts derived from TSVM parametrisation in the Pauli basis. The angles ⌥c and ↵c, if the target is sym-metric (⌧m = 0), correspond, respectively, to ⌥TVSM = (⇡/2−Φ↵s)/4 and ↵s. Helicity HelTVSM is deﬁned as a function of ⌧m and the Huynen con-eigenvalues polarizabilityγH [7, 10]:
HelTVSM = cos 2γHsin 2⌧m
cos4γH(1 +tan4γH). (III.21) On one side, as it is demonstrated in Fig. III.8 and in Table III.3, we obtain the perfect matching in terms of Hel (if we do not apply Eq. I.31). On the other side, even for the symmetric classes (⌧m ⇡0), we don’t have a perfect matching of ⌥, which is justiﬁed by the values of ↵c, which converge either to 0 or ⇡/2, when this parameters becomes meaningless . The angle ↵c agrees perfectly with ↵s in case of symmetric target. However, in order to reinforce this robustness proof, we have as well implicated↵−β−γ−δ parametrization, given in Eq. I.26 .
As it can be seen in Table III.3 and Fig. III.9, the derived ↵p parameter, as expected, matches perfectly↵c, regardless of symmetry. Aside from this we compared the↵−β−γ−δ parameters derived conventionally (using PCA) with the ones obtained using our approach.
It is the angle↵c (or↵p) which fortiﬁes the conclusion arising from the TSVM parameters -theﬁrst dominant components are quite similar, but the second (non-orthogonal in our case) contains undoubtedly diﬀerent information. This diﬀerence is certainly related to the removal of orthogonality constraint, which imposes conventionally ↵p1+↵p2 ⇡⇡/2.