Unsupervised Classification Based on H/A/α Parameters

In any classifier, the choice of parameters is important, and in the case of polarimetric radar data, content-independent scattering models can be used to get parameters that provide reasonable class separation. A current example is the H / A / Greek small letter alpha set of parameters derived from an eigenvalue decomposition of the coherency matrix. The H / A / Greek small letter alpha algorithm was developed by Cloude and Pottier, who showed that terrain classes sometimes produced distinct clustering in the H / Greek small letter alpha plane Cloude & Pottier 1997.

The H / Greek small letter alpha plane is drawn in Figure 7-1. The observable alpha values for a given entropy are bounded between curves I and II (i.e. the shaded areas are not valid). This because the averaging of the different scattering mechanisms (i.e. averaging of the different eigenvectors) restricts the range of the possible a values as the entropy increases. As H and Greek small letter alpha are both invariant to the type of polarization basis used, the H / Greek small letter alpha plane provides a useful representation of the information in the coherency matrix Cloude et al 2002.

figure 7.1
Figure 7-1: The H / Greek small letter alpha plane showing the model-based classes and their partitioning. A description of the classes (Z1 - Z9) is given in the text.

The bounds shown in Figure 7-1 (Curve I and Curve II) show that when the entropy is high, the ability to classify different scattering mechanisms is very limited. An initial partition into nine classes (eight usable) has been suggested by Cloude and Pottier Cloude & Pottier 1997, and is shown in Figure 7-1. Classes are chosen based on general properties of the scattering mechanism and do not depend up on a particular data set. This allows an unsupervised classification based on physical properties of the signal. The class interpretations suggested by Cloude and Pottier are as follows (see Cloude & Pottier 1997 for more details):

  • Class Z1: Double bounce scattering in a high entropy environment
  • Class Z2: Multiple scattering in a high entropy environment (e.g. forest canopy)
  • Class Z3: Surface scattering in a high entropy environment (not a feasible region in H / Greek small letter alpha space)
  • Class Z4: Medium entropy multiple scattering
  • Class Z5: Medium entropy vegetation (dipole) scattering
  • Class Z6: Medium entropy surface scattering
  • Class Z7: Low entropy multiple scattering (double or even bounce scattering)
  • Class Z8: Low entropy dipole scattering (strongly correlated mechanisms with a large imbalance in amplitude between HH and VV)
  • Class Z9: Low entropy surface scattering (e.g. Bragg scatter and rough surfaces)

It is important to note, however, that the boundaries are somewhat arbitrary and do depend upon the radar calibration, the measurement noise floor and the variance of the parameter estimates. Nevertheless, this classification method is linked to physical scattering properties, making it independent of training data sets. The number of classes needed as well as the usability of the method depends upon the application. Additional interpretation of the classes is given in Cloude et al 2002, where a small change in the class boundaries is proposed.

The third variable of polarimetric anisotropy has been used to distinguish different types of surface scattering. The H / A-plane representation for surface scattering is given in Figure 7-2, where the shaded region is not feasible. The line delineating the feasible region can be calculated using a diagonal coherency matrix with small minor eigenvalues Greek small letter lamda2 and Greek small letter lamda3, with Greek small letter lamda3 varying from 0 to Greek small letter lamda2.

figure 7.2
Figure 7-2 Types of surface scattering in the Entropy/Anisotropy plane.

Introduction of the anisotropy to the feature set represents a third parameter that can be used in the classification. One approach is to simply divide the space into two H / Greek small letter alpha planes using the green plane shown in the 3-D space of Figure 7-3, one side for A 0.5 the other side for A > 0.5. This introduces 16 classes if the H / Greek small letter alpha planes are divided according to Figure 7-1. Note that the upper limit of H is restricted when A > 0, as shown in Figure 7-2.

The H / A / Greek small letter alpha -classification space, given in Figure 7-3, now provides additional ability to distinguish between different scattering mechanisms. For example, high entropy and low anisotropy (Greek small letter lamda2almost equal toGreek small letter lamda3) correspond to random scattering whereas high entropy and high anisotropy (Greek small letter lamda2 >> Greek small letter lamda3) indicate the existence of two scattering mechanisms with equal probability.

figure 7.2
Figure 7-3: Illustration of how an A = 0.5 plane (green) creates 16 classes from the original 8 H / Greek small letter alpha classes shown in Figure 7-1. This gives 16 regions in the Entropy / Anisotropy / Greek small letter alpha space for use in an unsupervised classifier.

The three parameters H, A and Greek small letter alpha are based on eigenvectors and eigenvalues of a local estimate for the 3x3 Hermitian coherency matrix ("Hermitian" means a square matrix that has conjugate symmetry - it has real eigenvalues). The basis invariance of the target decomposition makes these three parameters roll invariant, i.e. the parameters are independent of rotation of the target about the radar line of sight. It also means that the parameters can be computed independent of the polarization basis.

Estimation of the three parameters H, A and Greek small letter alpha allows a classification of the scene according to the type of scattering process within the sample (H, A) and the corresponding physical scattering mechanism (Greek small letter alpha). The data need to be averaged in order to allow an estimation of H, A and Greek small letter alpha (without averaging, the coherency matrix has rank 1), which has the benefit of reducing speckle noise Lee et al 1999b.

An example of the clustering of pixels from a sea ice SIR-C scene is shown in Figure 7-4a Scheuchl et al 2001b. The H/A plane shows evidence of clustering into two and possibly three classes. Figure 7-4b shows the distribution of H/Greek small letter alpha values for a white spruce field; the target shows a dominant dipole scattering (Greek small letter alpha about 45°), with a high value of entropy H of about 0.8, indicating rather heterogeneous scattering. Figure 7-4b was produced on CCRS's Polarimetric Workstation (PWS).

Figure 7-4a Scatter plots showing distribution of SIR-C ice data over the H / A / Greek small letter alpha classification space (Scheuchl)

Figure 7-4b Scatter plots showing distribution of SIR-C ice data over the H / A / Greek small letter alpha classification space.