Polarimetric Decomposition

Polarimetric Decomposition Operator

   This operator performs the following polarimetric decompositions for a full polarimetric SAR product:

Sinclair Decomposition

   Let 
   

    be the complex scatter matrix. The Sinclair decomposition produces (R, G, B) bands with the following intensities
   
    The main drawback of this decomposition is the physical interpretation of the resulting RGB image.

Pauli Decomposition

   The (R, G, B) bands produced by the Pauli decomposition correspond to the following intensities [1]:

Freeman-Durden Decomposition

   The Freeman decomposition models the covariance matrix as the contribution of three scattering mechanisms [1]:
  1. canopy scatter from a cloud of randomly oriented dipoles, forest for example;
  2. even- or double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants;
  3. Bragg scatter from a moderately rough surface.
    The power scattered by the components of the above three scattering mechanisms are employed to generate a RGB image as the following:

Yamaguchi Decomposition

   The three-component Freeman-Durden decomposition [1] can be successfully applied to SAR observations under the reflection symmetry assumption. However, there exists areas in an SAR image where the reflection symmetry condition does not hold. Yamaguchi et al. proposed, in 2005, a four-component scattering model by introducing an additional term corresponding to nonreflection symmetric cases. The fourth component introduced is equivalent to a helix scattering power. This helix scattering power term appears in heterogeneous areas (complicated shape targets or man-made structures) whereas disappears for almost all natural distributed scattering. Therefore, Yamaguchi decomposition models the covariance matrix as the following four scattering mechanisms:
  1. volume;
  2. double-bounce;
  3. surface; and
  4. helix scatter components.

H-A-Alpha Decomposition

   The H-A-Alpha decomposition [1] is based on the eigen decomposition of the coherency matrix [T 3 ]. Let λ 1 , λ 2 , and λ 3 be the eigenvalues of the coherency matrix ( λ 1 > λ 2 > λ 3 > 0), and u 1 , u 2 and u 3 be the corresponding eigenvectors which can be expressed as the following:


 Then three secondary parameters are defined as the follows:
  1. Entropy:
  1. Anistropy: 
  1. Alpha:

Touzi Decomposition

   In 2007, for the monostatic scattering case, Ridha Touzi has proposed in [2] a new Target Scattering Vector Model (TSVM). Based on the Kennaugh-Huynen decomposition, this model allows to extract four roll-invariant parameters:
  1. Kennaugh-Huynen maximum polarization parameter: orientation angle (Ψ);
  2. Kennaugh-Huynen maximum polarization parameter: helicity (τ);
  3. Symmetric scattering type magnitude (α);
  4. Symmetric scattering type phase (Φ).
    The roll-invariant incoherent target decomposition, i.e. Touzi decomposition, is as the following:
  1. Compute target coherency matrix [T3] with a sliding window;
  2. Perform eigendecomposition on the coherency matrix;
  3. Apply the new target scattering vector model to each eigenvector to extract four parameters (Ψk, τk, αk, Φk, k = 1, 2, 3).
  4. Compute averaged parameters (Ψ, τ, α, Φ):

Van Zyl Decomposition

   The Van Zyl decomposition [1] assumes that the reflection symmetry hypothesis establishes and the correlation between co-polarized and cross-polarized channels is zero. The assumption is generally true in case of natual media such as soil and forest. With such an assumption, the eigen decomposition of the averaged covariance matric C 3 can be given  analytically  and C 3 can be expressed in the following manner:


The van Zyl decomposition thus shows that the first two eigenvectors represent equivalent scattering matrices that can be interpreted in terms of odd and even numbers of reflections.

Cloude Decomposition

   The Cloude decomposition [1] is an eigenvector based decomposition. It idetifies the dominant scattering mechanism via the extraction of the largest eigenvalue.

Generalized Freeman-Durden Decomposition

   The Generalized Freeman-Durden decomposition in [3] is also a model-based decomposition. Similar to the Freeman-Durden decomposition, it assumes that the main  backscattering components are direct backscatter from the underlying surface, double-bounce from a pair of orthogonal surfaces, and direct volume scattering from the top layer. With the 2-layer model and the assumptions above, we have more parameters than the measurements. Generally, to invert the modle, more assumptions on the parameters are needed.With the Generalized Freeman-Durden decomposition, it is assumed that the surface and dihedral mechanisms are orthogonal. Then the power  scattered by the three components can be computed by inverting the model.

Model-free 3 Component Decomposition (MF3CF)

    Model-free 3 component decomposition technique [4] does not consider any volume model for the computation of the three scattering power components. The scattering power components are roll-invariant. The total power is conserved after decomposition and all the scattering power components are non-negative. The target scattering type parameter is represented as:

where, &#920 FP is the target characterization parameter which varies from -45 to 45 degrees, m FP is the 3D Barakat degree of polarization, T 11 , T 22 and T 33 are the elements of the coherency matrix and Span is the total power of coherency matrix T 3 .

The scattering power components are:

P d FP is the even bounce power component, P s FP is the odd bounce power component and P v FP is the diffused power component.

Model-Free 4-Component Decomposition (MF4CF)

    Model-free 4 component decomposition technique [5] does not consider any volume model for the computation of the four scattering power components. The scattering power components are roll-invariant. The total power is conserved after decomposition and all the scattering power components are non-negative. However, in this decomposition a scattering asymmetry parameter is introduced which captures the helicity.

The scattering type parameters are:

where, &#932 FP is the target asymmetry parameter and &#920 FP is the target characterization parameter. K 11 , K 14 and K 44 are the elements of the Kennaugh matrix.

The four scattering power components are:

P d is the even bounce power component, P s is the odd bounce power component, P v is the diffused power component and P v is the asymmetry (helix) power component.

Input and Output

Parameters Used

   For all decompositions, the following processing parameter is needed (see Figure 1):


                                                       
                 Figure 1. Dialog box for Polarimetric Decomposition operator


For Freeman-Durden decomposition, an extra parameter is needed (see Figure 2):


                 Figure 2. Dialog box for Freeman-Durden decomposition


For Yamaguchi decomposition, the following parameters are needed (see Figure 3):


                 Figure 3. Dialog box for Yamaguchi decomposition

For H-A-Alpha decomposition, the following extra parameters are needed (see Figure 4):


                 Figure 4. Dialog box for H-A-Alpha decomposition


For Touzi decomposition, the following extra parameters are needed (see Figure 5):



                 Figure 5. Dialog box for Touzi decomposition

For Van Zyl decomposition, the following parameters are used (see Figure 6):


                 Figure 6. Dialog box for Van Zyl decomposition


For Model-free 3 component decomposition (MF3CF) decomposition, the following parameters are used (see Figure 7):



                 Figure 7. Dialog box for Model-free 3 component decomposition (MF3CF)


For Model-free 4 component decomposition (MF4CF) decomposition, the following parameters are used (see Figure 8):


                 Figure 8. Dialog box for Model-free 4 component decomposition (MF4CF)


Reference: 

[1] Jong-Sen Lee and Eric Pottier, Polarimetric Radar Imaging: From Basics to Applications, CRC Press, 2009

[2] R. Touzi, “Target Scattering Decomposition in Terms of Roll-Invariant Target Parameters,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 1, pp. 73–84, January 2007.

[3] S.R. Cloude, “Polarisation: Applications in Remote Sensing”, Oxford University Press, ISBN 978-0-19-956973-1, 2009.

[4] S. Dey, A. Bhattacharya, D. Ratha, D. Mandal and A. C. Frery, 2020. “Target characterization and scattering power decomposition for full and compact polarimetric SAR data”, IEEE Transactions on Geoscience and Remote Sensing, 59(5), pp.3981-3998.

[5] S. Dey, A. Bhattacharya, A. C. Frery, C. López-Martínez and Y. S. Rao, 2021. “A Model-free Four Component Scattering Power Decomposition for Polarimetric SAR Data”, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 14, pp.3887-3902.