4.1. Decomposition Model
The commonly used model-based PolSAR decomposition method was originally the three-component decomposition method proposed by Freeman-Durden. Under the framework of this model, the coherency matrix is decomposed into a linear combination of volume scattering, double-bounce scattering, and odd-bounce scattering. Mathematically, after OAC, the three-component model for the coherency matrix can be written as:
where
are the coherency matrices accounting for volume scattering, double-bounce scattering, and odd-bounce scattering, respectively.
are the expansion coefficients.
In practice, OAC processing has minimized the cross-polarization power of the initial coherency matrix . Entire cross-polarization power is assigned to the volume scattering category in many decomposition methods, which should represent the scattering from the vegetation only.
Based on the relationship between the volume scattering component and cross-polarization power, whether it is the FDD method or its improvement methods, cross-polarization power is usually completely assigned to volume scattering [
13]. In this case, the expansion coefficient
of volume scattering in Equation (17) can be written as:
Further, when the coherency matrix
is without the assumption of scattering symmetry, this means
,
, in which case Equation (17) does not hold. At this point, Equation (17) should be rewritten as
where
is called the remainder matrix. Obviously, if it is only to ensure the effectiveness of the above equation, regardless of the occurrence of negative energy,
can be expressed as [
36]:
4.2. Model-Based Optimization Problem
To ensure the validity of Equation (17), negative numbers will inevitably appear in
and
, which is clearly not in line with actual physical phenomena. If negative power is not required, the expression of remainder matrix
in Equation (19) does not always hold. At this point, there will always be other parts remaindered through decomposition. Therefore, without considering the limitation of scattering symmetry conditions, the generalized remainder matrix
after three-component decomposition can be written as:
In the above equation, each term can be represented as:
To ensure that the decomposition results meet physical reality, the basic scattering mechanisms (volume scattering, double-bounce scattering, and odd-bounce scattering) must satisfy the following criteria:
- (1)
After subtracting the volume scattering contribution from the observed coherency matrix, the remainder coherency matrix must be at most rank-2;
- (2)
After subtracting any linear combination of basic scattering mechanisms, the remainder coherency matrix must be Hermitian positive semi-definite, i.e., the eigenvalues of the remainder coherency matrix must be real and nonnegative.
For Equation (20), if the coherency matrix is sufficiently decomposed, then the remainder coherency matrix should be , but in general, . Thus, the criteria (2) should meet, that is, the energy of the remainder matrix should be as small as possible and must be Hermitian positive semi-definite.
The matrix energy is related to its eigenvalues. Therefore, minimizing the remainder matrix can be translated into minimizing the maximum eigenvalue of the remainder matrix. Considering a decomposition model that includes volume scattering, double-bounce scattering, and odd-bounce scattering, we need to solve the optimization problem as follows:
where
is the eigenvalues of the remainder matrix
;
is a notation for
being positive semi-definite,
are the expansion coefficients of volume scattering, double-bounce scattering, and odd-bounce scattering, respectively, so
are nonnegative. The constraint conditions in Equation (23) ensure the nonnegative properties of basic scattering components as volume scattering, and consider the positive-semi properties of the remainder matrix
, which accords with the actual physical scene.
4.3. Solution of the Optimization Problem
The particular form of the objective function in Equation (23) is difficult to minimize directly; at this point, we shall thus analyze the characteristics of optimization problem Equation (16) and perform corresponding equivalent transformations.
Define the objective function in Equation (23) as
, whose domain is
and
. Consider the following rearrangement of the function:
Let for , then is a linear function of , that is, is a convex function.
If both
and
are convex functions with respect to
, then the point-by-point supremum of
and
can be expressed as:
Let
and
, then:
The above equation illustrates the convexity of the function . By extending the above property of finite point-by-point supremum to the supremum of wireless convex functions, it can be proved that the function is still convex. Therefore, function , is a convex function.
In other words, optimization problem Equation (16) is a convex optimization problem. For the convex optimization problem, the global optimal solution can be obtained.
Further, to quickly solve the optimization problem Equation (23), perform an equivalent transformation on Equation (23), with the aim of enabling the transformed optimization problem to have well-developed numerical methods to be solved.
is a positive semi-definite matrix, which implies
is a Hermitian matrix, and
therefore,
is also a positive semi-definite matrix, and the eigenvalues of
are nonnegative real numbers. Let
; according to the definition of spectral norm there is
In addition, for the positive semi-definite matrix
, there is a non-singular matrix
, which makes
, and
is Jordan form. Further,
notice that
is a Hermitian matrix, thus
, which implies the eigenvalues of
are
. This leads to the following:
That is, for a Hermitian and positive semi-definite matrix, its largest eigenvalue is the spectral norm. Then Equation (16) can be equivalently transformed into the following optimization problem.
where
is the spectral norm of
.
The optimization problem above is also a convex optimization problem; the objective function is to minimize the spectral norm of matrix
. The power constraint leads to the following condition:
Using
Appendix A, it can be further represented as:
Therefore, the above optimization problem can be recast as:
According to the Schur Complement introduced in
Appendix A, the above inequalities
can be changed into the following convex form:
Eventually, the following convex optimization problem is obtained.
This is a semi-definite programming (SDP) [
37]; SDP is a particular class of convex optimization problem. Therefore, the optimization procedure enjoys all the advantages of convexity [
37]. There are well-developed numerical methods to solve a general convex optimization problem, among which the most well-known is the interior point method. In the numerical example, we adopt an optimization toolbox, called SeDuMi [
38], to solve the SDP formulated in Equation (36). Note that the globally optimal scattering components of
can be obtained by solving the SDP problem Equation (36).
For the FDD method, the dominant scattering mechanism can be determined by the sign of . Dominant volume scattering means that ; for , which represents the dominant double-bounce scattering, and similarly for the rotated matrix .
The detailed flowchart of the proposed decomposition process is shown in
Figure 1.
From the above, it can be seen that this article can globally optimize the energy of each scattering component. Compared with the commonly used method of remainder matrix Equation (17), this paper minimizes the remainder matrix based on the physical reality of nonnegative scattering components. Therefore, the residual of the proposed method is often higher than that of traditional methods. At the same time, this article only uses decomposition based on three basic models; the limited use of basic models cannot guarantee that the remainder matrix is zero after using the method proposed in this article. Additionally, the selection and accuracy of basic scattering models also have an impact on the residual.