Radar polarization decomposition, polarization target decomposition correlation

Recently, I am studying the target detection problem of polarimetric radar, polarization decomposition and other knowledge, and recorded some basic theories.

basic concept

A summary of the basic knowledge from some previous blogs, which is more friendly to beginners.

(1) Sinclair matrix (polarization matrix, scattering matrix): Usually, the electromagnetic scattering characteristics of the radar target in the far field area is a linear process, if the scattering space coordinate system and the corresponding polarization base are selected, then the radar There is a linear transformation relationship between the polarization components of the irradiated wave and the scattered wave of the target. Therefore, the variable polarization effect of the target can be expressed in the form of a complex two-dimensional matrix, called the Sinclair matrix, which represents the specific attitude and The full polarization information of the target at the observation frequency. If the horizontal and vertical polarized waves are transmitted, and the horizontal and vertical polarized waves are received, then Es=S.Ei, at this time, the Sinclair matrix can be expressed as S=[Shh Shv; Svh Svv], and Shv physically corresponds to the vertical Polarized The horizontally polarized component of the backscattered wave when illuminating a target. It can be seen that only the relative phase is considered, and the Sinclair matrix has 7 independent variables. In the case of a single station, the radar target whose propagation medium satisfies the reciprocity condition is called a reciprocal target. At this time, Shv=Svh, that is, the Sinclair matrix has only 5 independent variables (absolute phase is not considered).

(2) Mueller matrix: If the incident wave and scattered wave are represented by Stokes vectors gi and gs respectively, then gs=M.gi, where M is called the Mueller matrix. The Mueller matrix is ​​a real matrix, and in general it is not symmetric.

(3) Coherence matrix: If the Pauli base matrix is ​​used to vectorize the Sinclair matrix: kp=(1/sqrt(2)).[Sxx+Syy; Sxx-Syy; Sxy+Syx; i.(Sxy-Syx )], then T=kp.kp' (" ' " means conjugate transpose) is called coherence matrix. It can be seen that the coherence matrix is ​​a 4 4 Hermit semi-positive definite matrix. For the reciprocity target, the dimension of the coherence matrix is ​​reduced to 3 3.

(4) Covariance matrix: If the vectorization of the Sinclair matrix can be obtained directly: kl=[Sxx; Sxy; Syx; Syy], then C=kl.kl' ("'" means conjugate transposition) is called covariance matrix variance matrix. It can be seen that the covariance matrix is ​​also a 4*4 Hermit semi-positive definite matrix. For the reciprocal target, the dimension of the covariance matrix is ​​also reduced to 3*3. In addition, the coherence matrix and the covariance matrix are a pair of similar matrices, and one can obtain the other through similar transformation.

(5) When describing the polarization scattering matrix, many literatures are used to using the two affixes "polarization" and "scattering". For example, the Mueller matrix is ​​called the Mueller scattering matrix, and the coherence matrix is ​​called the polarization coherence matrix or coherent scattering matrix. The covariance matrix is ​​called the polarization covariance matrix. I personally think that in the field of PSAR, it is not necessary to use the two affixes "polarization" and "scattering". On the one hand, the description can be simplified, and on the other hand, the description can be unified without causing confusion.

(6) Some documents call the Sinclair matrix a 2*2 coherent scattering matrix. I personally think that the meaning of "coherence" here is relative to "within a resolution unit". Since the resolution unit size is much larger than the radar wavelength size, there are many strong scattering points in a resolution unit, and the target echo is positive. is the coherent superposition of echoes from these strong scattering points. It can be seen that the meaning of "coherent" here is consistent with the meaning of "coherent" in the formation principle of coherent spots.

(7) Some documents also call Mueller matrix and covariance matrix as coherence matrix. Personally think that the meaning of "coherence" in Mueller matrix, coherence matrix and covariance matrix is ​​relative to "between pixels", because the important significance of using coherence matrix (including Mueller matrix and covariance matrix) is to find The biggest difference between the set average and the Sinclair matrix at this time is that the coherent information between pixels is used, which is why the coherence matrix (including Mueller matrix and covariance matrix) is usually used for filtering, target decomposition, etc. For example, in the Huynen-type target decomposition, the coherence matrix {T} ("{.}" means seeking ensemble averaging) is decomposed into a rank-1 coherence matrix and an N target coherence matrix, where the rank-1 coherence matrix has a corresponding Sinclair matrix.

(8) To sum up, the meaning of "coherence" in the Sinclair matrix is ​​relative to "within a resolution unit", but the meaning of "coherence" in the Mueller matrix, coherence matrix and covariance matrix is ​​relative to "pixel In terms of "between pixels", the meanings of the two are quite different, so they should be treated separately when using the polarized scattering matrix.

Keywords: coherency matrix, covariance matrix, target decomposition

PS:
1. The scattering matrix S can be used to describe the variable polarization effect of the target, but its applicable object is the deterministic target. For the undulating target, the electromagnetic scattering characteristics are no longer fixed, but have certain randomness. At this time, the Mueller matrix is ​​used to Characterize its scattering properties.
2. There is also a matrix that is very similar to the M matrix—the Kennaugh matrix, only the sign of the last row is different from the Mueller matrix (under the BSA convention), and in some places these two matrices are mixed, and they contain the electromagnetic scattering characteristics of the target The messages are exactly the same, but their physical meaning is different. The M matrix is ​​used to describe the variable polarization effect of the target. The Kennaugh matrix (abbreviated as K) reflects the dependence of the radar receiving power on the transmitting and receiving antennas. There is a formula: P=1/2*(Jt * K*Js), where Jt, Js are the Stockeds vectors of incident wave and scattered wave.

polarization decomposition

Mainly three doctoral dissertations written in more detail.
Research on Polarimetric SAR Image Classification Method Zhou Xiaoguang National University of Defense Technology
Research on Polarimetric Decomposition and Scattering Feature Extraction Based on Polarimetric SAR An Wentao Research on
Polarimetric SAR Image Classification Technology Tsinghua University Wu Yonghui National University of Defense Technology

Jones matrix

Then
$$\begin{gathered} E(z,t)=({\mathbf{u}}_{\mathbf{x}}{E}_x+{\mathbf{u}}_{\mathbf{y}}{E}_y)e^{j(\omega t-k_{0}z)} \\ =(u_x{A}_x{e}^{j{d}_x}+u_y{A}_y{e}^{j{d}_y})e^{j(\omega t-k_{0}z)} \end{gathered}$$
where$A_x=|E_x|,A_y=|E_y|$。$k_0=2\pi f/w$为波束
得
$${\mathbf{E}}_{\text{Jones }}=\left[\begin{array}{l}{A}_x\exp \left\{j{d}_x\right\}\\ A_y\exp \left\{j{d}_y\right\}\end{array}\right]$$
令$d=d_y-d_x$, $A=\sqrt{A_x^2+A_y^2}$,$c=ta{n}^{-1}(A_y/A_x)$ equation: E Jones = [ A x A y exp ⁡ { j δ } ] = A [ cos ⁡ γ sin ⁡ γ ⋅ exp ⁡ { j δ } ] \boldsymbol{E}_{\text {Jones }}= \left[\begin{array}{c}A_{x}\\A_{y}\exp\{\mathrm{j}\delta\}\end{array}\right]=A\left[\begin{ array}{c}\cos \gamma \\\sin \gamma \cdot \exp \{\mathrm{j}\delta\}\end{array}\right]EJones ​=[Ax​Ay​exp{ jδ}​]=A[coscsinc⋅exp{ jδ}​]

Stokes vector

The Jones vector is only suitable for representing fully polarized electromagnetic waves . At this time, the average amplitude of the components of the electric field vector on the x and y axes is constant, and the phase difference is constant. The polarization ellipse drawn by the end of the electric field vector in the xoy plane has constant parameters . However, in practice, the trajectory of the electric field vector of the electromagnetic wave in the xoy plane is generally not a time-invariant ellipse, but a curve similar to an ellipse whose shape and direction change with time. This kind of electromagnetic wave is called partial polarization. Wave. It can be seen that fully polarized waves are only a special case of partially polarized waves. Obviously, other description methods need to be introduced for partially polarized waves.
Therefore, for partially polarized waves, the Stokes vector is introduced:
$$\begin{gathered} J=\begin{array}{c}\left[\begin{array}{c}{g}_0\\ g_1\\ g_2\\ g_3\end{array}\right]\end{array}=\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& -1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& \mathrm{j}& -\mathrm{j}\end{array}\right]\left[\begin{array}{c}{E}_x{E}_x^{\ast }\\ E_y{E}_y^{\ast }\\ E_x{E}_y^{\ast }\\ E_y{E}_x^{\ast }\end{array}\right] \\ =\left[\begin{array}{c}|E_x{|}^2+{|E_y|}^2\\ |E_x{|}^2-{|E_y|}^2\\ 2|E_x|\cdot |E_y|\cdot \cos d\\ 2|E_x|\cdot |E_y|\cdot \sin \end{array}\right] \end{gathered}$$

In the above formula, the constant matrix is ​​R, and g can be expressed as:

$$g=R\tilde{C}=R(E\otimes E^{\ast })$$
C is the wave coherence vector.
$g_{2}and{g}_3$Possible display is $2Re\left[E_x{E}_y^{\ast }\right]$和$-2Im[E_x{E}_y^{\ast }]$

It is easy to know that only three of the four Stokes parameters are independent, and their relationship is:

$$g_0^2=g_1^2+g_2^2+g_3^2$$
Stokes vector $J$ and polarization elliptic parameters,$E_{jones}$Definition:
$$\mathbf{J}=A^2\left[\begin{array}{c}1\\ \cos 2ps\cos 2x\\ \sin 2ps\cos 2x\\ \sin 2\end{array}\right]=A^2\left[\begin{array}{c}1\\ \cos 2c\\ \sin 2c\cos d\\ \sin 2c\sin \end{array}\right]$$
The Stokes vector of a partially polarized wave can be written as:
$$\mathbf{J}=\left[\begin{array}{c}{g}_0\\ g_1\\ g_2\\ g_3\end{array}\right]=\left[\begin{array}{c}\sqrt{g_1^2+g_3^2+g_3^2}\\ g_1\\ g_2\\ g_3\end{array}\right]+\left[\begin{array}{c}{g}_0-\sqrt{g_1^2+g_2^2+g_3^2}\\ 0\\ 0\\ 0\end{array}\right]$$
In the formula, the first term on the right is a completely polarized wave, and the second term is a completely unpolarized wave.

Scattering Matrix and Scattering Vector

The backscattering coordinate system is generally used in the case of a single station. At this time, the z-axis of the coordinate system whose origin is located at the antenna is opposite to the propagation direction of the target scattered wave. The forward coordinate system is generally used in the case of two stations. The radar transmits and receives Electromagnetic waves can be expressed as:
$$\{\begin{array}{l}{\mathbf{E}}_t=E_V^t{\mathbf{v}}_t+E_H^t{\mathbf{h}}_t\\ E_r=E_V^r{v}_r+E_H^r{h}_r\end{array}$$
In the formula, $E_t$和$E_r$Represent the transmitting and receiving electromagnetic wave vectors, respectively, and v and h represent the orthogonal polarization base. The relationship between the transmitted and received polarized electromagnetic wave vector formula is:
$${\mathbf{E}}_r=\left[\begin{array}{c}{E}_H^r\\ E_V^r\end{array}\right]=\frac{e^{j{k}_{o}R}}{R}\mathbf{S}\cdot E_t=\frac{e^{j{k}_{o}R}}{R}\left[\begin{array}{cc}{S}_{\text{HH}}& S_{\text{HV}}\\ S_{\text{VH}}& S_{\text{VV}}\end{array}\right]\left[\begin{array}{c}{E}_H^t\\ E_V^t\end{array}\right]$$
where$r$ is the distance between the scattering target and the receiving antenna,$k_0$is the electromagnetic wave number.
The scattering matrix is ​​defined as:
$$\mathbf{S}=\left[\begin{array}{cc}{S}_{\mathrm{HH}}& S_{\mathrm{HV}}\\ S_{\mathrm{VH}}& S_{\mathrm{VV}}\end{array}\right]$$
The scattering matrix S can be vectorized into a scattering vector:
$$\mathbf{x}={\left[\begin{array}{c}{S}_{HH}{S}_{HV}{S}_{VH}{S}_{VV}\end{array}\right]}^{\mathsf{T}}$$

From S, $G=S^{H}S$，即：
$$G=S^{H}S=\left[\begin{array}{cc}{\left|S_{\mathrm{HH}}\right|}^2+{|S_{\mathrm{VH}}|}^2& S_{\mathrm{HH}}^{\ast }{S}_{\mathrm{HV}}+S_{\mathrm{VH}}^{\ast }{S}_{\mathrm{VV}}\\ S_{\mathrm{HV}}^{\ast }{S}_{HH}+S_{\mathrm{VV}}^{\ast }{S}_{VH}& {|S_{HV}|}^2+{|S_{VV}|}^2\end{array}\right]$$
matrix G is called Graves matrix, because it is directly related to the power density of scattered waves, it is also called power matrix or power density matrix. In fact, assuming that the Jones vector of the incident wave is E , the power density of the scattered echo of the target is$P_s=E^{HGE}$。$\_$

Monostatic radar imaging satisfies the reciprocity theorem, that is, when the positions of the transmitting and receiving antennas are exchanged, the backscatter measurement value remains unchanged. At this time, $S_{HV}=S_{VH}$, the scattering vector is:
$$\mathbf{x}={\left[\begin{array}{ccc}{S}_{\mathrm{HH}}& \sqrt{2}{S}_{\mathrm{HV}}& S_{\mathrm{VV}}\end{array}\right]}^T$$
uses Pauli basis to decompose S to get:
$$\mathbf{k}=\frac{1}{\sqrt{2}}{\left[S_{HH}+S_{VV}{S}_{HH}-S_{VV}2S_{HV}\right]}^{\mathsf{T}}$$
Pauli基为：
$$p_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],p_2=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],p_3=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],p_4=\left[\begin{array}{cc}0& \text{j}\\ -\text{j}& 0\end{array}\right]$$
$x$ records the backscatter measurements for each polarization channel, and$The\; elements\; of\; k$ represent surface scattering, dihedral scattering, and 45° inclined dihedral scattering, respectively.

Muller matrix