# Principle and Implementation of Stokes Vector Polarization Imaging Technology
## 2 The Principle of Stokes Vector Measurement  
In 1852, [PERSON] discovered that any polarization state of light can be expressed in terms of four measurable quantities, the Stokes vector. The Stokes vector can describe any polarization state and being able to measure the Stokes vector means being able to measure any polarization state. The instrument for measuring the Stokes vector is also known as a polarizer (PSA). The polarizer should be able to detect arbitrary polarization states. In a real polarizer, the detector needs to \"directly\" detect at least four independent complete polarization states. Any other polarization states can be measured \"indirectly\".  
The structure of the Stokes vector measuring instrument is shown in the dotted line box in Figure 1. A beam of light with unknown polarization state enters the polarizer PSA and optical intensity detector D to calculate an unknown Stokes vector \(S\), m independent polarization components need to be detected by the polarizer, that is, m detection skewness is used as the basis to detect the projection of any unknown polarization states on each basis. For some unknown polarization state, the polarizer needs to measure the light intensity m times. These original light intensity signals can be written in vector form \(I\), whose matrix elements are \(\textit{Ij}\), \(\textit{j}\) = 0,1,2..., m \(-\) 1, m \(\geq\) 4. M measurements constitute a linear system of equations, which can be expressed in matrix form as  
\[I=A\cdot S \tag{1}\]  
In the above formula, \(A\) is the modulation matrix of the polarizer, and each row of \(A\) is filled by the polarizer vector \(S_{\text{Tj}}\). \(I\) is the intensity vector of the m column. In order to measure all Stokes parameters, the polarizer and the polarizer must be \"complete\", that is, both have at least four independent base polarization states. When m = 4, the  
Figure 1: Stokes vector measurement system (in dotted box). L: light source, PSG: polarizer, S: sample; PSA: polarizer, D: light intensity detector.  
matrix operation of Equation (1) represents a linear system of equations composed of four independent equations, which can be obtained by inverting the matrix \(A\).  
\[S=(inv(A)\cdot A)\cdot S=inv(A)\cdot(A\cdot S)=inv(A)\cdot I=B\cdot I \tag{2}\]  
In the above formula, \(B=inv(A)\) represents the inverse matrix of \(A\), which can be called the reconstruction matrix. When m > 4, the matrix operation of Equation (1) represents a linear system of equations composed of M independent equations. The number of equations is greater than the number of unknowns, so it is an overdetermined system of equations. Generally, there is no strict solution.  
\[S=(pinv(A)\cdot A)\cdot S=pinv(A)\cdot(A\cdot S)=pinv(A)\cdot I=B\cdot I \tag{3}\]  
\(B=pinv(A)\) in the above formula represents the pseudo-inverse matrix of \(A\). Equations (2) and (3) are similar except for the number of equations. In measurement, the known quantities \(A\) and \(I\) in each equation contain measurement errors. The process of solving overdetermined equations is similar to a least square method fitting, which can often reduce the influence of measurement errors.