## 2 Materials and Methods
### Vectorial Retardance Imaging  
To obtain the birefringence of the sample, the measured Mueller matrix is then decomposed by the MMPD method proposed by [PERSON] and [PERSON] [69], defined as follows:  
\[M=M_{\Delta}M_{\mathrm{R}}M_{\mathrm{D}} \tag{5}\]  
where \(M_{\Delta}\), \(M_{\mathrm{R}}\), and \(M_{\mathrm{D}}\) are the depolarization, retardance, and diattenuation matrices, respectively. Then, the vectorial retardance (\(\overset{\rightarrow}{\mathbf{R}}\)), which completely characterizes the birefringence properties, is derived from \(M_{\mathrm{R}}\)[69]:  
\[\overset{\rightarrow}{\mathbf{R}}= \begin{pmatrix}R_{\mathrm{H}}\\ R_{45}\\ R_{\mathrm{C}}\end{pmatrix} =R \begin{pmatrix}a_{1}\\ a_{2}\\ a_{3}\end{pmatrix}  \tag{6}\]  
where \(R_{\mathrm{H}}\), \(R_{45}\), and \(R_{\mathrm{C}}\) represent the horizontal, \(45^{\circ}\)-linear, and circular retardance components, respectively. The normalized Stokes vector \((1,a_{1},a_{2},a_{3})^{T}\) indicates the fast axis of \(\overset{\rightarrow}{\mathbf{R}}\). The magnitude of \(\overset{\rightarrow}{\mathbf{R}}\) (i.e., \(R\)), also called scalar retardance, can be deduced from the three components by the square root of the sum of squares method [69]. In addition, the orientation of linear retardance (\(\theta\)) can be obtained as follows:  
\[\theta=\frac{1}{2}\tan^{-1}(\frac{a_{2}}{a_{1}}) \tag{7}\]  
Recent reports have successfully proven birefringence is a useful marker for bio-diagnostics. As the basic functional indicator of birefringence, scalar retardance has been widely utilized benefiting from the ability to visualize the density of ultrastructural organization (e.g., the density of aligned collagen fibers) [46]. It has been used for differentiating MM from BE using an MMIP [31; 60]. Yet, due to the inherent limitations resulting from the square root of the sum of squares method, the scalar retardance lacks the fast axis information. The orientation of the fast axis is used to show the orientation information. Thus, they cannot fully describe the birefringence characteristics and shows relatively low contrast for internal anisotropic structures [47; 69]. Conversely, the vectorial retardance, including the horizontal, \(45^{\circ}\)-linear, and circular retardance components, contains not only the magnitude but also the fast axis of retardance. It is of gaining increasing interest in pathology research [31; 70; 71]. Corresponding clinical value using the components has been demonstrated in the diagnosis of diseases of the colon [71], skin [31], cervix [70], etc. Unfortunately, it is inconvenient and time-consuming for clinicians to read and analyze all three components simultaneously in the clinic. As a result, such clinical challenge severely hinders the application of advanced polarization imaging techniques in clinical practice. To address this clinical challenge, a true-color vectorial retardance imaging technique is utilized by encoding the three components of vectorial retardance into the color space. The key post-processing steps are shown in Figure 2. Specifically, \(R_{\mathrm{H}}\), \(R_{45}\), and \(R_{\mathrm{C}}\) are encoded as the primary colors of green, red, and blue, respectively, in the RGB color space. The brightness of the integrated image is then modulated in the RGB color space according to the scalar retardance. Thus, the true-color vectorial retardance image can be reconstructed to completely characterize the birefringence properties of probed samples. Figure 3 graphically illustrates the color-encoded vectorial retardance imaging technology. Figure 3a,b show all possible states of the vectorial retardance mapped on a color-encoded sphere. The color is determined by the orientation of the fast axis. Figure 3c lists the relationship between several typical vectorial retardance states and the colors, where different colors correspond to different fast axis orientations. Figure 3d,i show several typical cross-sections of the color-encoded sphere. The radius is determined by the scalar retardance that regulates the brightness.