## 2 Materials and Methods
### Frequency Distribution Histograms and Mueller Matrix Transformation Parameters  
Among the available polarization imaging methods, Mueller matrix polarimetry has some unique advantages for tissue measurement. As a label-free and non-contact technique, Mueller matrix polarimetry is sensitive to the changes in subwavelength micro-structures and contains rich optical information of the samples, which have shown great application prospects in biomedical studies recently [19]. For bulk tendon tissues whose light scattering property is prominent, the 3 \(\times\) 3 Mueller matrix can be used to conveniently obtain most of the structural information [20]. The 3 \(\times\) 3 Mueller matrix is calculated by analyzing the relationship between the SOPs of scattered light and the incident light as shown in Equation (1):  
\[S_{\text{out}}=M_{\text{sample}}\times S_{\text{in}}=\begin{bmatrix}M11&M12&M13 \\ M21&M22&M23\\ M31&M32&M33\end{bmatrix}\times S_{\text{in}} \tag{1}\]  
Figure 1: (**a**) Schematic of the experimental setup for the backscattering Mueller matrix measurement. P1: polarizer; L1, L2: lens. The oblique incident angle \(\theta\) is about 7 deg to avoid the surface reflection from the sample. The diameter of the illumination area is about 2.3 cm. (**b**) Schematic diagram of four stretching states of tendon tissue samples: origin means unstretched, TDO means transverse stretched, MDO means longitudinal stretched and ODO means 45 deg stretched. The light orange lines represent the fibers.  
where \(S_{\text{in}}\) represents the Stokes vector of the incident light, \(S_{\text{out}}\) represents the Stokes vector of scattered light, and \(M_{\text{sample}}\) represents the Mueller matrix of the sample.  
The backscattered 3 \(\times\) 3 Mueller matrix images of the bovine tendon tissue samples in different states are shown in Figure 2, from which we can clearly notice that the change in applied external force affects each matrix element differently, benefiting us in the qualitative collection and evaluation of the microstructural information related to the mechanical properties of the tissue. For instance, in the M12, M21, M13 and M31 elements, the linear dichroism induced by the tissue fibers is recorded; the values of diagonal elements M22 and M33 reflect the linear depolarization ability of the tissue; and the differences between the M22, M33, M23 and M32 elements reveal the total anisotropy of the tissue [9; 21; 22]. However, though the 2D Mueller matrix images contain abundant tissue microstructure information, it is difficult to find a direct and quantitative connection between a certain microstructure and the Mueller matrix elements [12; 23]. To solve this problem, here we adopt the frequency distribution histogram (FDH) processing on the individual Mueller matrix elements to calculate their central moments [24]. Furthermore, we combine the Mueller matrix transformation (MMT) parameters and neural network (NN) to quantitatively evaluate the tendon tissues with different mechanical properties.  
Equations (2) and (3) define two MMT parameters used in this study: \(t_{1}\) is a rotation invariant to characterize the total anisotropy induced by both dichroism and birefringence of the sample, while \(\alpha_{1}\) represents the orientation of the general anisotropy [21; 25]. For bulk tissue samples with prominent fibrous structures, it was found that the statistical information of the MMT parameters \(\alpha_{1}\) and \(t_{1}\) have the potential to detect external forces applied to the sample at different angles.  
\[t_{1}=\frac{1}{2}\sqrt{\left(M22-M33\right)^{2}+\left(M23+M32\right)^{2}} \tag{2}\]  
\[\alpha_{1}=\frac{1}{4}atan2(M23+M32,M22-M33)\ \ \ \ \ if\ \ t_{1}\
eq 0 \tag{3}\]