## 2 Experimental Details
### Characterization of Structure of Fabric Face Layer  
The color attributes of woven fabric depend on not only the dyed-yarn components for mixing but also the arrangement of yarn floats in the fabric face layer. Therefore, in order to elucidate the relationship between the color attributes of fabric and the characteristics of structure of fabric face layer, some key geometrical parameters were identified to represent the structure of fabric face layer. Figure 1a,b display the diagrams of structures of the fabric by mixing blue and red color and blue, red, and black color on the fabric surface, respectively. In this study, one repeating unit is composed of ten warp yarns and twenty-eight or twenty-four weft yarns for the weft-backed structure with or without the regulating layer, respectively, wherein the face layer of fabric consists of eight wefts and twelve wefts for the blue-red and blue-red-black mixed fabrics, respectively. In terms of the arrangement of face weft yarns, the fabric face layer can be divided into four equal segments while each segment comprises of two and three face weft yarns for the blue-red and blue-red-black mixed fabrics, respectively. Considering the weave repeat unit, seven and eight key geometrical parameters, that is, labelled as parameters d\({}_{\rm e}\), d\({}_{\rm p1}\), d\({}_{\rm p2}\), d\({}_{\rm p3}\), W, L, P and S, were proposed for describing the geometry of fabric face layer for two and three dyed-yarn color mixing, respectively. d\({}_{\rm e}\) is the warp yarn diameter. d\({}_{\rm p1}\), d\({}_{\rm p2}\), and d\({}_{\rm p3}\) are the diameters of face differently dyed-weft yarns for mixing. W is the width of one repeating unit. L is the length of one repeating unit. P is the distance between two corresponding face-weft yarns in the two adjacent segments (length of segment), and S is the spacing between two adjacent face-weft yarns in different segments. Parameters d\({}_{\rm e}\), d\({}_{\rm p1}\), d\({}_{\rm p2}\), and d\({}_{\rm p3}\) represent the fineness of warps and face wefts. Parameters W and L describe the size of one repeating unit in terms of width and length. Parameter P represents the length of one segment, which is assumed to be equal to the 1/4 length of the repeating unit. Parameter S focuses on the spacing between segments in the warp direction, which reflects the alignment of face weft yarns under a same binding unit by warps. It is assumed that all the weft yarns are in parallel position, and the diameter of the yarn with same yarn count should be uniform. The values of parameters W, L, and P are always positive theoretically while the value of the parameter S can be positive, zero, or negative. For the parameter S, a positive value represents that there is a spacing between two adjacent face weft yarns; zero means two adjacent face weft yarns are aligned with each other without overlapping and spacing; and a negative value indicates that there are overlapping between two adjacent face weft yarns. To obtain the value of geometrical parameters from the produced woven fabric, pictures in microscopic view of the fabric were captured and scaled using the Leica M165 C Microscope (Leica Microsystems, Wetzlar, Germany); Figure 2 shows the application of this approach \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Weave Code** & **N\({}_{\text{(BL+R+BK)}}\) (1)** & **N\({}_{\text{(BL+R+BK)}}\) (2)** & **N\({}_{\text{(BL+R-BK)}}\) (3)** & **N\({}_{\text{(BL+R-BK)}}\) (4)** \\ \hline Face weave & & & & \\ \hline Back weave & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fabric weaves designed for BLU, RE, and BLK wefts mixing on the fabric surface (N represents the weave without regulating layer). for characterizing the geometry of fabric face layer based on the microscopic images. The values of parameters were calculated and compared to investigate the effect of regulating layer, fabric density, weft yarn count, and fabric weave on the characteristics of structure of the fabric face layer, and thus, the resultant color attributes of fabrics.