# Analysis of Compound Stained Cervical Cell Images Using Multi-Spectral Imaging
## 2 Design and Principle
### Multi-Spectral Absorbance Unmixing Model  
There are three kinds of absorbent materials in the compound-stained cervical smears of this system; Figure 2 shows their absorbance curves in the spectral range of from 460 nm to 700 nm [27]. Traditional DNA quantitative analysis generally uses a wavelength of 570 nm because it is performed on Feulgen-stained specimens [28,29]. As shown in Figure 2, there are spectral overlaps of these absorbent materials in all spectral wavebands in the compound staining method. Therefore, the DNA content of cells cannot be measured directly.  
By the linear superposition character of absorbance, if there are n kinds of absorbent materials in the specimen, the total absorbance of the specimen is:  
\[A_{total}=\sum_{i=1}^{n}A_{i}^{\lambda}=\sum_{i=1}^{n}\epsilon_{i}^{\lambda}bc _{i} \tag{1}\]  
where \(A_{total}\) represents the total absorbance of the light-receiving medium. \(A_{i}^{\lambda}\) is the absorbance of each component in the medium, \(c_{i}\) is the concentration vector of the dyeing substance, and \(\epsilon_{i}^{\lambda}b\) is the spectral characteristic of the dyeing substance.  
Figure 1: Schematic diagram of microscopic multi-spectral imaging system.  
Figure 2: The absorbance curves of absorbent materials.  
If there are monochromatic lights with wavelengths of \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{m}\) in the system, the equation above can be described as:  
\[\left\{\begin{array}{l}A^{\lambda_{1}}=\sum\limits_{i=1}^{n}A_{i}^{\lambda_{1 }}=\sum\limits_{i=1}^{n}\varepsilon_{i}^{\lambda_{1}}bc_{i}\\ A^{\lambda_{2}}=\sum\limits_{i=1}^{n}A_{i}^{\lambda_{2}}=\sum\limits_{i=1}^{n} \varepsilon_{i}^{\lambda_{2}}bc_{i}\\ \vdots\\ A^{\lambda_{m}}=\sum\limits_{i=1}^{n}A_{i}^{\lambda_{m}}=\sum\limits_{i=1}^{n} \varepsilon_{i}^{\lambda_{m}}bc_{i}\end{array}\right. \tag{2}\]  
Equation (2) can be expressed by the matrix as follows:  
\[A=X\times C \tag{3}\]  
In Equation (3), \(A=\left[A^{\lambda_{1}},A^{\lambda_{2}},\cdots,A^{\lambda_{m}}\right]^{T}\) is the absorbance vector, \(X=\left[X_{1},X_{2},\cdots,X_{n}\right]\) is the feature Matrix where \(X_{i}=\left[\varepsilon_{i}^{\lambda_{1}},\varepsilon_{i}^{\lambda_{2}}, \cdots,\varepsilon_{i}^{\lambda_{m}}\right]\) is the feature vector, \(i=1,\ 2,\cdots,\ n\), and \(C=\left[c_{1},\ c_{2},\cdots,c_{n}\right]^{T}\) is the concentration vector of the absorbent materials in the specimen. By using the multiple linear regression method, the least square solution of the concentration vector is:  
\[C=\left(X^{T}\times X\right)^{-1}\times X^{T}\times A \tag{4}\]  
From the absorbance curves, the matrix \(X\) in Equation (3) can be obtained through collecting the absorbance image \(\left(A^{\lambda}\right)\) based on the microscopic multi-spectral imaging system. Then, the concentration of each absorbent material can be calculated by Equation (4).  
Four color bands are adopted for imaging analysis in this study. To be specific, the band of red color is 620 nm (central wavelength) \(\pm\) 2.5 nm (half width), while the band values of green color, blue color, and yellow color are 520 nm \(\pm\) 2.5 nm, 460 nm \(\pm\) 2.5 nm, and 570 nm \(\pm\) 8 nm, respectively. Fulgan presents the largest absorption in the 570 nm band, while eosin and brilliant green have a small amount of absorption. After subtracting the absorption values of eosin and bright aluminum from the absorption values of 570 nm, the absorption value of Forggen is obtained, which can be adopted to calculate the accurate DNA index.