# Development of a Calibration Strip for Immunohromatographic Assay Detection Systems
## Methodology
### Image Acquisition and Processing
#### 3.1.3 Image Processing  
The acquired image of the test strip can be segmented into three parts, which are strip shell, background, and test line part. Nevertheless, only the test line part contains the detection information. Therefore, image segmentation and test line extraction should be conducted. Basically, RGB, HSV, and YUV are three representative color spaces which are commonly used in the image processing field [19]. However, RGB and YUV are mainly applied in raw data and coding standards, whereas HSV is more closer to human perceptions [20]. In this regard, the HSV color space is selected to perform the fuzzy c-means (FCM) clustering algorithm and maximin-distance algorithm for strip image segmentation.  
The FCM clustering algorithm, which is an unsupervised clustering technique, has been widely used in biomedical image segmentation [21; 22; 23; 24; 25]. In the FCM clustering algorithm, fuzzy data is classified into a proper subset by minimizing the objective function. Assuming the given sample set is \(X=\{x_{1},x_{2},\cdots,x_{n}\}\subset R^{s}\), sample space dimension is \(s\), sample amount is \(n\), and \(c\) (\(1<c<n\)) is subset amount after classifying. In this case, the FCM can be described as follows:  
Figure 3: (**a**) Structure of the image acquisition device; and (**b**) structure of the light source.  
Figure 2: Schematic diagrams of the ICG assay test strip.  
\[J(U,V)=\sum_{i=1}^{c}\sum_{j=1}^{n}u_{ij}^{m}d_{ij}^{2} \tag{6}\]  
where  
\[\sum_{i=1}^{c}u_{ij}=1,\quad 1\leqslant j\leqslant n \tag{7}\]  
\[\sum_{j=1}^{n}u_{ij}>0,\quad 1\leqslant i\leqslant c \tag{8}\]  
\[u_{ij}\geqslant 0,1\leqslant i\leqslant c,1\leqslant j\leqslant n \tag{9}\]  
In Equation (6) to Equation (9), \(m\) (\(m>1\)) is the fuzzy parameter, \(U=u_{ij}\) is a \(c\times n\) fuzzy partition matrix, \(u_{ij}\) indicates the membership value of \(x_{j}\) that belongs to class \(i\), \(V=[v_{1},v_{2},\cdots,v_{c}]\) is a \(s\times c\) matrix which is composed of \(c\) clustering center vectors, and \(d_{ij}=\ ||\,x_{j}-v_{i}\,||\) is the distance between sample point \(x_{j}\) and center point \(v_{i}\). Therefore, the fuzzy clustering algorithm is evolved into the optimization of the restrained argument of \((U,V)\). Then, the iterative equation is acquired by the necessary conditions of the extreme point.  
\[v_{i}=\frac{\sum_{j=1}^{n}u_{ij}^{m}x_{j}}{\sum_{j=1}^{n}u_{ij}^{m}},\quad i=1,2,\cdots,c \tag{10}\]  
Assuming \(I_{j}=\left\{(i,j)|x_{j}=v_{i},1\leqslant i\leqslant c\right\}\), if \(I_{j}=\varnothing\), then  
\[u_{ij}=\left[\sum_{r=1}^{c}\left(\frac{d_{ij}}{d_{rj}}\right)^{\frac{2}{m-1}} \right]^{-1}i=1,2,\cdots,c;j=1,2,\cdots,n \tag{11}\]  
If \(I_{j}\
eq\varnothing\), then \(u_{ij}\) is the arbitrary non-negative real number that satisfies the following condition:  
\[\sum_{i=1}^{c}u_{ij}=1;u_{ij}=0,d_{ij}\
eq 0 \tag{12}\]  
The equation of membership degree shows the mapping relationship from point to set and the membership degree can be updated by Equation (13). Now, the implementation steps of the FCM algorithm are: firstly, initialize the clustering center or membership degree matrix; then, Equations (10) and (13) are iterated until the inequality (14) is satisfied. Specific steps of the FCM algorithm are explained in Table 1.  
\[u_{ij}=\left\{\begin{array}{l}\left[\sum\limits_{r=1}^{c}\left(\frac{d_{ij} }{d_{rj}}\right)^{\frac{2}{m-1}}\right]^{-1},I_{j}=\varnothing\\ \frac{1}{|I_{j}|},I_{j}\
eq\varnothing,i\in I_{j}\\ 0,I_{j}\
eq\varnothing,i\
otin I_{j}\end{array}\right. \tag{13}\]  
\[||\,V^{(k)}-V^{(k-1)}\,||\leqslant\varepsilon,k\geqslant 1 \tag{14}\]A maximin-distance algorithm is a simple heuristic procedure which can be used in initializing the cluster center in the FCM algorithm to assure stability of the result and to avoid random initialization [26; 27; 28; 29]. The specific steps of algorithm is described in Table 2.