## 4 Feature Extraction
The difference between slopes of two adjacent straight line segments is a good measure of the curvature at that point of intersection [19]. The slope is defined by:  
\[\varphi(t)=\tan^{-1}\left(\frac{\dot{y}(t)}{\dot{x}(t)}\right) \tag{7}\]  
Figure 5: **(a)** raw sputum cells; **(b)** nucleus and cytoplasm segmentation; **(c)** nucleus region; **(d)** the perimeter of the nucleus area. where \(\dot{x}(t)\) and \(\dot{y}(t)\) denote the derivative of \(x(t)\) and \(y(t)\). We computed the difference between adjacent slopes (\(\delta\theta\)) for each point in the nucleus contour. In our data, we found that in the case of malignant cells, \(\delta\theta\) goes above a threshold estimated to 50. Figure 7 shows the curvature extraction (\(\delta\theta\)) of the benign cell. Figure 7a shows the benign sputum cell and Figure 7d shows the boundary, which is used to compute the curvature. Figure 7e depicts the curvature where the \(x\)-axis represents the slope between the points in the image boundaries and the \(y\)-axis represents the curvature. As can be seen from this figure, all the lines are between \(0^{\circ}\) and \(45^{\circ}\), and no line exceeds our \(\delta\theta=50\). Figure 6: Intensity variances for the nuclei cells, the blue represents the malignant cells (MD) and the red represents the benign cells (BD). Figure 7: The curvature extraction of a benign cell. (**a**) Sputum cell; (**b**) Nucleus and cytoplasm segmentation; (**c**) Nucleus extraction; (**d**) Nucleus boundary and (**e**) reflects the curvature plot for the image boundary in (**d**). On the other hand, Figure 8 shows the curvature extraction of a malignant cell. As can be seen from Figure 8e, there are some lines in the curvature (\(y\)-axis) which exceed \(\delta\)0 = 50, and this is a clear reflection of the irregularities in the boundary of the cell. The fifth feature is the circularity, which is a feature that describes the roundness of the nucleus, and is defined as  
\[\text{Circularity}=\frac{4\pi\text{Area}(\text{Nucleus})}{\text{ Perimeter}(\text{Nucleus})^{2}} \tag{8}\]  
Cells in cleavage are normally round, so their roundness value will be higher. On the other hand, normal growing cells are irregular so their roundness value will be lower. For the circularity, it should be less than or equal to 1 within the threshold ratio, to distinguish between normal and abnormal cells, where 1 means a perfect circle [20]. The last feature is the Eigen ratio [21]. In our system, irregular cells are long, thus expected to have a higher Eigen ratio than that of round floating cells. Thus, using a proper threshold value we can distinguish between benign and malignant cells with this feature. The Eigen ratio is computed as follows:  
\[\text{Eigen\_ratio}=\frac{\frac{a}{b}+\frac{b}{a}}{2} \tag{9}\]  
where (\(a\), \(b\)) are the eigenvalues of the covariance matrix \(C\), defined by:  
\[C=\frac{1}{N}\sum_{i=1}^{N}p_{i}\,p_{i}^{T} \tag{10}\]  
Figure 8: The curvature extraction of a malignant cell. (**a**) sputum cell; (**b**) nucleus and cytoplasm segmentation; (**c**) nucleus extraction; (**d**) nucleus boundary and (**e**) reflects the curvature plot for the image in (**d**).