## 2 Methodology
Limited to the numerical aperture of the imaging system, the reflected light from the top area of the bump forms a light spot on the imaging device, with \(T^{\prime}\) as the center of the light spot. Therefore, the height of the bump can be calculated based on the parameters mentioned above. The parameter \(k\) is assumed to be stated in Equation (3); thus, \(h\) can be expressed in Equation (4). \[k=\frac{1}{2\beta\sin\theta} \tag{3}\]  
Figure 4: The measuring principle of the height of the bump. \[h=k(x^{\prime}-r)+rk\sin\theta \tag{4}\]  
When the parameters \(k\), \(r\) and \(\beta\) are calibrated, the height of the bump can be deduced. The magnification of the imaging system \(\beta\) can be calculated using a 2D calibration plate composed of circular marks in standard size for implementing the camera calibration. The incident angle \(\theta\) is calculated from a pair of steps with known height formed by two reflecting planes. Additionally, the radius of the arc can be deduced by fitting the circle with edge points of the arc formed by the shadow of the bump. However, when the width of the light stripe is comparatively smaller than the diameter of the spherical cap of bump, the arc formed by the shadow of the bump in the image without the point S' leads to larger fitting errors of the radius \(r\). An efficient method to obtain the parameter \(r\) is to detect the distance of point \(P\) and point \(Q\), which are both intersection points of line and arc formed by the shadow of the bump. As the light scans the wafer surface, the distance between the point \(P\) and \(Q\) changes. When the left edge of the light bar passes through the center point O in Figure 4, the distance reached its maximum. At this time, the distance of \(P\) and \(Q\) equals the diameter of the bump cap. Considering the magnification of the imaging system, \(PQ\) equals \(2\beta R\). Therefore, the radius \(r\) can be deduced from the maximal value of \(PQ\) and the calibrated parameter \(\beta\). Additionally, the coordinate of \(T^{\prime}\) can be deduced by extracting the imaging point center of the laser spot. It is worth noting that due to the limitation of the entrance pupil of the receiving imaging system, only the reflected beam in a small circle (represented by DTC) at the top of the hemisphere can enter the receiving system, where \(T\) is the vertex of the convex point. The light formed by the parallel light illuminating the hemispherical surface is reflected to the image detector at different angles through the receiving system. When the incident light fills the area of DTC, the center position of the light spot is represented by point \(T\). However, as the light moves through the wafer, the light beam cannot always fill the area of DTC. When the light beam fills the area, the spot area is the largest. Only at this time can the coordinate of the point \(T\) be extracted accurately. Therefore, the spot area size needs to be detected in real time. When the size reaches its maximum, the center coordinate can be extracted. As the light beam moves, it can be seen that the spot area can be kept at maximum when the light area covers the DTC area. As the light intensity distribution of light stripe is usually Gaussian distribution, the centroid extraction method is not adopted because the light distribution can affect the extraction accuracy; the circle fitting method is preferred. If the light intensity distribution is a uniform distribution, the centroid extraction method can be adopted. The light beam cannot simultaneously cover the area of DTC while \(PQ\) reaches its maximal length. So, the coordinate of \(T^{\prime}\) and the length of \(PQ\) can be recorded at different times, and the height of the bump can be calculated using the parameters mentioned above. In addition, the bump height also affected the measurement accuracy. As shown in Figure 4, to accurately measure the height of the bump, it is necessary to collect high-definition pictures of the top and bottom of the bumps simultaneously.