## 5 Application Discussion
### An Ultra-Low-Noise Image Sensor  
Recently, a type of ultra-low-noise image sensor [44] was developed for applications requiring high-quality imaging under extremely low light conditions. Such a type of sensor is ideally suited for a variety of low light level cameras for surveillance, industrial, and medical applications. In application, the sensor and other components are assembled on a printed circuit board (PCB). Due to the mismatch in the thermal expansion coefficient of the various materials, thermal deformation occurs on the PCB under the combined action of self-heating and thermal environment. As a result, the imaging quality of the sensor is reduced. Moreover, to acquire more image information under low-light conditions, the sensor is designed in a large format. Thus, the imaging quality is more susceptible to deformation. This issue has become a challenging problem in this field and needs to be solved urgently. A robust optimization problem is considered for the camera module as Figure 6, in which the image-sensor-mounted PCB is fastened with the housing. The sensor is designed as a 4/3-inch optical format and features an array of five transistor pixels on a 6.5 um pitch with an active imaging area of \(2560(\mathrm{H})\times 2160(\mathrm{V})\) pixels. It delivers extreme low light sensitivity with a read noise of less than 2.0 electrons root mean square (RMS) and a quantum efficiency above 55%. In order to analyze the thermal deformation of the sensor under the operating temperature (\(20\,^{\circ}\mathrm{C}\sim 45\,^{\circ}\mathrm{C}\)), the finite element model (FEM) is created as shown in Figure 7, in which the power dissipation \(\mathbf{P}=(P_{1},P_{2})\) of the codec chip and the converter is given as 1.2 W and 0.2 W, according to the test data. It can be observed that a certain deformation appears on the sensor die, and the peak-peak value (PPV) of the displacement response achieves about 3.0 um. Consequently, the image quality of the sensor will decrease. To address the issue, the design objective is set to minimize the PPV, and the design variables of \(\mathbf{X}=(X_{1},\,X_{2},\,X_{3})\) are the normal positions of the PCB-fixed points. In engineering, manufacturing errors are unavoidable and power dissipation fluctuates with changing loads, and thereby \(\mathbf{X}\) and \(\mathbf{P}\) are treated as evidence variables. Their BPAs are summarized on the basis of limited samples, as listed in Table 2. This robust optimization is constructed as follows:  
\[\min v,\,\max Bel(PPV(\mathbf{X},\mathbf{P})\leq v) \tag{23}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Results**} & \(Bel_{1}^{T}=80\%\) & \(Bel_{2}^{T}=85\%\) & \(Bel_{3}^{T}=90\%\) & \(Bel_{4}^{T}=95\%\) & \(Bel_{5}^{T}=99.9\%\) \\ \hline Proposed & \(\mathbf{X}_{j}^{*}\) (mm) & 0.937, 0.086, & 0.937, 0.086, & 0.937, 0.086, & 0.937, 0.086, & 0.937, 0.086, \\ method & 0.072 & 0.072 & 0.072 & 0.072 & 0.072 & 0.072 \\ Reference solution & \(v_{j}^{*},Bel_{j}^{*}\) & 22.1 mV, 80.4\% & 20.0 mV, 85.3\% & 18.2 mV, 90.6\% & 15.7 mV, 95.5\% & 9.6 mV, 99.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal set of the micro-force sensor problem. Figure 6: The camera module (**a**) with an ultra-low-noise image sensor (**b**). PCB: printed circuit board. As mentioned above, the performance function of PPV is implicit and based on the time-consuming FEM, which consists of 88,289 8-node thermally coupled hexahedron elements. The computational time for solving the FEM is about 0.1 h, if using a computer with the i7-4710 HQ CPU and 8 G of RAM. To realize the parameterization and reduce the computational cost of obtaining reference solutions, a second-order polynomial response surface is created for the performance function by sampling 200 times on the FEM. \[ \begin{array}{c}PPV=\ 2.396-9.924X_{1}-6.495X_{2}+14.178X_{3}+0.311P_{1}-0.226P _{2}+8.564X_{1}^{2}+16.960X_{2}^{2}\\ +14.104X_{3}^{2}-0.019P_{1}^{2}+0.794P_{2}^{2}-1.540X_{1}X_{2}-13.168X_{1}X_{3} -13.822X_{2}X_{3}-0.074P_{1}P_{2}\end{array}