# Research on Performance Evaluation and Optimization Theory for Thermal Microscope Imaging Systems
## 3 Results  
### MRTD Model  
Due to the different considerations and processing methods, MRTD has some different expressions, but its basic ideas are the same. As the derivation process of MRTD is quite complicated, we derive the MRTD model of the TMISs based on the points and results of the derivation by [PERSON] [25]. According to the basic idea of the MRTD derivation, when the image signal-to-noise ratio perceived by the human eye is greater than or equal to the visual signal-to-noise ratio, the temperature difference between the blackbody target and the background is the MRTD. The signal-to-noise ratio \(SNR_{d}\) of the stripe pattern at the output end of the display is  
\[SNR_{d}=R(f)SNR\left[\frac{\Delta f_{n}}{\int_{0}^{\infty}s(f)\cdot MTF_{i}{}^{2 }(f)\cdot MTF_{m}^{2}(f)df}\right]^{2} \tag{15}\]  
where \(R(f)\) is the square wave response of the system; \(MTF_{i}(f)\) is a modulation transfer function for electronic amplifiers and video processing circuits; \(MTF_{m}(f)\) is the modulation transfer function of the display; \(\Delta f_{n}\) is the equivalent power spectrum of noise. When the observer observes the target, the \(SNR_{d}\) will be modified in the following four aspects to obtain the visual \(SNR_{v}\):  
(1) The eye extracts the stripe pattern. In the case of distinguishable signals, the higher harmonics are filtered and the first harmonics are maintained. At this time, the signal peak attenuation is \(\frac{2}{n}R(f)=\frac{8}{\pi^{2}}MTF(f)\), \(MTF(f)\) is the system modulation transfer function  
(2) As a result of time integration, the signal will be sampled and integrated independently once according to the human eye integration time \(t_{e}(=0.2s)\), while the noise is superimposed in the form of the square root, and so the signal-to-noise ratio will be improved by \((t_{e}f_{p})^{1/2}\), \(f_{p}\) is the frame frequency. (3) In the vertical direction, the eye will carry out signal space integration, and take the root mean square value of noise along the line, using the vertical instantaneous field of view \(\beta\) as the noise correlation length, to obtain the improvement of visual signal-to-noise ratio  
\[\left(\frac{L}{\beta}\right)^{1/2}=\left(\frac{\varepsilon W}{\beta}\right)^{ 1/2}=\left(\frac{\varepsilon}{2f_{T}\beta}\right)^{1/2} \tag{16}\]  
where, \(L\) is the length of the strip; \(W\) is the stripe bandwidth; \(\varepsilon\) is a stripe length to width ratio, \(\varepsilon=7\); \(f_{T}\) is the stripe space-frequency \((c/mrad)\). (4) For the periodic rectangular line target with frequency \(f_{T}\), the filtering effect of the human eye on the narrowband space can be equivalent to a matched filter, filtered by the frequency domain of the human eye, in the horizontal direction, and the integral effect of the human eye can be captured by replacing the actual system bandwidth with the noise bandwidth \(\Delta f_{eye}\), which takes into account the effect of eye matching filter  
\[\Delta f_{eye}(f_{T})=\int_{0}^{\infty}s(f)\times MTF_{i}{}^{2}(f)\times MTF_ {m}^{2}(f)sinc^{2}(\pi f/2f_{T})df \tag{17}\]  
that is, the SNR is improved to \(\left[\frac{\int_{0}^{\infty}s(f)\cdot MTF_{i}{}^{2}(f)\cdot MTF_{m}^{2}(f)df }{\Delta f_{eye}(f_{T})}\right]^{2}\). Thus, combining the above four effects with \(SNR_{d}\), the visual \(SNR_{v}\) is obtained  
\[SNR_{v}=\frac{8}{\pi^{2}}MTF(f)\times SNR\times(t_{e}f_{p})^{1/2}\times(\frac {\varepsilon}{2f_{T}\beta})^{1/2}\times(\frac{\Delta f_{n}}{\Delta f_{eye}})^{ 1/2} \tag{18}\]  
by substituting Equation (14) into Equation (18), let \(s(f)=1\), and \(MTF_{i}(f)=MTF_{m}(f)=1\).