# Optimized Passive Defense Measures via IR Imaging
## 2 Review of the Model  
Historically, IR techniques are known since [PERSON]'s historical experiment in the 18 th century regarding refrangibility of the invisible spectrum of the Sun. Further, IR technologies are among the important controlling factors concerning night visioning. This is realized by optimizing the IR image formation over variations of the temperature and emission wavelength. Emissivity helps recognizing and surveillance of various defense instruments, such as tanks, anti-tanks and missile systems, Hull's military ship, and air-to-air missiles. Detection and concealing of such instruments are achieved largely on the basis of IR technology. Our proposed optimization of thermal energy radiation finds peaceful applications including industry, health, earth resource management, and energy conversation. Particular applications include a detection of cancer and other trauma, satellite IR images, global monitoring of the environment, pollution control and climate changing. Other technical aspects of this research include chemical monitoring, IR spectroscopy via frequency mode oscillations, IR astronomy, and monitoring of road accidents. As far as fluctuations in the radiative energy is concerned, the optimization analysis is equally supported at microscopic level by vibrations of atoms, their excitation to high energy states, their descending to low energy states, vibrations of charge particles, electronic vibrations, and all other instruments generating electromagnetic radiation. In this regard, the Planck diffusion law as a real valued map from the space of the wavelength and temperature \((\lambda,T)\) to the set of real numbers \(\mathbb{R}\) is defined by  
\[(\lambda,T)\to f(\lambda,T)=\frac{A}{\lambda^{5}}\left(e^{\frac{b}{ \lambda T}}-1\right)^{-1}. \tag{1}\]  
Here, we consider an estimation of the presence of enemies instrument, sensors or other sources of a danger via an optimized detection of the radiation of an emitting body. Let us consider an enemy's instrument that emits radiation with certain wavelength \(\lambda\) in a given background temperature \(T\). Ourreceiver detects the radiation from an existing source with a damped wavelength \(\lambda^{\prime}\) as shown in Figure 3. In the light of the fluctuation theory, we would like to emphasize that a thermal detector works well whenever we have \(\lambda\approx\lambda^{\prime}\) and \(T\approx T^{\prime}\), where \(T^{\prime}\) is the temperature corresponding to \(\lambda^{\prime}\). Notice that the Wien's radiation configuration is formed whenever \((\lambda,T)\) lie on a rectangular hyperbola \(\lambda T=b\), where \(b\) takes the value of 2808 \(\upmu\)m K [2]. Herewith, whenever we minimize the uncertainty  
\[\Delta\lambda=\lambda^{\prime}-\lambda \tag{2}\]  
in measuring the emitter's wavelength \(\lambda\), the uncertainty \(\Delta T\) in the temperature measurement becomes maximized, and vice-versa. In order to study extrema of the radiation distribution function \(f(\lambda,T)\) under variations of \(\lambda\) and \(T\), we need to examine its stationary points. In particular, we are interested in variations of \(\lambda\) and \(T\) as schematically represented with the dashed curve in Figure 3. This offers an intrinsic description of fluctuations of the thermal radiation distribution profile \(f(\lambda,T)\). As it is known, a stationary point \((\lambda_{0},T_{0})\) is an extremum, if the fluctuation discriminant  
\[D:=\partial_{\lambda}^{2}f(\lambda_{0},T_{0})\partial_{T}^{2}f(\lambda_{0},T_ {0})-(\partial_{\lambda}\partial_{T}f(\lambda_{0},T_{0}))^{2}>0. \tag{3}\]  
Moreover, there is a local minimum at \((\lambda_{0},T_{0})\) whenever, in addition to the positivity of \(D\), one of the pure capacities \(\partial_{\lambda}^{2}f(\lambda_{0},T_{0})\) or \(\partial_{T}^{2}f(\lambda_{0},T_{0})\) takes a positive value. We have a local maximum for the case of the negative sign of the either of the pure capacities at \((\lambda_{0},T_{0})\).