# An Efficient Numerical Approach for Field Infrared Smoke Transmittance Based on Grayscale Images
## 3 Principles of Numerical Approach  
### Grayscale Smoke Transmittance Model  
The fundamental equation of smoke transmittance is shown below, according to the theory of infrared radiation transmission [19]. \[\tau_{smk}=\frac{L_{re}}{L_{tar}} \tag{1}\]  
where \(\tau_{smk}\) means smoke transmittance, \(L_{tar}\) refers to the infrared radiance of the target that the infrared detector receives before the attenuation of smoke, \(L_{re}\) stands for the residue target radiance that infrared detector receives under the attenuation of the smoke screen. Figure 1: Schematic diagram of field trial. Theoretically, smoke transmittance can be accurately calculated based on Equation (1). However, it is impractical to measure \(L_{tar}\) and \(L_{re}\) directly in the field trial due to environmental factors. As evident, every nonzero temperature object in nature will emit electromagnetic energy spontaneously [20]. Thus, targets, background, and smoke cloud itself are all electromagnetically radiant. The incident radiation that thermal infrared imager receives is the total amount of radiation in line of sight, and the infrared detector cannot quantitatively distinguish the amount of radiation from the target. In order to obtain the true value of \(L_{tar}\) and \(L_{re}\) in Equation (1), some other parameters are introduced to represent them as below. The total radiance that the infrared detector receives before smoke interference in line of sight can be obtained from  
\[L(r_{0})=L_{tar}+L_{sur} \tag{2}\]  
where \(L(r_{0})\) stands for the total radiance that the infrared detector receives before smoke interference in line of sight. This not only includes the radiation from the target \(L_{tar}\) but also consists of radiation from the surrounding \(L_{sur}\), such as the background, the sun, and the sky. The total radiance that the infrared imager receives after the release of smoke in line of sight can be obtained from  
\[L(r)=L^{\prime}_{tar}+L^{\prime}_{sur}+L_{smk} \tag{3}\]  
where \(L(r)\) refers to the total radiance that the infrared imager receives after the release of smoke in line of sight, and it contains the target remaining radiance after attenuation by the smoke \(L^{\prime}_{tar}\) and surroundings radiance \(L^{\prime}_{sur}\). In addition, the radiation from the smoke itself \(L_{smk}\) is also included. Substituting Equations (2) and (3) into Equation (1), the smoke transmittance calculation equation is expressed as follows:  
\[\tau_{smk}=\frac{L(r)-L^{\prime}_{sur}-L_{smk}}{L(r_{0})-L_{sur}}. \tag{4}\]  
For the purpose of calculating smoke transmittance, five radiance variables in Equation (4) have to be determined first. With the field trial images provided by the infrared imager, the gray value of each pixel in the image can be extracted by an image processing algorithm. Consequently, a method of converting the radiance to a corresponding gray value in infrared images is proposed here. According to the infrared radiance calibration and the relevant infrared physics information, the relationship between the target's radiance in the field and its corresponding gray value in the image is as follows [21]:  
\[L=aG+L_{off} \tag{5}\]  
where \(L\) stands for target's radiance, \(G\) is the corresponding gray value in the infrared image, \(a\) is a coefficient relating radiance to gray level, and \(L_{off}\) refers to the radiance offset. For the same model of thermal infrared imager equipment, \(a\) and \(L_{off}\) are constants.