# An Efficient Numerical Approach for Field Infrared Smoke Transmittance Based on Grayscale Images
## 3 Principles of Numerical Approach
### Selections of Positions  
Once the coordinates of the points in an image are determined, the gray value of each points can be obtained by the image processing method [22]. In Equation (6), \(G(r_{0})\) and \(G(r)\) can be directly obtained by extracting the gray value of the corresponding target pixel point in the image. \(G_{sur}\) and \(G_{sur}^{\prime}+G_{smk}\) need to be estimated by extracting the gray value of the surrounding points of the target, which will be disscussed in this section.  
The principle of selecting positions of \(G_{sur}\) and \(G_{sur}^{\prime}+G_{smk}\) is based upon the theory that choosing the surrounding points of targets that can represent the background conditions of targets. The midpoint between targets was set to be twice the geometrical resolution of the system (IFOV) away from the target. In this way, the midpoint is close enough to the target to represent its background condition.  
Figure 2 below shows the schematic picture for the selection of positions without a smoke screen. In Figure 2, the red points stand for standard blackbodies used as targets, and blue points represent the midpoints near targets. The total gray value of each target \(G(r_{0})\) can be directly obtained by extracting the pixel gray value of the corresponding red point in the infrared image. In addition, the environmental condition of nearby blue points is similar to that of the target, so the gray value from the environment \(G_{sur}\) of each target can then be estimated by the gray value of a certain nearby blue point, or the mean gray value of the surrounding blue points in the image. By the method mentioned above, \(G(r_{0})\) and \(G_{sur}\) can be obtained, and the denominator in Equation (6) remains a constant value during the calculation process.  
Figure 3 shows the schematic picture of the selection of positions under the influence of smoke. After being released, smoke spreads out gradually and starts to obscure certain targets. In this circumstance, the total gray value of target \(G(r)\) can be acquired by extracting the gray value of the red point in the infrared image. In addition, the gray value of the environment and smoke itself under the influence of smoke \(G_{sur}^{\prime}+G_{smk}\) can also be estimated by the gray value of a certain nearby blue point, or the mean gray value of the surrounding blue points in the image.  
Figure 3: The selection of positions of \(G(r)\) and \(G_{sur}^{\prime}+G_{smk}\) (with smoke).  
Figure 2: The selection of positions of \(G(r_{0})\) and \(G_{sur}\) (no smoke).  
Provided that all the variables required in Equation (6) are all obtained by the method mentioned above, the infrared smoke transmittance of each obscured target during the smoke diffusion process can then be calculated.  
Moreover, the number of surrounding midpoints to represent target background condition also needs further study. In this paper, we propose three different algorithms to calculate the background condition of the target. The three value computing methods are shown in Figure 4 as follows:  
1. choose one midpoint beside the target;
2. choose the mean value of four midpoints surrounding the target;
3. choose the mean value of the points circled around the target.