For the “zero”-distance infrared image ($L=L0$), we can determine the temperature range ($Tmin$, $Tmax$) and can calculate the gray level range ($Gmin$, $Gmax$). The relationship between temperature and the gray values can be approximated by a linear relationship in a particular temperature range.^{12} Therefore, the temperature $T$ in the “zero”-distance infrared image is defined as Display Formula
$T=Gmax\u2212GminTmax\u2212Tmin\xb7G+Tmin,$(16)
where $G$ is the pixel gray level. The temperature difference $\Delta Tij$ of a given pixel ($i,j$) is defined as the temperature difference between the given point and its neighboring points: Display Formula$\Delta Tij={\u2211p=\u221211\u2211q=\u221211[T(i,j)\u2212T(i+p,j+q)]}/9,$(17)
where $T(i,j)$ is the temperature at pixel ($i,j$). The mean temperature difference $\Delta Tavg$ of the whole image at distance $L0$ is given by Display Formula$\Delta Tavg(f1)=\u2211i,j=1i,j=m,n\Delta Tijmn,$(18)
where $m$ and $n$ are the pixel numbers of the “zero”-distance infrared image in the horizontal and vertical directions, respectively, and $f1$ is the highest spatial frequency of the infrared image at $L0$. The temperature difference between neighboring pixels has the highest frequency at $L0$. Therefore, the average temperature difference of the image at $L0$ corresponds to the highest frequency $f1$. Sceneries at different distances have different highest spatial frequencies, each of which is less than $f1$. For example, at the distance $L=2L0$, the highest spatial frequency for the scenery on the detector is $f2=(1/4)f1$, indicating that each pixel on the detector represents the average temperature of the four pixels in the “zero”-distance infrared image. Therefore, the temperature of a pixel in the infrared image at distance $L=2L0$ is Display Formula$T(i,j)=[T(2i\u22121,2j\u22121)+T(2i\u22121,2j)+T(2i,2j\u22121)+T(2i,2j)]/4.$(19)