# Spacecraft and Asteroid Thermal Image Generation for Proximity Navigation and Detection Scenarios
## 2 Thermal Modeling
### Radiometry
#### 2.4.2 DN-Heat Flux Sensor Function  
The second approach follows the same steps as the previous one but using the following formula for the pixel output as [13]:  
\[DN=a\cdot F(T)+b\,, \tag{14}\]  
where \(F(T)\) is used to keep the same notation of [13], and it is the emissive power:  
\[F(T)=q_{out}(T)=\pi\int_{0}^{\infty}\varepsilon(\lambda)\,B(\lambda,T)\,R( \lambda)\,d\lambda\,.\]  
As for the previous case, \(a\) and \(b\) coefficients are computed by calibration using black bodies at different temperatures. For the exact coefficient values, please refer to [13]. The steps are summarized as follows:\[q_{fc}\to F(T)=\frac{q_{fc}\,\pi\,|\mathbf{r}_{fc}|^{2}}{\Delta A_{c}} \to DN=a\cdot F(T_{rad})+b\to \tag{15}\] \[\to DN_{noisy}=DN+Noise\to F(T)_{noisy}=\frac{DN_{noisy}-b}{a}\,.\]  
With the above formulation, the radiative temperature can be recovered using the sensor calibration curve [13], where each \(F_{noisy}(T_{rad})\) value corresponds to a radiative temperature value. As a recap, the workflow is pictured in Figure 5, where the pipeline starts from the Blender rendered image, which is in 16 bit, so each pixel has a DN value that goes from 0 to 216. Since we know the maximum heat flux value of the scene and the minimum one, the pixel with the maximum DN value can be associated with the maximum heat flux value, and the same goes for the minimum one. In this way, any DN value of a pixel of the image can be associated with a \(q_{fc}\) value. In other words, the scale of the image is converted from DN values to \(q_{fc}\) values; therefore, every pixel now corresponds to a \(q_{fc}\) value. With such values, it is possible to compute \(F(T)\) for each pixel, which is the input variable for the camera model described in Equation (14); therefore, by knowing the gain and the offset of the thermal sensor function, i.e., \(a\) and \(b\), it is possible to compute the exact DN value that the given thermal camera will produce. In conclusion, the DN value of each pixel is converted in the actual DN values produced by the given thermal camera by using Equation (14). With such an image, the noise of the given thermal camera can be added to produce the realist output of the thermal sensor. For more details on the noise model and characterization of thermal sensors, please refer to [17]. The final step is now possible, and that is to reconvert the DN just computed for each pixel and with the noise added in \(F(T)_{noisy}\) values so that by knowing the dependency with the temperature [13], it is possible, for each pixel, to obtain the radiative temperature values.  
In both approaches, in the inversion of Equation (10), the part relative to the faces-camera orientation is left out in order to retain the view factor effects in the final output, and it can be observed in the results in the next section. Indeed, it is not impossible to know the exact shape of the object a priori and compensate for the view factors effects caused by the orientation of each face.  
The temperature field can always be corrected with the real emissivity of the material, if known, thus recovering the kinetic temperature. Starting from the radiative temperature, the kinetic one can be computed as:  
\[T_{kin}=\frac{T_{rad}}{\varepsilon^{1/4}}\,. \tag{16}\]  
Figure 5: Radiometry camera model workflow.