# Small Target Radiometric Performance of Drone-Based Hyperspectral Imaging Systems
## 2 Background
### Imaging Point Targets: Radiometric and Spatial Response  
Imaging point targets (i.e., SPARC) with remote sensing instruments has been in development since 2010 [8; 9; 15; 16] with more recent engineering advances, by Labsphere Inc., to bring an automated calibration service to the satellite community [17; 18; 19; 20; 21]. Further advancements have been explored by the Rochester Institute of Technology on the application of point targets for drone-based HSI systems [22; 23]. In this section, we will summarize the radiometric and spatial response of an imaging system to a point target. A full derivation related to the radiometric properties of convex mirrors is presented in Appendix A.  
A point target is created by the virtual image of the solar disk created by a convex mirror. The convex mirror has intrinsic properties defined by the surface reflectance \(\rho_{m}(\lambda)\), radius of curvature \(R_{m}\), and clear aperture diameter \(D_{m}\). For remote imaging systems, a convex mirror (i.e., point targets) gives us both a radiometric and spatial calibration target, which is discussed below.  
An imaging system's spatial response to point targets is defined by the Sampled Point Spread Function (SPSF). The total radiometric energy from a point target spreads over the SPSF. The radiometric integral or summation over the SPSF defines the ensquared energy. Because the SPSF can be measured with convex mirrors during field experiments, the SPSF estimate will include not only the optical system and detector contributions but also motion blur, jitter and post-processing artifacts from orthorectification. Post-processing artifacts will be defined as orthorectification radiometric errors and will contribute to the small target radiometric performance analysis. Theoretically, the SPSF has an infinite spatial extent but the imaging system's Noise-Equivalent Irradiance (NEI) and point target signal intensity will ultimately limit the spatial extent of the SPSF. As long as the observed signal from a point target is bright enough, the energy lost to noise will be negligible.  
When downwelling irradiance, \(E_{T}(\lambda)\) illuminates a convex mirror, and a virtual solar image is formed at the focal length (i.e., \(f=-R/2\)). In addition to reflected solar irradiance, a considerable amount of sky irradiance (defined by \(G(\lambda)\)) is observed. A point target is radiometrically defined by its spectral radiant intensity \(I_{m}(\lambda)\) reflecting off the surface of the mirror. The mirror's spectral radiant intensity is propagated to the entrance aperture-reaching spectral radiance \(L_{EAR}(\lambda)\) as  
\[L_{EAR}(\lambda)=\
icefrac{{1}}{{4\rho_{m}(\lambda)R_{m}^{2}}}\bigg{[}1-G( \lambda)\cos{(2\theta_{m})}\bigg{]}\frac{E_{T}(\lambda)}{GSD^{2}} \tag{3}\]  
where \(G(\lambda)\) is the Global-to-Diffuse Ratio, \(2\theta_{m}\) defines the angle between the virtual solar image and optical axis, and \(GSD\) is the imaging system's ground sampling distance. In theory, Equation (3) is a point target's total energy when summed over the SPSF, and it will be connected to the measured data in Section 3.3. This equation has the following assumptions: the atmospheric transmission loss from the target to the sensor is assumed to be negligible for UAV altitudes [3], both the solar irradiance and \(G(\lambda)\) are measured throughout the experiment, and the ground sampling distances are related to square pixels. In addition, path radiance is negated from a point target's signal during the image extraction; this will be further discussed in Section 3.3. Note that if \(\cos{(2\theta_{m})}\) goes to zero (i.e., the mirror reflects the entire hemisphere), there is no need to measure the sky irradiance or estimate \(G(\lambda)\) because the ASD field spectroradiometer will be measuring the same solid angle that the mirror reflects.