# Automatic Detection of Marine Litter: A General Framework to Leverage Synthetic Data
## 2 Materials and Methods
### Methodology and Modelling
#### 2.1.2 Optical Modeling  
For a visual illustration of the quantities used in the optical model, see Figure 2.  
Figure 1: Definition of the methodology. From an optical point of view, satellite acquisition process may be modeled as a spatial filter for each wavelength [31]. Neglecting the radiative sources that are predominant for wavelengths greater than 2.5 um, the radiance received by the satellite sensors is, for each spectral band \(s\) and wavelength \(\lambda\in s\):  
\[L_{SAT}^{s\lambda}(p,t)=\int_{\Omega}\text{psf}^{\prime}(u)L_{TOA\uparrow}^{s \lambda}(p-u,t)\,\text{d}u \tag{1}\]  
where \(\Omega\) is the surface of the detector, \(p\) is the position on the surface of the detector of a point \(g\), observed on the surface of the Earth; \(t\) represents the picture acquisition time. Moreover:  
* \(\text{psf}^{\prime}\) is the Point Spread Function for band \(s\). It defines the local response of a Dirac delta impulse in the resulting image [32]. It is thus the convolution kernel of the filter. * \(L_{TOA\uparrow}^{s\lambda}(p,t)=\pi_{atm}^{s\lambda}(p,t)L_{g\uparrow}^{s \lambda}(p,t)+L_{atm}^{s\lambda}(p,t)\) is the ascending top of the atmosphere radiance, with \(\pi_{atm}^{s,\lambda}\) being the atmospheric transmittance and \(L_{atm}^{s,\lambda}\) being the atmosphere in-scattering term. All these terms are defined for each band. * \(L_{g\uparrow}^{s\lambda}(p,t)=\rho_{g}^{\lambda}(p)L_{g\downarrow}^{s\lambda }(p,t)\) where \(\rho_{g}^{\lambda}(p)\) is the reflectance of the object on the ground, and \(L_{g\downarrow}^{s\lambda}(p,t)\) sums the flux of light received by the object from the sun taking into account all atmospheric effects. To simplify the model, we consider the following hypotheses:
* \(\text{H1}\): All the terms are constant in time when compared to the timescale of image acquisition. * \(\text{H2}\): The observed scene is considered small enough so that \(\tau_{atm}^{s,\lambda}\) and \(L_{atm}^{s,\lambda}\) are constant with respect to \(p\), i.e., the atmosphere is assumed to be the same in a neighborhood. * \(\text{H3}\): To simplify further, the effects of the atmosphere are neglected. The term \(L_{atm}\) is overlooked and \(\tau_{atm}\) is set to 1. This is in accordance with real satellite images corrected of the effect of the atmosphere, as they are made available by ESA, supposing that this correction is perfect. * \(\text{H4}\): Keeping in mind our disregard of the atmosphere, we have: \(L_{g\downarrow}^{s,\lambda}(p)\simeq L_{TOA\downarrow}^{s,\lambda}(p)\). This implies that \(L_{TOA\uparrow}^{s,\lambda}(p)\simeq\rho_{g}^{\lambda}(p)L_{TOA\downarrow}^{s,\lambda}(p)\). * \(\text{H5}\): \(L_{TOA\downarrow}^{s,\lambda}(p)=\dfrac{I_{0}(\lambda)}{\tau_{ES}^{s }}\), the descending _top of the atmosphere_ radiance, is considered constant for band \(s\). \(I_{0}\) is the solar constant and \(r_{ES}\) the distance between the earth and sun. With this, we ignore the geometric parameters of the sun and satellite. Figure 2: Simplified illustration of the interaction between the main quantities taken into account in the model. While these hypotheses are certainly only approximations of the real conditions, our aim is not to obtain the best possible satellite model, but a proof of concept of the transferability of a classifier. Under these hypotheses, the radiance received by the detector for a given band is:  
\[L_{SAT}^{s}(p)=\int_{s}S^{s}(\lambda)\Big{(}\text{psf}^{s}*\rho_{g}^{\lambda}(p )\Big{)}L_{TOA\downarrow}^{s}\,\mathrm{d}\lambda \tag{2}\]  
where \(S^{s}(\lambda)\) is the spectral sensitivity of band \(s\) and \(*\) is the convolution described in Equation (1). We consider that \(S^{s}(\lambda)\) integrates to 1. If that is not the case, a normalization step is also required. In addition, we consider the internal noise of the measuring system:  
* H6: The noise \(N(p)\) follows a normal law \(\mathcal{N}(\mu,\,\sigma^{2})\), with \(\mu\) being the mean and \(\sigma^{2}\) the variance.