# Automatic Detection of Marine Litter: A General Framework to Leverage Synthetic Data
## 2 Materials and Methods
### Methodology and Modelling
#### 2.1.3 Sampling Modelling  
Once the landscape has been acquired by the different optical sensors of the system, the corresponding image in the focal plane is spatially discretized over a given integration time (pose) by the CMOS detectors into a digital image. Among all the steps, modeling the sampling process is critical to ensure the reliability of the data given the scale of the target. Depending on the GSD, aliasing might introduce distortion or artifacts on the reconstructed signal, in addition to the noise. According to the Nyquist-Shannon sampling theorem, the sampling process is reversible if the sampling frequency of a signal, \(f_{s}\), is at least double the frequency of its highest frequency component [33]. If \(f_{c}\) is the cut-off frequency of a signal, i.e., the inverse of the smallest dimension that the optical apparatus can discriminate, the theorem can be written as:  
\[f_{s}>2f_{c}\iff\frac{1}{2\text{GSD}}>f_{c} \tag{4}\]  
\(1/(2\text{GSD})\) being the Nyquist frequency, \(f_{n}\). In the case of an optical imaging system, the value of the spatial cut-off frequency is calculated as follows:  
\[f_{c}\simeq\frac{D}{\lambda H} \tag{5}\]  
with \(D\) being the diameter of the pupil in meters, \(\lambda\) the wavelength of the band in meters, and \(H\) the altitude of the satellite in meters. For instance, in the case of S2, the diameter of the pupil is \(150\,\mathrm{mm}\), the altitude of the satellite is \(786\,\mathrm{km}\), and the wavelength depends on the spectral band [7]. Table 1 shows the values of the cut-off and Nyquist frequencies for each band. As highlighted by Table 1, the \(\frac{f_{c}}{f_{n}}\) ratio is always superior to 1, meaning the the Shannon-Nyquist theorem is never verified. If we consider B02, which has the best spatial resolution and the shortest wavelength, respecting the Shannon-Nyquist condition without modifying the characteristics of the satellite (diameter of the pupil, altitude of the satellite) would imply having a GSD about 10 times smaller. Thus, the choice of design of S2, which privileged a large pupil diameter to enhance the contrast and improve the Signal to Noise Ratio (SNR), induces some potentially strong aliasing effects that the end-user must be aware of, especially when the size of the elements of interest are close to the GSD. Similar phenomena have been observed to have a strong impact on other satellites, such as SPOT [31]. For a simulation to be correct and take into account all the effects introduced by aliasing, we must consider the situation where the Shannon-Nyquist is not respected by the satellite we are modeling. In such a situation, we need to simulate the landscape at a resolution compatible with the maximum cut-off frequency of the satellite. By generating the landscape at a resolution where the Shannon-Nyquist theorem is respected and then resampling it with the same GSD as the satellite, we can faithfully reproduce the aliasing artifacts that would be observable in real images. To determine a suitable pixel size for the landscape generation, we propose the following method:  
1. Calculate the GSD under which no detail can be perceived by the optical system: \(\text{GSD}_{min}=\frac{1}{2\max(f)}\). 2. Based on the actual GSDs of the satellite bands, find a number, \(d\), that evenly divides into all of them and that is smaller than \(\text{GSD}_{min}\). 3. When building the initial image, before any other processing, consider a pixel to be a square of size \(d\,\times\,d\). In order to model the relative position of the satellite with the target, we consider the target as fixed at the center of the image, and the grid of sensors (evenly spaced spaced by the GSD of the band under consideration) is regarded as rigidly moving with sub-pixel translations, thus accounting for geometric uncertainty and inherent variability, module one pixel, from one acquisition to another. These rigid geometric shifts are hereinafter named jitters.