# Preliminary Results of Ship Detection Technique by Wake Pattern Recognition in SAR Images
## 2 Method Description
### Method Details  
The flow-chart of the proposed wake-based detection process is shown in Figure 1.  
The SAR image is scanned pixel-by-pixel or defining a proper pixel grid, in order to detect which pixels are crossed by wakes. An image tile is identified around each pixel under analysis, whose size is set on the basis of turbulent wake extent, i.e., few kilometers, and, then, the identification of the turbulent and narrow-V components of the wake is carried out.  
For the convenience of the reader, the main steps of the detection can be summarized as follows: (1) a tile is selected around its center, (2) the Radon Transform is applied to the tile properly masked to account for land and ship echoes, (3) the wake components are selected in the Radon domain as peaks/toughs satisfying particular criteria derived from wake geometry, and (4) the presence of each wake feature is validated by merit indexes, based on the estimation of how much their appearance is different from the background. The merit index of the bright component is calculated along the detected half-line, discarding the 5% brighter pixels of the half line to filter out the effect of localized brighter spots along dimmer lines. It is estimated by Equation (1) [24].  
\[F_{\text{p}}=\frac{CDF^{-1}(0.95)}{0.95\;n\;\overline{I}}-1 \tag{1}\]  
where \(CDF\)(xi) is the cumulative distribution function of the image intensity (xi) along the identified half-line. In addition, \(n\) is the number of pixels of the half-line, and \(\overline{I}\) is the intensity mean value over the whole tile.  
The merit index of the turbulent wake is estimated by Equation (2) [24], in which \(\overline{I}_{i}\) is the intensity mean value over the turbulent half-line.  
\[F_{t}=\frac{\overline{I}_{t}}{\overline{I}}-1 \tag{2}\]  
How such indexes are used to validate the ship/wake presence is an original contribution of this manuscript. In [24], they are exploited to confirm that the related wake component is really imaged on the basis of the single index value, i.e., each index validates the presence of each wake component. Instead, since the proposed approach implies that the wake is considered as really imaged when both  
Figure 1: Flow-chart of the proposed methodology.  
turbulent and narrow-V wake is shown, a new merit index of the wake is introduced and defined by Equation (3).  
\[F_{w}=F_{w}|F_{l}| \tag{3}\]  
High values of \(F_{w}\) imply that the wake components both appear as very dark and very bright linear features, validating the ship/wake presence. On the contrary, low values of \(F_{w}\) imply a low confidence level that the wake is really imaged. If \(F_{w}\) is negative, the wake is not present.  
Then, a global threshold detector is implemented as a segmentation algorithm. The threshold value is set based on the statistical distribution of the wake index \(F_{w}\). Even if no theoretical proofs are provided herein, the assumption of a bell-shaped Gaussian can be empirically verified (see Section 3). This implies that a list of candidate wakes can be identified as the pixels whose wake merit index satisfies Equation (4), in which \(\mu\) and \(\sigma\) are the mean value and the standard deviation of the \(F_{w}\) distribution, respectively.  
\[F_{w}>\mu+k\sigma \tag{4}\]  
Finally, the wake presence in neighbor pixels is investigated in the second level of validation. In details, around each candidate pixel, the half-line representative of the turbulent component is reconstructed by means of (\(\theta\),\(s\)) pair estimated in the previous phase. Then, a null matrix with the same size of the tile is built (Figure 2): when the image pixel is crossed by the reconstructed wake, the value of the matrix cell in the same location is increased by 1. Since the turbulent component extends for kilometers, the same wake component is detected in tiles centered in neighbor pixels. Hence, the cells corresponding to pixels affected by the wake show a score larger than 1 and its score is a measure of the confidence level of the wake presence.