Sergio García Ramírez**
5. Jurisprudencia consultiva de la Corte Interamericana
This section is dedicated to summarize the main research activities carried to pursue the Objective 2 defined in Chapter 1. The first part is dedicated to an overview of polarimetric detectors. Next to the theoretical part, the proposed approaches developed in the framework of this thesis are discussed showing the obtained results and validation.
4.2.1.
Overview of target detection using multi polarization SAR data
In this sub-section, an overview of marine target detection algorithms that make use of multi polarization SAR data is given. The first part introduces algorithms that need a full scattering matrix (quad-pol data), while in the second part are discussed the one that can perform well also for a simpler SAR configuration as for dual-pol acquisitions or that can be adapted to it.
Considering multi polarization SAR data as the contemporaneous availability of different image layers, a very simple strategy is to apply a single-pol pre-screener separately to each polarimetric channel. The final pre-screener has the task to properly combine the individual results. This strategy has been proven to be a solution to reduce the number of false alarms in non-homogeneous ocean clutter in CFAR approaches (Sciotti et al., 2002). On the other hand, fusing the polarimetric channels and then apply a pre-screener algorithm is a strategy that introduces polarimetric knowledge into the detector. One approach is to extract the total backscattered power from the scattering matrix. The detector in this case is provided by
𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚([𝑆𝑆]) = ��𝑆𝑆 ̇ℎℎ�2� + 2��𝑆𝑆 ̇
ℎ𝑣𝑣�2� + ��𝑆𝑆 ̇𝑣𝑣𝑣𝑣�2� > 𝑇𝑇ℎ𝑟𝑟 (4.10)
which should furnish a way to reduce the rate of missed detection due to possible nulls of the target in one polarization. However, the 𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚 detector doesn’t ensure an increase of detection rate for targets with low backscatter power.
(Novak et al., 1993) firstly demonstrated the Polarimetric Whitening Filtering (PWF) as a technique able to optimally reduce the standard deviation of the backscattering intensity associated with the speckle in PolSAR data. It was mathematically proved that via manipulation of the PolSAR basis, it is possible to have an equally distributed power making the clutter appear as white noise. The new basis is provided by the vector in Equation (4.11)
�𝑆𝑆 ̇ℎℎ 𝑆𝑆 ̇√ℎℎ 𝜀𝜀 𝑆𝑆 ̇𝑣𝑣𝑣𝑣− 𝜌𝜌̇∗√𝛾𝛾𝑆𝑆 ̇ ℎℎ �𝛾𝛾(1 − |𝜌𝜌̇|2) � 𝑇𝑇 (4.11) where 𝛾𝛾 and 𝜌𝜌̇ are the polarization ratio and complex coherence defined in Equation (3.13) and 𝜀𝜀 = 〈�𝑆𝑆̇ℎ𝑠𝑠�2〉 〈�𝑆𝑆̇� ℎℎ�2〉. Similarly with the Lexicographic basis and the 𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚 detector, the
4 Marine Target Detection and Discrimination |𝑖𝑖𝑃𝑃𝑃𝑃𝑃𝑃| = �𝑆𝑆 ̇ℎℎ�2+�𝑆𝑆 ̇ℎ𝑣𝑣�2 𝜀𝜀 + �𝑆𝑆 ̇𝑣𝑣𝑣𝑣− 𝜌𝜌̇∗√𝛾𝛾𝑆𝑆 ̇ ℎℎ�2 𝛾𝛾(1 − |𝜌𝜌̇|2) > 𝑇𝑇ℎ𝑟𝑟 (4.12)
(Liu et al., 2005) adapted the G-GLRT to PolSAR data, namely PO-LRT. Assuming that the scattering features vector 𝑘𝑘𝐿𝐿 = [𝑆𝑆 ̇ℎℎ 𝑆𝑆 ̇ℎ𝑣𝑣 𝑆𝑆 ̇𝑣𝑣ℎ 𝑆𝑆 ̇𝑣𝑣𝑣𝑣]𝑇𝑇 is characterized by its elements being
jointly complex Gaussian variables, 𝑘𝑘𝐿𝐿 is a random vector with pdf 𝑃𝑃 (𝑘𝑘𝐿𝐿) = 1
(2𝜋𝜋)2�det (C 4)
𝑚𝑚𝑥𝑥𝑆𝑆�−1 2⁄ (𝑘𝑘𝐿𝐿−𝜇𝜇)𝐻𝐻C4−1(𝑘𝑘𝐿𝐿−𝜇𝜇)� (4.13) where 𝜇𝜇 is the mean vector and C4 is the 4x4 polarimetric covariance matrix obtained by the
outer product of the features vector. (Liu et al., 2005) assumes that both ocean and ship backscatter have zero mean vector (𝜇𝜇 = 0), therefore, the detector in the G-GLRT sense given in Equation (4.5) can be written as
𝑘𝑘𝐿𝐿𝐻𝐻(C 4 𝑏𝑏)−1𝑘𝑘 𝐿𝐿− 𝑘𝑘𝐿𝐿𝐻𝐻(C4𝑡𝑡)−1𝑘𝑘𝐿𝐿 > 𝑇𝑇ℎ𝑟𝑟 (4.14) where C4𝑏𝑏 and C 4
𝑡𝑡 are the polarimetric covariance matrix of background and target,
respectively. Furthermore, it is typically found that the elements of the covariance matrix for target samples are much larger in magnitude than those of the background samples. Taken this into account the PO-LRT detector can be further simplified with the following approximation
𝑘𝑘𝐿𝐿𝐻𝐻(C 4 𝑏𝑏)−1𝑘𝑘
𝐿𝐿 > 𝑇𝑇ℎ𝑟𝑟 (4.15)
In (Marino, 2013) the Geometrical Perturbation-Polarimetric Notch Filter (GP-PNF) is developed and applied to detect ships in PolSAR images. The general idea of the GP-PNF is a filter able to reject sea clutter returns and extract the remaining features. Therefore, it is applied for ship detection but it can be generalized as detector of marine target. Considering the scattering features vector 𝑘𝑘𝐿𝐿 = [𝑆𝑆 ̇ℎℎ 𝑆𝑆 ̇ℎ𝑣𝑣 𝑆𝑆 ̇𝑣𝑣𝑣𝑣]𝑇𝑇 (with 𝑆𝑆 ̇
ℎ𝑣𝑣 ≈ 𝑆𝑆 ̇𝑣𝑣ℎ) of fully polarimetric
SAR data, the features partial scattering vector is introduced as: 𝑘𝑘 =𝑇𝑇𝑟𝑟𝑚𝑚𝑐𝑐𝑚𝑚([C3]Ψ)=���𝑆𝑆 ̇ℎℎ�2� ��𝑆𝑆 ̇
ℎ𝑣𝑣�2� ��𝑆𝑆 ̇𝑣𝑣𝑣𝑣�2� �𝑆𝑆 ̇ℎℎ𝐻𝐻𝑆𝑆 ̇ℎ𝑣𝑣� �𝑆𝑆 ̇ℎℎ𝐻𝐻 𝑆𝑆 ̇𝑣𝑣𝑣𝑣� �𝑆𝑆 ̇ℎ𝑣𝑣𝐻𝐻𝑆𝑆 ̇𝑣𝑣𝑣𝑣�� 𝑇𝑇
(4.16) where Ψ is a set of 3x3 basis matrices under a Hermitian inner product and therefore the 𝑇𝑇𝑟𝑟𝑚𝑚𝑐𝑐𝑚𝑚(⋅) operator is applied to a vector of six matrices. The partial scattering vector 𝑘𝑘 ∈ ℂ6 and
has the first three elements real positive and the second three complex numbers. The final version of the detector is given by
𝛾𝛾𝑛𝑛 = 1
�1 +𝑘𝑘𝐻𝐻𝑘𝑘 − �𝑘𝑘𝑅𝑅𝑚𝑚𝑠𝑠𝑅𝑅𝐻𝐻𝑘𝑘̂
𝑏𝑏�2
> 𝑇𝑇ℎ𝑟𝑟
where 𝑅𝑅𝑚𝑚𝑠𝑠𝑅𝑅 stands for reduction ratio, as nomenclature inheritance of the Partial Target Detector (PTD) which inspired (Marino et al., 2012). The term 𝑘𝑘𝐻𝐻𝑘𝑘 in the Equation (4.17)
is the total power, whereas the term �𝑘𝑘𝐻𝐻𝑘𝑘̂
𝑏𝑏�2 is the background power (sea clutter).
Therefore, 𝑘𝑘𝐻𝐻𝑘𝑘 − �𝑘𝑘𝐻𝐻𝑘𝑘̂
𝑏𝑏�2 represents the power of marine targets. For sea surface 𝛾𝛾𝑛𝑛 is
proximal to zero, while in the presence of a target it approximates the unity (Marino, 2013). It is important to note that both approaches proposed by (Liu et al., 2005) and (Marino, 2013) can be adapted for dual-pol SAR data. Both studies agree on the fact that the proposed detectors, PO-LRT and GP-PNF, obtain the best ship detection performances when applied to full polarimetric SAR or for dual-pol configuration when the co-pol channels (HH and VV) are available.
(Shirvany et al., 2012) investigated the complementary of the Degree of Polarization (DoP), thus defined as Degree of Depolarization (DoD), as a potential detector of ships under different linear, hybrid/compact dual-pol SAR configuration. Here we consider the results obtained in (Shirvany et al., 2012) only regarding ship detection using DoD in linear dual- pol SAR. Defined the scattering vectors for dual-pol radar configuration as
𝑘𝑘𝐷𝐷1 = [𝑆𝑆 ̇ℎℎ 𝑆𝑆 ̇ℎ𝑣𝑣]𝑇𝑇
𝑘𝑘𝐷𝐷2 = [𝑆𝑆 ̇ℎℎ 𝑆𝑆 ̇𝑣𝑣𝑣𝑣]𝑇𝑇
𝑘𝑘𝐷𝐷3 = [𝑆𝑆 ̇𝑣𝑣ℎ 𝑆𝑆 ̇𝑣𝑣𝑣𝑣]𝑇𝑇 (4.18)
which provides three different combination depending on the transmitted and received linear polarization, the DoD is related to the Stokes vector [𝑡𝑡0 𝑡𝑡1 𝑡𝑡2 𝑡𝑡3]𝑇𝑇 by the following relation
(Cloude, 2009; Lee and Pottier, 2009)
DoD = 1 −�𝑡𝑡12+ 𝑡𝑡22+ 𝑡𝑡32
𝑡𝑡0 (4.19)
The elements of the Stokes vector are called Stokes parameters, which are four real values capable to characterize the polarization state of a wave. If any of the Stokes parameters has a nonzero value, it indicates the presence of a polarized component in the plane wave. Being 𝑡𝑡0 equal to the total power (density) of the wave, DoD assumes values between zero and one quantifying the amount of de-polarization in the EM wave. A depolarizing interaction causes totally polarized states to emerge with DoD > 0 and this can be related to presence of a ship or marine target on the ocean surface. Hence, the detector is summarized as
1 −�𝑡𝑡12+ 𝑡𝑡22+ 𝑡𝑡32
𝑡𝑡0 > 𝑇𝑇ℎ𝑟𝑟 (4.20)
Based on their experimental results, (Shirvany et al., 2012) concluded that linear HH-VV dual-pol configuration deliver better detection performance compared to the other two possible combinations.
4 Marine Target Detection and Discrimination