## 2 WI-SV-Related Work and Overview
### Overview of the Proposed Method  
This section is devoted to a brief description of the proposed system, with emphasis on the learning stage. Figure 1 provides the key point RDV concept of the proposed method through a graphical depiction of a toy example. Let us consider a pair of static handwritten signature images and the corresponding covariance descriptors \(\mathbf{X},\mathbf{Y}\) of either the entire images or any equimass segment (denoted by an index \(a\)). Both \(\mathbf{X_{a}},\mathbf{Y_{a}}\) are regarded as points on the SPD manifold \(\mathbf{P}_{10}\). Subsequently, two distinct frameworks are established in order to contextualize the concept of dissimilarity between the aforementioned covariance matrices. In the first one (local approach), the Riemannian dissimilarity vector is a symmetric matrix \(\Psi\epsilon\mathbb{R}^{10\times 10}\), which lies on the local tangent plane \(\mathcal{T}_{\mathbf{X}}\). The evaluation of the RDV \(\Psi\) inherently utilizes (a) a specific manifold metric \(\delta\) and (b) the manifold equivalent \((-)_{\mathcal{M}}\) of the Euclidean subtraction operator \((-)_{Euc}\). Details regarding \(\delta\) and \((-)_{\mathcal{M}}\) shall be provided in the subsequent paragraphs. A \(vec(\cdot)\) operator aligns the symmetric matrix \(\Psi\) (i.e., the tangent vector) to a typical vectored form \(v_{\mathbf{X},\mathbf{Y}}\epsilon\mathbb{R}^{55}\). In the second approach, the evaluation of the RDVs \(\Psi_{\mathbf{I},\mathbf{X}}\) and \(\Psi_{\mathbf{I},\mathbf{Y}}\) is made with respect to the \(\mathbf{\mathrm{I}}_{10}\) common pole followed by (a) the Euclidean-based Dichotomy Transform defined as \(\Psi_{\mathbf{X},\mathbf{Y}}^{\mathbf{I},\mathbf{Y}}=|\Psi_{\mathbf{I},\mathbf{X}}-\Psi_{\mathbf{I},\mathbf{Y }}|\) and (b) a \(vec(\cdot)\) operator, which creates the \(dv_{\mathbf{X},\mathbf{Y}}\epsilon\mathbb{R}^{55}\) vectored forms. The aforementioned elementary building blocks (a, b) of Figure 1 create the \(v_{\mathbf{X},\mathbf{Y}}\) or \(dv_{\mathbf{X},\mathbf{Y}}\) vectors, which in their turn inherently express the dissimilarity between a pair of SPD matrices. Figure 2 illustrates the process of decomposing a static signature image into an array of fourteen covariance matrices, with each matrix corresponding to either the entire image or a specific segment indexed by \(a=1:14\). It is easy to conceive that for a pair \((Img_{X},Img_{Y})\) of handwritten signatures, two arrays of covariance matrices \(\mathbf{X_{a}},\mathbf{Y_{a}}\) are extracted; consequently, for each segment pair \(\mathbf{X_{a}},\mathbf{Y_{a}}\), a Riemannian dissimilarity vector \(v_{\{\text{{kind of measure}}\}}^{a}\) or \(dv_{\{\text{{kind of measure}}\}}^{a}\) can be created. According to the discussion in the introduction, the pairs of \(\mathbf{X_{a}},\mathbf{Y_{a}}\) can originate either from the positive or the negative class \(\mathbf{X}_{a}^{\pm},\mathbf{Y}_{a}^{\pm}\) and create the corresponding \(\Psi_{\mathbf{X}_{a}^{\pm},\mathbf{Y}_{a}^{\pm}}\) and \(\
u_{\{\cdot\}}^{\pm}\) or \(\mathbf{Y}_{\mathbf{X}_{a}^{\pm},\mathbf{Y}_{a}^{\pm}}\) and \(d\
u_{\{\cdot\}}^{\pm}\). In this work, two individual learning frameworks are explored according to the fusion strategy followed. Figure 3 depicts the first one, tagged hereafter as the local _LFW1_. Sets comprising positive (G-G) and negative (G-RF or G-SF) pairs form the corresponding sets of RDVs and \(v_{\mathbf{X}_{1}^{\pm},\mathbf{Y}_{1}^{\pm}}\) or \(dv_{\mathbf{X}_{1}^{\pm},\mathbf{Y}_{1}^{\pm}}\). Then, two different classifiers, a binary support vector machine and a Decision Stump Committee, are trained and validated in order to select the optimal parameters of each one. It must be made clear here that the LFW1 learning procedure of both the SVM and DSC classifiers involves only the first \(\mathbf{X}_{1}^{\pm},\mathbf{Y}_{1}^{\pm}\) covariance matrix, which corresponds to the entire image; i.e., the remaining segments are not utilized during the learning stage.