Offline signature verification necessitates the involvement of machine learning visual recognition techniques. Efficient signature e-verifiers in machine learning and data analysis methods often inherently assume that the visual representation of the signature data can be modeled by vectors of Euclidean nature. However, as scientists become increasingly aware, ignoring an intrinsically non-Euclidean, or geometric, structure of the data can lead to sub-optimal results. The objective of this research is to enhance offline signature verification by incorporating non-Euclidean representations, particularly symmetric positive definite matrices (SPD). Specifically, this work proposes a geometrically inspired solution to the challenging task of verifying a questioned signature sample given a reference set of template signatures. Thus, a pair of covariance matrices is intrinsically embedded into a vector, denoted as the Embedded Riemannian Pyramid (ERP) by exploring a set of pairwise SPD measures, between sub-SPD subsets originating from a pair of covariance matrices. In conclusion, the SigmML, a meta-optimization network in the space of the SPD matrices, is introduced as a novel metric learning framework with low-dimensional ERP embeddings as inputs. Evaluation is conducted under two blind scenarios, namely blind intra-dataset and cross-lingual testing, utilizing a multitude of signature datasets or different origins.
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