## 1 Introduction  
The adoption of biometric technology has become crucial in modern security and authentication systems [1, 2, 3]. Specifically, handwritten signatures hold particular significance due to their long-standing use in financial, legal, administrative, and forensic contexts [4, 5, 6, 7]. They have been a key focus of biometric research for many years [8, 9] due to their historical role in validating document authorship [10]. To this day, they remain a legally acknowledged method for verifying human identity in numerous types of transactions. Their sustained popularity is largely attributed to their simplicity and familiarity; individuals are accustomed to signing documents, whether through traditional pen-and-paper methods or modern electronic interfaces, such as touchscreens on smartphones and tablets [11]. However, the behavioral nature of handwritten signatures introduces a critical limitation: they are more vulnerable to forgery (F) compared to inherent physical characteristics, such as fingerprints or iris patterns. Skilled human forgers or computer-based presentation attackers can exploit the learned motor patterns of signatures to create convincing imitations. Signature verification (SV) poses significant challenges, particularly for forensic-based applications. There is an increasing focus on developing automated signature verification (ASV) systems in order to accurately and efficiently differentiate between authentic and forged signatures as e-assistance for human intervention [5,12]. This binary classification task falls under the traditional scientific disciplines of computer vision and pattern recognition, requiring the ability to discern natural variations in handwriting from deliberate forgeries. It is a challenge compounded by factors such as the limited availability of handwriting features, the small number of genuine or bona fide (G) reference signatures, and the variability that exists (a) within an individual's signatures samples (defined as the intra or the positive class \(\omega^{+}\)) and (b) between those of different individuals (defined as the inter- or negative class \(\omega^{-}\)). To begin with, ASVs are categorized according to the way that the handwritten signature is acquired. There are two modes for signature acquisition: offline or static and online or dynamic. The online methods employ dynamic (i.e., time series) features by capturing multidimensional properties of the signature trace such as speed, pressure, pen inclination data, etc., all collected on appropriate platforms such as tablets and smart pens [13-17]. On the other hand, offline signature verification utilizes static signatures, represented and processed as grayscale, or binary, digital images. These signature images are compared through a series of procedures such as feature extraction, which is typically a mapping of the raw image to a mathematical vector space (i.e., the feature space), followed by the learning (i.e., training and validating) and testing stages of a binary classifier, or verifier. It is a fact that, nowadays, the use of contemporary machine learning and computer vision methods, such as those in [18-25], made OfSV systems both an efficient and practical aid, particularly for forensic applications and other related domains [4,6,12]. Another important categorization for SV systems is whether they follow the writer-dependent (WD) or the writer-independent (WI) framework. In the WD case, every new user enrollment needs a separate binary classifier to learn from the writer's positive and negative samples [19,26,27]. The WI approach learns a universal model verifier, which tries to distinguish between similar and dissimilar pairs of images of a relatively large set of users, even with a few reference samples each [28-30]. It is a fundamental tenet of this framework that any verifier must compare pairs of signatures in order to ascertain their relative (dis)similarities. This process typically occurs in two stages.