Region adjacency graph based GNN approach for static signature classification

Abstract

Graph Neural Networks (GNNs) have gained popularity across various research fields in recent years. GNNs utilize graphs to construct an embedding that includes details about the nodes and edges in a graph’s neighborhood. In this work, a set of Region Adjacency Graphs (RAG) derives the attribute values from Static Signature (SS) images. These attribute values are used to label the nodes of the complete graph, which is formed by considering each signature as a node taken from the sample of signatures of a specific signer. The complete graph is trained by using GraphSAGE, an inductive representation learning method. This trained model helps to determine any newly introduced node (static signature to be tested) as genuine or fake. Standard static signature datasets, notably GPDSsynthetic and MCYT-75 are used to test the prevailing model. Experimental results on genuine and counterfeit signature networks demonstrate that our computed model enables a high rate of accuracy (GPDSsynthetic 99.91% and MCYT-75 99.56%) and minimum range of loss (GPDSsynthetic 0.0061 and MCYT-75 0.0070) on node classification.

Keywords

Signature verification GNN region adjacency graph complete graph GraphSAGE Node classifications

1 Introduction

The oldest known evidence of the use of signatures to identify the individual in 3100 BC. The scribe Gar Ama inscribed symbols on Sumerian clay tablets in 3100 BC, which are the earliest examples of using symbols to establish a person’s identity [1]. Globally, a wide range of transactions use static signatures for verification. Although several systems for verifying online (dynamic) signatures have recently been developed by researchers [2], off-line (static) signature recognition systems [3] continue to comprise greater social acceptance, particularly in developing nations. But these static signatures are often misused by miscreants through signature forgery. Signature forgery is the practice of impersonating someone’s signature to create a forged document or gain unauthorized access to someone’s identity.

Static signature verification (SSV) has historically been a particularly lively area of research due to its significance in authentication context [4] as well as the diversity of open challenges that are still being researched today [5]. An SSV systems are classified into two categories in the literary work: (1) writer dependent systems, developed for a single signer, and (2) writer independent systems, not prepared for each signer of the system but to a certain extent, are built for the authentication of all signers being considered [6]. This model was chosen as the initial strategy because it aids in resolving intra-personal conflicts. Researchers like Hafemann et al., 2017 [7], Bertolini et al., 2010 [8], and Mwangi, 2008 [9] have developed writer-dependent static signature verification systems. Each of these systems works well when there are many reference samples available but fails for fewer reference samples.

In real-time applications for identification, only a small number of legitimate signatures are taken from signers. As a result, it becomes required to run an SSV model using less data. Accessions including few-shot learning [10, 11], self-supervised learning [12, 13], zero-shot learning [14], and semi-supervised learning [15, 16] are the different strategies used to analyze the data. The zero-shot learning model requires a large sample for training; hence it is considered as an unreliable strategy [14]. It extends the learning model to invisible classes by learning from known samples. An account of self-supervised learning technique, a hybrid of supervised and unsupervised learning approaches, samples without labels are required to train the model. This strategy absorbs knowledge from data structures and apply it to foresee unknown data [11].

A type of database called a graph which depicts a group of things (called nodes) and their connections (called edges). Graph analysis extends on problems like node classification, link prediction, and clustering especially plays a crucial role in designing distance-based Region Adjacency Graph (RAG) to get potential attributes for SSV. RAG is a graphical representation of an input static signature image in which superpixels are vertices with edges that convey the resemblance or distinction between neighboring superpixels. Merging functions are used in RAG to group adjacent segments hierarchically. By using graph analysis, all the characteristic nodes are then classified which can be used to specify the key features of static signatures to verify them as authentic or counterfeit. The graphs can be used to represent a wide variety of systems in different field of study comprise social science [17], natural science [18, 19] and protein interconnected networks [20], knowledge graphs [21], and many others [22]. Recent research on reading graphs using machine learning has paid a lot of interest as it can be easily applied to graph analysis.

Resent research shows cutting-edge performance in several supervised learning tasks using supervised node representation learning algorithms [23, 24]. Many developments have also been made in the field of unsupervised node representation learning which uncover low-dimensional network nodes which can be useful in a variety of downstream applications [25 –28].

1.1 A detailed account of the present work

Goal of the study: The GraphSAGE node categorization strategy serves as the foundation for a static signature verification method which is being used for an SSV system. Region Adjacency Graph based node attributes are obtained from each static signature image. It is utilized to build a complete graph that shows how all authentic and fake static signature attributes are connected to one another. The freely available signature datasets GPDSsynthetic and MCYT-75 are used in this present study.

Uniqueness: GraphSAGE model is computationally effective and appropriate for real-world scenarios that incorporates RAG characteristics, such as node attributes.

Novelty: The proposed method performs superior in terms of fewer valid signatures than advanced models and also achieves maximum accuracy and minimum loss while considering the greatest compactness.

The remaining sections of the paper are categorized as follows. A few of the earlier studies proposed in the field of signature recognition are discussed in Section 2. An outline of proposed method stated in Section 3. Section 4 summarizes the experimental results and discussion using the proposed model. Finally, conclusion of research findings and future goals in Section 5 to wrap up our study.

2 Review of literature

Signature verification is a process of identifying the original signature in a document. It is divided into two categories: offline (static) and online (dynamic), depending on the process of acquisition [3]. In Static Signature Verification (SSV), the information extracted from the original copy of the signature image took over from a paper document can be utilized for verification. Verification has been carried out considering both static and dynamic aspects. Online signature verification is reliable than an offline method since these dynamic elements are exclusive to an individual writer [5]. Due to a lack of robust features, the verification system is more difficult. Additionally, it can be challenging to distinguish between authentic and fake signatures due to differences in an individual’s signature triggered by illness, ageing, or psychological factors, which leads to inaccuracy in offline systems. Furthermore, a pretender has the ability to create an incredible signature forgery [6].

The fact that Graph Neural Networks (GNNs) [29], more potent tool for comparable node classification tasks is important to note in this context. Through the use of graph structure, GNNs learn the embedding of a specific node (in this case, a sample of a signature). Because of this, GNN models have tremendous performance in a number of tasks such as node, edge and graph classification. However, because they are fundamentally designed for transductive learning, the majority of node classification techniques [30] only function with a single fixed network. Such models are difficult to generalize across networks and do not transfer well to unseen nodes implanted after training. For node classification issues, Hamilton et al., 2017 [15]; Li et al., 2018a, b [16, 31] have devised inductive learning algorithms where trained node features effectively build representations for unseen data. On the other hand, Li et al., 2018a, b [16, 31] built their models to conduct spatial clustering-based categorization in image processing. The researchers suggested that node embedding methods also excel at image classification tasks. It uses a GNN-based node classification model to differentiate authentic signatures from fake data. In this present work, each signature is initially noted as a node, and each node is associated to every other node by an edge. Hamilton et al., 2017 [15] suggested Graph SAmple and aggreGatEGraphSAGE node classification architecture is used to categorize each signature as either authenticate or counterfeit.

In recent decades, significant advancements have been noticed in the field of an SSV systems, which are typically presented as a categorization challenge. Authors commonly employ two strategies to carry out this work: feature-based categorization and template-based matching. In this study, some recently advanced feature-based categorization and template-based approach methodologies were summarized.

The extraction of attributes from given signatures forms the foundation of the feature-based categorization method. The features used in literature are mostly responsible for conveying color, shape, texture, and other information. Offline signature verification systems have used color information as features as the draft of chromatics (Ganar et al., 2014) [32], Color Co-existence Matrix (Lin et al., 2009) [33], and Dominant Color Descriptor (DCD) (Shao et al., 2008) [34]. The primary drawback of such features is the possibility of identical color information existing in two or more completely different images. Additionally, the form and texture of the image have been completely disregarded. Some research made the decision to use the texture-based elements of the distinctive static signature as a result.

Use of shape-based traits like contour, slant angle, texture-based features like GLCM, Local Binary Pattern (LBP), Speeded Up Robust Features (SURF), Histogram of Oriented Gradients (HOG) and the Scale Invariant Feature Transform (SIFT) noted in the literature (Deng et al., 1999 [35], Guerbai et al., 2015 [36], Batool et al., 2020 [37], Yilmaz et al., 2011 [38] & Nasser and Dogru, 2017 [39] and Mwangi, 2008 [9]). Several research discovered that when using classifiers like Artificial Neural Networks (ANN) [37] and Support Vector Machines (SVM) [38], texture-based features outperformed shape-based features. At present, literature has also used deep learning-based models such as the Siamese network and VGG-19. The Siamese network was trained by Ruiz et al., in 2020 [40] using synthetically created datasets. A node classification approach was taken into consideration when Ray et al., 2021 [41] built an off-line signature authentication system based on writer-dependent set utilizing GNN based binary classification. Elashry et al., 2022 [42] suggested the significance of discarding erroneous pairs from the set of coordinated point pairs. Here, two distinct approaches were investigated, GNN and GN, and both produced positive findings that led to the development of the RANdom SAmple Consensus (RANSAC) algorithm. Zhang et al., (2022) [43] suggested graph neural network called Siam-GNN incorporates Siamese structure and performs better than the original graph network.

A novel GNN model with adaptive mechanism was suggested by Bi et al., in 2022 [44]. It comprises of two modules: the interest-based Moment Adaptor module and the Multimode Moments Embedding (MME) module. Joshi et al., 2022 [45] recommended the method makes use of Knowledge Graph (KG) embedding. They developed and practiced a customized Deep Neural Network (DNN) called KGDNN (Knowledge Graph DNN) for forecasting ADRs using KG embedding. The six categories of entities that make up a KG are: target proteins, indications, pathways, drugs, ADRs, and genes. The Node2Vec approach was used to implant each vertex into a characteristic space. With the aid of these embeddings, the KGDNN model categorizes ADRs. According to empirical evidence, node-based embeddings are inferior to embedding edges in neural Skip Gram Learning with Negative Sampling (SGNS) methods [46]. They postulated that this was because line graphs have longer mixed periods for random walks. Eventually, they provided a solution to the standard graph conundrum and contributed to the understanding of how random walks behave when surfing on line graphs.

A template-based approach used When the majority of the template images make up the matching image. As many template-based matching requires sampling a large number of signature images, it is not possible to lessen the number of reference images or the resolution of the template images. Both images are pre-processed, and then template matching techniques that include dynamic time warping, Bipartite Graph Edit Distance, and polar graph embedding distance are employed in the literature. The most difficult step in template-based comparing is matching of objects affected by shift and/or scale change. Only minor shifts and rotations are required from a geometric standpoint. Only a few shifts and rotations are permitted geometrically. Furthermore, template matching demands a high level of computational power [41].

3 Proposed method

The steps involved in proposed method are illustrated in Fig. 1. Signatures from artificially generated models might also be a counterfeit. With the aforementioned information in mind, the node categorization strategy of GraphSAGE, practiced using both small and large sample sizes used in this work to construct a signature verification system. In present study, the features including distance-based Maximum, MaxStandard deviation, Sum (Maximum), Minimum, MinStandard deviation, Sum (Minimum), number of nodes and edges are employed for complete graph node attributes. Because the GraphSAGE node classification model is computationally demanding. Furthermore, RAG also generates high and low distance characteristic feature, unlike other texture-based features.

Fig. 1

Procedure for feature extraction and node classification.

3.1 Pre-processing

3.1.1 Edge detection

A fundamental tool for feature extraction and detection in image processing, edge detection seeks to locate discontinuities and pinpoint’s locations in a digital image where brightness abruptly changes. Edges in images are placed with high intensity contrasts; a pixel’s intensity moves from one to the next. Edge detection minimizes the quantity of data and filter out unnecessary information while maintaining an images’ crucial structural qualities [30].

The edge of a grey level image is a local feature that, within a neighborhood, divides regions into which the grey level is more or less uniform with slightly different values on the two sides of the edge. Since both edges and noise have high frequency components that cause the result to be blurry and distorted, it is challenging to detect edges in noisy images.

3.1.1.1 Types of edge detectors

This data set’s images contain some noise, which produces sudden shifts in pixel values. As a result, the preprocessing steps of image resizing and edge detection is employed using gradient-based Sobel filter operator with an aid of Gaussian filter in order to noise removal (Fig. 2). Rationale to select Sobel filter operator in edge detection of present study comprises appropriate detection of edges and their orientations, better approximation to gradient magnitude, easy to implement, fast to compute and competent noise removal [30].

Fig. 2

Edge detection operators.

3.1.1.2 Edge detection procedure

Convolution has several applications, one of which is edge detection. Thus, edges are defined for,

A location in the image when the intensity/color of pixels changes abruptly.

A transition between elements or between an object and its backdrop.

It draws attention from the perspective of human visual perception.

The edge detection procedure typically consists of three steps: (i). Noise cancellation Reduce as much noise as feasible while retaining edges. (ii). Edge improvement accentuate the edges while weakening the rest (high pass filter). (iii). Localization of the edge Examine probable edges (the maximum output from the preceding filter) and delete any bogus edges (often noise related).

For image I, the intensity gradient at a pixel in the X and Y directions are estimated using Equations (1) and (2) as follows: $\frac{\partial I}{\partial X} I (X + 1, Y) - I (X - 1, Y)$ (1) $\frac{\partial I}{\partial Y} I (X, Y + 1) - I (X, Y - 1)$ (2)

By convoluting with a low pass filter, noise smoothing can be achieved (e.g., mean, Gaussian, etc.) The gradient computations are GX and GY using Equations (3) and (4) as follows, $G_{X} = H_{X} * I (X, Y)$ (3) $G_{Y} = H_{Y} * I (X, Y)$ (4)

The Sobel filter is used to detect edges. It computes the pitch of picture strength at each pixel in the image. It determines the direction with the greatest increase from light to dark, as well as the change in direction. The outcome indicates how suddenly or smoothly the picture changes at each pixel, and hence how likely that pixel represents an edge. It also indicates how that edge is most likely to be oriented. When the filter is applied to a pixel in a consistent intensity zone; the outcome is a zero vector. When applied to a pixel on an edge, it produces a vector that points diagonally the edge from darker to brighter values. Two 3 x 3 kernels are used in the Sobel filter. One is for horizontal alterations, and the other is for vertical changes. To calculate derivative approximations, the two kernels are convolved with the original picture. The computations are simplified if we define H_x and H_y as revealed in Equations (5) and (6) both pictures containing the horizontal and vertical derivative approximations, respectively: $H_{x} = (\begin{matrix} 1 0 - 1 \\ 2 0 - 2 \\ 1 0 - 1 \end{matrix}) * A$ (5) $H_{y} = (\begin{matrix} - 1 - 2 - 1 \\ 0 0 0 \\ 1 2 1 \end{matrix}) * A$ (6)

In this case, A is the original source image. The x organize rises in the right direction, whereas the y organize rises in the downward direction. To compute H_x and H_y, the appropriate kernel (window) shifted across the input picture, in row-wise and column-wise through which all the pixel values are derived starting from the first row and end up with last row by taking each and every pixel in it. Positive and negative coefficients are present in the kernels. This implies that the final image will have both positive and negative values. The gradient of zero should be mapped onto a half-tone grey level. Negative gradients get darker, whereas positive gradients become brighter. Use the gradient map’s absolute values (stretched between 0 and 255). This makes gradients that are extremely negative or extremely positive look brighter. Horizontal and vertical transitions are detected by the kernels. The amplitude and angle of an edge are used to calculate its size. These may be easily determined using H_x and H_y. The gradient approximations produced by H_x and H_y are combined to obtain the gradient magnitude at each pixel in the picture, using Equation (7) $H = \sqrt{H_{x}^{2} + H_{y}^{2}}$ (7)

The gradient’s direction is computed using Equation (8) as follows, $⊖ = \arctan (\frac{H_{y}}{H_{x}})$ (8)

A ⊖ value of 0 indicates that the vertical edge is darker on the left side.

3.2 Feature extraction

3.2.1 Graph-based boundary merging

This work’s segmentation is divided into three phases: initial segmentation by means of simple linear iterative clustering (SLIC) algorithm [47], Region Adjacency Graph (RAG) representation and hierarchical segment merging. SLIC is used to build a preliminary segmentation based on spectral-spatial distance. RAG is a graphical explanation of an input static signature images in which superpixels are vertices associated with edges which convey the resemblance or distinction between neighboring superpixels. Merging functions are used to hierarchically arrange neighboring segments (superpixels in the first stage).

RAG is a graphical explanation of an input static signature images in which superpixels are vertices associated with edges which convey the resemblance or distinction between neighboring superpixels. RAG is mainly employed to achieve each and every characteristic boundary of all adjacent vertices and to calculate distance between the neighboring nodes also. As a result, those derived attributes are essential to classify the static signatures with the greatest accuracy level. RAG works faster to perform and better when the signature image and its background have high contrast. RAG can be processed the datasets efficiently by using graph algorithms and can perform complex operations on larger datasets quickly.

3.2.1.1 Simple Linear Iterative Clustering (SLIC)

SLIC is a clustering approach that generates well-ordered and short superpixels based on color similarity and image plane proximity. SLIC works similarly to the k-mean clustering technique [48]. Based on the lowest Euclidean distance between a pixel and all nearest k-centroids, each pixel is frequently to the nearest k-centroid in color space, upto any other changes occur. SLIC segmentation employs a novel distance related with five-dimensional [lcdxy] space. [lcd] is the pixel colour vector in Lcdcolour space, where L represents the brightness of the color and c and d represent the colour beside a red/green and blue/yellow axis, respectively [49]. The pixel location is represented by [xy] . [Lcd] distance compares the colour solidarity of superpixels, with a lower distance indicating more similar colour. The [xy] distance between superpixels gauges their closeness, with a lower distance indicating more neighboring pixels.

The distance between pixels is expressed using Equations (9)–(11) as follows: $ED (i, j) = D_{lcd} (i, j) + \frac{m}{s} E D_{xy} (i, j)$ (9) $E D_{lcd} (i, j) = \sqrt{{(l_{i} - l_{j})}^{2} + {(c_{i} - c_{j})}^{2}} + {(d_{i} - d_{j})}^{2}$ (10) $E D_{x, y} (i, j) = \sqrt{{(x_{i} - x_{j})}^{2} + {(y_{i} - y_{j})}^{2}}$ (11) where, ED_lcd (i, j) is the spectral [lcd] space whereas ED_x,y is the spatial [xy] space of Euclidean distance. ED (i, j) is the sum of [lcd] distance and normalized [xy] distance by the grid interval s. Here, m represents a variable that controls the compactness of superpixels. When m is great, the weight of [xy] distance is superior. For an image of M pixels and K superpixels, the size of each superpixel is about $\frac{M}{K}$ .

The distance between each two adjacent superpixel blocks is $s = \sqrt{\frac{M}{K}} .$ As a result, spatial proximity is more significant and the consequential superpixels are more compressed. When m is a tiny, equal weight are allocated to both [xy] and [lcd] distances. The importance of both [Lcd] spectral and [xy] spatial proximity is approximately the same, and superpixels comprise less standard size and shape. The spatial resolution of the input image is used to choose SLIC settings. The number of segments (K) is set to a reasonable quantity to allow for extinct-segmentation. To find the best compactness m, we strive to explore the homogeneity inside a superpixel. The distinction between mean and median is frequently given to determine segment heterogeneity. Equation (12) is solved using various n (number of pixels in each superpixel) values in the range. $E D_{n} = \max (μ (d_{n}) - ρ (d_{n}))$ (12)

Where, d_n is the [lcdxy] distance of each pixel from the centre of its corresponding superpixel. The mean and median functions are μ (d_n) and ρ (d_n), respectively. μ (d_n) - ρ (d_n) is employed to describe heterogeneity within a segment. As a result, limiting the greatest heterogeneity within all superpixels in the whole image yields the most optimum compactness parameter.

The benefits of employing primitive SLIC superpixels are twofold: the segments convey more information in characterizing the spatial arrangement of the regions, and the number of primal regions is significantly smaller than pixels in an image, allowing for much faster region merging procedures. SLIC considers both spectral and spatial similarities between pixels. Nevertheless, the output may include a large number of over and under segmented areas. Following that, the input picture is designated in a Region Adjacency Graph (RAG) to ease superpixel merging and splitting.

\paragraph \Secno 3.2.1.2Region Adjacency Graph (RAG)

After segmentation, taking nodes from superpixels and unite them from K neighbors to produce RAG. Figures 3 and 4 depict the different compactness and segmentation applied to the static signature images. By considering these parameter changes to produce the RAG generation to the original image. There are several methods for generating node characteristics i.e., coordinate information (superpixels’ center of gravity), mean and variance of each superpixel RGB channel. Figures 3 and 4 shows that once the SLIC has segmented the original image, the target features are first retrieved from the resultant superpixel image, and the superpixels are described as nodes based on the coordinates.

Fig. 3

RAG with different compactness and segmentation for GPDSsynthetic.

Fig. 4

RAG with different compactness and segmentation for MCYT-75.

Both coordinates and mean RGB color are utilized for initialization since the characteristics are not especially evident when the nodes simply use coordinate information for connection, and the plain contour of the target can be seen. The feature Y of the superpixel picture may be described by the Equations (13) and (14) for each superpixel where the coordinates are its center of gravity (2). $Y = {(y_{1}, y_{2}, \dots, y_{n})}^{T}$ (13) $y_{i} = {[\frac{1}{M_{i}} \sum_{j = 1}^{M_{i}} (R_{j}, G_{j}, B_{j}, c_{j}, d_{j})]}^{T}$ (14) where y_i is the node feature of a pixel i in a superpixel, and M_i is the number of pixels in a superpixel, c_j and d_j is pixel’s position in the image.

3.2.1.3 Hierarchical merging regions

RAG is a graphical description of connectivity-view of neighboring SLIC superpixels that takes spectral similarity and spatial closeness into consideration; nonetheless, these properties are insufficient to entirely differentiate area from background. Furthermore, the SLIC segmentation results several features over/under segmented areas. To balance, a multi-level construction is proposed in which the number of segments is lowered hierarchically from the bottom to the top-level such that segments can be united to form a complete network. Weighting and difference functions are two fundamental metrics for constructing multi-level structures. As seen in Equation (15), the weighting function establishes the spectral and spatial link between nearby vertices. $W (v_{i}, v_{j}) = ED (v_{i}, v_{j})$ (15) $W (v_{i}, v_{j}) = \sqrt{{(a_{v_{i}}, a_{v_{j}})}^{2} + {(b_{v_{i}}, b_{v_{j}})}^{2}}$ (16)

Where W (v_i, v_j) is the weighting function between vertices v_i and v_j. ED (v_i, v_j) is the Euclidean distance between vertices v_i and v_j. This distance is determined by a collection of spectral or spatial v_i and v_j properties that distinguish static signature image regions from the background. Assuming a and b are attributes that distinguish static signature image regions from image background. $Diff (R I_{1}, R I_{2}) = min_{v_{i} \in R I_{1}, v_{j} \in R I_{2}, (v_{i}, v_{j}) \in E} W (v_{i}, v_{j})$ (17)

The distinction function using Equation (17) between two nearby sections RI₁, RI₂ ⊆ V is the smallest weight edge linking them. Diff (RI₁, RI₂) has a value between 0 and 1. If surrounding regions are more homogenous, it is close to 0. As a result, Diff (RI₁, RI₂) functions as a threshold function to join neighboring areas. To get an estimated border, contiguous sections are hierarchically combined. Different spectral and spatial feature combinations are empirically investigated to provide a generic relation to extract border. Merging criteria M (RI₁, RI₂) between two regions RI₁ and RI₂ at various hierarchical levels are based on the following features: $\begin{matrix} M (R I_{1}, R I_{2}) \Rightarrow \min (max (R I_{1}), max (R I_{2})) \leq \\ Diff (R I_{1}, R I_{2}) \end{matrix}$ (18) Where, $max (R I_{1}) = ma x_{v_{i}, v_{ii} \in R I_{1}} (w (v_{i}, v_{ii}))$ (19) $max (R I_{2}) = ma x_{v_{j}, v_{jj} \in R I_{2}} (w (v_{j}, v_{jj}))$ (20)

Diff (RI₁, RI₂) is quite large when compared to internal differences across areas. The integration function, which functioned as a threshold utility, demonstrates that the distinction Diff (RI₁, RI₂) across areas must be greater than the minimal internal difference min (max(RI₁) , max(RI₂)).

The RAG after hierarchical merging for different data sets is seen in Figs. 3 and 4. As seen in Equation (21), After hierarchical merging, distance values from RAG’s adjacency matrix are evaluated. $G = RAG after Hierarchical merging$ (21) $D = Adjacency (G)$ (22)

The attributes taken are, number of nodes, number of edges and from RAG’s adjacency matrix using Equation (22), Max (maximum distance), Standard deviation (maximum distance), Sum (maximum distance), Min (minimum distance), Standard deviation (minimum distance), Sum (minimum distance) are calculated. All of these attributes are taken into consideration for node features.

3.2.2 Formation of a network

Complete Graphs reflect the relationship between real-world things owing to provide a comprehensive graphical perspective of the universe. It is more proficient to describe resemblance and association between nodes by using static signature images as vertices and linking these nodes to construct a network. Furthermore, the prerequisite for training any GNN-based categorization model is to build a network, say, G = (V, E), where V and E stand for a collection of nodes and edges respectively in the graph G [44]. To train the GNN model, a StellarGraph object is created with n nodes. Every node is a static signature sample, n denotes the number of static signatures employed for training, and these nodes are labelled using RAG features.

3.3 Classification

3.3.1 Static signature verification

GNN-based models used in a variety of functions, including social networking node cataloging [50], community detection [51], network similarity measurement [52], disease classification [53], and traffic network evaluation [54]. GNN models outperform typical deep learning models for scam revealing tasks because they obtain information openly from linkages created among distinct samples [41]. As a result of message transmission and aggregation operations, GNNs not only employ sample characteristics but also efficiently capture sample relations. GNNs are more proficient of identifying scam measures than typical deep learning models since they capture relationship information, making them effective in the challenge of signature forgery detection. They are an ideal combination of graphs and deep learning ideas.

A GNN disseminates information over the graph and recomputed node properties, forming a tree structure in the computation graph by unfolding around the currently calculated node. GNN models’ major learning goal is to study neighborhood embedding by transmitting and in receipt of information between nodes inside a neighborhood area. As a result, after a few rounds, each node has some characteristics that might disclose a lot about the network. As an input to the model, a GNN framework requires node characteristics x_u, ∀ u ∈ V. RAG features are considered as node features. GraphSAGE samples a fixed size subset (uniformly randomly). The number convolutional layers K of graph, which sets the number of hops through which node information is pooled at each iteration, is a critical hyperparameter of the GraphSAGE algorithm. Another key feature of GraphSAGE is a use of a distinct aggregator function AGG_k, ∀ k ∈ { 1, …, K } to cumulate information from neighbour nodes. The method is executed in forward and backward propagation stages.

3.3.2 Node embedding

Information about a vertices’ immediate vicinity is gathered and utilized to work out the node embedding, as same as convolution procedure in CNNs. The GraphSAGE approach begins with the assumption that the model has already been skilled and that the weight matrices and aggregator function attributes are fixed. The method iteratively accumulates information from the node’s neighbors, neighbors-neighbors, and so on for each node. At each iteration, the node’s immediate surroundings are sampled, and the information from the sampled vertices is aggregated into a single vector. The aggregated information $H_{N (v)}^{k}$ at a node v, depending on the sampled region N (v), at the k^th layer as Equation 23 [15]. $H_{N (v)}^{k} = AG G_{k} ({H_{u}^{k - 1}, \forall u \in N (v)})$ (23)

In this case, $H_{u}^{k - 1}$ denotes node u’s embedding in the preceding layer. These embeddings of entire nodes u in v’s neighbourhood are pooled into node v’s embedding at layer k. The topological and node characteristics in the graph, as gathered and aggregated from the k-hop vicinity of each graph node, are depicted in Fig. 5 (right).

Fig. 5

A given graph (left) and the GraphSAGE architecture with depth-2 convolutions and complete neighborhood sampling (right).

GraphSAGE gathers information from neighbor node features at the target node utilizing mean-based aggregator convolution that collects information and static signature attributes from a node’s k-hop neighborhood. The sampled neighborhood’s aggregated embeddings $H_{N (v)}^{k}$ are then concatenated with the embedded nodes from the preceding layer $H_{v}^{k - 1}$ . The layer k node v embedding is derived by pertaining the model’s parameters (W^k weight matrix) and assisting the result via a non-linear activation function σ (e.g., ReLU), as indicated in Equation 24 [15].

$H_{v}^{k} = σ (W^{k} . CONCAT (H_{v}^{k - 1}, H_{N (v)}^{k}))$ (24)

As indicated in Equation 25, the ultimate representation (embedding) of node v is represented as z_v, which is effectively the node’s embedding at the last layer K. z_v can be routed via a sigmoidal neuron layer for node categorization. $z_{v} = H_{v}^{k}, \forall v \in V$ (25)

After training, a new static signature node’s label is assessed by embedding the new node first in the previously learnt network embedding space and then cumulating the characteristics from k-hop neighboring vertices. In this case, a noted static signature is treated as a novel node that must be identified as either fabricated or authentic after being embedded in the learnt network. A GNN is differentiated by the use of a type of neural message transmission through which vector messages are exchanged between nodes and altered by means of neural networks. Furthermore, GNN gives extra neighborhood properties to the nodes. It collects messages from a node’s immediate neighbors in the graph. The communications from these neighbors are related to the information pooled from their own neighborhoods and so on, making even RAG distance-based features meaningful. When a novel node (signature image) is introduced into the graph, rather rerunning the entire model, it simply engages the embeddings of the neighbors to decide the zone of the new static signature, using previously learnt embeddings and neighborhood aggregation that foresee the label and embedding for the unknown vertex. This feature increases the model’s scalability and robustness to novel signature images. The model grows as the graph increases with each signature introduced.

3.3.3 t-Distributed Stochastic Neighbor Embedding (t-SNE)

The visualization of graphs and data with more than two dimensions, assessing and understanding graphs and high-dimensional data sets is a vital task. For example, [55] provide novel strategies for sampling and exhibiting high-dimensional extreme datasets in order to decrease computer intricacy and memory use while upholding accuracy. t-SNE approach [56] is used to minimize the dimensionality for better display. This method is a very effective nonlinear dimensionality reduction approach for displaying data sets with higher dimensions in 2D and 3D maps (2D maps are focused in this present work). Although principal components analysis (PCA) is used to construct 2D graph visualizations [57], one disadvantage of employing PCA for visualization is that data samples are closely packed together [58]. As a result, this paper using t-SNE, a strong visualization tool, to generate 2D graph representations. In this situation, static signature node embeddings also used to discover communities and other hidden structures with an aid of t-SNE.

3.4 Data set and experimental setup

GNN-based binary classifier is used to determine if a questioned static signature is authentic or counterfeit using RAG features. We selected samples from two easily accessible static signature datasets to assess the proposed model’s performance: GPDSsynthetic [59] and MCYT-75 [60]. GPDSsynthetic is made up of 100 user signatures. The collection comprises 30 skillfully forged signatures and 24 genuine signatures for each user. MCYT-75 has 75 signers’ signatures, followed by 30 signature images each signer, separated into two groups, genuine (15) and skillfully forged (15) for each user. Table 1 displays the distributions of training and testing data.

Table 1
Image sets under three categories: training, testing, and validating

Data sets Users Genuine Forgery Total Training (80%) Validating &

Testing (20%)

GPDSsynthetic 100 30 24 5400 4320 1080

MCYT-75 75 15 15 2250 1800 450

Data sets	Users	Genuine	Forgery	Total	Training (80%)	Validating &
GPDSsynthetic	100	30	24	5400	4320	1080
MCYT-75	75	15	15	2250	1800	450

The models used in present study were built with GPU (Graphics Processing Unit) support in mind. Experiments imposed in Google Colab pro using a 64-bit operating system, an Intel(R) Core (TM) i3-7020U CPU running at 2.30 GHz, and 8 GB RAM. The Python scripts are applied with Tensor Flow environment and Stellar- Graph library (61).

3.4.1 Training

Static signature data gives the function of GraphSAGE node classification to a complete graph, with in-node and out-node neighborhoods sampled independently. In Keras, the GraphSAGE model is developed. To feed data from the graph to the model, a data generator that supplies data from the graph to the Keras model is mandatory. The GraphSAGE Node Generator is chosen since need to forecast node characteristics using a GraphSAGE model and the generators are tuned to the model. Two further parameters are required: the batch size (training) and the number of nodes (sample) at each level. Batch size taken = 16 (GPDSsynthetic) and 32 (MCYT-75) and a two-level model with 15 nodes sampled in the first layer 10 and 5 in the second layer.

A GraphSAGE Node Generator object is required to send the node characteristics of sampled sub graphs to Keras. To train, evaluate, or analyze the model, the generator flow () function is used to create iterators across nodes. For training, we solely utilize the training nodes and target values returned by our splitter. The flow technique is given the shuffle = True option to increase training. We create our machine learning model by include the following parameters: The model’s layers contain a list of hidden feature sizes known as layer sizes. In present study, 64 (GPDSsynthetic) and 32 (MCYT-75) dimensional hidden node features are used at each layer. The bias and dropout of the model is internal attributes. Where, bias = True, dropout = 0, activations = Sigmoid. During the training phase, we employed the binary cross-entropy loss utility to decide the variance of the class’s predicted value from the real value.

The Adam optimization algorithm, an extension of the stochastic gradient descent process, was used. It’s used to keep network weights up to date. The activation function in the last layer was Sigmoid. Using Keras Sigmoid layers, the model built to predict the two classes. During training, the model checkpoint callback stores the model or weights in a checkpoint file at regular intervals. This includes stopping training when a specific accuracy score is achieved, saving the model as a checkpoint after each successful epoch, and progressively modifying the learning rates. As a result, these models employ the Adam optimization strategy. Furthermore, while the Adam method’s learning rate at 0.005, the justification stage set to the length of the validation set for both models. Figures 6 and 7 demonstrates how to track its loss, accuracy, and generalization performance on the validation set for the both data sets.

3.4.2 Statistical measures

The following standard statistical metrics were used to estimate the performance of this model: Mean Square Error (MSE), Root Mean Square Error (RMSE), Accuracy, Precision and F1 Score. $MSE = \frac{1}{N_{si}} \sum_{j = 1}^{N_{si}} {(t_{j} - {\hat{p}}_{j})}^{2}$ (26) $RMSE = \sqrt{\frac{1}{N_{si}} \sum_{j = 1}^{N_{si}} {(t_{j} - {\hat{p}}_{j})}^{2}}$ (27)

In Equations (26) and (27), N_si, t_j, ${\hat{p}}_{j}$ , denotes the number of static signatures, specified target value, predicted value respectively. $Acc = \frac{TP + TN}{TP + FN + TN + FP}$ (28) $Pre = \frac{TP}{TP + FP}$ (29) $F 1 Score = \frac{2 TP}{2 TP + FP + FN}$ (30)

Accuracy refers to the percentage of samples that are properly identified among all data. Precision is defined as the percentage of successfully predicted positives from all identifications. The F1-score (F-score or F-measure) is a weighted combination of accuracy and sensitivity. The following are the measurements and their enlarged calculations for binary-class categorization.

Here, the metrics in Equations (28)–(30) [62] are computed using variables such as TN (True Negative), TP (True Positive), FN (False Negative) and FP (False Positive) which are based on the values obtained by such classifications in the confusion matrix.

TP represents the number of properly identified static signature images in each class, whereas TN signifies the total of properly categorized static signature images in all other groups except the relevant category. FN provides the number of incorrectly classified static signature from the appropriate category. FP presents the total amount of incorrectly classified static signature images in all classes, with the exception of the most significant.

4 Results and discussion

The experiments were conducted on both the static signature datasets. These models were evaluated with five different compactness and N_segments for GPDSsynthetic and MCYT-75 data set. On the training and testing dataset, MSE, RMSE, accuracy, precision and F1 score values were recorded in Tables 2 and 3.

Table 2
Evaluation of Training and testing GPDSsynthetic data set

Data sets (SYNTH)

Training set Testing set

Compactness n_segments MSE RMSE Acc % loss MSE RMSE Prec F1 Score Acc % loss

10 100 0.011 0.106 99.75 0.0109 0.016 0.127 0.9996 0.9947 99.54 0.0149

20 200 0.009 0.096 99.84 0.0089 0.020 0.140 0.9999 0.9905 99.35 0.0199

30 300 0.006 0.077 99.81 0.0062 0.015 0.121 0.9999 0.9948 99.54 0.0148

40 400 0.005 0.074 99.86 0.0053 0.009 0.095 0.9999 0.9958 99.63 0.0081

50 500 0.005 0.073 99.93 0.0051 0.005 0.074 1.0 0.9989 99.91 0.0061

Data sets (SYNTH)
		Training set	Testing set
Compactness	n_segments	MSE	RMSE	Acc %	loss	MSE	RMSE	Prec	F1 Score	Acc %	loss
10	100	0.011	0.106	99.75	0.0109	0.016	0.127	0.9996	0.9947	99.54	0.0149
20	200	0.009	0.096	99.84	0.0089	0.020	0.140	0.9999	0.9905	99.35	0.0199
30	300	0.006	0.077	99.81	0.0062	0.015	0.121	0.9999	0.9948	99.54	0.0148
40	400	0.005	0.074	99.86	0.0053	0.009	0.095	0.9999	0.9958	99.63	0.0081
50	500	0.005	0.073	99.93	0.0051	0.005	0.074	1.0	0.9989	99.91	0.0061

Tables 2 and 3 gives the accuracy values as well as loss values for 10–50 compactness of N segments to analyze the training and testing data sets. The range of accuracy and loss values for each compactness was slightly different. According to the results illustrated in Table 2, the level of accuracy was gradually increased whereas loss values were decreased when compared with the previous compactness values. In case of testing data set, the range of accuracy is about 99.91%, 99.63%, 99.54%, 99.35% and 99.54% for compactness 50, 40, 30, 20 and 10 respectively. Here, the compactness 50 was found to be efficient in terms of comparatively increased level of accuracy (99.91%) and also decreased loss value (0.0061) with others. The substantial precision of the GPDSsynthetic data set was achieved by the compactness 50 which is considered to be an ideal classification strategy for static signature.

GPDSsynthetic data set was trained with 50 epochs. Metrics such as values of functional accuracy (or success rate), MSE and RMSE were essential to evaluate the training and testing process of any data set. These metrics enable the classification’s capacity, accuracy and also the network’s ability to assess the authentication of static signature and their degrees of over-adjustment. The MSE and RMSE indicate that how much the model is a well-adjusted and hardly over-fitting issue. In addition, rate of precision and F1 score metrics performed a vital role in evaluating the reliability of prevailing results.

The performance under each compactness for both training and testing data sets depicted in Fig. 6(a). It demonstrates the model’s accuracy with the parameters specified in each epoch, utilizing both the training and testing data sets. The rate of accuracy is accompanied by the proportion of precisely identified static signature to the overall static signatures. Error range for training data set was observed in gradually diminishing order up to the minimum saturation which is similar in case of testing data set that demonstrated the curve remains parallel as represented in Fig. 6(a).

Confusion matrix depicted in Fig. 6(b) that helps to estimate the variables include TN, TP, FN and FP which can be used to drive precision and F1 Score. Hereof the proportion of forgery signatures incorporated with genuine models and vice versa is exhibited in Fig. 6(b) as plotted on true and predicted label using confusion matrix. In accordance with this matrix, compactness 50 was proved to be the best with respect to properly identified static signatures and also recognized a least proportion of forgery (0.0017) from genuine signatures precisely. In addition, node embedding shown in Fig. 6(c) also proposed as an activation of the GraphSAGE layer stack’s output in which the outputs are expressed by 2D plotted dichromatic nodes based on their category labels.

Fig. 6

Monitor (a) training and testing loss during validation, (b) confusion matrix and (c) visualization of GraphSAGE embedding for GPDSsynthetic data set.

In accordance with results exhibited in Table 3, the range of accuracy is about 99.56%, 99.78%, 99.56%, 99.33% and 99.56% for compactness 50, 40, 30, 20 and 10 respectively. An evaluation with testing data set, considerable accuracy range (99.78%) was obtained from compactness 40 itself but the expected output in terms of minimal loss (0.0070) was recorded from compactness 50 so that it can be concluded as the best fit to recognize static signatures reliably. The remarkable precision (0.9999) of the MCYT-75 data set was attained by the compactness 50, found to be the greatest attempt to classify static signature models. MCYT-75 data set run with 100 epochs. The performance of each compactness for both training and testing data sets depicted in Fig. 7(a). Error range for training data set has minimum variation in a descending order for each compactness which is in line with testing data set that represented as a parallel curve as shown in Fig. 7(a). An instance of confusion matrix depicted in Fig. 7(b), compactness 50 was prevailed to get better output in recognizing perfect static signatures and also observed even a minute proportion of forgery (0.0044) from properly identified genuine signatures. Node embedding shown in Fig. 7(c) represents a clear visualization of classified static signatures by multidimensional reduction into two-dimensional display.

Table 3

Evaluation of Training and tesingMCYT-75 data set

Data sets (MCYT-75)
		Training set				Testing set
Compactness	N_segments	MSE	RMSE	Acc %	Loss	MSE	RMSE	Prec	F1 Score	Acc %	Loss
10	100	0.001	0.030	100	0.0009	0.025	0.058	0.9998	0.9955	99.56	0.0252
20	200	0.001	0.031	100	0.0010	0.024	0.156	0.9998	0.9911	99.33	0.0231
30	300	0.001	0.038	100	0.0015	0.024	0.156	0.9996	0.9955	99.56	0.0245
40	400	0.001	0.034	100	0.0011	0.015	0.122	0.9998	0.9977	99.78	0.0148
50	500	0.001	0.030	100	0.0009	0.008	0.089	0.9999	0.9955	99.56	0.0070

Fig. 7

Monitor (a) training and testing loss during validation, (b) confusion matrix and (c) visualization of GraphSAGE embedding for MCYT-75 data set.

As per Table 4, the proposed method i.e., Region adjacency based GNN model of GPDS synthetic was observed with remarkable accuracy (99.91%) when associated with Li et al. (2019) who reported that the level of accuracy was 96.46% by using two-channel CNN and Ghosh et al. (2021) stated the accuracy level of about 96.08% while using Recurrent Neural Network. The accuracy of proposed method is in line with a result of Liu et al. (2021) who described 99.93% accuracy in the presence of region based deep learning network. In this regard of MCYT-75, prevailed method has the greatest level of accuracy (99.56 %) when compared with the outcomes of Masoudnia et al. (2019), Ghosh et al. (2021) and Davood Keykhosravi et al. (2022) where the accuracy was about 98.01%, 99.39% and 99.47% respectively. The present results of MCYT-75 dataset are in consonance with Jampour et al. (2021) with the accuracy level of 99.66%.

Table 4

Comparison of the proposed method with recent studies

Reference	Features	Data sets	Accuracy
Li et al. (2019) [63]	two-channel CNN network	GPDS synthetic	96.46%
Sam et al. (2019) [64]	Inception v1 and v3		83%, 75%
Ghosh et al. (2021) [65]	Recurrent Neural Network		96.08%
Liu et al. (2021) [66]	region based deep metric learning		99.93%
	network
Sharma et al. (2022) [67]	Siamese Neural Network		92%
Proposed method	Region adjacency based GNN		99.91%
Masoudnia et al. (2019) [68]	CNN	MCYT-75	98.01%
Shariatmadari et al. (2019) [69]	Hierarchical CNN		94.54%
Ghosh et al. (2021) [65]	Recurrent Neural Network		99.39%
Jampour et al. (2021) [70]	ResNet		99.66%
Davood Keykhosravi et al. (2022) [71]	developed deep neural network_V3		99.47%
Proposed method	Region adjacency based GNN	99.56

5 Conclusion

In this current study, static signature verification system has been designed following a writer dependent setup by utilizing a GNN based binary classifier, where node categorization also performed using graphSAGE model. Region adjacency graph-based features extracted from each static signature to get the model computationally efficient. Readily accessible two datasets used namely, GPDS synthetic and MCYT-75 exhibited outstanding results in terms of accuracy and loss values compared with recent studies having same datasets. It reflects the prominence of GNN towards static signature verification process. Random graph models are widely applied in molecular biology and social networks. Since, there is a chance that a random graph may be designed with completely all edges or without an edge of forged ones at all which can be negatively influenced the performance of specific model. It can be refined by using extracted features where each and every node has been interconnected to get a complete graph followed by GraphSAGE node classification. Hereof, compactness performed the best part in designing a region adjacency graph thereby better segmentation leads to precise recognition of static signatures. At present study, among the compactness used were 10, 20, 30, 40 and 50, the substantial results recorded with compactness 50 with respect to per cent accuracy (GPDSynthetic-99.91% and MCYT-75-99.56%) and loss (GPDS synthetic-0.0061 and MCYT-75-0.0071). However, while considering the recent studies, further investigation has to be essential to sort out some flaws encountered in identifying genuine signatures from forged ones.

Footnotes

Acknowledgment

The authors are very grateful to extend their sincere thanks and gratitude to the Management of Shiv Nadar Foundation for their ongoing support and encouragement in carrying out this research.

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Region adjacency graph based GNN approach for static signature classification

Abstract

Keywords

1 Introduction

1.1 A detailed account of the present work

2 Review of literature

3 Proposed method

3.1.1 Edge detection

3.2.1 Graph-based boundary merging

3.3 Classification

3.3.1 Static signature verification

3.3.2 Node embedding

3.4 Data set and experimental setup

Table 1 Image sets under three categories: training, testing, and validating Data sets Users Genuine Forgery Total Training (80%) Validating & Testing (20%) GPDSsynthetic 100 30 24 5400 4320 1080 MCYT-75 75 15 15 2250 1800 450

3.4.2 Statistical measures

Footnotes

Acknowledgment

References

Table 1
Image sets under three categories: training, testing, and validating

Data sets Users Genuine Forgery Total Training (80%) Validating &

Testing (20%)

GPDSsynthetic 100 30 24 5400 4320 1080

MCYT-75 75 15 15 2250 1800 450