**1. Introduction**

Biometric verification technology has aroused a lot of interest due to its reliability, effectiveness, and convenience in verifying personal identity [1]. Verification techniques based on face [2], fingerprint, and some such physiological biometric attributes have brought extra convenience and changed our lifestyle [3]. Although behavioral biometric attributes are slightly inferior to physiological ones in stability and uniqueness, they are more accessible and less intrusive to users. Voice, signature, gait, etc. are all typical behavioral attributes. Among them, signature remains the most widespread and recognized socially and legally verification approach in our day-to-day life [4]. Signing is a customary and fast movement driven by long-term nervous system and writing habit. Therefore, signature verification techniques can have more potential applications in the real world.

Depending on the different methods of signature acquisition, signature verification technique can be split into two categories: offline and online. In the system of offline signature verification [5], images containing signatures are collected after finishing the signing process. For online signature verification, signatures are captured by sensor-based devices while the user is signing and represented by a set of temporal functions, from which both static and dynamic features are extracted and then used to make a decision on whether the signature belongs to its claimed user. Compared with offline signature verification, the dynamic information collection of online signature ensures its uniqueness and higher

difficulty to forge, so online signature verification technique usually owns better performance in accuracy and security.

There are two parts of an online signature verification system: enrolment and verification. Several signatures are provided as reference signatures by the users during enrolment and their extracted features along with calculated thresholds would be stored in the knowledge base. In verification, the authenticity of a test signature is evaluated by matching its features with those from reference signatures of its claimed user [6].

Online signatures are collected by electronic devices such as tablets, smart phones, and so on. Most of them use sensors to capture various real-time data such as coordinates, pressure, timestamp, etc. during signing. After collection, the signatures are represented as time series and then undergo preprocessing and feature extraction modules successively.

Online signature verification methods can be categorized based on the feature extraction process and matching strategy [7]. According to the employed features, there are broadly two groups: parametric and function features-based approaches. In the framework of parametric features-based methods, a signature is characterized as a vector of elements and each one is a representative of the value of one feature [8]. Examples of such attributes are width, height, average speed, etc. The dimensions of parametric features of signatures are all equal. In the function features-based method, a signature is represented by a multi-dimension feature set constituted by several time functions. Coordinate, timestamp, pressure, etc. are commonly used function features. Generally, the function features-based approaches perform better due to more dynamic information application, but these kinds of method consume more computational time and memory.

With regards to the matching methods, distance-based and model-based approaches are two main techniques [9]. Dynamic time warping (DTW) has been often adopted in distance-based methods [10]. DTW is a well-known approach for aligning vectors of different lengths. For application in signature verification, a set of features at each sample point is extracted and the similarity between the test signatures and enrolled reference signatures is then computed using dynamic programming. Point-based warping technique is a variant of DTW, wherein only selective points are warped. Extreme point [11] and stroke point [12] are often used. In addition, some works make a fusion of DTW with other methods. Sharma and Sundaram [9] propose a method that uses the information from DTW cost matrix and warping paths alignments. The decision is made by the conjunction of warping path score and DTW score. Yanikoglu and Kholmatov [13] fuse the Fast Fourier Transform with DTW and the fusion system lowers the error rate by up to about 25%. Chen and Xia [14] extract a set of function features for comparing the dissimilarity-based DTW between the test signature and the template database. In addition, the nearest template and majority vote are proposed to classify. Model-based approaches employ either generative-based classifiers such as hidden Markov model (HMM) [15–17] or discriminative ones such as neural network (NN) [18–20] and support vector machine (SVM) [21,22]. Also, there are some hybrid methods that combine different methods mentioned above. Multi-stage cascade framework [23], multi-stage decision-level score fusion [24,25] or a multi-expert system for signature verification [26,27] have been reported in the literature. Recently, inspired by the great success of recurrent neural networks (RNNs) in sequential modeling, several verification methods based on RNNs are proposed. Lai et al. [28] propose a novel descriptor called the length-normalized path signature (LNPS) for feature representation and then features are fed into the GRU (Gated recurrent unit) network. Triplet loss and center loss were used to train the network with the BP algorithm. The method proposed in [29] extracts 23 hand-crafted time function features and uses the bidirectional LSTM (Long short-term memory) and GRU networks with Siamese architecture to learn a dissimilarity metric from the pairs of signatures.

Although it is not that easy for a forger to fake a signature that is exactly the same as the genuine one, due to the large intra-class variations from one person and small inter-class variations between forgeries and genuine ones, accurate online signature verification still remains a challenging problem.

In real applications, the forgeries are usually classified to be two types, named skilled forgery and random one. A skilled forgery is signed by a person who had access to the genuine signatures and practiced for a while. A random forgery is signed without with any information about the signature, or even the name of the person whose signature is forged [30]. Compared with skilled forgeries, the random forgeries are more common in our daily life. Obviously, the skilled forgeries are more difficult to verify. In addition, the loss brought by accepting forgeries is higher than that by rejecting genuine signatures, which means accepting a signature as genuine should be stricter. Considering these factors, we propose a two-stage method using shape contexts and function features for accurate online signature verification. Features of shape contexts are extracted from the input firstly and classification of this stage is based on shape distance metric. Only the inputs passing by the first stage are represented by a set of function features and verified. To improve the matching accuracy and efficiency, we employ a shape context-dynamic time warping (SC-DTW) to compare the test signature with the enrolled reference ones based on the extracted function features. An interval-valued symbolic representation-based classifier is proposed to decide if the test signature is a genuine one.

The contributions of this paper are as follows:


This paper is an extended version of the one published in proceedings of PRCV2018 [31]. In this paper, more details on feature extraction and matching methods are given. Moreover, to further improve the performance method in the paper of PRCV2018, more effective features are extracted. Instead of distance metric classification, an interval-valued symbolic representation-based classifier is employed to enhance classification ability. Besides, more detailed experimental results are reported.

The rest of this paper is organized as follows. Section 2 details the methodology we proposed. Signature preprocessing is presented in Section 2.1. Section 2.2 presents the shape context descriptor and online signature verification method based on it. The function features extraction, feature alignment, and symbolic classifier are showed in Section 2.3. Section 2.4 discusses the two-stage verification protocol. The database used in our experiment, experimental results, and performance analysis are provided in Section 3. The conclusion is offered finally in Section 4.
