*1.1. Related Work*

Although a signature can show individual behavior features, it is more unstable and diverse than other biometric verification technologies such as fingerprint recognition, iris recognition, face recognition and so on. Due to changes in the internal and external environment, there are fluctuations in the size, location and rotation angle of signatures with the same signer at different input times. In addition, signatures will not maintain high consistency for a long time as writing habits and the external environment change. As a result, two repetitions of a signature from the same writer never have an identical appearance [28]. Each person can even have several signatures of diverse styles, and one style of signature is obviously not suitable for the verification of another style of signature. Of course, the style variability also makes signature verification better for privacy protection than fixed biometric recognition technology.

In a traditional signature verification system, a large number of samples are mandatory when building a reliable statistical classifier and many algorithms even also require skilled forgery samples [29]. Figuring out a stable signature region is also a hot topic in recent research. Similarly [19,29], extraction of a stable signature region also depends on availabiliy of a large number of training signatures. In practice, it is often impractical to obtain various signature samples from a signer, which limits the applicability of a signature verification system.

How to reduce the enrollment signature size is a crucial issue. Another problem is how to reduce the differences between different signatures, that is, the problem of signature alignment is also a key problem to be solved.

It is the most widely used and recommended method for size alignment by max-min normalization [30]. Some template matching methods, such as DTW [14], LCSS [22] and so on, also apply for alignment. Recently, alignment methods based on Gaussian mixture model (GMM) have been developed [17], but training a Gauss model requires a large number of samples.

For selecting effective reference signatures, the intra-class variation of genuine signatures can be quantified with a correlation-based criterion which detects and recovers non-linear time distortions in different specimens as described in [31].

A single reference signature system (SRSS) for training with only a single reference signature has been proposed in [28], which followed the strategy of duplicating the reference signature to enlarge the training set. In this work, the strategy consists of duplicating the given signature a number of times and training an automatic signature verifier with each of the resulting signatures and the duplication scheme is based on a sigma lognormal decomposition of the reference signature.

Nevertheless, in a real situation, it is sometimes difficult to obtain enough signatures from a signer, especially in commercial applications and forensic covers. Therefore, this paper discusses the model and method of designing an automatic signature verification system using only one real reference signature per enrolled signer. Moreover, in this study, it is vitally important to effectively align the test signature to the reference signature for verification in order to cut down the influence of fluctuations caused by variances of size, location and rotation angle, which may deteriorate the performance of verification.

The signature trajectory can be viewed as a 2D/3D curve. The similarity between two signatures can be measured by curve similarity [32]. Curve similarity is a major category of similarity measure and a large number of similarity problems can be transformed or abstract into curve similarity problems.

Measuring curve similarity is a common method for curve matching. The curves are usually assumed to be represented as polygonal chains in the plane and to be measured by distance such as DTW or Fréchet distance as in [33]. The Fréchet distance, which relies on fewer features, can be applied for signature verification as proposed in [33].

By computing cumulative distance, DTW provides normalization and alignment as a computational technique to determine the best match between two curves, which might produce different sample points. The Fréchet distance belongs to a general class of distance measures that are sometimes called "dog-man" distances and is a max measure which is outlined in terms of the maximum leash length over a parameterization. However, two classical curve similarity measures are sensitive to data anomaly points and cannot adapt to changes in the translation and scaling of the curve.

## *1.2. This Paper*

Our goal of this paper is to design an automatic signature verification system for a SRSS. To this end, a new curve similarity measure model and calculation method has been established, which we call the curve similarity model (CSM). The curve similarity draw lessons from geometric similarity, and can be adapted to various transformations such as translation, scaling and rotation, and can better be adapted to the inconsistency of signatures such as signature size, position and rotation angle in the signature curve.

The procedure presented in this paper considers a rigorous and adaptive CSM to build a robust SRSS. Therefore, we completed the exploratory work reported in [32], proposed a continuous and discrete curve similarity model based on transformation, and accomplished the curve optimal matching calculation based on evolutionary computation (EC) [32]. Based on the characteristics of the SRSS, a differentiated fusion feature named local similarity score (LSC) is designed for the difference calculation between two signatures.

The paper is organized as follows: Section 2 introduces the relevant definitions of CSM. Section 3 describes the proposed curve similarity calculation method and process. The fourth section describes the optimal sectional matching of signature curves and local matching feature extraction for SRSS. The experimental results will be presented and discussed in Section 3. Conclusions are drawn in the last section.
