**2. Methodology**

The diagram of the proposed method is shown in Figure 1. The input signature is first preprocessed for smoothing and normalization, and then it is fed into the shape context-based verification module, which does well in quickly distinguishing the random forgeries owing to their manifest differences in shape. Most obvious forgeries can be rule out in this stage. The signature passed through the first module is verified by function features-based verification module. This module achieves more accurate verification results due to the application of details in signature and decision fusion by interval-valued symbolic representation-based classifier.

**Figure 1.** Diagram of proposed verification system.

#### *2.1. Preprocessing*

Captured by electronic devices, the time series of a signature are mixed with noises and fluctuations unavoidably. In addition, the acquired signatures of one individual vary with time or places, with the result that there are differences in size and location between signatures. Therefore, we firstly let the acquired signatures pass the preprocessing module to address those issues. The preprocessing module includes smoothing and normalization. Gaussian smoothing is employed to filter the artifacts and smooth the data. Then we adopt moment normalization technique [32] to standardize the size and location of acquired signatures.

Set the signature as *S* = (*s*1,*s*2, ...,*si*, ...,*sN*), *si* = (*xi*, *yi*). N is the number of sample points, (*xi*, *yi*) is x and y coordinates information.

In the moment normalization technique, the size of a signature is not the difference between maximum and minimum in horizontal and vertical directions, but the width and height of the window derived from its moment, as is show in Figure 2. Denote the width and height of window as W and H, given by

$$\mathcal{W} = 4\sqrt{\mu\_{20}},\\H = 4\sqrt{\mu\_{02}} \tag{1}$$

*μpq* denotes the center moment, and (*xc*, *yc*) denotes the signature's centroid.

$$\mu\_{pq} = \sum\_{\mathbf{x}} \sum\_{\mathbf{y}} (\mathbf{x} - \mathbf{x}\_c)^p (\mathbf{y} - \mathbf{y}\_c)^q \tag{2}$$

After window calculation, the size normalization technique is implemented as follows. The heights of the signatures are normalized to a predetermined value that in this paper is 300. Moreover, the aspect ratio of before and after preprocessing remains consistent to keep the signature shape unchanged.

$$\begin{aligned} \mathbf{x'} &= \boldsymbol{\mathfrak{a}} \times (\mathbf{x} - \mathbf{x\_c}) + \mathbf{x'\_c} \\ \mathbf{y'} &= \boldsymbol{\mathfrak{g}} \times (\mathbf{y} - \mathbf{y\_c}) + \mathbf{y'\_c} \end{aligned} \tag{3}$$

where *x* and *y* are smoothed originate coordinates. *x* and *y* are normalized coordinates. *x <sup>c</sup>* and *y c* are the centroid of normalized signature. *α* and *β* are the ratio of the normalized signature size to its original size, given by

$$\begin{aligned} \alpha &= \frac{\mathcal{W}\_{norm}}{\mathcal{W}},\\ \beta &= \frac{H\_{norm}}{H},\\ \frac{\mathcal{W}\_{norm}}{H\_{norm}} &= \frac{\mathcal{W}}{H} \end{aligned} \tag{4}$$

where *Wnorm* and *Hnorm* denote the normalized width and height.

Signatures are centered at (0, 0) to normalize their locations. After preprocessing, the signatures have the same size and location. In this paper, we did not employ translation normalization since we believe signature's angle is an out-of-habit feature. Figure 2 shows some examples of original signatures and corresponding preprocessed signatures.

**Figure 2.** Examples of signature preprocessing. (**a**) Four English or Chinese examples of original signatures. (**b**) Window calculated by moment of signatures. (**c**) Preprocessed results of corresponding signatures.

#### *2.2. Shape Context-Based Online Signature Verification*

In the methods proposed for online signature verification, the dynamics properties of the signatures, for example, velocity, pressure, acceleration, etc. are widely applied. However, the shape of signature contains very useful details, which is critical for distinguish a signature between forgery and genuine one. The method proposed by Gupta and Joyce [33] extracts the dynamics properties of position extreme points of signatures and achieved better performance. Features based on shape also have been successively applied in offline signature verification [34].

In this paper, we propose a verification method based on shape context features. Specifically, shape context descriptor [34,35] is used to extract features of a signature and a cost matrix is computed. After finding the best one-to-one matching between two signatures' shape and modeling transformation, the measurable shape distance is used for classification. To further improve the efficiency, only trend-transition-points (TTPs) that can represent the shape of a signature roughly are used for calculating distance.

#### 2.2.1. Shape Context Feature Extraction

Shape context descriptor captures the distribution over relative positions of shape points and the connectivity properties between features points along curves. Therefore, shape context features not only provide global characterization of shape but also contain more contextual information within a certain range of a signature. Besides, shape context descriptor is designed in a way of describing shapes that allows for measuring shape similarity and the recovering of point correspondences. Traditionally, the first step is to randomly select a set of points that lie on the edges of two shapes separately. Here the shape of an online signature is represented by a set of sampled points which in this work is (*xi*, *yi*), *i* = 1, 2, ..., *N*. *N* is the number of sampled points.

Figure 3 shows the shapes and shape context histograms of a reference signature, a genuine signature and a skilled forgery of one user. Because the writing speed is a kind of relatively fixed and unique information, the number and distribution of sample points between genuine signature and reference signature are more similar. Taking one point as the origin of polar coordinate, the shape contexts of this point can be represented using log-polar histogram. We set five bins for *logr* and 12 bins for *θ*. The number of neighboring points that fall into the very bin is just the histogram value. Figure 3d–f present the corresponding histograms for certain points. We can see that the difference in shape context histograms between genuine signature and reference signature is relatively small, while the histogram of skilled forgeries is quite dissimilar to the reference ones.

**Figure 3.** Examples of shape context feature extraction. (**a**) A reference signature of one user. (**b**) A corresponding genuine signature from the same user with reference one. (**c**) A skilled corresponding forgery from the same user with reference one. The green square points represent selected trend-transition-points. (**d**–**f**) Shape context histograms for chosen trend-transition-point in the signatures of (**a**–**c**), respectively.

Considering a point *pi* on the first shape and a point *qj* on the second shape, denote *Cij* = *C*(*pi*, *qj*) as the matching cost of these two points, given by

$$\mathcal{C}\_{ij} = \mathcal{C}(p\_i, q\_j) = \frac{1}{2} \sum\_{k=1}^{K} \frac{[h\_i[k] - h\_j[k]]^2}{h\_i[k] + h\_j[k]} \tag{5}$$

where *hi*[*k*] and *hj*[*k*] denote the *kth* bin histogram at *pi* and *qj* respectively.

For all pairs of points *pi* on the first shape and *qj* on the second shape, calculate the cost as Equation (5) and then we got a cost matrix. The next step is to find the optimal alignment between two shapes that minimizes total cost. This can be done by the Hungarian method with time complexity of *O*(*N*3). The cost between shape contexts is based on the chi-square test statistic that is not a suitable distance metric. Thin plate spline (TPS) model is adopted for modeling transformation [35]. After that, we get the measurable distance of two shapes. The smaller the distance, the more similar these two shapes, or vice versa. So, if the average distance between test signature and reference signatures is lower than a threshold, it would be accepted as a genuine signature of its claimed user.

#### 2.2.2. Trend-Transition-Point Selection

The shape context representation of signature should not only capture specific shape features but also allow considerable variations. Besides, the computational load of distance calculation is closely related to the number of points. Therefore, with the hope of efficiency improvement and variances tolerance, a few representative points are selected. Only selected points can participate in shape distance calculation. In this paper, we propose the TTP selection method.

Trend-transition-points are the points where the curve trends before and after them are completely different while the trends between two successive TTPs do not have obvious change so that the curve shape of the segment approximates to a straight line. So, the signature could be re-constructed with

these points. In our method, local extreme points and corner points are all defined as TTPs. The local extreme points are selected depending on its value greater or smaller than its neighborhood. The corner points selection we adopted is proposed in [36,37], which makes use of the smaller eigenvalues of covariance matrices of regions of support.

Let *Sk*(*si*) denotes the region of support (ROS) of point *si*, a small curve segment containing itself and *k* points in its left and right neighborhoods. That is

$$S\_k(s\_i) = [s\_j = (x\_{j'}y\_j)|j = i-k, i-k+1, \cdots, i+k-1, i+k]$$

where (*xj*, *yj*) are the Cartesian coordinates of *si*.

Therefore, the 2 × 2 covariance matrix for points in the segment *Sk*(*si*) is calculated. *λ<sup>L</sup>* and *λ<sup>S</sup>* are two eigenvalues corresponding to the covariance matrix. The smaller eigenvalues *λ<sup>S</sup>* can be used to measure prominence of corners over its ROS. In other words, sharper corner points have the large *λ<sup>S</sup>* and weaker corners have small one. When the points are on a straight line or on a flat curve, the *λ<sup>S</sup>* will be very small, even approximate to zero. So, corners can be determined if its *λ<sup>S</sup>* exceeds a predetermined threshold.

Shape contexts are calculated on every point, but only TTPs are used to distance calculation. For every sample point of signatures, the algorithm is implemented as follows.

