2.2.3. Segmentation Matching

The process of the optimal segmentation matching is as follows:

*Step 1*: Take the reference signature curve as a template, and divide it into *K* segments according to Equation (9).

*Step 2*: The comparison curve should be divided into *K* segmentations according to Equation (10).

*Step 3*: For the *i*-th segmentation of the comparison curve, search the optimal matching with the corresponding the *i*-th segmentation of the reference curve by EC algorithm, and get the similarity distance *di.* Meanwhile, the matching distance *dxi* and *dyi* of the corresponding *X*, *Y* curves can be calculated based on the current matching result.

*Step 4*: Set σ*xi*, σ*yi* which are the standard deviation of *X*, *Y* data points in *i*-th segmentation of the reference curve, compare it with the matching distance *dxi* and *dyi*, and calculate the similarity score *sxi* and *syi* of this segmentation, respectively, as in Equations (11) and (12).

$$\begin{cases} s\mathbf{x}\_i = h(d\mathbf{x}\_i, \sigma \mathbf{x}\_{i\prime}\boldsymbol{\alpha}) + h(d\mathbf{x}\_{i\prime}\sigma \mathbf{x}\_{i\prime}\boldsymbol{\beta}) + h(d\mathbf{x}\_{i\prime}\sigma \mathbf{x}\_{i\prime}\boldsymbol{\gamma})\\ s\mathbf{y}\_i = h(d\mathbf{y}\_i, \sigma \mathbf{y}\_{i\prime}\boldsymbol{\alpha}) + h(d\mathbf{y}\_{i\prime}\sigma \mathbf{y}\_{i\prime}\boldsymbol{\beta}) + h(d\mathbf{y}\_{i\prime}\sigma \mathbf{y}\_{i\prime}\boldsymbol{\gamma}) \end{cases} \tag{11}$$

$$\begin{cases} \begin{array}{c} h(d,\sigma,\delta) = 100 \cdot \exp\left(-0.5 \delta \cdot d^2 / (\sigma/4 + 10)^2\right) \\ a = 0.25 \sim 0.5, \beta = 1 \sim 2, \gamma = 2 \sim 5 \end{array} \end{cases} \tag{12}$$

*Step 5*: Repeat step 3 until all segmentation curves parameters are calculated.

*Step 6*: Calculate the average of *sx*, *sy* as the result outputs of *X*, *Y* curves' similarity measure, and use the weighted average as the result output of the similarity measure *Score* of the two curves, as shown in later Formula (14).

It should be noted that when performing the optimal matching segmentation calculation, the similarity distance between the two curves can be obtained, which are considered as 2D curves. Next, the matching distances of the corresponding 1D curves *X* and *Y* can be separately calculated. Obviously, the calculated matching distances are absolute values, and if a similarity evaluation is to be performed, one threshold is needed to discriminate. For this reason, the similarity average is calculated using Gaussian functions of three different widths and the absolute distance measure is converted to a relative measure between 0 and 100. Thus, the discrimination threshold can be unified to a value between 0 and 100.

A pair of genuine signature curves is adopted for the optimal segmentation matching as seen in Figure 5. The matching results of *X*, *Y* curves are shown in Figure 6, respectively.

A pair of genuine and forged signature curves is adopted for the segmentation matching as seen in Figure 7. The matching results of *X*, *Y* curves are shown in Figure 8, respectively.

**Figure 5.** The matching results of two genuine signatures.

**Figure 6.** The matching result of *X* and *Y* curves between two genuine signatures.

**Figure 7.** The matching results of the genuine and the forged signatures.

**Figure 8.** The matching result of *X*, *Y* curve between the genuine and forged signatures.

The optimal segmentation matching results between three curves are shown in Tables 1 and 2.


**Table 1.** Results of segmentation matching between a pair of genuine signature curves (*K* = 10).

**Table 2.** Results of segmentation matching between a pair of genuine and forged signature curves (*K* = 10).


It can be seen from Tables 1 and 2 that for the similarity of *X*, *Y* curves is calculated by the same template signature and segmentations, the similarity between two genuine signatures is usually higher than that between the genuine and forged signatures. On each segmentation, the similarity calculation of the *X*, *Y* curves depends on the standard deviations of the respective segmentations in the template signature, as shown in Equation (11). To accurately estimate this deviation, a large number of genuine and forged signatures are needed for matching calculations and statistics. Obviously, this is difficult to obtain in practical applications. Here we use the intra-segmentation standard deviation of the signature segmentation itself and a deviation as the empirical value. In addition, the three control parameters α, β, and γ are equivalent to controlling the width of the Gaussian function, and are also an empirical value. Here, α = 0.4, β = 1.6, γ = 3.2 are selected. The changes in these parameters are not sensitive to the correct rate of the final evaluation results. Due to space limitations, this article will not discuss them.
