2.2.2. Segmentation

*SignR* is divided into *K* sections and there are *m* data points of each the reference segmentation curve. Each the reference segmentation curve can be defined as

$$\begin{aligned} \left( \text{Sign}R \right)\_i &= \left\{ \left( \mathbf{x}\_t, y\_t \right) \middle| t \in \left[ R\_i, R\_i' \right], R\_i' - R\_i = m \right\} \\ m &= \text{INT}(M/K) \\ R\_0 &= (M - m \cdot K)/2 \\ R\_i &= R\_0 + (i - 1) \cdot m \\ R\_i' &= R\_0 + i \cdot m \\ i &\in \left[ 1, K \right] \end{aligned} \tag{9}$$

where, INT(*x*) is the integral function.

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Likewise for the comparison curve, each possible matching interval corresponding to the reference segmentation curve can be defined as

$$\begin{cases} (\text{Sign}\mathbf{C})\_i = \left\{ (\mathbf{x'}\_i, \mathbf{y'}\_i) \middle| t \in [\mathbf{C}\_i, \mathbf{C'}\_i], \mathbf{C'}\_i - \mathbf{C}\_i = 2n + m \right\} \\ n = \text{INT}(N/K) \\ \mathbf{C}\_0 = (N - n \cdot \mathbf{K})/2 \\ \mathbf{C}\_i = \mathbf{C}\_0 + (i - 1) \cdot n - n/2 - m/2 \\ \mathbf{C'}\_i = \mathbf{C}\_0 + (i + 1) \cdot n - n/2 + m/2 \\ 0 \le \mathbf{C}\_i < \mathbf{C'}\_i \le N, i \in [1, K] \end{cases} \tag{10}$$

Here, considering the positional correlation and the deviation between the signature curves, when the reference signature length of each segmentation is *m*, the interval to be matched of the comparison signature is at least *m* in length, and is offset by *n* before and after the corresponding segmentation position. Equivalently, the reference curve swims within the interval to be matched on the comparison curve to obtain the optimal matching position.
