2.1.2. Improvement Definition

**Definition 3.** *Given functions f* <sup>1</sup>*(x) defined on [R*1*, R*2*] and f* <sup>2</sup>*(x) defined on [C*1*, C*2*], and make transform T* → *f*- <sup>1</sup>(*x*) = *k*· *f*1((*x* − *b*)/*a*) − *h, then the minimum distance of all the matching distances between f'1(x) and f2(x) with di*ff*erent transformation T is as follows:*

$$\begin{array}{lcl} \text{dis}(a,b,b,h) &= \int\_{aR\_1+b}^{aR\_2+b} [T\_1(\mathbf{1}(\mathbf{x})) - f\_2(\mathbf{x})] d\mathbf{x} \\ &= \int\_{aR\_1+b}^{aR\_2+b} [f\_1'(\mathbf{x}) - f\_2(\mathbf{x})] d\mathbf{x} \\ &= \int\_{aR\_1+b}^{aR\_2+b} [k \, f\_1((\mathbf{x}-b)/a) - h - f\_2(\mathbf{x})] d\mathbf{x} \\ &= \int\_{R\_1}^{R\_2} a \cdot [k \, f\_1(t) - h - f\_2(a \cdot t + b)] dt \\ &= \int\_{R\_1}^{R\_2} a \cdot [k \, f\_1(\mathbf{x}) - h - f\_2(a \cdot \mathbf{x} + b)] d\mathbf{x} \end{array} \tag{1}$$

$$d(f\_1, f\_2) = d(f'\_{1'}, f\_2) = \min\{\text{dis}(a, b, k, h)\}\tag{2}$$

*which is called the curve similarity distance of f2(x) to f1(x) under the transformation T, where f1(x) is called the reference function or the reference curve, f2(x) is called the comparison function or the comparison curve, f'1(x) is called the transform function or the transform curve, T is called a function similarity transformation or a curve similarity transformation (CST), and dis(a,b,k,h) is called the distance of the two curves under the curve similarity transformation T.*

Obviously, the distance between the two curves is different under different similarity transformations *T*, and the curve similarity distance (CSD) is the distance after the optimal matching of the reference curve for the comparison curve. After the curve similarity transformation, the curve similarity distance is denoted by min{*dis(a,b,k,h)*}.

Once the optimal matching of the curves is obtained, the corresponding curve similarity distance can be obtained, as shown in Figure 2.

**Figure 2.** The curve similarity transformation between the two curves.

From Figure 2, *a*, *k* are translational transformations in the horizontal and vertical directions, respectively, and *b*, *h* are scaling transformations in the horizontal and vertical directions, respectively.

Therefore, the similarity of curves based on curve similarity transformation is defined as follows:

**Definition 4.** *Given a reference curve f1(x) and a comparison curve f2(x), for a given threshold* ε*, if d(f1,f2)* ≤ ε*, then the curve f2(x) is called similar to curve f1(x), and vice versa.*
