In recent years, different methods for ERG signal processing have been proposed with some advantages and disadvantages. However, the goal of all methods is to find the best way to analyze the ERG signals. The standard characteristics of ERG signals, which have been evaluated in most research, include ERG waves' amplitude and implicit time.

The ERG length is normally 200 milliseconds, and the first 80 milliseconds are the most critical part of the signal; because it contains the main components. Therefore, one of the fundamental challenges of processing ERG is its short length. Nonlinear methods could be the right choice in such signals to represent the signal's dynamic, which cannot be extracted through conventional methods.

## 2.2.1. Mapping Method

In this method, ERG is mapped to a parabolic curve. Therefore, the pairs of \(\left({X}_{i},{\left(\stackrel{-}{X}-{X}_{i}\right)}^{2}\right)\) are used to rearrange the points of ERG. The parabolic curve could provide a high resolution for distinguishing between different signals. The distribution of points in the parabolic curve is obtained through a two-degree polynomial equation as follow [28]:

$$Y=\alpha {x}^{2}+\beta x+\gamma$$

1

The three main parameters of this equation are \(\alpha\), \(\beta\), and \(\gamma\). The distribution of points in a parabolic curve can have three states. The accumulation of points may be more to the left, right, or approximately divided into both sides equally. An illustration of the possible distribution of signal points on the axis is shown in Fig. 1.

The time series (ERG in our study) may include negative or positive amplitudes. Therefore, based on the \(\left({X}_{i},{\left(\stackrel{-}{X}-{X}_{i}\right)}^{2}\right),\)the square of the signal causes these parts of the signal to accumulate on the left or right side of the curve. In this way, we can even interpret the positive and negative amplitudes of ERG.

A criterion for measuring the distribution of points on the right or left side of the curve is the angles made by drawing a line from the beginning to endpoints distributed on the parabolic and horizontal axis. If the accumulation of points were on the parabolic curve's right, it indicates that the angle is less than 90 degrees. Such a result means the signal contains positive waves with higher amplitudes. On the other hand, if the accumulation of points were on the left of the parabolic curve, it leads to the angle, which is less than 90 degrees.

Negative angles confirm that signals contain negative waves with higher amplitudes. Signals with small angles (close to zero) indicate the balance between positive and negative waves [28]. The calculation of angles and measuring accumulated points on the curve's left or right side is shown in Fig. 2.

The tangent is defined as the ratio of the opposite side to the adjacent side. Theta is defined as the inverse of the tangent function, which is the angle whose tangent is a given number:

$$\theta =arctan\frac{opposite\left(a or b\right)}{adjacent}$$

2

In positive angles, the opposite side is "a," which is a portion of b-wave amplitude, and in the negative angle, the opposite side is "b," which is a portion of a-wave amplitude. In both cases, opposite sides contain more amplitudes than other parts of a-wave and b-wave [28].

There might be cases where several dispersed data far away from the rest of the points open the angle's arc and show a larger Theta. In principle, most of the points have accumulated elsewhere, and if the upper arc angle is considered to be the accumulation of those points, the Theta angle may be smaller. A density criterion can be a solution to this problem. The density of points in the Theta arc can be calculated and multiplied by the Theta angle. Therefore, the density criterion decreases if the accumulation of points is less and the angle is large. On the contrary, the higher the accumulation of points and the smaller the angle, the more robust the density criterion [28].

$$Density=\theta \times Number of points in the arc of \theta$$

3

Figure 4 shows the different densities of points in the Theta angle arc for the two sample time series. As can be seen, Figure 3(a) shows the proportional distribution of points and the formation of a corresponding logical angle. However, in Figure (b), two points are located far from the concentration of points, making it a more significant angle. This false magnification of the angle can lead to misjudgment.