Keywords

1 Introduction

Advanced driver-assistance systems have been developed by automobile manufacturers and suppliers considering an automated driving scenario in the near future [1, 2]. Accordingly, driving simulators (DSs) are being used to develop such systems [3]. DSs provide a sense of driving to users/drivers while being in a limited space and offer the following advantages [4]:

  1. (1)

    Ease of ensuring driving safety,

  2. (2)

    Ease of establishing a traffic environment and events with high repeatability,

  3. (3)

    Several types of information can be measured more easily than a real automobile, and

  4. (4)

    Independent of seasons, weather, and time of day.

Therefore, the demand for DSs is increasing [5]. The images used in DSs are live-action or computer graphics (CG) [6]. A live-action image is actually a video recorded on a real automobile, and a CG image is actually a virtual traffic scene [7]. Although the realistic sensation provided by CG images is lower than that provided by live-action ones, the former offers an advantage in that the traffic environment can be created without limitations. Therefore, CG images predominate in DSs. However, the subjective driving speed in DSs is lower than that in real automobiles [8,9,10]. Thus, the experimental results obtained using DSs become unreliable because some participants drive at fairly high speeds while using them [11]. Therefore, drivers must be provided with an environment as similar as possible to a real-automobile one to increase both the fidelity of their behavior and effectiveness of the experiments [12]. Therefore, increasing the subjective speed in DSs is a challenge. There exist two major approaches to improve the subjective speed: introducing a motion system and expanding the field-of-view (FOV) of the image. The motion system provides inertial acceleration to the drivers according to the real-automobile behavior [3]. A wide FOV is implemented using wide screens or displays. However, both the approaches are not preferred in the case of small research-and-development projects because they require high costs and large spaces [13, 14]. Drivers on a real automobile rely on their subjective speed based on visual cues from the optic flow of their surroundings, vestibular and somatosensory cues of acceleration and vibration, haptic cues of reaction force by a steering wheel and pedals, and auditory cues of engine sound and wind noise [15]. Particularly, the visual information gained by drivers comprises 90% of their driving information [11, 16]. In a previous study, the subjective speed was improved by correcting CG images with respect to the characteristics of a person’s visual space [17]. In this study, empirical examinations were conducted to control the subjective speed to be the same as that on a real automobile by using an image-correction method.

2 Image Correction to Improve Subjective Speed

In a previous study [17], a correction coefficient f(z), defined using Eq. (1), was used to distort the CG image with respect to the human visual space to improve the subjective speed. One has

$$ f\left( z \right) = \beta \times 10^{ - 5} \times \left| z \right| + \left( {100 - \beta } \right) \times 10^{ - 2} $$
(1)

where z denotes the depth coordinate of a CG object from a DS driver. The constant β denotes the degree of distortion and is determined how the position of a CG object becomes inward from its original location in the same depth as the driver is. The values of β were set at 0%, 40%, 60%, or 80% in following experiments.

The CG image was distorted using Eq. (2), where the coordinates of the position of the object before and after the correction are (x, y, z) and (X, Y, Z), respectively. One has

$$ \left( {\begin{array}{*{20}c} X \\ Y \\ Z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {f(z)} & 0 & 0 \\ 0 & {f (z)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) $$
(2)

3 Experiment 1: Collection of Subjective Speed

The relation among the degree of distortion β, subjective speed Vp, and running speed Vg should be clarified. The objective of this experiment is to collect subjective speeds under certain conditions on both the degree of distortion and running speed.

3.1 Apparatus and Participants

Figure 1 shows the outline of the experimental environment. A computer (CERVO, Applied Corp.) equipped with a graphic board (Quadro K620, video memory: DDR3, 2 GB) was used to draw the CG images, which were projected using a liquid-crystal projector (EMP-1825, EPSON. Refresh rate: 60 Hz) on a screen (SRMS3D-100, Solidray). Each participant sat on a chair in front of the screen at a viewing distance of 1.5 m, and he/she put his/her head on a chin rest to match the line joining his/her eye points with the horizontal and vertical center of the image. The horizontal and vertical FOVs were 49.1° and 36.9°, respectively. The experimental apparatus was surrounded with partitions, and the light was turned off. The luminance at the eye point was 21.9 lx in average.

Fig. 1.
figure 1

Outline of the experimental environment.

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Figures 2 and 3 show the examples of CG images, which were created using OpenGL and depict a situation of running along a lone straight asphaltic road on a lawn. A dashed centerline was putted on the road with a length of 5 m and an interval of 5 m, and continuous side lines were also putted on right and left sides of the road. Moreover, mountains were placed at a significant distance and covered with a sky to augment the reality. A red circle is displayed on the middle of the image as a fixation point where participants were asked to look. The image had a resolution of 1024 × 768 pixels, and it was vertically synchronized with the projector.

Fig. 2.
figure 2

Example of a CG image with the degree of distortion β = 0%. (Color figure online)

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Fig. 3.
figure 3

Example of a CG image with the degree of distortion β = 60%. (Color figure online)

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Fifteen university students (20–25 years of age) participated after providing informed consent. Each of them had a valid driving license and normal or corrected-to-normal binocular vision.

3.2 Subjective Speed

The absolute method of magnitude estimation was adopted to measure the subjective speed. It is one of the methods used to construct a psychological scale of a physical stimulus [18]. This means that the participants verbally answered the subjective speed in the unit of km/h with respect to the image they viewed. Moreover, they were instructed that they could use any numbers that were integers, decimals, and fraction values.

3.3 Conditions

Table 1 lists the experimental conditions. The subjective speeds were not different between the images with 20% and 40% distortions, according to a previously conducted study [17]. Therefore, in this experiment, 80% distortion was adopted instead of 20%. The four distortion levels were 0%, 40%, 60%, and 80%. The values used for running speed Vg were 20, 40, 60, 80, and 100 km/h. Moreover, the speeds of 1 and 120 km/h were added as dummy conditions to eliminate the regression bias [19], which is a tendency to answer neither high nor low values. A block of 28 conditions (four image-distortions × seven running-speeds) was repeated five times.

Table 1. Conditions for Experiment 1
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3.4 Procedure

In one trial, a gray image was displayed for 5 s, and then stimulus images were presented for 5 s. This trial was repeated 28 times as one block of experiment. The participants were asked to verbally answer the subjective speed while the gray image was displayed, and they were instructed not to compare the stimulus image with the previous one. The order of the 28 conditions was completely randomized in each block for each participant.

At the beginning, the participants were instructed regarding the procedure of the experiment, following which they provided informed consent. Subsequently, they were tested for visual acuity. Next, they performed one block of training session. Furthermore, a test session, which comprised five blocks, was performed. The participants rested for 2 min between the blocks. After the test session, they answered questionnaires regarding the trials. The accuracy of the subjective speed that they answered was not fed back in either the training or test session.

3.5 Results

Figure 4 shows the relationship between subjective speed Vp and running speed Vg. Each plot represents the geometric mean of the subjective speeds among the participants, and the error bar denotes the standard error of the geometric mean. Evidently, the subjective speed becomes high with respect to the degree of distortion. This trend is the same as that in a previously conducted study [17].

Fig. 4.
figure 4

Relationship between subjective and running speeds in Experiment 1.

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The two-way analysis of variance was conducted to confirm the effect of the degree of distortion and running speed on the subjective speed. Consequently, the main effects of the running speed (F(4, 280) = 80.29; p < 0.05) and distortion (F(3, 280) = 98.16; p < 0.05) were significant. However, the interaction effect between the running speed and distortion was not significant (F(12, 280) = 1.26; p = 0.244). Certainly, the distortion of CG images affects the subjective speed.

4 Derivation of Appropriate Distortion to Control Subjective Speed

An equation that represents the relationships between the subjective speed, running speed, and degree of distortion is derived from the results of Experiment 1.

A power law, popularly called Stevens’ power law, represents the relationship between physical and psychological quantities. The running speed is denoted as Vg, degree of distortion as β, and subjective speed as Vp. Accordingly, the following relationships hold:

$$ {\text{V}}_{\text{p}} \propto {\text{V}}_{\text{g}}^{\text{A}} $$
(3)
$$ {\text{V}}_{\text{p}} \propto\upalpha^{\text{B}} $$
(4)

where A and B denote constants. The constant β is replaced by dis-distortion α defined as follows to avoid the value from being zero:

$$ \alpha = \left( {100 - \beta } \right) \times 10^{ - 2} $$
(5)

Equations (3) and (4) can be transformed as follows by taking the common logarithm on both the sides of them:

$$ { \log }_{ 1 0} {\text{V}}_{\text{p}} \propto {\text{Alog}}_{ 1 0} {\text{V}}_{\text{g}} $$
(6)
$$ { \log }_{ 1 0} {\text{V}}_{\text{p}} \propto {\text{Blog}}_{ 1 0}\upalpha $$
(7)

Combining Eqs. (6) and (7), log10Vp can be expressed as follows:

$$ { \log }_{ 1 0} {\text{V}}_{\text{p}} = {\text{Alog}}_{ 1 0} {\text{V}}_{\text{g}} + {\text{Blog}}_{ 1 0}\upalpha + {\text{C}} $$
(8)

where C denotes a constant.

The experimental data collected in Experiment 1 were approximated using Eq. (8). Thus, constant values A = 0.754, B = −0.617, and C = 0.0438 were obtained (R2 = 0.988). Figure 5 shows this three-dimensional approximation. Solving Eq. (8) for β using Eq. (5), an appropriate distortion with respect to running speed Vg and subjective speed Vp can be obtained as follows:

Fig. 5.
figure 5

Three-dimensional approximation of subjective speed using dis-distortion and running speed.

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$$ \beta = \left( {1 - 10^{{ - \frac{C}{B}}} \cdot V_{p}^{{\frac{1}{B}}} \cdot V_{g}^{{ - \frac{A}{B}}} } \right) \times 10^{2} $$
(9)

A desired subjective-speed value is assigned to Vp in Eq. (9). Thus, distorted images would be generated, as a driver would feel as though he/she is running at a speed equal to Vp. In the next experiment, it is validated that the subjective speed can be controlled using Eq. (9).

5 Experiment 2: Controlling the Subjective Speed

The objective of this experiment is to verify whether the subjective speed can be controlled, as mentioned in the previous section. The apparatus, measurement method of subjective-speed, and participants were the same as those in Experiment 1.

5.1 Conditions

Table 2 summarizes the experimental conditions. The running speed had five conditions, same as those in Experiment 1, namely, 20, 40, 60, 80, and 100 km/h. In this experiment, the objective was to reproduce the subjective speed as follows [21]:

Table 2. Experimental conditions for Experiment 2.
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$$ V_{p} = 0.324\;V_{g}^{1.15} $$
(10)

where Vp denotes the subjective speed and Vg the driving speed. The degree of distortion β was expressed by equating Vp in Eq. (10) to Vp in Eq. (9).

$$ \beta = \left( {1 - 7.31\;V_{g}^{ - 0.634} } \right) \times 10^{2} $$
(11)

The CG-image distortion was changed for each condition of driving speed Vg according to Eq. (11). Moreover, two dummy trials were added to the conditions, and they had distortion β = 0% at the speed of 1 km/h and β = 80% at the speed of 120 km/h.

5.2 Procedure

The experiment was conducted similarly to Experiment 1. However, one block comprised seven trials.

5.3 Results

Figure 6 shows the relationship between subjective speed Vp and running speed Vg. Each plot represents the geometric mean of subjective speeds for the participants; the error bar denotes the standard error of the geometric mean; the red solid line represents the target speed to be perceived, as shown in Eq. (10). If the plot lies on the red line, it is indicated that the subjective speed can be controlled according to Eq. (10).

Fig. 6.
figure 6

Relationship between subjective and running speed in Experiment 2. The red solid line denotes target speed to control the subjective speed. (Color figure online)

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The two-tailed t-test was conducted between the measured subjective speed and target speed Vp in Eq. (10) for each running speed. Consequently, the difference was not significant when the level of significance was 5% (Vg = 20 km/h: t(14) = 0.83, p = 0.420; Vg = 40 km/h: t(14) = 0.94, p = 0.364; Vg = 60 km/h: t(14) = 0.51, p = 0.621; Vg = 80 km/h: t(14) = 0.44, p = 0.664; Vg = 100 km/h: t(14) = 0.93, p = 0.366).

Moreover, the confidence interval (CI) was calculated at the level of significance of 95%. The results were as follows: Vg = 20 km/h: M = 9.04, CI = [3.47, 1.53];

Vg = 40 km/h: M = 24.33, CI = [2.48, 6.33]; Vg = 60 km/h: M = 36.70, CI = [4.81, 7.77]; Vg = 80 km/h: M = 48.96, CI = [5.56, 8.46]; Vg = 100 km/h: M = 66.84, CI = [4.70, 11.96]. The mean subjective speed for all running speeds was within a 95% CI.

5.4 Discussion

The difference between the subjective and target speeds was not significant. Moreover, the subjective speed while driving was within the 95% CI of the experimental results for all the running-speed conditions. Therefore, the subjective speed could be controlled through the adaptive distortion of the CG image. However, there are four considerations.

The first consideration is the personalization of the distortion. Although only the mean subjective speed among the participants was evaluated in this experiment, the individual subjective speed significantly varied among the participants. Therefore, a mechanism to consider individual variation must be proposed.

The second consideration is road alignment. Although only the scenes that depicted running along a straight road were examined, several types of road alignments exist in reality. Therefore, the validation of controlling the subjective speed must be conducted for a situation of running on a curved road.

The third consideration is the static change in image distortion. Although the distortion technique described in Sect. 4 can be applied when the running speed dynamically changes, the image actively changes according to the dynamic changes in distortion. Therefore, the subjective feeling of experiencing dynamic changes in image distortion must be evaluated.

The fourth consideration is the surrounding environment in a virtual scene. In these experiments, the scenes only simulated a rural-like simple field. Therefore, evaluations in various other surrounding environments must be performed.

6 Conclusions

Two experiments were conducted with the aim of controlling the subjective speed of DSs. The first experiment was conducted to collect the subjective speed with four conditions of distortion of CG image at five running-speed conditions. The results showed that the subjective speed became higher with larger image distortion at all the running-speed conditions. The second experiment was performed to control the subjective speed in a DS to be the same as that on a real vehicle. Consequently, it was indicated that the subjective speed could be controlled through CG-image distortion. This image-distortion technique can contribute to the development of DS images.