Error Estimation

Factors influencing the accuracy of range estimation will be discussed and their impact will be estimated in this section.

2.3.2.1 Quantization errors

Image digitization may causes quantization errors, errors in range estimation are particularly caused by spatial quantization, and are within ±½ pixels [79][80]. The results of range estimation are dominated by the projective v-coordinate of P1. Therefore, the largest quantization error in mapping to the Z-coordinate can be estimated with the condition that the errors of v are within ±½ pixels. Based on (2.26), when Y=0, the range of Z should be between the largest range ZL and the smallest range ZS as shown in (2.27)( 2.28) and eq the percentage of the largest quantization error is displayed in (2.29).

Table 2-3 Error analysis of range estimation caused by change of tilt angles

Table 2-2 displays the largest quantization error in the range Z=[10, 60] m with specific α and λ. As can be seen from Table 2-1, the relation between quantization errors and the N-coordinate can be derived from the relation between Z and N-coordinate. In Table 2-2, the largest quantization error grows with an increasing Z. The larger the focal length of the camera is, the smaller the quantization errors become. The tilt angle of the camera will not influence the largest quantization error according to the analysis results shown in Table 2-2.

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2.3.2.2 Influence of changes in translation

The analyses of translation can be divided into the directions of X, Y and Z. The origin of the world coordinates is on the ground below the camera, so the Z-coordinate is the range between the preceding vehicle and the camera. Therefore, the subsection will analyze how the changes of X and Y translation influence the range estimation on the Z-coordinate.

X-translation: in (2.1), the projective position of P1 onto the v-coordinate determines the Z-coordinate. Figure 2-2(b) shows that the changes of X-translation rarely affect the mapping

position of P1 onto the v-coordinate. So X-translation seldom influences the accuracy of range estimation.

Y-translation: if the ground is flat, the Y-translation of every point on the ground is zero.

When the ground is uneven or when the height of the camera is changed because of vibrations, then the initially determined camera height h may be influenced. Let h denote the initially determined height, and h2 denote the actual height. According to (2.26), the Z-coordinate mapping result can be obtained by (2.30). If the original height h is adopted, then the error coming from changes of height will be Zdh in (2.31) and the error ratio is erh in (2.32).

Accordingly, errors caused by the Y-translation can be suppressed by increasing the camera height or making the changes of height smaller.

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2.3.2.3 Influence of changes in camera tilt angles

If vibrations cause the tilt angle of the camera to change from α to α1, the result of mapping is computed by (2.33). Therefore, if the original α is applied, the error ratio of range estimation caused by changes of tilt angles is erα in (2.34).

( )

To estimate errors caused by tilt angle changes of the camera during the range estimation, let h be 1.3 meters, and focal length λ be 8 mini-meters. The analysis of errors is shown in Table 2-3. As depicted in Table 2-3, when both α=0° and α=2° have a variation of 1°, the obtained errors are the same. So the initially set tilt angle does not influence the errors of results. However, errors increase when changes of tilt angle grow larger. The error ratio is about 40% at Z=50 meters with a change of 1° on the tilt angle, revealing that changes of angles significantly affect the results of range estimation. With the same camera parameters but the focal length being changed to 16mm, the result will remain unchanged, which demonstrates that the focal length is not related to errors arising from changes of tilt angles.

This is because when the focal length varies, the estimated ranges Z and Zα1 will still remain the same, representing that the error ratio will still keep constant.

2.3.2.4 Influence of changes in camera pan angles

Table 2-4 Variation ratio between P1 and P3 on the Z-coordinate

Figure 2-2(c) shows the condition that the Z-axis parallels the preceding direction of vehicles, denoted by Sur

. However, the condition may not be always valid. For example, in Fig. 2-4, the pan angle between Sur

and the Z-axis is θ, the variation between P1 and P3 on the Z-coordinate is Zdp as expressed in (2.35) and the variation ratio is modeled by (2.36). When the distance between P1 and P3 is 1.4m, the related value of Zdp and the variation ratio are shown in Table 2-4. In Table 2-4, the influence turns smaller with a smaller pan angle or a larger range. Even when θ=10°and the range is 30m, the variation ratio is still less than 1%, which shows that pan angles have little influence on the range estimation.

1 cos

2.3.2.5 Influence of changes in camera swing angles

The swing angle, i.e. the u-v image plane rotation angle, denotes the angle between the u-axis in the image coordinates and the X-axis in the world coordinates. As shown in Fig. 2-5, let P1 project onto i1 and let i1 be (u, v) on the u –v plane and (u’, v’) on the u’ –v’ plane. (u, v) and (u’, v’) are the coordinates when ψ≠0 and ψ=0, respectively. The transformation of the two coordinates can be computed by (2.37).

, If ψ≠0, from (1), we can obtain the results of range estimation by using (2.38).

Table 2-5 shows that the variation between the two coordinates grows with the increasing ψ, u and v. Even if ψ is very small, it still has a great influence when the coordinates are far away from the image center.

(100,200) (98.24,101.73) (96.45,103.43) (200,200) (196.48,203.46) (192.90,206.86)

Fig. 2-4 The relation between the Z-axis and the direction of movement of vehicles, denoted by Sur

.

Fig. 2-5. Relation between the Coordinates (u, v) and (u’, v’)