Fig. 3.10 (Left) The standard deviation obtained from the first-order approximation and the Mon- te-Carlo simulation points respectively display in black and blue. (Right) Confidence interval size computed for many range values using a first-order approximation. The Monte-Carlo simulation points are displayed at the top of the confidence interval. The reconstruction volume covers a range from 450 to 700 mm and is shown in Fig. 3.9(Left). The confidence interval for Z = 510 mm is given. Figure courtesy of NRC Canada

(i.e. Z-axis) is an important design characteristic. Figure 3.10(Left) illustrates the confidence interval for different range values for the geometric configuration at the left of Fig. 3.9. As the range increases, the size of the confidence interval also increases. This is a characteristic of triangulation systems and this explains why triangulation systems are often used for short range measurements [10]. Note that the results presented in Fig. 3.10 ignore many things such as the effect of optical components (see Sect. 3.8).

The usable measurement volume is the portion of the reconstruction volume for which the error on the position of a 3D point is expected to be smaller than a maximum permissible error. The depth of field of a scanner is the size of the interval of Z values inside this usable measurement volume. Note that because of optical limitations, such as blurring effects, the usable measurement volume can be significantly smaller than the reconstruction volume (see Sect. 3.8).

Since a first-order approximation is used when computing the uncertainty with Eq. (3.44), it is important to validate the results using Monte-Carlo simulations. Figure 3.10(Right) shows the results of one such simulation. The simulation process consisted of the following steps:

•50000 values of x2 were generated using a uniform distribution.

•Zero-mean Gaussian noise was added to each point.

•The points were triangulated using Eq. (3.31) at [x1, y1]T = [0, 0]T .

•The variance of the range values obtained previously was compared with the one computed using the first-order approximation.

3.6.4 Uncertainty as a Design Tool

For design purposes, it is useful to fix the position of a 3D point and examine how the uncertainty varies when modifying the parameters of the system. In order to do so, it