A sheet-of-light system such as the one illustrated at Fig. 3.3(right) can be calibrated similarly by replacing the tables ti (x1, y1) = x2 and ti (x1, y1) = Z by ti (α, y2) = x2 and ti (α, y2) = Z where α is the angle controlling the orientation of the laser plane, y2 is a row of the camera and x2 is the measured laser peak position for the camera row y2. Systems that use a Gray code with sub-pixel localization of the fringe transitions could be calibrated similarly. Note that tables ti and ti can be large and the values inside those tables may vary smoothly. It is, therefore, possible to fit a non-uniform rational B-spline (NURBS) surface or polynomial surface over those tables in order to reduce the memory requirement. Moreover, different steps are described in [25] that make it possible to reduce the sensitivity to noise of a non-parametric calibration procedure.

3.6 Measurement Uncertainty

In this section, we examine the uncertainty associated with 3D points measured by an active triangulation scanner. This section contains advanced material and may be omitted on first reading. Some errors are systematic in nature while others are random. Systematic errors may be implementation dependent and an experimental protocol is proposed to detect them in Sect. 3.7. In the remainder of this section, random errors are discussed. This study is performed for area-based scanners that use phase shift. An experimental approach for modeling random errors for the Gray code method will be presented in Sect. 3.7. Moreover, because the description requires advanced knowledge of the image formation process, the discussion of random errors for laser-based scanners is postponed until Sect. 3.8.

In the remainder of this section, we examine how the noise in the images of the camera influences the position of 3D points. First, the error propagation from image intensity to pixel coordinate is presented for the phase shift approach described in Sect. 3.4.2. Then, this error on the pixel coordinate is propagated through the intrinsic and extrinsic parameters. Finally, the error-propagation chain is used as a design tool.

3.6.1 Uncertainty Related to the Phase Shift Algorithm

In order to perform the error propagation from the noisy images to the phase value associated with a pixel [x1, y1]T , we only consider the B1(x1, y1) and B2(x1, y1) elements of vector X(x1, y1) in Eq. (3.23). Thus, Eq. (3.23) becomes

where M is the last two rows of the matrix (MT M)−1MT used in Eq. (3.23). First, assuming that the noise is spatially independent, the joint probability density function p(B1(x1, y1), B2(x1, y1)) must be computed. Finally, the probability density

from Cartesian to polar coordinates and integrating over the magnitude. Assuming that the noise contaminating the intensity measurement in the images is a zero-mean Gaussian noise, p(B1(x1, y1), B2(x1, y1)) is a zero-mean multivariate Gaussian distribution [27, 28]. Using Eq. (3.33), the covariance matrix ΣB associated with this distribution can be computed as

where ΣI is the covariance matrix of the zero-mean Gaussian noise contaminating the intensity measured in the camera images [27, 28].

We give the details for the case θi = 2π i/N when the noise on each intensity measurement is independent with a zero mean and variance σ 2. One may verify that

This is the special case of the work presented in [53] (see also [52]).

Henceforth, the following notation will be used: quantities obtained from measurement will use a hat symbol to differentiate them from the unknown real quanti-

When σ is small and B is large, p( φ) can be approximated by the probability