Thus, for a given α, the angle ρ can be computed such that any point along the laser beam path is in-focus on the position detector. This condition allows one to design a system with a large aperture to increase light collection power. As will be explained next, for a laser-based scanner, this allows the reduction of noise without affecting the focusing range.

3.8.4 Speckle and Uncertainty

In Sect. 3.6.1, the error propagation from image intensity to pixel coordinate was examined for area-based scanners that use phase shift and an expression for the variance σx22 was provided. As shown, the variance σx22 can then be used to compute the uncertainty on the 3D points computed by a triangulation scanner. Here, we give the result of a similar analysis performed for point-based systems that use a laser.

For laser-based system, the value of σx22 depends on the type of laser spot detector used (e.g. CMOS, CCD, lateral-effect photodiode, split diodes), the laser peak detector algorithm, the signal-to-noise ratio (SNR) and the imaged laser spot shape [11]. The laser spot shape is influenced by lens aberrations, vignetting, surface artifacts, etc. In the case of discrete response laser spot sensors, assuming both a high SNR and a centroid-based method for peak detection, the dominant error source will be speckle.

Speckle is the result of the interference of many light waves having the same wavelength but having different phases. Different waves emitted by the projection system are reflected on the object at slightly different positions and thus reach the detector with slightly different phases. The light waves are added together at the detector which measures an intensity that varies. The speckle depends on the surface micro-structure or roughness (of the order of the source wavelength) of the object which is scanned. Note that, speckle noise is more a multiplicative noise source than an additive source. Explicitly, the variance σx22 can be approximated as

where Φ is the lens aperture diameter, λ is the laser wavelength and d is the distance between the laser spot detector and the collection lens [16]. The effects of speckle on the peak detection have also been studied by [8, 30, 45]. Note that when substituting Eq. (3.53) back into Eq. (3.41), one can verify that the presence of speckle noise caused by a laser does not depend on d . When λ is reduced or when Φ is increased, the uncertainty is reduced. While a large lens diameter reduces the uncertainty, it also limits the focusing range when a Scheimpflug condition is not used.

Note that speckle can be an important error source even for a fringe projection system that uses a low-coherence light source with a relatively long coherence length. The coherence length is proportional to the square of the nominal wavelength of the source and inversely proportional to the wavelength range [45].

Fig. 3.20 The focus of an actual laser beam is governed by diffraction. We show the case of a focused laser beam with a Gaussian shape transversal profile. The minimum beam diameter is 2w0 and R0 is the distance from√the lens to the point at which the beam diameter is minimal. A maximum beam diameter of 2 2w0 is used to compute the depth of field, Df , of the laser. Figure courtesy of NRC Canada

3.8.5 Laser Depth of Field

Until now, the laser beam and collected ray were assumed to be infinitely thin. Though convenient to explain the basic principles, this is an over-simplification. Taking into account the properties of Gaussian beams associated with lasers is fundamental to understanding the limitations of some 3D laser-based vision systems [17, 20, 55]. As a result of diffraction, even in the best laser emitting conditions, a laser beam does not maintain focus with distance (see Fig. 3.20). Note that in many close-range 3D laser scanners, a focused laser beam is the preferred operating mode. This is a complex topic whose details fall outside the scope of this chapter. Nevertheless, because it is fundamental to understanding the resolution limitations of some 3D imaging systems, we give two important results concerning Gaussian beam propagation. Using the Gaussian beam propagation formula, the beam radius measured orthogonally to the beam axis, denoted by w(R) at the e−2 irradiance contour in the direction of propagation R is

where the distance R0 is the distance from the lens to the point at which the beam radius is minimal. The minimum radius is denoted by w0 and λ is the wavelength of the laser source. More details information concerning Eq. (3.54) can be found

For a point-based scanner, the depth of field Df , the distance R0 and the angular interval for the scanning angle α can be used to compute the usable measurement