3.7.4 Application-Based Characterization

Fig. 3.17 The image of the point at distance Z from the optical center is in focus on the image plane, while the point at distance Z from the optical center is imaged as a circle of diameter c on the image plane. The lens aperture is Φ and the distance between the image plane and the optical center is d . Figure courtesy of NRC Canada

negligible thickness. Given the distance Z between an object and the optical center of the lens and the focal length f of this lens, one may compute, using the thin lens equation the distance between the optical center and the image plane needed in order to obtain a sharp image of the object. Since optical engineering falls outside the scope of this chapter, we provide the thin lens equation without derivation (see for details [61]). The thin lens equation is

where f is the lens focal length, Z is the distance between the optical center and the object plane and d is the distance between the optical center and the image plane (i.e. the CCD or CMOS sensor).

Since d ≈ f when the distance between the camera and the object is sufficiently large, Chap. 2 and other textbooks use f (for focal length) rather than using d in their camera models (see Sect. 3.3.1).

Usually, 3D imaging systems are used for applications that require the scan of non-planar objects. Thus, Eq. (3.45) is not fulfilled for all the points on the surface of the object. As will be explained next, this induces out-of-focus blurring in some parts of the image.

3.8.2 Depth of Field

The point located at Z in Fig. 3.17, will be imaged as a circle of diameter c on the image plane. This circle is named a circle of confusion. Using simple trigonometry and the thin lens equation, the diameter of this circle can be computed as

where Φ is the size of the lens aperture. The proof is left as an exercise to the reader. Given a maximum diameter cmax for the circle of confusion and assuming

Fig. 3.18 (Left) The circle of confusion acts as a box filter. The diameter of the first circle of confusion is two units, while the diameter of the second one is one unit. (Right) Effect of blurring induced by the circles of confusion (shown left) on the magnitude of a sinusoidal pattern having a period of three units. Figure courtesy of NRC Canada

that Z > Z > f , the depth of field can be computed, using Eq. (3.46), as

Thus, a large lens diameter will induce a small focusing range, even though more light is captured by the imaging system.

Figure 3.18 illustrates the impact of blurring on the magnitude of sinusoidal patterns used by a phase shift scanner. As the ratio between the circle-of-confusion diameter and spatial period of the pattern increases, the magnitude of the signal is reduced. In Eq. (3.40), it can be seen that reducing the spatial period reduces the uncertainty. However, once the optical components of the system are taken into account, one can see that reducing the spatial period, may also reduce the magnitude of the sinusoidal pattern, possibly increasing the uncertainty rather than reducing it. Furthermore, because the blurring depends on the distance of a 3D point, the magnitude of the sinusoidal pattern also depends on the distance. Thus, one should expect that the curve of standard deviation shown at Fig. 3.10, should look more like a U shape when taking into account the optically-induced blurring. It is, therefore, important to factor in the optically induced blurring and other optical related degradations when designing a system because those define the usable measurement volume, which is generally smaller than the reconstruction volume. Note than even when a system is perfectly in-focus, diffraction and aberrations induce a degradation of the image which is similar to out-of-focus blurring [61].