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Table 3.2 Approximate depth of field as a function of a few beam radii. The laser wavelength is 0.633 μm. Table courtesy of NRC Canada
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Beam radius (w0) |
Approximate depth of field (Df ) |
10 μm |
1 mm |
100 μm |
100 mm |
1 mm |
10 m |
volume of the scanner. Moreover, the usable measurement volume of a sheet-of-light scanner could be computed similarly.
3.8.6 Lateral Resolution
Intuitively, the lateral resolution is the capability of a scanner to discriminate two adjacent structures on the surface of a sample. A formal definition can be found in [3]. For some applications such as the one presented in Sect. 3.7.4, it is critical to use a 3D scanner with sufficient lateral resolution. The lateral resolution is limited by two factors which are the structural and the spatial resolution [3].
For a phase-shift system, when working out-of-focus, the lateral resolution of a system is not limited by the camera resolution (spatial resolution), but by the optical resolution (structural resolution) of the camera lens. Thus, to increase the lateral resolution, one may have to reduce the depth of field of the scanner or the lens aperture size. When a digital projector is used, artifacts induced by inter-pixel gaps and discretization may limit the lateral resolution of the system. Note that it is possible to alleviate those artifacts by using the hybrid hardware-software solution presented in [33].
For a laser spot scanner, the knowledge of the beam radius on the scene allows one to determine the structural component of the lateral resolution of the system. The spatial resolution is the smallest possible variation of the scan angle α. Increasing the angular resolution of α can improve the lateral resolution as long as the spatial resolution does not exceed the structural one. Thus, reducing the beam radius may be the only way to increase the lateral resolution. When the beam radius is reduced, the depth of field is also reduced unless an auto-focusing method is used while measuring. Thus, there is a trade off between lateral resolution and the depth of field. Table 3.2 gives some numerical examples of beam radii.
3.9 Research Challenges
In Sect. 3.6 we presented the error propagation from the image formation to the 3D points for some area scanners. To the best of our knowledge, no commercial scanner associates to each 3D point a covariance matrix that can be used for performing a first-order error propagation. An important research issue is the understanding and
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modeling of error propagation from the calibration step to the visualization step of the modeling pipeline. This is challenging because the modeling pipeline can contain a significant amount of geometric processing such as the fusion of multiple scans, the transformation of point clouds into meshes, the decimation of triangles, and the fitting of geometric primitives.
As the individual components of 3D imaging systems continue to improve, it is expected that the spatial resolution of 3D imaging systems will increase up to the limits imposed by physics. As an example, in recent years the resolution of cameras has significantly increased. This had a significant impact on the performance of fringe projection systems; however, there are physical limitations that make further improvement of a 3D scanner impossible. As an example, a laser point scanner can be designed to reduce the effect of speckle, but speckle cannot be removed as it is a physical limit of any system that uses coherent light. Another example of physical limitations is the diffraction introduced by the finite size of a lens aperture. Thus, one of the main challenges in the development of 3D imaging systems is to combine the improvements in commercially available components with innovative new designs and algorithms in order to bring the performance of the system as close as possible to the physical limits. Another interesting area of research is the design of systems for niche applications which are required to work in harsh environments or that must scan very challenging objects, such as translucent objects, objects with grooves or other surface concavities, and underwater objects.
3.10 Concluding Remarks
Many of the traditional measurement instruments like theodolites and CMMs are being replaced by non-contact optical scanners based on triangulation, time-of-flight, or interferometry technology. This sudden change in process design and quality assurance practices needs to be addressed by research organizations and companies. When the goal of a business is to make a quality product for a profit, then metrology will have a direct impact on that business. The quality of measurements planned in the design stage, applied during manufacturing and performed during inspection directly affect the quality of a product. Poor measurements (those without an accuracy statement) may even lead to creating waste with scrapped products. Conversely, precise measurements (those with an accuracy statement) lead to superior products. The dimensional deviations between as-designed, as-built and as-measured devices can only be understood and controlled if traceable measurements can be made in compliance with clear standards. While 3D imaging systems are more widely available, standards, best practices and comparative data are limited. In the near future, we expect to see more comparative data in scientific publications and industrial standards aimed at active 3D imaging systems.
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3.11 Further Reading
One of the earliest papers on triangulation-based spot scanning for the capture and recording of 3D data was published by Forsen in 1968 [36]. Kanade presents a collection of chapters from different authors that describe a number of close-range active 3D imaging systems [44]. Many survey papers that review range sensors have been published [7, 16]. The geometric description of the point and profile based system presented a simple scanner. Mirrors can be used to fold the baseline such that the baseline of a system is larger than the physical scanner and where the mirrors dynamically modify the field of view of the camera such that the sensor only sees a small area around the laser spot [54]. The calibration of point-based triangulation scanner is discussed in [12, 13, 25].
An article published by Salvi et al. [57] presents an in-depth classification of different types of structured light patterns. Davis et al. [29] present a unifying framework within which one can categorize 3D triangulation sensors, for example on the basis of their coding within the spatial and temporal domains. Moreover, an analysis of the uncertainty of a white light fringe projection based on Gray codes is presented in [63]. Many analyses of the impact of random noise on phase shift methods have been conducted [31, 40, 53, 62].
The two authoritative texts on the matter of uncertainty and vocabulary related to metrology are the Guide to the Expression of Uncertainty in Measurement (GUM) and the International Vocabulary of Metrology (VIM) [1, 5]. The document designated E 2544 from the American Society for Testing and Materials (ASTM) provides the definition and description of terms for 3D imaging systems [6]. Moreover, the VDI 2634 is a document from a standardization body that addresses the characterization of optical distance sensors [2]. Error propagation in the context of multiple-view geometry is discussed in [28]. The characterization of active 3D imaging systems is discussed in [19, 24, 26, 27, 35, 38, 45–47].
3.12 Questions
1.Name and explain three categories of triangulation scanner, based on different methods of scene illumination.
2.In recent years, the resolution of cameras has significantly increased. What are the impacts of this on each type of triangulation scanner?
3.What are the impacts on the 3D data of varying the baseline of a laser stripe scanner without recalibrating the system?
4.What are the impacts on the 3D data of varying the distance, d (the distance between the camera center and image plane) of a laser stripe scanner without recalibrating the system?
5.What are the values of Rp , Tp , x1, y1 and x2 for which the three constraints in Eq. (3.30) are not linearly independent?
6.What are the elements that can limit the lateral resolution of a stripe scanner? Classify those elements as belonging to the spatial or structural resolution.
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3.13 Exercises
1.Using a programming environment of your choice, develop a 2D simulator of a phase shift fringe projection scanner that can reproduce the intensity artifact shown in Fig. 3.15. Assume that optical-induced blurring is only present in the camera images and that the camera has an infinite spatial resolution. Repeat the experiment for a fringe projection system that uses a Gray code.
2.Modify the previously developed prototype in order to apply it to a stripe scanner. Plot a graph that shows the variation of error due to the width of the stripe.
3.For a stripe-based scanner, an occlusion occurs when the linear detector does not see the laser spot. However, since the spot size is not infinitesimal, there are intermediate situations where only a fraction of the spot is seen by the detector. Using a prototyping environment, develop a model to evaluate the impact of this on the recovered geometry.
4.Perform the error propagation computation for a stripe scanner that includes uncertainty on the angles α.
5.Using Fig. 3.17, trigonometry and the thin lens equation, give the derivation of Eq. (3.46).
6.Modify the camera model presented in Sect. 3.3.1 to incorporate a Scheimpflug condition.
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