New Range Sensors at the Physical Limit of Measuring Uncertainty

Transcription

1 New Range Sensors at the Physical Limit of Measuring Uncertainty G. Häusler, S. Kreipl, R. Lampalzer, A. Schielzeth, B. Spellenberg Chair for Optics, University of Erlangen, Staudtstraße 7/B2, D Erlangen tel: , fax: ABSTRACT Coherent noise is the major source of the measuring uncertainty of triangulation at rough surfaces. This is well known for laser based systems. We will discuss coherent noise in phase measuring triangulation systems that work with white, large incandescent lamps. Surprisingly, the remaining small amount of coherence still limits the distance uncertainty of all triangulation sensors. We will demonstrate two new sensors that utilize this knowledge, and display extremely small distance uncertainty. One sensor is completely incoherent and can be applied to on-line measurement during laser material processing, the other sensor is a real time 3D-video camera. 1. Introduction Active triangulation is essentially based on the projection of some pattern onto the object, via a certain direction, and on the localisation of this pattern by observation from a different direction. The angle between these direction is called the triangulation angle θ. The uncertainty of triangulation methods is essentially limited by the uncertainty of the pattern localization. An illustrative example is the ubquitous laser triangulation with a small spot that is projected onto the optically rough surface (we assume here technical surfaces such as ground, milled, turned... surfaces). In the spot image on the camera chip we will observe a speckle pattern [3]. As a consequence, the locus of the spot image be determined with only a statistical uncertainty δx [2] of λ δx = C 2π sin u obs where sin uobs is the aperture of the observation, and λ is the wavelength. C is the speckle contrast. Taking into account the triangulation angle θ, we get the uncertainty of distance measurement δz λ δz = C 2π sin sinθ u obs (1) Figure 1 illustrates the influence of the coherence on the measuring uncertainty: In Figure 1 we see the (laser-triangulation-) profile across a milled surface that displays considerable noise originating from the coherent illumination. in Figure 2 we covered the surface by a thin fluorescent layer, and utilized the fluorescent light for measurement, instead of the laser light. Since fluorescence is completely (spatially) incoherent, there is no speckle noise at all and the measurement uncertainty is reduced dramatically. We learn from this experiment that at least for triangulation laser illumination is not the best choice. In the following sections we will use this knowlegde. (2)

2 Fig. 1: The influence of coherent noise in laser triangulation illustrated by the profile across a milled surface which is measured by laser light. Fig. 2: The object surface is covered by a thin fluorescent layer and the measurement is done by the fluorescent light which shows no spatial coherence. The measuring uncertainty is reduced dramatically. 2. A noise free sensor for laser material processing It is not always possible to prepare the object surface by fluorescent layers. In laser material processing however we can make use of the thermal radiation of the heated material. This radiation is again completely spatial incoherent. The sensor is sketched in Figure 3 [4]. It is a so-called focus sensor, based on nothing but triangulation. The plasma spot on the object surface is imaged through a double slit aperture. The unsharp projection of the aperture displays two peaks. The distance of these peaks is connected to the depth location of the plasma spot. Now the important trick is: Since there is no coherent noise on the image signal we can apply a hundredfold subpixel interpolation that allows a measuring uncertainty δz less than 10 µm, despite a lot of turbulence and spot instability, as displayed in Figure 4. Fig. 3: Principle of the focus sensor working with an unsharp projection of a double slit aperture. The distance information about the plasma spot is given by the mutual distance of the two image peaks.

3 Fig. 4: Because there is no coherent noise at all, a hundredfold subpixel interpolation of the camera signal results in a measuring uncertainty of less than 10 µm. 3. Is there coherent noise in phase measuring triangulation? Competing to laser triangulation, phase measuring triangulation (pmt) is well established. Here we do not project a single laser spot but a fringe pattern. The illumination is usually done by some incandescent (spatially incoherent) source. In the context of our noise considerations the question arises if we still have to take coherent noise into account. To keep it short: Yes, we have to, and the contribution is significant. This might be surprising for two reasons: Even though the source is incoherent, coherence theory tells us that we have some spatial coherence on the object that can easily be calculated by the van Cittert-Zernike theorem [1]. As a consequence we will see low contrast speckles in the image plane of our sensor. As the speckle contrast is usually low, one might not be aware of the speckle noise, but it can be measured. A beautiful experiment to convince people of the ubiquitous speckle is to look at your fingernail in sunlight. Since the sun is a small source (30 arcminutes in diameter), we can see wonderful speckles, if the observation aperture sin uobs is larger than the illumination aperture sin uill. Figure 5 displays the measured speckle contrast versus the incoherence ratio i = sin uill / sin uobs for quasi-monochromatic light. For i > 1, i.e. for uill > uobs, the speckle contrast C is u C 1 = sin i sinu obs ill This is an important result: Practically we cannot reduce the speckle contrast to extremely small values, unless we use camera pixels much larger than the (diffraction-based) resolution of the observation system. (3) incoherent illumination. The spatial incoherence is given by the ratio i = sin u ill / sin u obs. Fig. 5: Measured speckle contrast C for partially spatial

4 Now, as we know about the presence of speckle noise we might hope that this is approximately multiplicative for grating frequencies not too high and pmt is unsensitive against multiplicative noise. So why worry? But nature never gives us any presents. Figure 6 displays the results of a depressing experiment: We project a sinusoidal grating by spatially coherent illumination onto a rough planar surface. The phase ϕ across the surface should display a perfect saw tooth (0 ϕ < 2π) if there is only multiplicative noise. But figure 6 illustrates that there is much coherent noise in the phase. Fig. 6: After projecting coherent sinusoidal fringes onto a planar object the phase ϕ of the fringes is measured along one line across the surface. Instead of displaying a perfect saw tooth graph, the phase is noisy and will cause significant uncertainty of distance measurement. So, is speckle noise not multiplicative at all? To find out, we can make a different experiment: We measure the phase at one certain object location (one certain camera pixel) while we shift the phase of the illuminating grating. It turns out that the phase can be measured with at least ten times less noise than in the experiment of figure 6 and the speckle noise can be considered approximately multiplicative. The contradiction between these two experiments has a surprising solution: We measure the correct phase, but we measure it at the wrong location. The objective speckle pattern in the observation pupil introduces wedge shaped phase errors which cause a random lateral shift of image points with a standard deviation of about the speckle size. So we do not exactly know where we look at by each camera pixel. The consequence of this observation: Speckle noise limits the distance uncertainty of pmt as well as of laser triangulation. The limit δz of equation (2) that was derived for laser triangulation has to be modified just a little for pmt: We have to put the speckle contrast C of equation (3) into equation (2) and we find: λ δz = 2π sin sinθ u ill (In our experiments which include some experimental unreliability, the pmt noise limit is even twice as large.) As a consequence of those considerations we should design pmt sensors with high illumination aperture to reduce the physical noise limit. 4. The real time 3D- video camera (4) Finally, we introduce a pmt sensor with a new technology, that displays 3D data presently after 320 ms, and which has the potential to be at least hundred times faster [5]. The sensor is light weighed (400 g) and small enough to be applied for intraoral measurements of teeth, or to be mounted on a small robot (figure 9). The fringe generation is based on astigmatic projection of a binary mask as shown in figure 7. The mask

5 is a ferroelectric pattern of electrodes as sketched in figure 8. Since there are only 2 4 electrodes (two spatial frequencies, each with four electrodes for four phase shifts) the system is simple and not expensive. Fig. 7: Principle of astigmatic fringe projection electrode A = electrode B = electrode C = electrode D = Seperately adressable f = 0 f = 90 f = 180 f = 270 Fig. 8: Pattern of electrodes. The combination of different electrodes displays the desired fringe pattern including the phase shifts.

6 Fig. 9: Sensor mounted on a small robot. Fig. 10: Measured electronic circuit. In figures 9 and 10 we display one embodiment of the sensor and one measuring result. We achieved a measuring uncertainty of down to 6 µm within a volume of 15 mm 15 mm 15 mm, and this not far from what is physically possible. 5. Conclusions Coherent noise is limiting laser triangulation sensors as well as phase measuring triangulation. To avoid coherence we can work with fluorescent light or with thermal excitation (this is possible in laser material processing). To reduce coherence a good pmt sensor must have a large illumination aperture. With this knowledge and FLC-technology we have built a real time 3D-video camera working almost at the physical limit of measuring uncertainty. References

7 [1] M. Born, E. Wolf: Principles of Optics, 4th ed. (Pergamon Press, London, New York, 1970) [2] R. Dorsch, G. Häusler, J. M. Herrmann: Laser Triangulation: Fundamental Uncertainty in Distance Measurement, Appl. Opt. 33, 1306 (1994) [3] J. W. Goodman: In Laser Speckle and Related Phenomena, ed. by J. C. Dainty (Springer Verlag, Berlin, Heidelberg, New York, 1975) [4] G. Häusler, J. M. Herrmann: Optischer Sensor wacht über Lasermaterialabtrag, F&M 103, 540 (1995) [5] G. Häusler, R. Lampalzer, A. Schielzeth: Physi- kalische Grenzen von Triangulation mit strukturierter Beleuchtung - und wie man sie hinaus- schiebt, 2. ABW-Workshop 3-D Bildverarbeitung, Techn. Akademie Esslingen (1996) [6] J.M. Herrmann: Physikalische Grenzen von Optischen 3D-Sensoren, Thesis (Phys. Inst.), University of Erlangen (1994)