thus nearly cancels out. Again, since tilting the target corresponds to changing the image-forming rays themselves it would be impossible to do so. One could imagine that, if 40 µm of error is tolerable, two or three dewarping planes would be sufficient; the interpolation between planes would keep the error under control.
A scatter plot of the error distribution across the volume proves that this error is just a rotation of the data’sZ-plane. TheZ error is larger simply due to the axis of rotation (Y). The case of a 5°-tilt shows this nicely (figure 13.5-3).
Figure 13.5-2: Error inXfor dewarping done with target tilt of 1.0°. The bottom plot shows the actual locations of the dewarping planes.
−540 −520 −500 −480 −460 −440 −420
Z location (mm) 5
10 15 20 25 30 35
Error (µm)
X error, target tilt of 1.0 degrees
0 5 20 all No. of skipped planes
This type of error makes this case a great chance to see the effects of interpolation between planes when the error is very smooth. The very slight tilt case of 0.1°, when viewed from the point of view of the ratio between the out-of-plane and in-plane error (figure13.5-4), shows this exactly. Since a physical shift inZ corresponds to a larger pixel shift in xas one approaches the camera, the error in the spaces between dewarping planes increases. (In this figure the error ratio seems staggered because of numerical precision; the averageX error is extremely small.)
Even at this slight tilt, though, the averageZ error is a whopping 25µm. Absolute position mea- surements are extremely sensitive to the target quality and alignment. But it is important to realize also that velocity is a relative position measurement, and thus the velocity vectors obtained with tracking, though misplaced in space, will have nearly correct components, because the displacement between frames of the particles forming them will be small and thus the relative error induced by a bad or ill-positioned target between them will also be very small.
At this time it is appropriate then to analyze the method by which the target is positioned relative the camera. Third-generation cameras are equipped with two laser diodes—one which in
Figure 13.5-3: Scatter plot ofXversusZerror for the case of target tilt of 5.0°, 0-skip.
−5.23 −3.138 −1.046 1.046 3.138 5.23
−5.23
−3.138
−1.046 1.046 3.138 5.23
Scatter plot for target tilt of 5.0 degrees, 0−skip
X error in mm
Z error in mm
Figure 13.5-4: ZtoXerror ratio for dewarping done with target tilt of 0.1°. The bottom plot shows the actual locations of the dewarping planes.
−540 −520 −500 −480 −460 −440 −420
Z location (mm) 200
400 600 800 1000 1200
Error ratio
Z:X error ratio, target tilt of 0.1 degrees
0 5 20 all No. of skipped planes
theory corresponds to the optical axis, and another one projecting a beam at an angle to the axis equal to the angle of the sensor axis so that the reference plane location can be quickly found in multi-medium experiments. Both are glued into fixtures which are then bolted to the cameras.
Alignment is done in a jig, with holes that offer a snug fit to the fixtures and a target mounted on a stage. The target is a metal plate with a small dimple machined into it so that its height (Y-coordinate) coincides to theY-coordinate of the center of the fixture hole to within machining precision of the plate, the stage on which it sits, and the jig. The X-coordinates are aligned only to mounting precision on an optical table; this is done by applying pressure to the jig and the stage in the same direction during bolting. Thus the accuracy is dependent on the thread quality of the screws used.
The fixture must have a hole large enough for the diode to fit with enough space around it so that epoxy can be applied once it is in place, thus the diameter of the hole is nearly twice that of the diode. The diode is held in place with a 6-axis stage sandwich. It’s position is adjusted so that three criteria are simultaneously met. First, the diode must be concentric to its hole in the fixture; this is checked by eye only. Second, the beam must land on the dimple in the target; this is evaluated by observing the brightness of the beam from an off-axis location—since the dimple is conical, the spot looks much brighter off-axis once it is within the dimple. The dimple has diameter of less than 400 µm, and the diode’s beam is focused on the target surface, so at worst case the laser spot is some 200 µm larger than the dimple. The third, and most important criterion, is that when a precisely flat first-surface mirror is rested on the target face, the beam should reflect back “into” the laser diode.
The procedure obviously does not provide much X orY precision, but the angular precision is high. Assuming the mirror is perfectly flat and of even thickness, and that the target plate is straight within mounting precision (which could not be worse than a few thousands of an inch concentricity with average-quality screws), and that it is possible to evaluate the “beam going back into the diode” well within two millimeters, and, keeping in mind that the reference plane distance (L) for the Emilio Camera is 640 mm, then the angle of the beam of the diode cannot be worse than 0.09°. Moving the diode fixture from the jig to the camera should introduce no angular misalignment since all surfaces are relatively small (1-inch-diameter) and are all machined flat.
When positioning the target, the same method of watching the reflection of the center diode beam is used to check the angle of the target relative to the camera—so the same level of accuracy can be expected. The pinhole optics tests show then that the best-case scenario of image quality coupled with the worst-case scenario for target alignment should yield an averageZerror of around 20 to 30µm for these third-generation cameras. Again, it should be reinforced that, to this day, no measurement performed with any defocusing camera was intended to provide absolute position, and the error introduced by target misalignment do not affect relative measurements perceptively.