
- Journal of the European Optical Society-Rapid Publications
- Vol. 19, Issue 1, 2023017 (2023)
Abstract
Keywords
1 Introduction
When an object point is imaged to a detector, its lateral position (x, y) can be detected rather easily by calculating the Center of Gravity (CoG) of the PSF, whereas the axial dimension (z), thus the distance of the object point, is lost. It can be reconstructed by evaluating the increasing diameter of the defocused PSF, but the achievable accuracy is bad. The accuracy of this reconstruction can be improved by increasing object-space numerical aperture (NA) of the imaging system. However, a large NA limits the depth of field and, therefore, the measurement range. Another possibility to improve the depth reconstruction of object points is to modify the PSF of the imaging system. A classic way of PSF modification is, for example, a superimposed astigmatism by the use of two orthogonal cylindrical lenses [
In this contribution we address the question, to which extent the formerly published single-image depth reconstruction method [
The objective of the single-image 3D position sensor is to achieve both, good accuracy and large measurement range. Therefore, two measures are used: We combine a PSF modification method known from microscopy with a holographic replication technique called multipoint method (MPM) [
2 Principle and results
The MPM is a technique to improve the detection accuracy of a point light source imaged to a sensor. The wave nature of light fundamentally limits this accuracy, since the position, where a single photon impinges the camera sensor can only be described statistically (photon noise). Hence, the more photons are collected, the more precise the spot position can be localized. The number of photons, that can be collected by each pixel is limited by the quantum well capacity. It can be increased by temporal averaging, however error contributors like discretization and fixed-pattern-noise are not affected and the temporal resolution is reduced. The idea of the MPM is to use spatial averaging for single points by replicating the spot to a pattern of copies using a DOE. If the object moves, all copies move by the same amount. By making the object point brighter (light emitter) and using the DOE to replicate the spot to N copies, the number of pixels carrying useful position information of the object is increased by N. By averaging the centers of all spot copies, the accuracy of subpixel localization can be improved in theory by a factor of
For the PSF modification technique we use the DH-PSF introduced by Baránek and Bouchal in [
Figure 1.Combination of the MPM and the DH-PSF. The two rotating spots created by the SPM are replicated to four copies [
Simulations and experiments show, that the two rotating spots, created by the double-helix, form a tail, whose length is growing with the angle of rotation. More information regarding the simulations can be found in [
In a former publication we have shown, that the on-axis performance of the measurement principle is very good [
This is done in two steps:
Field evaluation with no offset to the calibration position xi.
Field evaluation with offset ∆x to the calibration position.
The scheme of the field measurements is illustrated in
Figure 2.Field evaluation of the proposed sensor. The blue/white crosses at x0, x1, x2 mark the image positions, where a reference image stack is acquired. The red cross marks the current measurement position. For ∆x′ = 0 the calibration and measurement position is identical.
At each position, the linear stage is positioned to K = 2000 equidistant points in the measurement range of 20 mm in order to acquire reference image stacks at those positions.
The results for all five field measurements are shown in
Figure 3.Field measurements at calibration positions x0 = 0 mm, x1 = 5 mm, x2 = 10 mm, x3 = 15 mm, x4 = 20 mm and x5 = 25 mm with ∆x = 0 mm, meaning that the measurements are taken at the same position as the calibration. To show all field measurements in one plot, the signals are shifted by an offset of 0.1 mm times index i of xi, so the measurements at field position x0 are shown in the lowest and at x5 in the highest plot.
The results show, that the standard deviation is getting smaller, the further away of the optical axis the measurement is performed. On the optical axis the standard deviation of the error is σ0 = 8.55 μm and at x5 = 25 mm it is σ5 = 3.54 μm. It is not clear, why the standard deviation is getting smaller with increasing field position. One reason could be the field dependent intensity distribution change of the spots, making the pattern more unique for correlation. In
Figure 4.Changing intensity distribution of one defocused DH-PSF spot depending on field position. Images (a) to (f) show one spot at calibration positions x0, x1, x2, x3, x4 and x5. The spot is always at the same defocus position of z = 20 mm.
As previously stated, the second step is, to examine the performance at different distances ∆x from a calibrated position. In this experiment we use the calibration position x3 = 15 mm. The offset distances are ∆x1 = 0.2 mm, ∆x2 = 0.4 mm, ∆x3 = 0.6 mm and ∆x4 = 1.0 mm. Each measurement consists of M = 180 equidistant points within the measurement range of 20 mm. The results are shown in
Figure 5.Field measurements at calibration position x3 = 15 mm with different offsets ∆x1 = 0.2 mm, ∆x2 = 0.4 mm, ∆x3 = 0.6 mm and ∆x4 = 1.0 mm. Blue dotted lines show the curves without jumps and detrended.
Those two occurrences obviously limit the accuracy of the method for 3D field measurements and are analysed in more detail. Therefore, in
Figure 6.2D correlation energy distribution for ∆x4. Each point (x, y) in the image represents the maximum correlation energy between measurement image #x and reference image #y. The blue dotted line illustrates a perfect linear relationship between measurement and reference.
However, with increasing offset ∆xi multiple peaks appear, as it can be seen in the magnified cross-section plot in
3 Discussion
In this article first field measurements of the proposed single image 3D position sensor are presented. The results are both, very promising and challenging. Promising is the fact, that field measurements on the calibrated positions achieve even better standard deviations than on the optical axis. On the other hand, as soon as the calibration position is left, two effects, that arise from the ambiguity of multiple correlation peaks, currently limit the accuracy of the measurement results. However, the fact, that not only the coarse distance of the light source (MP-DH-PSF reconstruction), but also the lateral position is known (CoG), should make it possible to handle those effects. Several measures can be taken to solve those problems, which will be analysed in detail in subsequent publications:
Low-pass filtering of the correlation energy signal. This would remove the multiple peaks and, therefore, the jumps.
Reduce the calibration grid period.
Use of a different PSF modification to combine it with the MPM
Investigation of a simulation based calibration of the sensor.
Furthermore, other reconstruction algorithms need to be analysed, such as neural network approaches. Another challenge of the proposed method is the amount of reference images and the computationally demanding task of cross correlation. However, better search algorithms can be used, so that instead of the whole reference stack just a few correlations have to be performed, in order to find the peak. Furthermore, the computation time of correlation can be accelerated considerably by processing it on the graphics board.
The advantages of the proposed sensor system are the cost-effective single-camera setup together with the ability, to be able to retrofit it to existing applications. The application scope of this kind of measurement system can be small machines, whose position is to be measured, like 3D printers and milling or turning machines. It is also possible to extend the measurement volume by increasing the distance between lens and light source. One has to keep in mind, that in this case the NA is getting smaller and, thus, the depth resolution is reduced.
At the calibration positions the ratio of accuracy to measurement range of the presented method is clearly above 2000. Comparable single image 3D methods are astigmatism (below 400), tetrapod PSF (around 600) and time of flight (up to 1000). Details to the named ratio values can be found in [
4 Summary
The presented single-image depth measurement system is based on the combination of a DH-PSF and a spatial replication method, which is both created by a phase modulating DOE placed in front of the imaging lens.
In this article the measurement system accuracy is analysed both, on the optical axis and in the field. The depth measurement range analysed in this article is 20 mm. On the optical axis, the standard deviation is 8.5 μm. When only a single DH-PSF is used for depth reconstruction (without MPM), the result is up to factor 3 worse. For the field measurements two scenarios are investigated. In the first, where calibration and measurement position match, the standard deviation of the error ranges between 8.3 μm and 3.5 μm, depending on field position. In the second scenario, calibration and measurement position are separated by an offset. Here two effects are observed: Superimposition of a linear slope and jumps on the error signal. The origin of both is analysed. Following publications will focus on the compensation of those effects to realise a 3D calibration.
References
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