Field evaluation of a novel holographic single-image depth reconstruction sensor

. A camera-based single-image sensor is presented, that is able to measure the distance of one or multiple object points (light emitters). The sensor consists of a camera, whose lens is upgraded with a diffractive optical element (DOE). It ful ﬁ ls two tasks: adding a vortex point spread function (PSF) and replication of the vortex PSFs to a prede ﬁ ned pattern of K spots. Both, shape and rotation of the vortex PSF is sensitive to defocus. The sensor concept is presented and its capabilities evaluated both on axis and off-axis. The achieved standard deviation of the error ranges between 8.5 l m (on-axis) and 3.5 l m (off-axis) within a measurement range of 20 mm. However, as soon as calibration and measurement position no longer match, the accuracy is limited. An analysis of the effects responsible for this are also part of the publication.


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 [1,2]. In the last two decades, mainly for the application in optical microscopy, other ways of PSF manipulation based on diffractive optical elements (DOE) were developed. The purpose is, to modulate the phase of the light in a way, that the shape of the PSF corresponds to the changing distance z. Popular examples are the corkscrew PSF (CS-PSF) [3], self-bending PSF (SB-PSF) [4], tetrapod PSF (TP-PSF) [5] and double-helix PSF (DH-PSF) [6]. The ratio between measurement range and accuracy reached by those techniques ranges between 280 and 560.
In this contribution we address the question, to which extent the formerly published single-image depth reconstruction method [10] can be used as a single image 3D position sensor. Therefore, field measurements are carried out to investigate its off-axis performance.
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) [7] and apply it to a low NA objective lens. The low NA lens ensures a large measurement range and the combination of the multipoint method and the PSF modification is used to increase the accuracy of depth detection [11].

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 Special Issue -EOSAM 2022 Guest editors: Patricia Segonds, Gilles Pauliat and Emiliano Descrovi As this paper is written for the topical issue of the EOSAM 2022 conference, a part of this paper can be found in the proceeding submitted for this conference (see [11] and https://jeos.edpsciences.org) well capacity. It can be increased by temporal averaging, however error contributors like discretization and fixedpattern-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 ffiffiffiffi ffi N p . The MPM has already been successfully applied to various applications to improve the lateral position measurement accuracy, like stereo 3D position measurement [8] and vibration measurement [9].
For the PSF modification technique we use the DH-PSF introduced by Baránek and Bouchal in [6]. The discrete spiral phase modulation (SPM) modifies the incoming light in a way, that the transversal component of the resulting intensity profile consists of two helixes rotating around each other. In the image plane, this forms two spots that rotate around a common axis. The angle of rotation is dependent on the defocus of the object point. The MPM is used to replicate this DH-PSF to a predefined pattern. Each copy consists of two rotating spots. So if the object point is moved in z, all DH-PSF replications show the same angle of rotation. By averaging all measured angles, errors caused by photon noise and discretisation are reduced and, therefore, the accuracy of the measured rotation angle can be improved theoretically by a factor equal to the square root of the number of replications. In Figure 1 the principle of the multipoint double-helix PSF is shown for four replications. The experimental setup consists of a linear stage (Walter Uhl GT6-BO01) that is used to move a point light source (fibre coupled laser, k = 633 nm) in x, y and z. The camera system has a low NA objective lens (f 0 = 50 mm, NA = 0.0595) with a DOE mounted in front, to perform the replication to N = 25 copies and induce the vortex phase modulation. The distance between light source and objective lens is 234 mm and the depth measurement range is 20 mm.
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 [10]. This tail makes it difficult to evaluate the rotation angle using the CoGs of both spots. Therefore, we use the method of cross correlation with a reference image stack. The reference images are acquired at K equidistant points in the measurement range of 20 mm. This measurement range is chosen for practical reasons, to avoid overlap of the increasing diameters of the PSF copies. It could be increased by some more millimetres, until the decreasing signal-to-noise ratio becomes the limiting factor. A measurement is performed by positioning the light source inside the measurement range, acquire an image, in the following this is called a measurement image, and cross correlate it with the reference image stack. The peak of the resulting correlation energy curve marks the measurement result. In order to get the peak position, we use the ±15 points around the maximum value to fit a parabola function. Subsequently, the position, where the reference image stack is acquired, is called calibration position.
In a former publication we have shown, that the on-axis performance of the measurement principle is very good [10]. The mean standard deviation of three measurements was r = 8.51 lm within a measurement range of 20 mm. This leads to a measurement range to accuracy ratio of 20 mm/8.51 lm = 2350, which is very high compared to other single-image measurement systems. However, the measurement result is acquired on the optical axis. In order to use the proposed method as a 3D position measurement sensor, its off-axis performance is to be evaluated. This is done in two steps: 1. Field evaluation with no offset to the calibration position x i . 2. Field evaluation with offset Dx to the calibration position.
The scheme of the field measurements is illustrated in Figure 2. In the first step, the actual measurement is executed at the same position where the reference image stack is acquired. The difference is now, that it is no longer on the optical axis. In the second evaluation step, the measurement position is moved away from the calibration position by Dx. In object space, the calibration positions are selected in different distances to the optical axis: x 1 = 5 mm, The results for all five field measurements are shown in Figure 3. The curves show the difference between the measured position of the MP-DH-Sensor and the linear stage position at 180 equidistant points within the measurement range. In order to show all measurements in one graph, they are separated by an offset equal to 0.1 mm times the index i of the field position x i , so the measurement result of x 0 is shown as the lowest graph and the one of x 5 as the highest. 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 r 0 = 8.55 lm and at x 5 = 25 mm it is r 5 = 3.54 lm. 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 one spot of the MP-DH-PSF Cluster is shown at the different field positions x 0 , x 1 , x 2 , x 3 , x 4 and x 5 . There the field dependent intensity distribution change to one side is shown. This effect can be ascribed to vignetting of the objective lens and the NA of the fibre.
As previously stated, the second step is, to examine the performance at different distances Dx from a calibrated position. In this experiment we use the calibration position  Figure 5. As in the previous case (Fig. 3), for visualization the signals are separated by an offset equal to 0.5 mm times the index i of the distance Dx 1 . So the error signal of Dx 1 has an offset of 0.5 mm and of Dx 4 an offset of 2.0 mm. Two things are conspicuous in those results. Firstly, with increasing distance Dx i jumps are appearing in the error signal. The number of jumps seems to vary linear with Dx i (Dx 1 = 0.2 mm has one and Dx 5 = 1 mm has five jumps). Secondly, an almost linear trend is superimposed on the actual signal.
Those two occurrences obviously limit the accuracy of the method for 3D field measurements and are analysed in more detail. Therefore, in Figure 6 the 2D correlation energy distribution is plotted for Dx 5 . Each row of the image shows the correlation energy curve of one measurement image. It is generated by cross-correlating one measurement image (in total M = 180 images, plotted as y-axis) with the whole reference image stack (K = 2000 images, plotted as x-axis) and storing the maximum correlation energy of each correlation result (plotted as colormap). In a perfect world, the measurement and reference images are connected in a linear relationship (illustrated as a blue dotted line in Fig. 6), so that each correlation energy curve has only one peak marking the measurement result.
However, with increasing offset Dx i multiple peaks appear, as it can be seen in the magnified cross-section plot in Figure 6. If those peaks are changing height, a jump in the measurement result will appear. The linear slope, which is superimposed to the measurement results of Figure 5 can also be explained by the existence of those multiple peaks, since the stripes representing the multiple peaks in Figure 6 no longer have the same slope as the

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 [10].

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 lm. 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 lm and 3.5 lm, 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.