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Abstract
To measure surface displacement on micro samples, a non-invasive method with both a low displacement measurement uncertainty below 100 nm and high spatial resolution of around 20 µm is required. In digital image speckle correlation, both requirements can be fulfilled individually but not simultaneously. To lower the displacement measurement uncertainty without deteriorating the spatial resolution, an ensemble averaging technique over multiple uncorrelated speckle patterns is presented. To generate and reproduce different speckle patterns, two concepts for the respective modulation of laser light illumination are investigated: a low-cost concept with a rotating glass diffuser, as well as a faster concept using a digital micromirror device combined with a stationary diffuser with a maximum pattern rate of 17.9 kHz. Both setups lead to a measurement uncertainty reduction by one order of magnitude over a wide range of spatial resolutions. As a result, displacements in the micrometer range are measured with a measurement uncertainty of 40 nm and spatial resolution of 20 µm.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
In material science and production technology, non-invasive displacement and strain measuring methods with a high spatiotemporal resolution are required to monitor material properties and loads during manufacturing processes and in particular material testing. Currently, the development of new structural materials such as metal alloys is both expensive and time consuming. One approach for experimental high-throughput development of new structural materials relies on testing microscopic samples to accelerate the development process [1]. For these samples, electrohydraulic extrusion is used as an incremental forming method for tensile tests where strain fields are measured on the samples at multiple stages during extrusion. To enable displacement field measurements on such microscopic samples, i.e., on areas that are only several 100 µm wide, a high spatial resolution of a few micrometers is required together with a low measurement uncertainty down to the nanometer range [2].
A. State of the Art
Two-dimensional digital image correlation (DIC) with incoherent light illumination is an established technique for contactless displacement measurements on a plane surface with high spatial and temporal resolution [3]. Naturally occurring or, on optically smooth surfaces, artificial surface markers are used to measure the local in-plane surface displacement of a sample through cross-correlation of before and after images. To obtain a two-dimensional displacement field, the image pairs are subdivided into square evaluation windows, and the two in-plane components of the local displacement are calculated for each evaluation window [4]. For DIC, both systematic errors and random errors of the displacement field have been extensively studied: Wang et al. first developed analytic formulas for both systematic and random displacement errors [5]. Xu et al. derived that the systematic errors are proportional to the second-order displacement gradients, and the ratio is determined only by the size of the evaluation window [6]. Su et al. finally proposed a theoretical model for the spatial uncertainty of measurement errors [7]. Additionally, the evaluation [8] and optimization [9,10] of speckle patterns were analyzed and theoretically modeled. Dong and Pan reviewed various speckle pattern fabrication techniques in [11] with spatial resolutions down to the micrometer range. However, the time-consuming and invasive pre-application of artificial speckle patterns is not practical for in-process applications in material development or production technology where high quantities of workpieces have to be tested. Digital image speckle correlation (DISC) uses the same image processing as DIC but also a coherent light source that produces laser speckles in the image. The laser speckles serve as virtual surface markers and also occur on optically smooth surfaces [12]. Hence, DISC allows non-invasive displacement field measurements without preparing the sample’s surface where the spatial resolution is limited only by the optical system’s diffraction limit [11]. Finally, an additional advantage of DISC over DIC is the quick adaptability of the speckle pattern through modulation of laser illumination in contrast to the fixed pattern of the naturally occurring or artificial surface markers in DIC. Generating different speckle patterns on the sample allows for multiple measurements and subsequent averaging to potentially reduce overall measurement uncertainty.
As a laser-optical measuring method, the DISC measurement uncertainty is ultimately limited by shot noise due to the quantum mechanical properties of light. The respective lower limit of the measurement uncertainty was derived in [13] for DISC with fully developed speckles and large evaluation windows of ${200} \times {200}\,\,\rm pixels$, demonstrating the potential for nanometer resolution in speckle displacement measurements. The theoretically achievable uncertainty is consistent with the simulation results from [4], where smaller evaluation windows of ${40} \times {40}\,\,\rm pixels$ were studied as well. According to these findings, DISC can resolve sub-pixel displacements down to 0.005 pixel, which corresponds to less than a nanometer for maximum image magnification. Therefore, a measurement uncertainty in the nanometer range is theoretically achievable with DISC.
Recent experimental studies also considered camera noise [14], primarily comprising read-out noise and dark noise, and speckle noise [2] as additional contributions to the measurement uncertainty budget. Speckle noise is caused by the image quantization due to the finite pixel size of the camera. A sub-pixel displacement of the speckle pattern alters the detected speckle image in addition to the expected displacement, and, thus, the correlation result deviates. Speckle noise peaks at half-pixel displacements and decreases with increasing speckle-to-pixel-size ratios and an increasing number of speckles in the evaluation window. Additionally, the DISC measurement uncertainty is influenced by the random spatial distribution of the speckle pattern with locally varying contrast. This varying contrast causes some evaluation windows to be more susceptible to speckle noise than others. Alexe et al. recently found that either speckle noise or camera noise dominates the measurement uncertainty budget, depending on the sub-pixel displacement, i.e., the measurand, and the number of speckles in the evaluation window [2]. Table 1 shows previous DISC experimental results [2,14,15]. Note that [2,14] defined the spatial resolution of the displacement field as the step size of half the evaluation window size, while in this paper and in [15], the spatial resolution is the full width of an evaluation window. In Table 1, the respective values of the spatial resolution $\Delta x$ were therefore accordingly adjusted to $\Delta x = {W_{{\rm eval}}}$, the width of the evaluation window. In the reported experiments, either the measurement uncertainty or the spatial resolution required for measurements on micro samples was achieved, but not both simultaneously. Additionally, when calculating the strain field, the measured displacement field must generally be spatially low-pass filtered [14], which deteriorates spatial resolution. This further highlights the need to measure the local displacement field at a high spatial resolution. For these reasons and to utilize the full potential of DISC, a method is necessary that reduces the measurement uncertainty from both speckle noise and camera noise without deteriorating spatial resolution.
Table 1. Previous DISC Experimental Results for Displacement Measurement Uncertainty $\sigma$ and Spatial Resolution $\Delta x$
Table 2 summarizes averaging strategies to reduce the effects of the different types of noise on the DISC measurement uncertainty. Speckle noise is caused by the speckle pattern, which is time independent. In contrast to shot noise or camera noise, it can therefore not be reduced by temporal averaging of sequential images. Spatial averaging over larger evaluation windows with a higher photon count and a larger number of speckles was shown to drastically decrease speckle noise while also lowering shot noise and camera noise [2]. That improvement, however, comes at the expense of the displacement field’s spatial resolution [16]. The requirement to make a compromise between measurement uncertainty and the spatial resolution of the displacement measurement is a fundamental limitation of the current DISC method that can be overcome only by an improved measurement approach.
Table 2. Features and Limitations of Temporal, Spatial, and Ensemble Averaging
Ensemble averaging is a noise reduction scheme where the multiply measured displacement fields with a multitude of different speckle patterns are averaged. Different speckle patterns are generated through the modulation of laser illumination. Thereby the temporal resolution is worsened, but not the spatial resolution, as the individual evaluation windows remain unchanged. If it were possible to generate statistically independent speckle patterns, the ensemble average could potentially reduce all noise sources in the same way as the spatial average does. This approach was proposed by Alexe et al. in [2], but has not yet been experimentally validated for DISC. However, a similar technique was tried in the related field of laser speckle contrast imaging (LSCI) [17]. Here, a continuously slow rotating diffuser was used to generate multiple statistically independent speckle images taken at exposure times fast enough not to be affected by the diffuser motion. The measurement uncertainty of the speckle contrast, which is analyzed to detect movements in the scattering medium, was thus lowered by one order of magnitude. The measurement uncertainty was shown to scale with $1/\sqrt N$, where $N$ is the number of evaluated pixels multiplied by the number of analyzed images. In other laser optical measuring techniques, rotating [18] or vibrating [19–21] diffusers are implemented to suppress unwanted speckles by averaging a large number of speckle images over time, thus greatly diminishing the effect of speckles on the image data, and then analyzing them. These studies have shown the possibility to generate multiple statistically independent speckle patterns with a movable glass diffuser. This raises questions as to in what way the measurement uncertainty of DISC displacement measurements can be reduced by introducing an ensemble average technique of multiple speckle patterns, how this reduction scales with the number of patterns, and what ultimate lower bound for the uncertainty principle-like relation between measurement uncertainty and spatial resolution is achievable.
B. Aim and Outline of the Paper
For this reason, the aim of the paper is to introduce DISC displacement measurements with ensemble averaging to significantly lower measurement uncertainty while retaining high spatial resolution, thereby surpassing the current limits of measurability. First, the DISC measurement principle with ensemble averaging is explained and also the theoretical lower bound for the uncertainty principle between measurement uncertainty and spatial resolution is considered in Section 2. Two measurement setups for the generation of different laser speckle patterns are presented in Section 3. This includes a ground glass diffuser in a precision rotation mount as a relatively low-cost approach as well as a stationary glass diffusor combined with a digital micromirror device (DMD) that improves the spatial control over the laser, the reproducibility of speckle patterns, and the temporal resolution. The experimental results of both setups are shown and discussed in Section 4. The final Section 5 draws the conclusions and gives an outlook on further research aspects.
2. MEASUREMENT PRINCIPLE
A. DISC Ensemble Average
For the standard DISC method, two images of a sample’s surface are taken before and after the studied deformation. Pairs of sub-images are cross-correlated to determine the local lateral surface displacement between the first and second images. In Fig. 1, a section of an image pair is shown together with the cross-correlation coefficient function for the evaluation window depicted by a white rectangle in both sub-images. To calculate the local displacement components with sub-pixel accuracy, the nine cross-correlation coefficient values around and at the local maximum are interpolated with a two-dimensional Gaussian function according to [22].
Fig. 1. (a) Sections of a speckle pattern image pair with corresponding evaluation windows (white rectangles). (b) Cross-correlation coefficient function plotted over the local displacement components ${d_x}$ and ${d_y}$.
For DISC with ensemble averaging, not only one but multiple image pairs are evaluated, and an ensemble average of the displacement measurements is calculated:
which is the weighted arithmetic mean of the $n$ individual displacement fields ${D_k}$, $k = 1, \ldots ,n$, with the weightings ${w_k}$.Note that the weighted arithmetic mean is a least squares solution. Using the inverse covariance matrix as the weighting matrix in a least squares solution, the Gauss–Markov theorem states that the least squares estimator minimizes the estimator variance (best linear unbiased estimator) [23]. The required covariance matrix consists of the variances of the individual displacement measurements on the principal diagonal and the respective covariance terms as the remaining matrix entries. If the individual displacement measurements are uncorrelated, the covariance terms are zero. The least squares estimator in Eq. (1) is based on this assumption, i.e., the covariance terms are not considered here. The estimation optimality is thus achieved when the weightings ${w_k}$ are directly proportional to the reciprocal of the corresponding variances $\sigma _k^2$, i.e., ${w_k} \sim \sigma _k^{- 2}$.
The ensemble average is applied to reduce the measurement uncertainty $\sigma$ of the displacement $D$. The variance of the measured displacement reads
To determine the spatial resolution $\Delta x$ of the ensemble averaged displacement measurement, the DISC method in Fig. 1(a) is reconsidered. The local displacement components determined for an evaluation window are spatial averages over the evaluation window’s size ${W_{{\rm eval}}}$. Spatially averaging means low-pass filtering with the spatial cutoff frequency ${f_{c}} = 1/(2{W_{{\rm eval}}})$. To measure all of the low-pass-filtered displacement field’s information, the Nyquist–Shannon sampling theorem specifies the minimum spatial sampling frequency ${f_{s}} = 2{f_{c}}$. The inverse of this minimal sampling frequency yields the maximum spatial resolution $\Delta x$:
As a result, the achievable spatial resolution is determined by the size of the evaluation window. Using a higher sampling frequency than $2{f_c}$ allows for increasing the point density in the measured displacement field, but the physical spatial resolution is not increased. The ensemble average neither improves nor deteriorates the spatial resolution since the size of the evaluation window is maintained.B. DISC Uncertainty Principle
By combining Eqs. (3) and (4), the product of measurement uncertainty and spatial resolution can now be expressed as
The measurement uncertainty $\sigma$ of the ensemble averaged displacement is ultimately limited by photon shot noise. For a fully developed speckle pattern, this lower bound of the measurement uncertainty is [13]
Note that Eq. (6) does not consider pixel quantization effects and assumes a large number of speckles $({\gt} 10)$ in the evaluation window. The lower bound depends on the total photon count ${N_{{\rm total}}}$ and the speckle size: defined by Goodman [12], with ${z_i}$ as the distance between aperture and sensor, and $\delta$ as the diameter of the circular aperture. For ensemble averaging, the total photon count is the sum of each evaluation window’s photon count ${N_k}$.The uncertainty principle formulated with Eqs. (6) and (8) reads
For the standard DISC method, $n = 1$, so that reducing the measurement uncertainty is possible only by spatial averaging, which reduces spatial resolution. This limitation of the standard DISC method is overcome by ensemble averaging, i.e., for $n \gt 1$. Using a large number of speckle patterns $n$, the measurement uncertainty can be reduced without the necessity to deteriorate $\Delta x$.
The theoretical lower bound for the spatial resolution $\Delta x$ is given by the diffraction limit [24]
which is expressed here using the Rayleigh criterion. Comparing Eqs. (7) and (10) without optical magnification, i.e., for ${z_i} = 2f$, the light diffraction obviously determines both the speckle size and the minimum of the evaluation window size in dependency of the laser wavelength $\lambda$ and $f/\delta$, the f-number of the optical system. For ${W_{{\rm eval}}} = \Delta {x_{{\rm min}}}$, an evaluation window contains only one speckle on average and theoretically yields the highest possible resolution of the desired displacement field. However, small numbers of speckles per evaluation window are applicable only if the speckle size is significantly larger than the pixel size (minimum speckle size of five pixels [14]) due to pixel quantization effects. Additionally, the speckle noise is high for speckle counts close to one. For this reason, standard DISC uses mostly ${W_{{\rm eval}}} \gt 3 \cdot \Delta {x_{{\rm min}}}$, which corresponds to a speckle count per evaluation window greater than nine [25]. If smaller evaluation windows are used, reaching the lower limit of the uncertainty principle from Eq. (9) is not possible. However, applying the ensemble average reduces the measurement uncertainty for any spatial resolution and, thus, should also allow measurements close to the lower limit of the spatial resolution.C. DISC Measurement Uncertainty Budget
The DISC measurement uncertainty budget consists of photon shot noise, camera noise, and speckle noise [2,14]. Table 2 shows the suitability of spatial, temporal, and ensemble averaging for reducing the measurement uncertainty budget. It is subsequently discussed how the uncertainty budget’s components are reduced by the ensemble average, i.e., how they scale with the number $n$ of uncorrelated speckle patterns.
According to Eq. (6), the measurement uncertainty due to shot noise is inversely proportional to the square root of the sum of ${N_k}$. The photon counts ${N_k}$ are assumed to be nearly equal and, thus, the measurement uncertainty component due to shot noise is inversely proportional to the square root of $n$.
Camera noise comprises the random image noise other than photon shot noise, such as read-out noise and dark noise. Like shot noise, the camera noise varies with time and space and therefore is reduced by averaging in the temporal or spatial domain. Because camera noise is a random error, the measurement uncertainty of the ensemble average due to camera noise is inversely proportional to the square root of the number of measurements.
Speckle noise is the influence of the spatially varying speckle pattern on the displacement measurement, and here occurs only due to pixel quantization effects [2]. Because speckle noise depends on the time-independent speckle pattern, multiple correlated measurements of the same pattern do not improve measurement uncertainty. However, for the ensemble average over different, uncorrelated speckle patterns, the measurement uncertainty due to speckle noise is inversely proportional to the square root of the number of speckle patterns.
To summarize, the ensemble average reduces all known components of the DISC measurement uncertainty budget with $\sigma \sim 1/\sqrt n$, while the spatial resolution remains unchanged.
3. EXPERIMENTAL SETUP
Figure 2 shows the measuring setup in two variants. In both setups, the laser light is scattered by a ground glass diffuser onto a paper screen. A monochrome CMOS camera with five megapixels photographs the sample surface through an $f/1.4$ lens with a focal length of 25 mm. The magnification of the optical system is increased to 1.4 with a 60 mm extension tube. The screen is moved parallel to the camera sensor by means of a piezo linear actuator with a linear position repeatability of ${\pm}5\;{\rm nm}$.
Fig. 2. Scheme and images of the two measuring setups used for the generation of different speckle patterns: (a), (c) rotating diffuser; (b), (d) DMD in combination with a stationary diffuser.
The two measuring setup variants differ in the way in which the laser light is modulated, i.e., how different speckle patterns are generated. In Fig. 2(a), the diffuser is mounted on a motorized precision rotation stage with a bidirectional angular position repeatability of ${\pm}{0.1^ \circ}$. Due to the diffuser’s rotation, the laser beam is directed at different positions of the diffuser, which changes the speckle pattern. A high angular repeatability is essential, since the same speckle patterns must be precisely reproduced before and after the displacement of the sample. The maximum rotation velocity of the rotation stage is 25°/s. Therefore, the achievable speckle pattern rate depends on the desired number of patterns that corresponds to the angular increment. For 180 speckle patterns, the maximum rate is 12.5 Hz.
Figure 2(b) shows the second measuring setup variant with a DMD. A DMD is a semiconductor-based array of micromirrors, each of which can be switched on and off by tilting them by ${+}{12^ \circ}$ or ${-}{12^ \circ}$ from the neutral position [26]. The incident light beam can thus be spatially modulated by removing parts of the wavefront in specified areas. The DMD type DLP9500 and the controller DLPC410 from Texas Instruments are used together as parts of the V-9501 module from ViALUX GmbH. The DMD comprises $1920 \times 1080$ micromirrors and has a switching rate of 17.9 kHz for 1 bit binary images. Two lenses are used to expand the laser beam to cover the DMD’s entire active area. The incidence angle on the DMD is adjusted to ${12^ \circ}$ to achieve the highest light efficiency [27]. Thus, the off-state mirrors are perpendicular to the incident beam. The DMD is used to direct the laser light at different positions of the stationary diffuser. For this purpose, a square of $70 \times 70$ on-state mirrors (about $0.76\;{\rm mm} \times 0.76\;{\rm mm}$) is moved over the illuminated area of the DMD while the remaining micromirrors are in off-state. The light intensity is significantly reduced by the beam expansion and subsequent spatial modulation. However, with a pattern rate of 17.9 kHz, the DMD allows a more than 1000 times faster measuring rate and more accurate adjustment of the laser beam position on the diffuser than the rotation stage of setup variant (a).
4. RESULTS
A. Rotating Diffuser Setup
To answer how the measurement uncertainty of DISC displacement measurements can be reduced by the ensemble average method, Fig. 3 shows a comparison of the temporal, spatial, and ensemble averages over the effective evaluation area. Five test series were conducted for each curve, and the error bars show the respective standard deviations of the mean. Note the double-logarithmic scales where the expected square root function from the theoretic prediction is represented as a straight line (black) with a slope of ${-}1/2$. The measurement uncertainty $\sigma$ is the standard deviation of the displacement field in the investigated field of view of $500 \times 500\; {\rm pixels}$. For the evaluation window area ${A_1}$ with ${W_{{\rm eval}}} = 20\; {\rm pixel}$, this corresponds to 625 evaluation windows in the field of view. Since the sample is displaced parallel to the camera sensor, the true values of the local displacements are equal across the entire field of view. The different averaging approaches are compared using the effective evaluation area, which is the total area, either in one image pair (spatial average) or over multiple image pairs (temporal and ensemble average), which is evaluated for each entry of the displacement field. The effective evaluation area divided by ${A_1}$ is shown on the abscissa as $n$. For the temporal and ensemble averages, the size of the evaluation window does not change from ${A_1}$, and $n$ corresponds to the number of displacement fields that are averaged. In the case of the spatial average, the value of the abscissa shows the ratio by which the area of the evaluation window is increased, i.e., ${A_n} = n \cdot {A_1}$. The effective evaluation area for the different averaging approaches is visualized in Fig. 4 for the example of $n = 3$.
Fig. 3. Comparison of temporal, spatial, and ensemble averaging according to the displacement field standard deviation $\sigma$ over $n$ the effective evaluation area divided by the original evaluation window area ${A_1} = 20 \times 20\,\,{\rm pixel}$. Error bars show the standard deviation of the mean of five test series. The dotted lines indicate that the resulting measurement uncertainty of the ensemble average corresponds to spatial averaging over $n = 29$ times the original evaluation area ${A_1}$. Note the double-logarithmic scales where the square root function from the theoretic prediction is represented as a straight line (black).
Fig. 4. Visualization of the effective evaluation area of temporal, spatial, and ensemble average for $n = 3$. Temporal average: evaluation window ${A_1}$ in three images of the same speckle pattern. Spatial average: evaluation window ${A_3} = 3 \cdot {A_1}$ in one image. Ensemble average: evaluation window ${A_1}$ in three images of three different speckle patterns.
To measure the temporal average, 60 images of the same speckle pattern were taken before and after displacement. The average is then calculated from the resulting displacement fields. While this technique maintains the original spatial resolution, it does not lead to a substantial reduction in displacement measurement uncertainty. For the spatial average, only one speckle image pair is evaluated multiple times while increasing the area of the evaluation window up to $60 \cdot {A_1}$. This worsens the spatial resolution, but the measurement uncertainty is reduced substantially. The reduction closely follows the theoretical model, according to which the measurement uncertainty is inversely proportional to the square root of the number of measurements when averaging uncorrelated measurements. To calculate the ensemble average, again 60 images were taken but of 60 different speckle patterns. These patterns were generated with a diffusor rotating in ${6^ \circ}$ increments. The ensemble average of 60 patterns has a significantly lower measurement uncertainty than the individual measurement or the temporal average, but does not achieve the result of the spatial average for an increasing effective evaluation area. Note that the different $\sigma$ at $n = 1$ for the temporal average and the ensemble average is due to being calculated from separate test series. Therefore, the first images in each series are different and have a slightly different measurement uncertainty $\sigma$.
The ensemble average’s minimum measurement uncertainty corresponds to the value of the spatial average of evaluation windows of $29 \cdot {A_1}$ as indicated by the dotted black lines. This means that with the ensemble average of 60 images at the same spatial resolution, the measurement uncertainty can be reduced by a factor of five. To achieve this reduction through spatial averaging, the spatial resolution must be reduced by a factor of $\sqrt {29} = 5.4$. The varying slope of the curve of the ensemble average suggests that not all generated speckle patterns are uncorrelated. Note that the ensemble average is calculated from the same evaluation window with differently modulated speckle patterns, i.e., the same surface region, while the spatial average is formed over a larger evaluation window, i.e., different surface regions. However, despite the restriction to a fixed surface region, the ensemble average significantly reduces the measurement uncertainty.
The scaling of the ensemble averaged displacement field’s measurement uncertainty $\sigma$ with the number of speckle patterns $n$ is further illustrated in Fig. 5. Multiple test series with different angular increments of the diffuser between each measurement are shown. For each test series, the diffuser is rotated by ${360^ \circ}$ in total, therefore the angular increment corresponds to the number of speckle patterns. The individual test series’ curves of $\sigma$ over $n$ for 10 angular increments between ${36^ \circ}$ ($n = 10$) and ${1^ \circ}$ ($n = 360$) are shown in red, while their resulting measurement uncertainty values (end point of each red curve) are indicated with black asterisks and connected by a thin black line. Thereby the trend in Fig. 3 is confirmed: first the curves follow the $1/\sqrt n$ proportionality (dotted-dashed black line), then the negative slope decreases asymptotically until the measurement uncertainty reaches a lower limit (horizontal dotted-dashed black line). This behavior is consistent across all test series of varying angular increments.
Fig. 5. Measurement uncertainty $\sigma$ of ensemble averaged displacement field over number of speckle patterns $n$. Red lines show multiple test series of varying diffuser angle increments, which corresponds to varying speckle pattern counts. Note the double-logarithmic scales where the square root function from the theoretic prediction is represented as a straight line.
Apparently, this measuring setup is able to generate only a limited number of uncorrelated speckle patterns, which limits the achievable lower bound of the ensemble averaged measurement uncertainty. Since the behavior of the individual curves is consistent independent of angular increment, i.e., overall speckle pattern count, the results suggest that the maximum number of uncorrelated speckle patterns is limited by the diffuser. More speckle patterns can be generated by decreasing the angular increment, but the number of uncorrelated patterns is determined by the characteristics of the diffuser. Here, the number of uncorrelated speckle patterns is about 30. Thus, the plot in Fig. 5 is divided into three sections indicated by two dashed vertical lines: in the first section of about 1–30 speckle patterns, the curve clearly follows the $1/\sqrt n$ proportionality. Then, from 30 to 200 images, the negative slope decreases, but the measurement uncertainty is still continuously reduced. In the third section above 200 images, the lower limit of the measurement uncertainty is reached. Smaller angular increments or more speckle patterns no longer yield any new information for the ensemble averaging. From this point on, the curve is comparable to the case of temporal averaging.
The ensemble average measurement uncertainty depends not only on the number of speckle patterns but also on the spatial resolution. Small evaluation windows negatively affect the detection of the cross-correlation’s peak. As the evaluation area is decreased, displacement measurement uncertainty increases [5,6]. In addition, small evaluation windows allow only small local displacements to be measured before decorrelation interferes with the analysis. Figure 6 shows the measurement uncertainty over the spatial resolution of a test series before and after the ensemble averaging of 180 speckle patterns. Additionally, the measurement requirements as well as the theoretical lower bounds for the measurement uncertainty and the spatial resolution are indicated. The requirements are given by the application of the measuring method. For the deformation measurement of the electrohydraulic extrusion of micro samples, a spatial resolution in the range of 20 µm and a measurement uncertainty of less than 100 nm are required. The diffraction limit $\Delta {x_{{\rm min}}}$ from Eq. (10) for $\lambda = 638\;{\rm mm}$, $f = 25\;{\rm mm}$, $\delta = 3.125\;{\rm mm}$ and the lower bound of measurement uncertainty ${\sigma _{{\rm min}}}$ from Eq. (6) for ${s_{{\rm speckle}}} = 7.14\;{\unicode{x00B5}{\rm m}}$ are also indicated. The number of photons ${N_{{\rm total}}}$ is estimated from the number of pixels per evaluation window, the mean pixel intensity of 50 divided by its 8 bit maximum of 255, the full well capacity of 11,000 electrons, and the camera sensor’s quantum efficiency of 0.56. The area between the measurement requirements and respective lower bounds is shaded in green. Furthermore, the average number of speckles per evaluation window is displayed on a second abscissa on top. Note that the average number of speckles is calculated by dividing the evaluation window area $W_{{\rm eval}}^2$ by the squared speckle size $s_{{\rm speckle}}^2$ from Eq. (7). Thus, the figure shows how ensemble averaging decreases measurement uncertainty for a wide range of spatial resolution. For single measurements (blue), the results do not meet measurement requirements, but through ensemble averaging, displacement fields from multiple different speckle pattern (red) measurements inside the required range (green) are possible. With an optical magnification of 1.44 and a pixel size of 3.45 µm, an evaluation window of ${6} \times {6}\;{\rm pixels}$ contains around five speckles on average and results in a $\Delta x = 6 \cdot 3.45\;{\unicode{x00B5}{\rm m}}/1.44 = 14.4\;{\unicode{x00B5}{\rm m}}$ and a measurement uncertainty $\sigma$ of 0.087 µm. Furthermore, the ${8} \times {8}\;{\rm pixel}$ evaluation window containing around nine speckles also meets the measurement requirements and yields $\Delta x = 19.2\;{\unicode{x00B5}{\rm m}}$ and $\sigma = 0.038\;{\unicode{x00B5}{\rm m}}$. Here, the measurement uncertainty without ensemble average is 0.71 µm, demonstrating a reduction of $\sigma$ by over an order of magnitude. For larger evaluation windows, the reduction is slightly below one order of magnitude. The ${40} \times {40}\;{\rm pixel}$ evaluation window results in $\Delta x = 95.8\;{\unicode{x00B5}{\rm m}}$, and the measurement uncertainty is reduced from 0.061 µm to 0.008 µm.
Fig. 6. Displacement measurement uncertainty over spatial resolution with and without ensemble average. Solid black line indicates ${\sigma _{{\rm min}}}$ from Eq. (6) and dashed line $\Delta {x_{{\rm min}}}$ from Eq. (10). The measurement requirements for displacement field measurements of micro samples generated by electrohydraulic extrusion are shaded in green. A second abscissa on top shows the average number of speckles per evaluation window. Note the double-logarithmic scales where the square root function of ${\sigma _{{\rm min}}}$ is represented as a straight line (black).
To investigate if varying the evaluation window size leads to systematic errors in the displacement measurement, the mean of the displacement deviation $\Delta D$ is analyzed for different evaluation window sizes. $\Delta D$ is calculated by subtracting the true displacement value that is set on the piezo actuator from the ensemble averaged displacement field $D$. $\overline {\Delta D}$ is the mean deviation over the displacement field. Figure 7 shows the mean of $\overline {\Delta D}$ of 10 test series plotted over the spatial resolution. Error bars indicate the double standard deviation ($k = 2$) of the mean of 10 test series. As in Fig. 6, the diffraction limit is shown by a dashed vertical line. The results indicate that varying the evaluation window size causes systematic errors. The behavior of $\overline {\Delta D}$ over $\Delta x$ is analogous in other test series with different speckle pattern counts, even if the absolute values of the mean deviations vary. This behavior may be due to the evaluation algorithm, but the exact cause is not yet known. The mean deviations for $\Delta x = [14\;{\rm mm},50\;{\rm mm}]$ are below 10 nm and thus close to the linear repeatability of the piezo actuator (gray). Together with the significant reduction of the measurement uncertainty for all evaluation window sizes, the ensemble average therefore enables the use of small evaluation windows down to speckle counts of around five.
Fig. 7. $\overline {\Delta D}$ is the mean deviation of a displacement field $D$. The mean of 10 test series of $\overline {\Delta D}$ with error bars of the double standard deviation ($k = 2$) of the mean is plotted over $\Delta x$. The piezo actuator’s linear repeatability range of ${\pm}5\;{\rm nm}$ is shaded in gray. Note the logarithmic scale on the abscissa.
B. DMD Setup
Here, the number of speckle patterns is determined by the usable DMD area and the laser spot size, i.e., the number of micromirrors that are switched on. As the position of the laser spot changes, the light intensity on the sample also varies. Depending on the position of the camera’s field of view, a mask is used to limit the possible range of the active micromirrors. The mask prevents overexposed or underexposed speckle patterns, which would lead to a high measurement uncertainty. Additionally, the exposure time of the camera is adjusted to account for the lower radiation intensity of a smaller laser spot size. The parameters laser spot size, DMD mask, and exposure time are optimized for the measurement uncertainty of the displacement for both the individual speckle patterns and the ensemble averaged result.
Figure 8 shows the results from multiple test series of different laser spot sizes. Each curve represents the average of 10 measurement series. The spot sizes indicate the active micromirrors of the DMD. At a mirror pitch of 10.8 µm, a spot size of $100 \times 100$ therefore corresponds to an area of $1.08\;{\rm mm} \times 1.08\;{\rm mm}$. Since the range of possible active positions on the DMD remains the same, a smaller spot size means a higher number of speckle patterns. The relation of $\sigma$ to $n$ exhibits an asymptotic behavior (horizontal dotted-dashed black line). In this measurement setup, there also seems to be a limit beyond which a further increase in the number of speckle patterns does not significantly improve the measurement uncertainty. In addition, the first three curves (100, 90, 80) all have almost the same slope proportional to $1/\sqrt n$, i.e., the theoretical behavior for averaging uncorrelated measurements (dotted-dashed black line). The curves of the smaller spot sizes have progressively smaller negative slopes, indicating that their speckle patterns are not uncorrelated. For 95 speckle patterns (spot size $70 \times 70$ micromirrors on the DMD, corresponding to $756 \times 756$ µm) at a spatial resolution of 20 pixels or 47 µm, the measurement uncertainty is reduced from 0.180 µm to 0.014 µm or from 0.080 pixel to 0.006 pixel, respectively. For the used setup and desired field of view, $70 \times 70$ micromirrors appears to be the optimal laser spot size. However, by using an optimized combination of different spot sizes, the same measurement uncertainty might be achieved with a lower number of speckle patterns.
Fig. 8. Measurement uncertainty $\sigma$ of ensemble averaged displacement field over number of speckle patterns $n$ for different DMD spot sizes. Note the double-logarithmic scales where the square root function from the theoretic prediction is represented as a straight line.
In Fig. 9, a comparison of the two measuring setups is shown. The measurement uncertainty results from Figs. 5 and 8 are presented with error bars that show the experimental standard deviation of the mean of $\sigma$ over five test series for both the diffuser the DMD setup. The results from both measuring setups show comparable behavior. Both the linear section and the lower uncertainty bound for a higher number of speckle patterns mostly lie within each other’s error margins. Thus, both measuring setups provide similar and consistent results. However, the measuring frequency of the DMD setup is about six times faster than the rotating diffuser setup with the potential of being 1000 times faster because of the DMD’s 17.9 kHz switching rate. In addition, the near-perfect reproducibility of the DMD patterns is superior to the limited angular repeatability of the rotation mount.
Fig. 9. Comparison of the resulting measurement uncertainty $\sigma$ from both setup variants when ensemble averaging over increasing numbers of speckle patterns $n$. Error bars show the experimental standard deviation of the mean of $\sigma$ over five test series for both the diffuser and the DMD setup. Note the double-logarithmic scales.
5. CONCLUSION
By ensemble averaging over multiple uncorrelated speckle patterns, a method for DISC was introduced that significantly reduces the displacement measurement uncertainty without deteriorating spatial resolution. Two measuring setups were investigated that allow the generation and accurate reproduction of uncorrelated speckle patterns. Different speckle patterns are generated by rotating a glass diffuser in the illumination path in the first setup and by a DMD directing the laser at different positions on the diffuser in the second setup. With both measuring setups, a maximum number of uncorrelated speckle patterns of around 30 was realized. Up to this limit, the uncertainty of the ensemble average is proportional to the reciprocal of the square root of the number of speckle patterns, which agrees with theoretical behavior. For higher numbers of correlated speckle patterns, the achievable measurement uncertainty asymptotically approaches a lower limit, which depends on the size of the evaluation window and thus depends on the spatial resolution of the displacement field. The lower limit for the ensemble average is about one order of magnitude smaller than measurement uncertainty of a single measurement. This reduction in measurement uncertainty is achieved over a wide range of spatial resolutions from 10 µm to 100 µm.
Additionally, the ensemble average allows measurements even with very small evaluation windows containing only a few speckles. The requirements for measurement uncertainty are achieved for speckle counts greater than five. In the used setup with an optical magnification of 1.4 and a pixel size of 3.45 µm, an evaluation window of ${6} \times {6}$ pixels results in a spatial resolution of 14.4 µm and a measurement uncertainty of 0.087 µm. Thus, the ensemble average allows DISC displacement measurements on micro samples that are only several 100 µm wide. Further research should focus on the performance of the ensemble averaging method in the case of a decorrelation of the speckle pattern, i.e., in applications with large displacements or out-of-plane displacements.
Both measuring setup variants are suitable for generating uncorrelated speckle patterns. Therefore, the ensemble averaging method can be applied to a multitude of DISC applications. The rotation mount is much cheaper than the DMD. However, the DMD can generate speckle patterns at a rate of 17.9 kHz and thus allows for a more than 1000 times higher measuring frequency than the rotating diffuser setup. By optimizing DMD control and high-speed camera triggering, hundreds of speckle patterns could be generated and measured in less than a second to enable in-process measurements.
Funding
Deutsche Forschungsgemeinschaft (DFG) (SFB/TRR136).
Acknowledgment
The authors gratefully acknowledge the financial support by the DFG for subproject C06 “Surface optical measurement of mechanical working material loads” within the Transregional Cooperative Research Center. They also thank F. Glandorf for his support in setting up and measuring with the DMD.
Disclosures
The authors declare no conflicts of interest.
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