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This paper presents a visual measurement method able to sense 1D rigid body displacements with very high resolutions, large ranges and high processing rates. Sub-pixelic resolution is obtained thanks to a structured pattern placed on the target. The pattern is made of twin periodic grids with slightly different periods. The periodic frames are suited for Fourier-like phase calculations—leading to high resolution—while the period difference allows the removal of phase ambiguity and thus a high range-to-resolution ratio. The paper presents the measurement principle as well as the processing algorithms (source files are provided as supplementary materials). The theoretical and experimental performances are also discussed. The processing time is around 3 μs for a line of 780 pixels, which means that the measurement rate is mostly limited by the image acquisition frame rate. A 3-

Numerous scientific fields such as micromechatronics and instrumentation for micro-, nano- and bio-technologies require a high accuracy in the tracking of objects as well as in the manipulation of actuators and stages. For such position and displacement control purposes at the microscopic scale, vision-based approaches are often found to be the best suited methods. The resolution of the camera combined with the imaging magnification is then an important performance limitation parameter with a general consequence: the higher the accuracy, the lower the range. Numerous sub-pixel motion detection algorithms have been proposed to relax this kind of trade-off. A common practice is the phase correlation method. To achieve the translation estimation at sub-pixel level, some researchers use peak interpolation methods in the spatial domain [

High accuracy can also be obtained through phase computations by means of Fourier-like processing applied to a periodic grid. Despite phase-to-displacement conversion providing high measurement resolution, a main drawback comes from the 2

Another way to overcome the 2

This paper presents the implementation of this principle for the visual measurement of 1D rigid body displacements with very high resolutions, large ranges and high processing rates. The method is based on a pattern made of twin periodic grids that allows for phase calculations while the period difference is used to extent the measurement range.

The next section introduces the measurement principle in detail. Section 3 presents the processing algorithms (source files are provided as supplementary materials). Afterwards, we discuss the theoretical capabilities of the method as well as the experimental results obtained.

The phase-to-displacement relationship is widely known. If an object _{i}_{f}

where _{i}_{f}_{i}_{f}^{2} = −1.

Thanks to this linear relationship, the object displacement is encoded in the phase of the Fourier transform and different approaches have been proposed for the measurement of displacements through this phase term. They differ mainly in the number of considered spectral components. If the whole Fourier spectrum is considered, an unambiguous value of Δ can theoretically be retrieved. However, the right combination of phase constants 2_{ν}π

In computer vision, the target displacements are retrieved through the processing of images captured by a static camera observing the moving object. The simplest way to apply phase computation to this task consists in associating some kind of periodic pattern to the target and thus to get periodically structured images for processing. This is illustrated in

where

The use of a second stripe set with a slightly different period provides complementary and independent phase data that can be used for the removal of phase ambiguities. The principle is illustrated in

where _{1}_{2}_{1} = 6 and _{2} = 10, the equivalent period is Λ = 15 whereas the perfect reproduction of the stripe sets is only obtained at 30.).

From a computational point of view, the phase relative to the synthetic period Λ is simply given by the subtraction between the two elementary phases (

in which ΔΦ_{1} is the phase shift for the chosen stripe set, _{1} the number of periods derived from the synthetic phase and _{x}_{1}.

Theoretically, Λ can be chosen as large as we want (_{1} and _{2}. We notice however that large values of Λ are obtained with small period differences. In such cases the effects of noise are significantly enlarged and, at some level, the determination of _{1} (

In

In the following, we present an implementation for 1D lateral displacement measurement in a micro-mechatronic context as well as the performances obtained. This principle and the software provided can however be applied to different purposes, especially at other dimensional scales.

We assume that a two-stripe pattern is attached to the target of interest in order to measure its displacement.

The determination of the phase associated with each image of this kind and representative of the target position at the recording instant of time assumes three successive steps described in the following subsections: extraction of intensity distributions from each stripe set; determination of the period in pixel of each stripe set; computation of the phase of the central pixel for both stripe sets

This preprocessing task aims to determine the image lines to be used for the phase computation for each stripe set. Since our aim is to measure only 1D displacements, the hardware is set in such a way that the stripes are perpendicular to the displacement and the camera lines are parallel to the stripes. With this setting, stripes move only in the horizontal direction of the recorded images and this task has to be performed only once. Since the used pattern is repetitive, there are several possibilities and for instance the stripes sets between the red lines can be chosen (_{1}(_{2}(_{1} and _{2} respectively.

This second step aims to determine the spatial frequency of each stripe set in the recorded images. This task has also to be performed only once since the imaging magnification is not affected by 1D lateral target displacements. Furthermore

Since the log of a Gaussian produces a parabola, we use a quadratic interpolation on the points around

The stripe period _{1} and _{2} are obtained.

Once the preprocessing described above is done, the aim of the phase computation is to determine the phase associated with both stripe sets as fast as possible. Then instead of performing a complete DFT we compute only the two phase terms of interest. For that purpose we use a complex analysis function _{1}(

We notice that the analysis functions have also to be defined only once. The expected phase Φ_{1} is then given by the argument of the sum Σ_{1} over all pixels of the product of this windowed analysis function _{1}(_{1}(

The phase Φ_{2} related to the second stripe set is computed in the same way by applying this procedure with _{2} instead of _{1} and _{2}(_{1}(_{1} − Φ_{2} which has to be unwrapped in the interval ]−

For the current image, after the computation of phases Φ_{1}, Φ_{2} and Φ, a coarse displacement Δ is given by _{1} of

We get

From a theoretical point of view, _{1} returned by _{1} (or _{2}) instead of that of the synthetic period Λ is obtained by rounding (_{1} = _{1}) where _{1} to the closest integer _{1}. The high-resolution displacement measurement is finally provided by inserting the computed value of _{1} in _{x}

Without considerations for the material used, the ultimate capabilities of the method would be determined by the physics phenomena involved and, in our case, would imply quantum statistics of the light source of illumination and of photon conversion on the image sensor. Such fundamental sources of noise are responsible for extremely low error levels as expected for instance in gravitation wave detection interferometers. In the case of applications based on scientific grade instrumentation as aimed here, the ultimate performances are determined by material and environmental specifications. The most influential error sources in such visual experiments are thus due to digitizing and to the signal-to-noise ratio of the detected images. Environmental disturbances are also detrimental to high resolution displacement measurements and may be found to be the most restrictive parameter. They remain however independent of the intrinsic method capabilities and thus form extrinsic error sources.

Our method capabilities were evaluated through computations aimed to reconstruct the position of sets of computer generated images with chosen and perfectly known grid positions. Perfect square grids were used in the image design and light diffraction effects were not considered. This point is not crucial since a low pass filtering due to light diffraction would not affect the first harmonic of the square grid spectrum that is actually used in image processing (

Differences observed between the reconstructed positions and the ones used for image generation provide the actual method detection errors. ^{−6} pixel between consecutive images. The green curve corresponds to the most favorable case involving only the computing noise. We can see that the expected linear displacement is perfectly reconstructed and in fact, the 10^{−6} pixel step is already too large to make the numeric noise effects visible. The blue curve corresponds to an image digitizing over 8 bit depth and we can see that the highest consecutive steps are of about 0.4 × 10^{−3} pixel (^{8}). The standard deviation of digitizing errors is of 10^{−4} pixel at this digitizing depth. The red curve was obtained without digitalization but by adding a gaussian noise to the generated grid with a standard deviation equal to 1% of the square amplitude. The resulting errors present a different distribution but approximately with the same peak-to-peak amplitude. The error standard deviation due to gaussian noise is equal to 9.6 × 10^{−5} pixel. These simulation results determine the best resolutions that can be expected experimentally with these parameters that correspond to typical specifications of usual image sensors.

If the period increase is due to an increase of the vision system magnification;

These numerical evaluations of the method capabilities can be used to match the setup design with the expected system performances. For instance, if sufficient free space has to be kept between the lens and the target, a larger grid period can be chosen to be suited with a low magnification objective and a large working distance. The resolution loss due to period increase can then be compensated by the use of a higher grade camera with improved signal-to-noise ratio and digitizing depth. Such an elementary period increase may also be chosen to enlarge the unambiguous measurement range as defined by

_{1} after compensation for 2

Finally the experimental setup was used to characterize the free oscillations of a compliant shuttle as shown in ^{3} Hz on this machine.

In order to permit the reproducibility of results and the development of new applications, we provide the source code of the presented algorithms within a library called VERNIER along with the paper. The library is distributed under the GNU General Public License in the hope that it will be useful. The package can be downloaded from the website of MDPI (

While developing VERNIER, our goal was to allow a portable (independent from the hardware), fast, and reliable code. We also wanted to provide a package that is suitable for real-time implementation and that allows to obtain good performances with both simulations and real experiments from the same code. Therefore, we chose the C++ language for the implementation. VERNIER is built upon the cross-platform OpenCV library that provides a large compatibility with many image processing applications. OpenCV (

VERNIER is written following C++ Coding Standards with the aim to be easily understandable and reusable. Describing the full implementation of the library is not the purpose of this article. For more information, a complete documentation is provided with the package.

The visual measurement of 1D displacements with a large range-to-resolution ratio as well as a high frequency rate suited for real time applications is presented and demonstrated in this paper. The subpixelic performances derive from phase computations applied to the images of periodic grids whereas the large unambiguous range is obtained thanks to twin stripe sets with slightly different periods. A 3-

The performance level can be adapted to the requirements of the final application in different ways. Firstly the method can be applied to the nanometer, the micrometer or the millimeter ranges (or even larger) by matching the actual values of the grid periods and of the vision system magnification accordingly. These conversion parameters from the object space domain to the image domain remains indeed ignored by the image processing routines that consider images only. Secondly as presented in the principle and numerical performance sections, the method resolution and unambiguous measurement range can be chosen almost independently from each other through the suitable choice of the grid period difference and of the camera signal-to-noise ratio and digitizing depth; provided that environmental disturbances remain sufficiently low. Thirdly, the vision system magnification does not have to be calibrated since the knowledge of the grid periods provides a size reference that is sufficient to convert pixels into actual distances in the reconstructed displacements.

These specifications make the method very attractive for a wide range of applications especially in robotics and automation. At present time a single direction is addressed and this point is one of the few limitations of the method. The extension of the proposed principle to measurements versus multi-degrees of freedom can however be envisaged for instance by using stereovison and/or multiple grid patterns. In this prospect, the capabilities of grid processing to comply with defocus is of particular interest [

This work was supported by the Smart Blocks project (ANR-251-2011-BS03-005) and by the Labex ACTION project (ANR-11-LABX-01-01). Authors acknowledge the French RENATECH network through its FEMTO-ST technological facility MIMENTO.

This paper is a joint work and all the authors have contributed.

The authors declare no conflict of interest.

Correspondence between lateral position and phase of a sinusoidal pattern. (

Extended unambiguous range by means of two stripe sets with different periods. (

Experimentally recorded image of the twin stripe set pattern on the target.

Example of intensity vector (

Analysis function used for phase computation.

Illustration of the quantization principle applied to a perfect square grid; case of 8 bits sensor, with pixels encoded between 0 and 255.

Simulated position reconstruction as a function of numeric noise (green), signal quantization (blue) and presence of noise (red); period of 51.123 pixels, computation step of 10^{−6} pixel.

Ultimate resolution in pixels as a function of grid period and quantization depth (At each period, maximum error observed on 10^{3} positions shifted by 10^{−6} pixel).

Ultimate resolution in picometers as a function of grid period and quantization depth (At each period, maximum error observed on 10^{3} positions shifted by 10^{−6} pixel, with a grid of period 8 μm).

Experimental setup used for method demonstration.

Measurement of linear displacement of the pattern. 780 × 580 pixels image. (

Shuttle used as target in real-time experiments.

Reconstructed shuttle displacements in response to a starting pulse. (