Optimal speckle size and signal processing for displacement measurements using a 4f optical setup

Fully developed laser speckle patterns are, due to their high contrast and statistical nature, well suited to measure strain and displacement via an appropriately designed measurement system. Laser speckle patterns are formed when a sufficiently coherent light source, such as a HeNe-laser, illuminates an optically rough surface. Therefore, methods based on laser speckle patterns can be applied to any surface scatterer with a minimum mean surface roughness of about a quarter of the laser’s wavelength. This includes also materials such as thin natural and technical fibres as well as foils, for which the presented measurement system, including the digital signal processing, was designed. In order to achieve the best possible resolution of a speckle-based measurement system, combined with a sufficiently small measurement uncertainty, all available design parameters must be optimised. One of these parameters is the speckle size, which is dependant on the properties of the imaging optics. In this paper a subjective laser speckle-based measurement system based on a so-called 4f-optical setup is presented. This setup allows the speckle size to be controlled in both axial and lateral dimensions separately, which is achieved with the help of an aperture in the Fourier plane of the optics. It is shown that the optimal speckle size for the presented measurement system, not only depends on the physical setup, but also on the signal processing applied. The signal processing routine estimates displacements of the speckle pattern, leading to an estimate for the strain. Additionally, it is demonstrated that the optimal speckle size can be lower than the commonly reported optimum between two and five pixel pitches, necessary to circumvent aliasing in the image data. While this is shown for a measurement setup using 4f-optics, the results are of general importance to speckle-based strain or displacement measurement systems and should thus be taken into account.


Introduction
Laser speckle patterns are randomly looking patterns which occur if sufficiently coherent light, such as the light of a HeNelaser, hits an optically rough surface.Examples of speckle patterns can be seen in figure 1 [1].In many applications, such as vibrometers, these patterns are considered as noise which needs to be sufficiently suppressed [2,3].However, they also carry very useful information about the scattering surface and can be used-among other possibilities-to track the displacement, rotation, or strain of the illuminated surface area [4][5][6].
To achieve the best possible resolution, when measuring one of the aforementioned quantities, it is necessary to look at every possibility to improve the measurement system.One important and optimisable speckle characteristic, which can be controlled by the measurement system, is the speckle size.Two simulated speckle-patterns with different speckle sizes can be observed in figure 1.The images show, that the directional speckle size can vary drastically.In figure 1(a) a 4f -optical system with a square aperture has been simulated, whereas figure 1(b) is the result of the same optical system implementing a rectangular aperture.Hence, when designing a speckle based strain or displacement measurement system, the question of an optimal speckle size-and how to achieve it-arises.In literature, such as in papers concerning the closely related domain of digital image correlation, often times a range in which the optimal speckle size lies, instead of a singular value for the optimum speckle size, is given.Following the literature, the optimal range-for strain and displacement measurement applications-seems to be the following: h Speckle = 2 . . . 5 • d PixelP,y [7].To minimise the dependency on a given imaging sensor, the speckle size is described as a multiple of the directional pixel pitch d PixelP,y .In this paper it is shown that the range of optimal speckle sizes can be extended below the aforementioned lower limit.It is also shown, that speckle size alone is a rather insufficient parameter and one should, depending on the signal processing routine used, consider the power spectral density of the intensity of the observed speckle pattern.The theory and simulation results presented in this article are adapted to a measurement system based on optical parameters 4f -that was developed to estimate the strain in thin fibres, including natural fibres such as human hair and technical fibres.However, we believe that the two key results: • optimal speckle size depends on the used signal processing algorithm • and an ideal speckle size might be smaller than two pixel pitches are of more general nature and might be useful for many other speckle based measurement systems.Throughout this paper fully developed, subjective laser speckle patterns are assumed.This requirement is met by most materials, since fully developed laser speckle patterns only require the surface roughness to be greater than a quarter of the laser's wavelength.This paper is structured as follows: first, the general measurement principle as well as the measurement setup is presented in section 2. In the subsequent section 3 a general definition for speckle size is presented and the theoretical foundation for its discussion within the 4f optical setup is given.In section 4 it is shown-with the help of simulation results-that the optimal speckle size depends not only on the measurement setup but also on the signal processing applied.Therefore, in order to optimise the measurement system as a whole, codevelopment of the optics and the signal processing is strongly advised.It is shown, that the optimal speckle size extends below the above mentioned minimum speckle size, also supported by aliasing concerns, of h Speckle = 2 • d PixelP,y .A summary and conclusion is given in section 5.

Measuring strain using a 4f-optical setup
The plethora of possibilities of measuring strain based on speckle patterns also lead to multiple different optimal speckle sizes [7,8].Speckle patterns used in cases where one is interested in different measurands do not necessarily share the same optimal speckle size [9,10].Hence, when referring to an optimal speckle size, describing the application in which the speckle patterns are involved is important.Therefore, this section deals with a brief description of the measurement setup and the application for which an optimal speckle size is determined in later parts of this article.
The overall goal of the measurement system is to measure strain in a wide range of fibres, including natural fibres such as human hair, but also technical fibres such as carbonbased ones.Laser speckle patterns are particularly interesting for these applications since they allow for non-contact strain measurement and therefore do not interfere with the sample itself even if it were of elevated temperature.Additionally, they offer the possibility to measure strain when the sample is placed in a high temperature environment.Technical fibres can be very stiff, therefore high strain resolution in the range of ∆ε yy = 5 µmm is necessary to be able to yield a decently resolved stress-strain curve.As will be motivated by the following paragraphs, this results, considering an initial spot distance d Spot = 20 mm, in a necessary displacement resolution of speckle patterns of ∆a K,y = 0, 1 µm.Here, the subscript K denotes the pattern displacement in the image plane.The displacement resolution needed to achieve our goal equates only to a fraction of a typical pixel pitch ∆a K,y = 1 70 d PixelP,y .Therefore optimising all available parameters, including speckle size, is critical.
Strain measurement based on laser speckle pattern can be achieved by a variety of approaches such as laser speckle interferometry or several different speckle imaging methods.Here, a speckle imaging technique is chosen, which makes use of indirect strain measurement utilising local surface displacements.The general idea of indirect speckle imaging methods is comparable to the one of video extensometers, in which-at least two-surface markers are applied to the sample and their displacements are tracked.The difference in displacement, as an approximation for the sought displacement gradient, with knowledge of the initial distance d Spot which is the distance separating the centres of two marked local surfaces, lead to an estimate of the engineering strain ε yy .The engineering strain is given by a P,y|1 and a P,y|2 denoting the displacements of two selected local surface elements in the direction of interest.Hence, to measure the strain, the displacements of both local surface elements need to be estimated with very low uncertainty.If laser speckle patterns are used as surface markers, then the displacement of the observed speckle pattern a K,y|1 and the displacement of the surface element a P,y|1 can be expressed by a function a P,y|1 = f(a K,y|1 , κ).Here, κ denotes a vector of parameters, such as the rotation, displacement, or elongation of the sample.The function as well as the parameters influencing the speckle behaviour changes depending on the optical setup.In his article [11] Yamaguchi presents three different optical setups and their respective functions f(a K,y|1 , κ).
One of these setups is the so called 4f -optical setup, depicted in figure 2. The optics consist of two lenses and an aperture/filter placed in the back-focal plane of the first lens L 1 , coinciding with the front-focal plane of the second lens L 2 .
If the sample is placed in the front-focal plane of the first lens L 1 , then the analogue two-dimensional spatial Fourier transform of the object appears in the back-focal plane of the same lens [12].This setup comes with the advantage of a fixed demagnification factor, independent of camera and sample position, of M = 1.000, as well as the possibility to adapt the speckle patterns power spectrum and therefore the speckle size.Additionally, due to its symmetry, no image distortions can occur.For this setup, assuming good sample positioning as well as a sufficiently large radius of curvature of the wavefront of the illuminating beam then leads to the following equation Equation ( 2) relates the observed shift of speckle patterns in the image plane a K,y|1 to the actual shift of the corresponding local surface a P,y|1 .A more complete and general discussion, without the aforementioned, simplifying assumptions, is given in [11].
The developed measurement setup consisting of a HeNelaser, illuminating the fibre sample, the 4f optics and a line scan charge-coupled device (CCD) camera is shown in figure 3. The corresponding schematic of the setup, is depicted in figure 4.
The measurement system makes use of stretched, elliptical speckles, with the semi-major axis being oriented in xdirection.Therefore, the systems sensitivity to any displacement or strain of the sample in x-direction is reduced.Only the directional strain ε yy in y-direction is of interest, since fibres are to be measured and loaded along this direction.These elongated speckles then allow for the use of a line scan camera, opposed to an area scan camera.This results in faster signal processing and the possibility of real-time measurements.Exemplary speckle images, taken with the help of the above described setup are shown in figure 5.The figure displays a subset of the Sony ILX526A 3000 pixel.Each horizontal line in the image corresponds to a single line scan image of the speckle pattern of a human hair.The evolution of the speckle pattern is tracked over time.Between each line scan image the sample was shifted using a high precision translation stage by a im = 0.1 µm.

Speckle size as a influencing parameter
The speckle size is an important measure describing the spatial dimensions of observed speckles.In this paper, we follow closely the definition given by Goodman in [1].The transversal speckle size, in which we are interested, is defined as the total area A c below the normalised correlation coefficient Γ UK of the field in the image plane U K .Here and in the following, fields always describe scalar fields according to scalar diffraction theory.The speckle size can then be expressed as in which the normalised correlation coefficient only depends on the difference in coordinates ∆y K and ∆x K and not the coordinates themselves.This is the case when the surface height profile is wide sense stationary, meaning neither the mean height nor the auto-covariance of the surface does depend on surface coordinates.Wide sense stationary surface height profiles are assumed in the following.The normalised correlation coefficient is given by with the autocorrelation function of the field in the image plane (5) In equation ( 5) it becomes apparent, that speckle size depends on the optical transfer function of the system h B and the field in the object plane U P .The operation • • • denotes the calculation of the expected value and (• • • ) ⋆ the complex conjugate.The coordinates with subscript K are the ones in the image plane, whereas the subscript P indicates the ones in the object plane.Therefore, to be able to achieve a specific speckle size, the knowledge of the transfer function of the optical system is of great importance.
The transfer function of the 4f optical system is given by the following equation if the following conditions and simplifications are assumed • constant scaling factors are neglected, • constant phase factors are neglected, • lenses are infinitely large, • Fresnel-approximation is applicable, • the light source is quasi-monochromatic, • and scalar diffraction theory is applicable.
The function P(f x , f y ) describes the, possibly complex valued, limiting aperture, placed in the Fourier plane of the setup expressed with transformed coordinates f x = xF λl f where x F denotes the x-coordinate in the Fourier plane and l f the focal length of both lenses.The coordinate transform is used to make the Fourier transform property of the system easily observable to the reader.The transfer function also depends on the wavelength λ as well as the misalignment d fP and d fK of the sample and the camera.In the following it is assumed that camera and sample are perfectly placed in their respective planes.In case of perfect alignment, the system performs two times the forward Fourier transform, one for each lens' operation.Hence, the resulting image is an inverted image of equal size of the original object.
One approach to describe the still missing part of equation ( 5) which is the autocorrelation function of the field in the object plane, is to make assumptions about both the lighting and the surface.These assumptions include, among others: the autocorrelation function and the probability density function of the height distribution as well as the general geometric setup.Following this approach is rather time consuming and depending on the surface parameters not analytically solvable.Therefore, often times a second approach, where one assumes directly the autocorrelation function of the field in the object plane, is employed [1,11].Simulations confirmed that, even for notable correlation lengths of the surface height distribution, the resulting fields in the image plane do not differ significantly between the two methods.Therefore, the direct approach is chosen in the following.It is assumed that the correlation length of the surface height distribution of the specimen is infinitely small and can be described by a twodimensional Dirac distribution δ(• • • ) [13] as follows R UP (∆x P , y P ) = δ (∆x P , ∆y P ) , (8) where constant scaling factors, including the intensity distribution of the light illuminating the surface-which for ease of derivation is assumed to be constant-, are omitted.In our measurement system, a simple rectangular aperture stop with height h F in the y-direction and width b F is placed in the Fourier plane.Mathematically, the real-valued aperture can be described by the following equation It is assumed that the aperture centre lies on the optical axis of the system and the cut-off frequencies in the corresponding directions are given by Using all of the above one arrives at the following expression for the autocorrelation function of the field in the image plane Using equations ( 3) and ( 4) the average speckle size A c follows In order to get two separable one-dimensional measures, here we propose to split the equation into its two principal components leading to directional speckle sizes of This result shows that by controlling the dimensions of the rectangular aperture one can control the speckle size.By decreasing the height or width of the aperture in the Fourier plane, one increases or decreases the speckle size in the corresponding direction and vice versa.So, the speckles displayed in figure 1 are the result of a horizontal slit placed in the aperture plane.Both the wavelength of the laser, λ, and the focal length of the lenses, l f , also influence the speckle size; however, they tend to be fixed values for a given measurement system and can not be changed as easily as the aperture dimensions.This result is in line with the classical, well known results given by Fourier optics and Fourier transform properties [12].While a definition of the speckle size and its evaluation for a 4f -optical system can be of use, the parameter itself lacks crucial information which can be revealed by taking a look at the power spectral density of the observed intensity G I (f K,x , f K,y ).Therefore, the power spectral density for the 4foptical setup is derived in the following paragraphs.For a wide sense stationary surface, one can-with help of the Wiener-Khinchin theorem [14]-express the power spectral density of the intensity with the following equation Here, f K,x and f K,y are the spatial frequencies of interest which are related to their respective direction of observation x K and y K .If fully developed speckle patterns are assumed, one can show [1] that the autocorrelation function of the intensity is linked to the autocorrelation function of the field in the image plane, described by equation ( 12), as follows Equations ( 12), ( 16) and ( 17) result in the power spectral density of the intensity in the image plane in case of the 4f -optical setup with rectangular aperture, given by f By (18) and displayed in figure 6.In equation ( 18) one can observe, that the power spectral density is dominated by a two dimensional, scaled triangular function in the 2D frequency plane.
The triangular functions are given by the following definition, interchanging only the directional variables As can be seen-in case of a rectangular aperture-the spectral contents of the power spectral density of the intensity in the image plane does not extend beyond the inverse of the directional speckle sizes.However, if the aperture's shape is changed, the shape of the power spectral density can drastically change as well.When optimising a laser speckle based system, this should be kept in mind, since also the shape of the power spectral density might be considered as a parameter to adapt.
In the following section it will be shown that if the shape of the spectral density is considered, contrary to just the speckle size itself, an ideal speckle size below h Speckle = 2 • d PixelP,y can result to be optimal.

Optimal speckle size for the 4f-optical setup
It is important to realise, that a general optimal speckle size for the estimation of the displacement of a speckle pattern does not exist.An optimal speckle size exists only for a specific set of parameters such as camera noise, the optical system, as well as the digital signal processing applied to estimate the shift.It follows that, ideally, when designing the optical system the dimensions of the aperture which is placed in the Fourier plane, the digital signal processing and the noise characteristics of the CCD camera are already known.In the following paragraphs, this statement will be elaborated upon.
In figure 7 the normalised amplitude of the power spectral density of three speckle patterns with varying speckle size is depicted.If the speckle size falls below two pixel pitches, which is equivalent to the limiting frequency of the spectrum f B (equating or) exceeding half of the sample frequency f samp , then-according to the Nyquist-Shannon sampling theorem-aliasing occurs.In case of the often times employed generalised cross correlation [15] to evaluate the displacement of speckle patterns, aliasing can reduce the accuracy of the resulting estimate.The same argument applies to Fourier domain methods if aliasing is not taken care of.This then leads to the reported [7,16] minimum optimum speckle size of h Speckle = 2 • d PixelP,y .However, if aliasing is taken into account, then it can be shown that choosing speckle sizes even below that minimum might be beneficial.This can be generally done in the Fourier domain by either ignoring bins which are known to be aliased or trying to de-alias them.For simplicity, the first method is chosen and discussed in this section.
However, before discussing the signal processing in detail the general idea of why it could be of benefit to chose an aliased speckle signal is described in the following.
The noise which predominantly limits the attainable shift resolution consists of shot noise, caused by the random nature of light quanta, and camera noise.Camera noise includes different noise components-for extensive noise models, see e.g.[17,18]-which are, among other parameters such as chip temperature, highly dependent on the exposure time.Since exposure time, chip temperature, and the number of light quanta collected by the camera can be kept constant by adjusting the light source, camera noise as well as shot noise do not change notably for different speckle sizes.Hence, also their power spectral density remains unchanged.However, when changing the speckle size, the power spectral density of the observed intensity changes.This effect can be seen in figure 7 and is described by equation (18).The broader the bandwidth of the imaged speckle pattern, the smaller the slope of the amplitude of its power spectral density.It follows that the signalto-noise ratio for lower frequencies increases.Therefore, the information in these, non-aliased, low-frequency bins is more reliable and potentially leads to better shift estimates.In figure 5 the range for which this applies is highlighted with the help of a light blue background.
In order to estimate the shift between two speckle images the following signal processing routine is applied: In equation ( 20) not all frequency bins are utilised, instead only these between k start and k end are considered.Choosing k end allows for excluding aliased bins, so in case of f B = 0.75f samp a reasonable choice could be k end = floor{0.25fsamp }.A more detailed description of the measurement system and the algorithm can be found in a previous paper [22].The optimal speckle size for the presented system and signal processing is determined with the help of the simulated speckle pattern.First the field in the object plane U P (x P , y P ) is established.The field is calculated with the assumption of equation ( 8) and of TEM00 illumination at λ = 632.8nm.The dimensions of the laser spot are chosen such that they fit those of the real laser.The field is then propagated through the system according to the point spread function given in equation (6).The intensity in the image plane I K (x K , y K ) of the speckle pattern is calculated and noise is added.Both shot noise and additive, white Gaussian noise (AWGN) are added.The shot noise is calculated for each pixel with the help of the estimated full well capacity of the sensor and the AWGN is chosen to mirror the sensor performance for a chosen exposure time of t Exp = 25 ms.The exposure time differs from the exposure time used in figure 5.This is due to an adaptation of the exposure time to a second HeNe-laser with different output power, which was used in the referenced experiment.Resulting speckle patterns with a Gaussian-shaped intensity profile are presented in figure 8.The Gaussian-shaped intensity profile with comparable width is also visible in figure 5.The presented region of the image sensor is smaller in case of the simulation results.Shifts between successively taken line scan images are 0.1 µm in both figures.Since the shift between each line is only about 1 70 d PixelP,y , which equals the aimed for resolution of the measurement system, the shift between the 101 line images depicted in figure 8 is only slightly notable.
In figure 9 the mean square error is used as a measure to determine the performance of the estimator for different speckle sizes.The mean square error is evaluated for all shifts for each speckle size.For each speckle size, five different speckle patterns were simulated.We would have liked to provide many more different speckle patterns per speckle size to reduce variance in the result, however the simulation time, using a standard PC, did not allow for many more.Furthermore, as can be seen in figure 9 the difference between the evaluation of a single speckle pattern and the result of the average speckle pattern is rather negligible.Both, the averaged mean squared error as well as the single realisation based one clearly show, that speckle sizes smaller than h Speckle = 2 • d PixelP,y can be considered to be in the optimal range.Here, a size in the range of h Speckle = 1.5 . . .4.5 • d PixelP,y is as a rather good choice for speckle size, when signal processing is accordingly adapted.With the noise model applied, the power spectral density expressed by equation (18) and the presented signal processing, the best performance has been achieved for speckles of size h Speckle = 1.6 • d PixelP,y .
Additionally, experimental results, presented in a previous paper [22], with fixed speckle size of approx.h Speckle = 2 • d PixelP,y , show a MSE which is, even though slightly higher at 0.027 µm 2 , in good agreement with the simulation.The difference is most likely caused by correlated noise present in the experimental data.

Summary
In this paper we have shown that, for displacement estimation from discretised image data of laser speckles, the optimal speckle size can be smaller than the Nyquist-limit, h Speckle = 2 • d PixelP,y , demands.Additionally, it has been shown that signal processing needs to be taken into account when evaluating the optimal speckle size and that rather than just looking at speckle size it is beneficial to consider the resultant power spectral density of the intensity of the observed speckle pattern.These results have been demonstrated with help of a realised measurement system for the measurement of strain in thin fibres.A core component of the measurement system is the 4f optics, used to perform optical Fourier filtering.Analysing the optical system is key in understanding speckle size and the results which can be obtained by varying this parameter.

Figure 1 .
Figure 1.Simulated speckle patterns, showing the important possibility to influence the speckle size by modifying the optical setup.Here, the speckle size h Speckle in y-direction is varied, whereas their size b Speckle in x-direction is kept constant.

Figure 2 .
Figure 2.In order to control speckle size, as well as assuring a constant demagnification M, a 4f -optical setup is used.The back-focal plane of the first lens L 1 is also called Fourier plane, since in this plane one can observe the analog spatial two-dimensional Fourier transform of the object placed in it is front-focal plane.

Figure 3 .
Figure 3.A HeNe-laser illuminates the sample (human hair) mounted on a piezo translation stage.The resulting speckle pattern is observed with help of a line scan CCD camera and a 4f optical system.

Figure 4 .
Figure 4. Schematic of the setup in figure with an angle of α = 45 • between the incoming laser and the optical axis.The laser is parallel to the x-z-plane.The focal length of both lenses is chosen to be l f = 102 mm.

Figure 5 .
Figure 5. Speckle images taken with the experimental setup.Each horizontal line is the result of a speckle pattern captured by an Eureca™ line scan camera which makes use of Sonys ILX526A image sensor.An exposure time of t Exp = 20 µs was set in the self-developed real-time linux driver.

Figure 6 .
Figure 6.Theoretical power spectral density of the observed intensity of a laser speckle pattern imaged through a 4f -optical setup with rectangular aperture in its Fourier plane.The spectrum includes no higher frequencies than f Bx and f By respectively.

Figure 7 .
Figure 7. Normalised amplitude of the power spectral density |GI( f )| |GI(0)| for different speckle sizes.In case of f B = 0.5fsamp the speckle size is smaller than two pixel and therefore aliasing in the spectrum occurs.
(i) Estimate centre and position as well as width of the laser spot.(ii) Limit data to region of interest.(iii) Calculate the cross power spectral density G im [f] of the reference image b i [n] and the image of the shifted pattern b m [n] using Welch's method, first presented in [19].(iv) Calculate weights according to the Hannan-Thompson window.(v) Estimate shift based on the slope of the phase of the cross spectral density.The slope is estimated using a method based on Kakarala & Cadzow's article [20].The actual shift estimator âim , (• • • ) denoting estimate, described by the last step is given byâim = d PixelP,y • N DFT 2π • arg max φ ∈[−π,π) Re k=k end k=kstart G 2 mod [k] exp [−ı2φ k](20) where the index K, y|1 used in equation (2) has been dropped for better readability.In equation (20) N DFT denotes the selected length of the discrete Fourier transform, typically

Figure 8 .
Figure 8. Simulation of a shifted speckle pattern with speckle size h Speckle = 2d PixelP,y .The shift between two neighbouring line scan images b i [n] and bm[n] is barely, if not at all, discernible since it equates to only a im = 0.1 • d PixelP,y to the left.

Figure 9 .
Figure 9. Mean squared error of the displacement estimation for different speckle sizes of simulated speckle patterns.It can be seen that with adequate signal processing, the optimal range of speckle size extends below the commonly reported minimum.