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Phase-shifting of correlation fringes created by image processing as an alternative to improve digital shearography
Optics Communications 380 (2016) 114–123
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Phase-shifting of correlation fringes created by image processing as an alternative to improve digital shearography Roberto A Braga Jra, Rolando J González-Peña b,n, Marlon Marcon c, Ricardo R Magalhães a, Thiago Paiva-Almeida a, Igor V.A. Santos a, Moisés Martins a a
Universidade Federal de Lavras, Dep. Engenharia, Lavras CP 3037, MG, Brazil Universidad de Valencia, Unidad de Biofísica y Física Médica, Departamento de Fisiología, Facultad de Medicina y Odontología, Valencia CP 46010, Spain c Instituto Federal Minas Gerais, Dep. Engenharia e Computação (DEC), Câmpus Bambuí (IFMG), Brazil b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 January 2016 Received in revised form 28 May 2016 Accepted 1 June 2016
The adoption of digital speckle pattern shearing interferometry, or speckle shearography, is well known in many areas when one needs to measure micro-displacements in-plane and out of the plane in biological and non-biological objects; it is based on the Michelson's Interferometer with the use of a piezoelectric transducer (PZT) in order to provide the phase-shift of the fringes and then to improve the quality of the final image. The creation of the shifting images using a PZT, despite its widespread use, has some drawbacks or limitations, such as the cost of the apparatus, the difficulties in applying the same displacement in the mirror repeated times, and when the phase-shift cannot be used in dynamic object measurement. The aim of this work was to create digitally phase-shift images avoiding the mechanical adjustments of the PZT, testing them with the digital shearography method. The methodology was tested using a well-known object, a cantilever beam of aluminium under deformation. The results documented the ability to create the deformation map and curves with reliability and sensitivity, reducing the cost, and improving the robustness and also the accessibility of digital speckle pattern shearing interferometry. & 2016 Elsevier B.V. All rights reserved.
Keywords: Phase-shift algorithm Speckle interferometry Piezoelectric transducer Digital speckle pattern shearing interferometry
1. Introduction Digital speckle pattern interferometry (DSPI) and digital speckle pattern shearing interferometry (DSPSI), or speckle shearography, represent a significant improvement in image acquisition compared to approaches using analogue devices, in spite of their reduced resolution compared to holographic media, and noise introduced by speckle grains [1]. One way to improve the quality of the images that come from the interference of the speckle patterns in digital application is the adoption of the phase-shifting process [2–5] and “the phase of the difference between two correlated speckle patterns can be determined. This is done by applying phase-shifting techniques to speckle interferometry, which will quantitatively determine the phase of double-exposure speckle measurements” [2]. Phaseshifting is also adopted to suppress the phase error caused by gamma nonlinearities in digital projectors and cameras [6]. Phaseshifting means the application of a shift in the phase of the waves n Correspondence to: Universitat de València, Facultad de Medicina y Odontología, Departamento de Fisiología, Avda. Blasco Ibáñez, 15, Valencia 46010, Spain E-mail address: [email protected] (R. González-Peña).
http://dx.doi.org/10.1016/j.optcom.2016.06.001 0030-4018/& 2016 Elsevier B.V. All rights reserved.
from the reference and from the object under analysis. The traditional assembly adopted by the digital shearography method is similar, and in Fig. 1 the arrangement of digital shearography is presented. A beam splitter, represented by an optical cube with a semi-transparent mirror, provides the separation and the union of the beams, creating their interference. The beam that comes from the object is divided in two by the cube, with both beams reaching mirrors. The scattered light falls on a Michelson interferometric system that generates two perpendicular beams of equal average irradiance I0, by means of a beam-splitter cube (Fig. 1). Both beams were reflected, respectively, on two perpendicular mirrors, one of them with a tilt to obtain the shear and the other with a piezoelectric transducer (PZT) for applying phase shifts [7–9]. There are some algorithms that carry out the shifting of phase, which is also known as phase stepping [10], and the aim is to create four images with a change of phase of π/2 between two images in sequence; for example, with one image as the reference before the deformation of an object and other four, after the deformation [11]. The beams go back to the centre of the interferometer, merge at the entrance of the photographic lens and record the image in the CCD camera. The resulting correlation intensity, IR, is obtained by
R.A Braga Jr et al. / Optics Communications 380 (2016) 114–123
(
(
IB = I0 1 + cos Ψ0+ΔΨ ( x, y)
115
))
⎛ ∆Ψ ( x, y) ⎞ ⎛ ∆Ψ ( x, y) ⎞ IR = IA − IB 2 = 4I0 sin ⎜⎜ Ψ0 + ⎟⎟ ⎟⎟ sin ⎜⎜ 2 2 ⎠ ⎠ ⎝ ⎝
⎛ I −I ⎞ Φ = arctan ⎜ 4 2 ⎟ ⎝ I1−I3 ⎠
Fig. 2. Cantilever beam sample.
Fig. 3. Image of fringes in a cantilever beam (a) with noise and (b) the profile of a longitudinal line in greyscale; (c) the image after a Gaussian blur filter (sigma¼ 10) and (d) the profile of a longitudinal line in greyscale.
subtracting the two speckle patterns belonging to the two states of the object (before and after deformation):
(
)
2
(3)
where IA and IB are the irradiances of the states before and after deformation, Ψ0 is the phase introduced by the light diffused by the object and ΔΨ(x,y) is the phase difference due to the out-ofplane deformation including the shear. [7]. In order to determine the distribution of the optical phase produced when different loads are applied to the object, four consecutive images are taken. Each one is generated by adding to the previous one a known displacement equivalent to a π/2 phase. The displacement is achieved by means of the piezoelectric element attached to one of the mirrors. Phase Φ, experienced by the points of the object due to the deformation, is obtained in accordance with Eq. (4).
Fig. 1. Digital speckle shearing pattern interferometry (DSSPI) setup.
IA = I0 1 + cos ( Ψ0 )
(2)
(1)
(4)
where I1, I2, I3, and I4 stand for the intensities (obtained by Eq. (3)) of the four dephased images [12,13]. In order to determine the deformation of all the points of the object being studied, a phase unwrapping process is performed so that the phase values between 0 and 2π are obtained. The creation of the shifting images using a PZT, despite its great use, has some drawbacks, such as the additional cost of the apparatus and, most importantly, the difficulties to apply the same displacement of the mirror by means of the PZT. Some causes of those difficulties are related to the connection of the PZT to the mirror, and the adjustment of the right voltage to move the PZT at the same interval in the three displacements to generate the shifting [14–17]. In addition, one can see that the PZT has nonlinearities [18] that can compromise the phase shifting. The nonlinearities, although their small errors are introduced in the positioning of the mirror [18], are still today under study, since the presence of the inherent hysteresis of the PZT can disturb the results if it is not compensated [19,20], which increases the complexity of the needed apparatus. Other materials such as PMN-PT (lead magnesium niobate - lead titanate) and PZN-PT (lead zinc niobate - lead titanate) have attracted a great deal of attention, because they exhibit superior piezoelectric properties and an electromechanical response significantly larger than conventional ceramics such as PZT [21–23]. The use of microelectromechanical systems (MEMS) can also replace the PZT actuator serving as the phase-shifting component [24,25]. Nevertheless, in this case, it always has nonlinearity problems similar to the conventional PZT actuator, as well as increasing the costs. Therefore, the challenge of the viability of the new material to substitute the PZT in phase shifting and, additionally, the challenge of bias and compensation for the PZT nonlinearities create some issues of repeatability and uncertainty, which can compromise the results, or even increase the costs, complexity and number of procedures needed to overcome the issue. Additionally, the limitation presented by the pulsed laser applications to record more than one image with known phase differences boosts the search for alternatives to the techniques using a spatial carrier based on Fast Fourier Transform (FFT) calculations [26]. The limitation of using the mechanical displacement of the mirror can be also observed when the measurement needs to be conducted in an object under movement. Thus, the aim of this work was to present a protocol to create digitally the phase-shift images avoiding the mechanical action of the PZT and other new
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Fig. 4. Parameters for phase shift algorithm.
Fig. 5. Centreline profiles of images after the digital phase shift, where the first profile represents 0° shift, the second 90° shift, the third 180° shift and the fourth 270° shift.
materials. The proposed protocol based on image processing from just one image was tested with a digital shearography method.
2. Materials and methods 2.1. Sample The sample used to prove the hypothesis of digital phaseshifting was a cantilever beam under deformation (Fig. 2). The cantilever beam had dimensions of 25 110 mm, manufactured in aluminium with a Young's modulus of E ¼70 GPa and geometrical inertial moment of I ¼5.63 10 11 m4. Two experiments were conducted in order to compare the proposed method to the traditional optical approach. The first assay was to check the variability of the methods under replications, and the second the sensitivity under a range of deformations, from 19 g to 90 g. 2.2. Experimental setup Fig. 1 presents the experimental setup to provide the deformation of the samples and the image acquisition. A linear polarised He–Ne laser beam (632.8 nm, 30 mW) was used. The beam size of the expanded laser light was 110 mm diameter over the sample. The images were acquired by a macro lens (SIGMA) with a focal length of 50 mm, numerical aperture of f/16, connected to an Allied Vision Technologies CCD Camera (AVTMarlinF-145B, pixel size of 4.65 μm). The speckle size in the image plane was 12.35 μm.
The theoretical phase-shift was carried out using a PZT (model STr-25, Piezomechanik GMBH; maximum voltage 500 V and travel 3 mm). The incidence angle of the laser beam on the PZT/mirror (in front of the CCD camera) was perpendicular to the mirror (Fig. 1). The shift necessary to create the displacement in the fringes of π/2 was 0.0791 μm, which means a voltage supply of 13.1833 V in the PZT. 2.3. Digital phase-shift The digital phase-shift process consists of two parts; the first responsible for the pre-processing of images and detection of fringes, and the second related to image generation of three images, corresponding to π/2, π and 3π/2 of phase changes. The images used in the process were cropped in order to consider only the portion that matters to the process. The cantilever beam imaged had the dimensions of 154 pixels width and 724 pixels height. The entire process was executed using ImageJ software [27] and automated using a macro feature. Eq. (4) represents the processing of the four images, after the comparisons [11,28]. 2.3.1. Fringe detection The first step before fringe detection was the pre-processing stage, which consisted of converting the images to 8-bit greyscale (i.e. 256 levels of grey) and running a Gaussian blur filter (with sigma equal to 10) in order to reduce the noise present in the input images mostly caused by the speckle grains. The detection of fringes was based on the information extracted from the input image, and a fringe was considered as a
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Fig. 6. Deformation curves of an aluminium cantilever beam deformed three times by a load of 10 g measured by digital speckle shearing pattern interferometry using (a) a PZT to provide the phase-shifting and (b) the digital phase-shifting generation.
portion between two dark stripes. A separation line between each fringe was obtained by finding the local minimum. Fig. 3(a) shows the noise over the fringes and in Fig. 3(b) it is possible to see the profile of a centreline from the image of Fig. 3(a). After application of a Gaussian Blur filter (Fig. 3(c)), it is possible to see the original image after the filter; and in Fig. 3(d) the profile of a centreline with the local minimum is pointed out by an arrow. After detection, each local minimum was considered as central point of a fringe (F) and used as a parameter for phase shifting. The concept of Fringe Size (FS) was created and defined as the distance between two fringes in pixels. Eq. (5) shows how this is calculated, considering Fi as a fringe F of index i.
FSi = Fi + 1−Fi
(5)
The shift of a phase (Δ) is π/2 and was related to each local minimum in pixels. However, the length of the shift is different between fringes, thus there is a shift to each fringe, where the index i of ∆i represents the distance the fringe (FSi) that was shifted on the newly generated image (Eq. (6)).
Δi =
FSi 4
(6)
The parameters used for the phase shift calculation are shown in Fig. 4. The election of the simplest way to detect the fringes is in
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Fig. 7. Variation of the three replicates of a 10 g cantilever using the boxplot, with the red line in the centre of the box meaning the average, and the box meaning the variation within 25–75% of the data. The whiskers are the extreme values of the (a) PZT application and (b) digital phase-shifting application.
accordance with the aim of this work, which was to simplify the method of phase-shifting of correlation fringes. The adoption of other advanced methods of identification would certainly improve our proposal, despite the reliable answer provided in the results achieved. 2.3.2. Image generation with digital phase shift All the processes were performed starting with an input image that corresponds to the image without shifting. After the detection and calculation of the parameters extracted from this image, the shifting process was carried out. The generation of a shift of the order jπ/2 (where j = {1,2,3}) is followed by displacing and resizing the new image. Eq. (7) shows the new fringe position ( NFi ) obtained after shifting, based on the delta previously obtained. As mentioned, each displacing factor of a fringe in the new image is different, and this may cause loss of information by overlaying fringes. Therefore, to avoid this problem, it is necessary to resize each fringe before its positioning on the new image. The resizing process takes into account the new fringe size ( NFSi ), calculated by Eq. (8). The resizing is produced digitally by a bilinear interpolation.
NFi = Fi+jΔi , to j = { 1,2,3}
(7)
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2.3.3. Finite element analysis additional comparison The deformation observed by the interferometer using a PZT and the proposed digital phase-shift was compared to a Finite Element Analysis (FEA) simulation. The cantilever beam was modelled using a three-dimensional hexahedral mesh of 128,600 elements and 494,503 nodes. The same loads were applied in the simulation, and the deformation was obtained from the neutral axis of the cantilever beam model.
3. Results and discussion 3.1. A cantilever beam In Fig. 6 it is possible to see the values of deformation in an aluminium cantilever beam measured by digital speckle shearing pattern interferometry using a PZT to provide the phase-shifting (Fig. 6(a)), and using the digital phase-shifting generation (Fig. 6 (b)) when a load of 10 g was applied thrice, in accordance with Fig. 1 and using Eq. (4). Also in Fig. 6, it is possible to see the theoretical curve of the deformed cantilever beam as a comparison. The experimental values of the phase differences ΔΨ(x,y) on the linear and isotropic cantilever were obtained from the phase map and phase unwrapping process at each point of the specimen with respect to the embedded cantilever point. Also, the theoretical values of the phase differences ΔΨ(x,y) on the cantilever beam were obtained from the calculation of deformations (∂w(x,y)/(∂y)) using Eq. (9) [29]:
∆Ψ( x, y) =
where
Fig. 8. Deformation curves of an aluminium cantilever beam deformed three times by a load of 20 g measured by digital speckle shearing pattern interferometry using (a) a PZT to provide the phase-shifting and (b) the digital phase-shifting generation.
(FSi−j∆i +j∆i +1 ) , to j = { 1, 2, 3} FSi
Each value of j generates an order of shifting (jπ/2), creating three new images at the end of the process (π/2, π and 3π/2 respectively). The whole process of digital phase-shift can be viewed in Fig. 5, starting from the original fringe pattern, without shifting, the profile of the centreline. The other three shifting profiles can be seen in Fig. 5, representing the centrelines of a digital shift of π/ 2, π and 3π/2 respectively.
(9) ∂w ( x, y) ∂y
is the out-of-
plane deformation at point (x,y) of the object, Δy is the shear introduced by the mirror (a constant value of 4.82 mm) and w(x,y) is out-of-displacement in the xy-plane. According to the theory of elasticity (Eq. (10)):
∂y (8)
)
λ is the wavelength of the laser beam,
∂w ( x, y) NFSi =
⎛ ⎞ 4π ⎜ ∂w ( x, y ⎟ ⎜ ⎟ ∆y λ ⎝ ∂y ⎠
=
P⎛ y2 ⎞ ⎜ Ly − ⎟ EI ⎝ 2⎠
(10)
where P is the load, E is the material Young's modulus, I is the geometrical moment of inertia, L is the beam length and y is the distance between the displacement measurement point and clamping position [30]. In the extremities of the cantilever beam, related to its free end and clamping points, there are greater variations of the values than in the middle regions of the cantilever, which is expected from the low and the high movements of the cantilever in those regions respectively that cause the uncertainties. Visually, one can see that the digital phase-shifting presents a
Fig. 9. Fringes in a cantilever beam with a load of 10 g from (a) PZT phase-shift and (b) digital phase-shift.
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Fig. 10. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting provided by a PZT.
Fig. 11. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting using digital phase-shifting with the curves tangled in some loads.
Fig. 12. Images of the fringes obtained by the proposed method with a line showing where the images were cropped.
Fig. 13. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting using digital phase-shifting after cropping the images of 60 g, 70 g and 90 g without the curve related to 19 g.
well-adjusted result, better than the results provided by phaseshifting using the PZT. The statistical analysis of these data using variation graphics can be seen in Fig. 7. In Fig. 7(a), the three replications of the cantilever beam deformation provided by phase-shifting using the PZT are presented in a graph of variations by means of a boxplot. In Fig. 7(b), it is possible to see the same variation provided by the boxplot graph, however showing the digital phase-shifting approach, where it is also possible to see its lower variation than the phase-shifting using the PZT. For each box, the line in the centre is the median, and the edges of the boxes are the 25th and 75th percentiles; the whiskers represent the most extreme data considering the outliers. It is possible to observe the greater variation created by the analogue phase-shifting provided by the PZT during the replications, which can be explained by the difficulties linked to the setup of the PZT and its actuation with repeatability in the three shifts of π/2. In Fig. 8, it is possible to see the same approach with a load of 20 g in the cantilever beam; it is clear that any small error in the positioning of the PZT can compromise the measurements during the replications. In Fig. 8(a), the three replicates of the cantilever beam deformed by 20 g show that statement of sensitivity, where one of the replications provided an outcome completely different to the other two, and different also to the outcome expected. The same results occurred for the load of 30 g not presented here. The adoption of PZT, in order to obtain four images displaced to create the phase-shifting process, can compromise the results expected in theory [18]. There are also many factors that can be
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Fig. 16. View of the cantilever beam presented during the simulation of finite element analysis.
Fig. 14. Phase difference for load 19 g with (a) thick dashed line representing the proposed digital shifting, continuous line representing the PZT phase shifting, and small dashed line representing the adjustment in the software to identify the fringes and thus to generate digital phase-shifting, and (b) thick dashed line representing the proposed digital shifting, continuous line representing the PZT phase shifting, and dotted line representing the crop in the bottom of the image to generate digital phase-shifting.
resolution of 14 bit. Since the magnitude is not exact, from the command to the complete action of the PZT, some error of positioning will occur; therefore the phase maps will vary randomly, which can create a complete decorrelation like that observed in one of the replications related to 20 g. The cantilever beam horizontal displacement caused by 20 g created more fringes compared to 10 g; thus as they are closest, one from the other, this could be the cause of the decorrelation. Otherwise, the adoption of digital phase-shifting presented better results (Fig. 8(b)) than using PZT, since the shifting is created using the centre of the fringes in the prime image, with the displacement limited only by the pixels of the camera. In addition, the setup adopted to carry out the experiments was based on the PZT procedure, which means that when only the digital phase-shifting is adopted, the setup can be improved in order to let the fringes be well positioned in the image. Thus, avoiding some errors mainly at the edges of the cantilever beam enhances even more the digital shifting proposed. In Fig. 9, the fringes on the cantilever beam caused by a load of 10 g presented visually the same behaviour for both procedures. 3.2. Test of limits and challenges of digital phase-shift
Fig. 15. Results in millimetres of the deformations obtained by phase difference using a PZT (continuous line) versus theoretical curve and a Digital Phase-Shift (dashed line) versus the theoretical curve, with the dotted line representing the ideal relation.
isolated or together influence the incorrect positioning of the mirror. One of the errors could be attributed to the size of the mirror related to the PZT force and its ability to restore the prime position. This is known as residual displacement, and it should be avoided or compensated for [21]. The other key factor linked to the PZT work is related to its calibration; in other words, the adjustment of its movement with respect to the voltage applied. Since one needs a voltage with a magnitude of 13.1833 V to tilt the mirror and shift the fringes by 90°, the values achieved by the source varied by 70.031 V ( 70.23%) and are related to a
The results of testing the proposed method to create digitally the phase-shift were also analysed concerning the challenges of image analysis, thus exploring the limits of its sensitivity when the cantilever received a range of deformations based on the loads from 19 g to 90 g. In Fig. 10, it is possible to observe the deformation (i.e. phasedifference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting provided by a PZT. It is clear that the sequence of the curves' area is directly related to the sequence of loads, which was expected. However, in Fig. 11, the same experiment in turn using digital phase-shifting presented the curves tangled in some loads; therefore, this is clearly a challenge regarding the use of the proposed method mainly related to image processing. In Fig. 12, the images of the fringes obtained by the proposed method are presented, and when they were compared to the PZT method, in the loads of 60–90 g, the upper fringes were placed in a different position. This is mainly related to the difficulty of the digital method to identify properly the upper fringes when they were spread and blurred, such as that observed in the load 50 g, thus shifting it to the bottom of the image. The solution to this problem was found by cropping the prime images before starting the digital shifting, eliminating the upper part of the image, as pointed out in Fig. 12. In Fig. 13, it is possible to see the result after the cropping, and then to identify a solution of the problem caused by undefined upper fringes. The cropping did not compromise the analysis of the cantilever regarding its deformation characteristics,
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Fig. 17. Results of the deformations obtained by phase difference using a PZT (dotted line), using a digital phase-shift (continuous line) and finite element analysis (dashed line) for the loads (a) 35 g, (b) 45 g, (c) 50 g, (d) 60 g, (e) 70 g and (f) 90 g.
considering that it is homogeneous. However, if there is information over the limit of the crop, then the analysis needs to be tailored to place the fringes, in order to let the proposed method, recognise those that are relevant in the analysis. Therefore, the curves were placed in the expected order of deformation, except for the curve linked to load 19 g, since the post-processing of the images did not provide the same result in this instance. The deformation created by the load 19 g, the lowest limit of our test of sensitivity, showed that additional attention must be taken into account with the lowest portion of the image, where an undefined fringe appears. Some changes in the analysis of 19 g images were performed, and the results are presented in Fig. 14. The first change was in the software when the routine identifies the fringes; the other was by a manual crop of the bottom portion of the image. The expected results prevenient by the correction of the images generated by the 19 g load were conducted using a different approach to the other loads from 60 g to 90 g. Thus, one can see that the creation of digital phase-shifting is reliable; however, particular attention must be taken to define the sensitivity limits, and then the necessary corrections and post-processing.
In order to overcome the subjectivity of manual cropping, the user must define the best positions of the fringes particularly concerned with the edges (bottom and top) and with the loads. Since the aim of this work was also to test the sensitivity of the proposed method, where we had to maintain the same configuration to obtain the deformation caused by loads from 19 g to 90 g, we could not adjust the best position of the fringes. This could be effected during the setup of the system, avoiding the crop. In Fig. 15 it is possible to see the results from the cantilever beam loaded with 90 g using a PZT and the proposed digital shift, compared to the result with the theoretical curve. The other experiments using all loads were also carried out; however, we only present that with the load 90 g, since they are all rather similar. The comparisons with the theoretical curve presented the ability of the proposed method to reproduce the expected phenomena. Some additional adjustments can be effected to calibrate the outcomes, i.e. using an off-set correction. The adoption of the methods using different loads with replications can be considered as confirmation of the expected potential of the proposed method. Here, the results of loads 10 g, 20 g and 30 g were used with three replications. The results of 30 g were not presented, since they
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evinced the same pattern as that of 10 g and 20 g. In addition, the other assay using seven different loads from 19 g to 90 g presented more replications of the proposed method, showing its reliability. The results from the PZT and from the proposed digital phaseshift were also compared to an FEA simulation (Fig. 16); the results can be seen in Fig. 17. The region in the cantilever beam chosen to compare the results was related to a portion where the deformation does not compromise the correlation of the speckle pattern. The results presented the ability of the proposed technique to follow the phenomenon with a similar behaviour compared to a simulation provided by an FEA, with a better correlation when compared to the PZT approach. The digital creation of phase-shifting presented reliable results with repeatability and sensitivity compared to the traditional method using the PZT or similar devices. The main limitation of using this proposed methodology is the restriction to parallel fringes, or quite parallel, such as in a cantilever under the Young's modulus test. The adoption of irregular fringes will demand processing in two dimensions that is feasible [31] if associated with a methodology such as that proposed here. The assumption that the interpolated fringes vary linearly does not present a great limitation, since the distances of the original fringes were respected. In addition, the results showed that the final quality of the curves was comparable to that using a PZT. In addition, the feasible adoption of the simplest interpolation enhances the potential of the proposed method when the most sophisticated methods of interpolation are adopted. Despite the reduced cost of the proposed technique, which makes it assessable to a greater number of users, its adoption when only one fringe can be acquired, such as in the case of pulsed laser application, or when it is necessary to monitor an object under movement, justifies its application.
4. Conclusions The digital phase-shifting approach was able to monitor the deformations of a cantilever beam as well as the traditional method using the PZT as the phase-shift generator. The proposed technique was able to reproduce the deformation of the cantilever beam in the intermediary region between the clamping point and its free end, presenting sensitivity, repeatability and robustness, which justify its use as an alternative to PZT in traditional applications, and mostly when the PZT cannot be used.
Acknowledgements This study was supported by the Physiology Department of Universitat de València, the Federal University of Lavras, and the Federal Institute of Minas Gerais, Campus Bambuí, and was also financed partially by CNPq (302805/2012-5), Capes, FINEp, and FAPEMIG (CAG - PPM-00075-15).
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Wyant, Phase shifter calibration in phase-shifting interferometry, Appl. Opt. 24 (1985) 3049–3052, http://dx.doi.org/10.1364/ AO.24.003049. [15] X. Chen, M. Gramaglia, J.A. Yeazell, Phase-shift calibration algorithm for phaseshifting interferometry, J. Opt. Soc. Am. A 17 (11) (2000) 2061–2066, http://dx. doi.org/10.1364/JOSAA.17.002061. [16] R. Zhu, J. Chen, L. Chen, L. Liu, M. Chen, Accuracy of phase-shifting interferometry: phase-shifter calibration and error compensation, Proc. SPIE 1553 (1992) 702–710, http://dx.doi.org/10.1117/12.135355. [17] C. De Veuster, P.R. Slangen, Y.L.M. Renotte, L. Berwart L, Y.F. Lion, In-line phaseshifter calibration and drift measurement and compensation for digital speckle pattern interferometry (DSPI), Proc. SPIE 2951 (1996) 117–124, http: //dx.doi.org/10.1117/12.262424. [18] C. Ai, J.C. Wyant, Effect of piezoelectric transducer nonlinearity on phase shift interferometry, App. Opt. 26 (6) (1987) 1112–1116, http://dx.doi.org/10.1364/ AO.26.001112. 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Roberto A. Braga Jr. graduated with a Master's degree in Electrical Engineering from UFMG, Brazil, and has a Doctoral degree in Agricultural Engineering UNICAMP, Brazil. He has been an Associate Professor at UFLA, Brazil, since 1996. He teaches electricity and automation and researches optical instrumentation. His sabbatical leaves include BIOSS Scotland (2005 and 2008), senior stage in BIOSS, and JHI Scotland in 2011. His publications include more than 40 papers, the majority in international journals; he has some patents, books chapters, and books published.
Thiago Paiva Almeida is a technician in informatics (core technical education) since 2005, BA in Information Systems UNIPAC (2009), Specialist in Computer Networks from ESAB (2013), and is currently a Master's student in Systems Engineering and Automation at UFLA (Federal University of Lavras). He currently teaches courses in IT and technical drawing at the Institution of Higher Education CEMES and is a temporary teacher at the Federal Institute of Minas Gerais (Bom Sucesso), where he teaches technical courses in computer areas.
Rolando J. González-Peña received his Master of Science degree in Optics from the High Technology Institute of Havana in 1998, and his PhD in Technical Science in 2001. His current research is focused on digital speckle interferometry and dynamic speckle lasers. Currently, he is Professor at the Physiology Department in the Faculty of Medicine and Odontology at the University of Valencia.
Igor V. A. Santos is an undergraduate student of Control Engineering and Automation at UFLA Federal University of Lavras. He has received a scholarship for research from FAPEMIG in the area of shearography.
Marlon Marcon graduated with a Master's degree in Systems Engineering from the Federal University of Lavras (2009). He has been a Professor at the Federal University of Technology - Paraná. He graduated with a Bachelor of Computer Science from the State University of West Paraná (2006). He has experience in Computer Science, with an emphasis on Computer Vision, Natural Language Processing (NLP), Machine Learning and Pattern Recognition, mainly in the following areas: construction of mathematical models based on image analysis, lexicography, computational linguistics and Support Vector Machines (SVM).
Moisés Batista Martins is an undergraduate student of Control Engineering and Automation at UFLA Federal University of Lavras. He is a technician in Mechanics at the Municipal Technical School of Sete Lagoas. He has taken a training course of Industrial Training in Machining Mechanical SeteLagoas/MG, and experience as a Maintenance Mechanic at the Ambev and Iveco companies.
Ricardo R. Magalhaes received his Master of Science degree in Mechatronics (2008) and PhD in Industrial Engineering (2011), both from the University of Bahia. He is a Professor at the University of Lavras. He has experience in automotive companies with an emphasis on product development. His current research is focused on measurement techniques, the finite element method, and stress analysis.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase-shifting of correlation fringes created by image processing as an alternative to improve digital shearography Roberto A Braga Jra, Rolando J González-Peña b,n, Marlon Marcon c, Ricardo R Magalhães a, Thiago Paiva-Almeida a, Igor V.A. Santos a, Moisés Martins a a
Universidade Federal de Lavras, Dep. Engenharia, Lavras CP 3037, MG, Brazil Universidad de Valencia, Unidad de Biofísica y Física Médica, Departamento de Fisiología, Facultad de Medicina y Odontología, Valencia CP 46010, Spain c Instituto Federal Minas Gerais, Dep. Engenharia e Computação (DEC), Câmpus Bambuí (IFMG), Brazil b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 January 2016 Received in revised form 28 May 2016 Accepted 1 June 2016
The adoption of digital speckle pattern shearing interferometry, or speckle shearography, is well known in many areas when one needs to measure micro-displacements in-plane and out of the plane in biological and non-biological objects; it is based on the Michelson's Interferometer with the use of a piezoelectric transducer (PZT) in order to provide the phase-shift of the fringes and then to improve the quality of the final image. The creation of the shifting images using a PZT, despite its widespread use, has some drawbacks or limitations, such as the cost of the apparatus, the difficulties in applying the same displacement in the mirror repeated times, and when the phase-shift cannot be used in dynamic object measurement. The aim of this work was to create digitally phase-shift images avoiding the mechanical adjustments of the PZT, testing them with the digital shearography method. The methodology was tested using a well-known object, a cantilever beam of aluminium under deformation. The results documented the ability to create the deformation map and curves with reliability and sensitivity, reducing the cost, and improving the robustness and also the accessibility of digital speckle pattern shearing interferometry. & 2016 Elsevier B.V. All rights reserved.
Keywords: Phase-shift algorithm Speckle interferometry Piezoelectric transducer Digital speckle pattern shearing interferometry
1. Introduction Digital speckle pattern interferometry (DSPI) and digital speckle pattern shearing interferometry (DSPSI), or speckle shearography, represent a significant improvement in image acquisition compared to approaches using analogue devices, in spite of their reduced resolution compared to holographic media, and noise introduced by speckle grains [1]. One way to improve the quality of the images that come from the interference of the speckle patterns in digital application is the adoption of the phase-shifting process [2–5] and “the phase of the difference between two correlated speckle patterns can be determined. This is done by applying phase-shifting techniques to speckle interferometry, which will quantitatively determine the phase of double-exposure speckle measurements” [2]. Phaseshifting is also adopted to suppress the phase error caused by gamma nonlinearities in digital projectors and cameras [6]. Phaseshifting means the application of a shift in the phase of the waves n Correspondence to: Universitat de València, Facultad de Medicina y Odontología, Departamento de Fisiología, Avda. Blasco Ibáñez, 15, Valencia 46010, Spain E-mail address: [email protected] (R. González-Peña).
http://dx.doi.org/10.1016/j.optcom.2016.06.001 0030-4018/& 2016 Elsevier B.V. All rights reserved.
from the reference and from the object under analysis. The traditional assembly adopted by the digital shearography method is similar, and in Fig. 1 the arrangement of digital shearography is presented. A beam splitter, represented by an optical cube with a semi-transparent mirror, provides the separation and the union of the beams, creating their interference. The beam that comes from the object is divided in two by the cube, with both beams reaching mirrors. The scattered light falls on a Michelson interferometric system that generates two perpendicular beams of equal average irradiance I0, by means of a beam-splitter cube (Fig. 1). Both beams were reflected, respectively, on two perpendicular mirrors, one of them with a tilt to obtain the shear and the other with a piezoelectric transducer (PZT) for applying phase shifts [7–9]. There are some algorithms that carry out the shifting of phase, which is also known as phase stepping [10], and the aim is to create four images with a change of phase of π/2 between two images in sequence; for example, with one image as the reference before the deformation of an object and other four, after the deformation [11]. The beams go back to the centre of the interferometer, merge at the entrance of the photographic lens and record the image in the CCD camera. The resulting correlation intensity, IR, is obtained by
R.A Braga Jr et al. / Optics Communications 380 (2016) 114–123
(
(
IB = I0 1 + cos Ψ0+ΔΨ ( x, y)
115
))
⎛ ∆Ψ ( x, y) ⎞ ⎛ ∆Ψ ( x, y) ⎞ IR = IA − IB 2 = 4I0 sin ⎜⎜ Ψ0 + ⎟⎟ ⎟⎟ sin ⎜⎜ 2 2 ⎠ ⎠ ⎝ ⎝
⎛ I −I ⎞ Φ = arctan ⎜ 4 2 ⎟ ⎝ I1−I3 ⎠
Fig. 2. Cantilever beam sample.
Fig. 3. Image of fringes in a cantilever beam (a) with noise and (b) the profile of a longitudinal line in greyscale; (c) the image after a Gaussian blur filter (sigma¼ 10) and (d) the profile of a longitudinal line in greyscale.
subtracting the two speckle patterns belonging to the two states of the object (before and after deformation):
(
)
2
(3)
where IA and IB are the irradiances of the states before and after deformation, Ψ0 is the phase introduced by the light diffused by the object and ΔΨ(x,y) is the phase difference due to the out-ofplane deformation including the shear. [7]. In order to determine the distribution of the optical phase produced when different loads are applied to the object, four consecutive images are taken. Each one is generated by adding to the previous one a known displacement equivalent to a π/2 phase. The displacement is achieved by means of the piezoelectric element attached to one of the mirrors. Phase Φ, experienced by the points of the object due to the deformation, is obtained in accordance with Eq. (4).
Fig. 1. Digital speckle shearing pattern interferometry (DSSPI) setup.
IA = I0 1 + cos ( Ψ0 )
(2)
(1)
(4)
where I1, I2, I3, and I4 stand for the intensities (obtained by Eq. (3)) of the four dephased images [12,13]. In order to determine the deformation of all the points of the object being studied, a phase unwrapping process is performed so that the phase values between 0 and 2π are obtained. The creation of the shifting images using a PZT, despite its great use, has some drawbacks, such as the additional cost of the apparatus and, most importantly, the difficulties to apply the same displacement of the mirror by means of the PZT. Some causes of those difficulties are related to the connection of the PZT to the mirror, and the adjustment of the right voltage to move the PZT at the same interval in the three displacements to generate the shifting [14–17]. In addition, one can see that the PZT has nonlinearities [18] that can compromise the phase shifting. The nonlinearities, although their small errors are introduced in the positioning of the mirror [18], are still today under study, since the presence of the inherent hysteresis of the PZT can disturb the results if it is not compensated [19,20], which increases the complexity of the needed apparatus. Other materials such as PMN-PT (lead magnesium niobate - lead titanate) and PZN-PT (lead zinc niobate - lead titanate) have attracted a great deal of attention, because they exhibit superior piezoelectric properties and an electromechanical response significantly larger than conventional ceramics such as PZT [21–23]. The use of microelectromechanical systems (MEMS) can also replace the PZT actuator serving as the phase-shifting component [24,25]. Nevertheless, in this case, it always has nonlinearity problems similar to the conventional PZT actuator, as well as increasing the costs. Therefore, the challenge of the viability of the new material to substitute the PZT in phase shifting and, additionally, the challenge of bias and compensation for the PZT nonlinearities create some issues of repeatability and uncertainty, which can compromise the results, or even increase the costs, complexity and number of procedures needed to overcome the issue. Additionally, the limitation presented by the pulsed laser applications to record more than one image with known phase differences boosts the search for alternatives to the techniques using a spatial carrier based on Fast Fourier Transform (FFT) calculations [26]. The limitation of using the mechanical displacement of the mirror can be also observed when the measurement needs to be conducted in an object under movement. Thus, the aim of this work was to present a protocol to create digitally the phase-shift images avoiding the mechanical action of the PZT and other new
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Fig. 4. Parameters for phase shift algorithm.
Fig. 5. Centreline profiles of images after the digital phase shift, where the first profile represents 0° shift, the second 90° shift, the third 180° shift and the fourth 270° shift.
materials. The proposed protocol based on image processing from just one image was tested with a digital shearography method.
2. Materials and methods 2.1. Sample The sample used to prove the hypothesis of digital phaseshifting was a cantilever beam under deformation (Fig. 2). The cantilever beam had dimensions of 25 110 mm, manufactured in aluminium with a Young's modulus of E ¼70 GPa and geometrical inertial moment of I ¼5.63 10 11 m4. Two experiments were conducted in order to compare the proposed method to the traditional optical approach. The first assay was to check the variability of the methods under replications, and the second the sensitivity under a range of deformations, from 19 g to 90 g. 2.2. Experimental setup Fig. 1 presents the experimental setup to provide the deformation of the samples and the image acquisition. A linear polarised He–Ne laser beam (632.8 nm, 30 mW) was used. The beam size of the expanded laser light was 110 mm diameter over the sample. The images were acquired by a macro lens (SIGMA) with a focal length of 50 mm, numerical aperture of f/16, connected to an Allied Vision Technologies CCD Camera (AVTMarlinF-145B, pixel size of 4.65 μm). The speckle size in the image plane was 12.35 μm.
The theoretical phase-shift was carried out using a PZT (model STr-25, Piezomechanik GMBH; maximum voltage 500 V and travel 3 mm). The incidence angle of the laser beam on the PZT/mirror (in front of the CCD camera) was perpendicular to the mirror (Fig. 1). The shift necessary to create the displacement in the fringes of π/2 was 0.0791 μm, which means a voltage supply of 13.1833 V in the PZT. 2.3. Digital phase-shift The digital phase-shift process consists of two parts; the first responsible for the pre-processing of images and detection of fringes, and the second related to image generation of three images, corresponding to π/2, π and 3π/2 of phase changes. The images used in the process were cropped in order to consider only the portion that matters to the process. The cantilever beam imaged had the dimensions of 154 pixels width and 724 pixels height. The entire process was executed using ImageJ software [27] and automated using a macro feature. Eq. (4) represents the processing of the four images, after the comparisons [11,28]. 2.3.1. Fringe detection The first step before fringe detection was the pre-processing stage, which consisted of converting the images to 8-bit greyscale (i.e. 256 levels of grey) and running a Gaussian blur filter (with sigma equal to 10) in order to reduce the noise present in the input images mostly caused by the speckle grains. The detection of fringes was based on the information extracted from the input image, and a fringe was considered as a
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Fig. 6. Deformation curves of an aluminium cantilever beam deformed three times by a load of 10 g measured by digital speckle shearing pattern interferometry using (a) a PZT to provide the phase-shifting and (b) the digital phase-shifting generation.
portion between two dark stripes. A separation line between each fringe was obtained by finding the local minimum. Fig. 3(a) shows the noise over the fringes and in Fig. 3(b) it is possible to see the profile of a centreline from the image of Fig. 3(a). After application of a Gaussian Blur filter (Fig. 3(c)), it is possible to see the original image after the filter; and in Fig. 3(d) the profile of a centreline with the local minimum is pointed out by an arrow. After detection, each local minimum was considered as central point of a fringe (F) and used as a parameter for phase shifting. The concept of Fringe Size (FS) was created and defined as the distance between two fringes in pixels. Eq. (5) shows how this is calculated, considering Fi as a fringe F of index i.
FSi = Fi + 1−Fi
(5)
The shift of a phase (Δ) is π/2 and was related to each local minimum in pixels. However, the length of the shift is different between fringes, thus there is a shift to each fringe, where the index i of ∆i represents the distance the fringe (FSi) that was shifted on the newly generated image (Eq. (6)).
Δi =
FSi 4
(6)
The parameters used for the phase shift calculation are shown in Fig. 4. The election of the simplest way to detect the fringes is in
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Fig. 7. Variation of the three replicates of a 10 g cantilever using the boxplot, with the red line in the centre of the box meaning the average, and the box meaning the variation within 25–75% of the data. The whiskers are the extreme values of the (a) PZT application and (b) digital phase-shifting application.
accordance with the aim of this work, which was to simplify the method of phase-shifting of correlation fringes. The adoption of other advanced methods of identification would certainly improve our proposal, despite the reliable answer provided in the results achieved. 2.3.2. Image generation with digital phase shift All the processes were performed starting with an input image that corresponds to the image without shifting. After the detection and calculation of the parameters extracted from this image, the shifting process was carried out. The generation of a shift of the order jπ/2 (where j = {1,2,3}) is followed by displacing and resizing the new image. Eq. (7) shows the new fringe position ( NFi ) obtained after shifting, based on the delta previously obtained. As mentioned, each displacing factor of a fringe in the new image is different, and this may cause loss of information by overlaying fringes. Therefore, to avoid this problem, it is necessary to resize each fringe before its positioning on the new image. The resizing process takes into account the new fringe size ( NFSi ), calculated by Eq. (8). The resizing is produced digitally by a bilinear interpolation.
NFi = Fi+jΔi , to j = { 1,2,3}
(7)
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2.3.3. Finite element analysis additional comparison The deformation observed by the interferometer using a PZT and the proposed digital phase-shift was compared to a Finite Element Analysis (FEA) simulation. The cantilever beam was modelled using a three-dimensional hexahedral mesh of 128,600 elements and 494,503 nodes. The same loads were applied in the simulation, and the deformation was obtained from the neutral axis of the cantilever beam model.
3. Results and discussion 3.1. A cantilever beam In Fig. 6 it is possible to see the values of deformation in an aluminium cantilever beam measured by digital speckle shearing pattern interferometry using a PZT to provide the phase-shifting (Fig. 6(a)), and using the digital phase-shifting generation (Fig. 6 (b)) when a load of 10 g was applied thrice, in accordance with Fig. 1 and using Eq. (4). Also in Fig. 6, it is possible to see the theoretical curve of the deformed cantilever beam as a comparison. The experimental values of the phase differences ΔΨ(x,y) on the linear and isotropic cantilever were obtained from the phase map and phase unwrapping process at each point of the specimen with respect to the embedded cantilever point. Also, the theoretical values of the phase differences ΔΨ(x,y) on the cantilever beam were obtained from the calculation of deformations (∂w(x,y)/(∂y)) using Eq. (9) [29]:
∆Ψ( x, y) =
where
Fig. 8. Deformation curves of an aluminium cantilever beam deformed three times by a load of 20 g measured by digital speckle shearing pattern interferometry using (a) a PZT to provide the phase-shifting and (b) the digital phase-shifting generation.
(FSi−j∆i +j∆i +1 ) , to j = { 1, 2, 3} FSi
Each value of j generates an order of shifting (jπ/2), creating three new images at the end of the process (π/2, π and 3π/2 respectively). The whole process of digital phase-shift can be viewed in Fig. 5, starting from the original fringe pattern, without shifting, the profile of the centreline. The other three shifting profiles can be seen in Fig. 5, representing the centrelines of a digital shift of π/ 2, π and 3π/2 respectively.
(9) ∂w ( x, y) ∂y
is the out-of-
plane deformation at point (x,y) of the object, Δy is the shear introduced by the mirror (a constant value of 4.82 mm) and w(x,y) is out-of-displacement in the xy-plane. According to the theory of elasticity (Eq. (10)):
∂y (8)
)
λ is the wavelength of the laser beam,
∂w ( x, y) NFSi =
⎛ ⎞ 4π ⎜ ∂w ( x, y ⎟ ⎜ ⎟ ∆y λ ⎝ ∂y ⎠
=
P⎛ y2 ⎞ ⎜ Ly − ⎟ EI ⎝ 2⎠
(10)
where P is the load, E is the material Young's modulus, I is the geometrical moment of inertia, L is the beam length and y is the distance between the displacement measurement point and clamping position [30]. In the extremities of the cantilever beam, related to its free end and clamping points, there are greater variations of the values than in the middle regions of the cantilever, which is expected from the low and the high movements of the cantilever in those regions respectively that cause the uncertainties. Visually, one can see that the digital phase-shifting presents a
Fig. 9. Fringes in a cantilever beam with a load of 10 g from (a) PZT phase-shift and (b) digital phase-shift.
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Fig. 10. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting provided by a PZT.
Fig. 11. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting using digital phase-shifting with the curves tangled in some loads.
Fig. 12. Images of the fringes obtained by the proposed method with a line showing where the images were cropped.
Fig. 13. Deformation (phase-difference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting using digital phase-shifting after cropping the images of 60 g, 70 g and 90 g without the curve related to 19 g.
well-adjusted result, better than the results provided by phaseshifting using the PZT. The statistical analysis of these data using variation graphics can be seen in Fig. 7. In Fig. 7(a), the three replications of the cantilever beam deformation provided by phase-shifting using the PZT are presented in a graph of variations by means of a boxplot. In Fig. 7(b), it is possible to see the same variation provided by the boxplot graph, however showing the digital phase-shifting approach, where it is also possible to see its lower variation than the phase-shifting using the PZT. For each box, the line in the centre is the median, and the edges of the boxes are the 25th and 75th percentiles; the whiskers represent the most extreme data considering the outliers. It is possible to observe the greater variation created by the analogue phase-shifting provided by the PZT during the replications, which can be explained by the difficulties linked to the setup of the PZT and its actuation with repeatability in the three shifts of π/2. In Fig. 8, it is possible to see the same approach with a load of 20 g in the cantilever beam; it is clear that any small error in the positioning of the PZT can compromise the measurements during the replications. In Fig. 8(a), the three replicates of the cantilever beam deformed by 20 g show that statement of sensitivity, where one of the replications provided an outcome completely different to the other two, and different also to the outcome expected. The same results occurred for the load of 30 g not presented here. The adoption of PZT, in order to obtain four images displaced to create the phase-shifting process, can compromise the results expected in theory [18]. There are also many factors that can be
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Fig. 16. View of the cantilever beam presented during the simulation of finite element analysis.
Fig. 14. Phase difference for load 19 g with (a) thick dashed line representing the proposed digital shifting, continuous line representing the PZT phase shifting, and small dashed line representing the adjustment in the software to identify the fringes and thus to generate digital phase-shifting, and (b) thick dashed line representing the proposed digital shifting, continuous line representing the PZT phase shifting, and dotted line representing the crop in the bottom of the image to generate digital phase-shifting.
resolution of 14 bit. Since the magnitude is not exact, from the command to the complete action of the PZT, some error of positioning will occur; therefore the phase maps will vary randomly, which can create a complete decorrelation like that observed in one of the replications related to 20 g. The cantilever beam horizontal displacement caused by 20 g created more fringes compared to 10 g; thus as they are closest, one from the other, this could be the cause of the decorrelation. Otherwise, the adoption of digital phase-shifting presented better results (Fig. 8(b)) than using PZT, since the shifting is created using the centre of the fringes in the prime image, with the displacement limited only by the pixels of the camera. In addition, the setup adopted to carry out the experiments was based on the PZT procedure, which means that when only the digital phase-shifting is adopted, the setup can be improved in order to let the fringes be well positioned in the image. Thus, avoiding some errors mainly at the edges of the cantilever beam enhances even more the digital shifting proposed. In Fig. 9, the fringes on the cantilever beam caused by a load of 10 g presented visually the same behaviour for both procedures. 3.2. Test of limits and challenges of digital phase-shift
Fig. 15. Results in millimetres of the deformations obtained by phase difference using a PZT (continuous line) versus theoretical curve and a Digital Phase-Shift (dashed line) versus the theoretical curve, with the dotted line representing the ideal relation.
isolated or together influence the incorrect positioning of the mirror. One of the errors could be attributed to the size of the mirror related to the PZT force and its ability to restore the prime position. This is known as residual displacement, and it should be avoided or compensated for [21]. The other key factor linked to the PZT work is related to its calibration; in other words, the adjustment of its movement with respect to the voltage applied. Since one needs a voltage with a magnitude of 13.1833 V to tilt the mirror and shift the fringes by 90°, the values achieved by the source varied by 70.031 V ( 70.23%) and are related to a
The results of testing the proposed method to create digitally the phase-shift were also analysed concerning the challenges of image analysis, thus exploring the limits of its sensitivity when the cantilever received a range of deformations based on the loads from 19 g to 90 g. In Fig. 10, it is possible to observe the deformation (i.e. phasedifference) curves of a cantilever beam exposed to a sequence of loads with the phase shifting provided by a PZT. It is clear that the sequence of the curves' area is directly related to the sequence of loads, which was expected. However, in Fig. 11, the same experiment in turn using digital phase-shifting presented the curves tangled in some loads; therefore, this is clearly a challenge regarding the use of the proposed method mainly related to image processing. In Fig. 12, the images of the fringes obtained by the proposed method are presented, and when they were compared to the PZT method, in the loads of 60–90 g, the upper fringes were placed in a different position. This is mainly related to the difficulty of the digital method to identify properly the upper fringes when they were spread and blurred, such as that observed in the load 50 g, thus shifting it to the bottom of the image. The solution to this problem was found by cropping the prime images before starting the digital shifting, eliminating the upper part of the image, as pointed out in Fig. 12. In Fig. 13, it is possible to see the result after the cropping, and then to identify a solution of the problem caused by undefined upper fringes. The cropping did not compromise the analysis of the cantilever regarding its deformation characteristics,
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Fig. 17. Results of the deformations obtained by phase difference using a PZT (dotted line), using a digital phase-shift (continuous line) and finite element analysis (dashed line) for the loads (a) 35 g, (b) 45 g, (c) 50 g, (d) 60 g, (e) 70 g and (f) 90 g.
considering that it is homogeneous. However, if there is information over the limit of the crop, then the analysis needs to be tailored to place the fringes, in order to let the proposed method, recognise those that are relevant in the analysis. Therefore, the curves were placed in the expected order of deformation, except for the curve linked to load 19 g, since the post-processing of the images did not provide the same result in this instance. The deformation created by the load 19 g, the lowest limit of our test of sensitivity, showed that additional attention must be taken into account with the lowest portion of the image, where an undefined fringe appears. Some changes in the analysis of 19 g images were performed, and the results are presented in Fig. 14. The first change was in the software when the routine identifies the fringes; the other was by a manual crop of the bottom portion of the image. The expected results prevenient by the correction of the images generated by the 19 g load were conducted using a different approach to the other loads from 60 g to 90 g. Thus, one can see that the creation of digital phase-shifting is reliable; however, particular attention must be taken to define the sensitivity limits, and then the necessary corrections and post-processing.
In order to overcome the subjectivity of manual cropping, the user must define the best positions of the fringes particularly concerned with the edges (bottom and top) and with the loads. Since the aim of this work was also to test the sensitivity of the proposed method, where we had to maintain the same configuration to obtain the deformation caused by loads from 19 g to 90 g, we could not adjust the best position of the fringes. This could be effected during the setup of the system, avoiding the crop. In Fig. 15 it is possible to see the results from the cantilever beam loaded with 90 g using a PZT and the proposed digital shift, compared to the result with the theoretical curve. The other experiments using all loads were also carried out; however, we only present that with the load 90 g, since they are all rather similar. The comparisons with the theoretical curve presented the ability of the proposed method to reproduce the expected phenomena. Some additional adjustments can be effected to calibrate the outcomes, i.e. using an off-set correction. The adoption of the methods using different loads with replications can be considered as confirmation of the expected potential of the proposed method. Here, the results of loads 10 g, 20 g and 30 g were used with three replications. The results of 30 g were not presented, since they
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evinced the same pattern as that of 10 g and 20 g. In addition, the other assay using seven different loads from 19 g to 90 g presented more replications of the proposed method, showing its reliability. The results from the PZT and from the proposed digital phaseshift were also compared to an FEA simulation (Fig. 16); the results can be seen in Fig. 17. The region in the cantilever beam chosen to compare the results was related to a portion where the deformation does not compromise the correlation of the speckle pattern. The results presented the ability of the proposed technique to follow the phenomenon with a similar behaviour compared to a simulation provided by an FEA, with a better correlation when compared to the PZT approach. The digital creation of phase-shifting presented reliable results with repeatability and sensitivity compared to the traditional method using the PZT or similar devices. The main limitation of using this proposed methodology is the restriction to parallel fringes, or quite parallel, such as in a cantilever under the Young's modulus test. The adoption of irregular fringes will demand processing in two dimensions that is feasible [31] if associated with a methodology such as that proposed here. The assumption that the interpolated fringes vary linearly does not present a great limitation, since the distances of the original fringes were respected. In addition, the results showed that the final quality of the curves was comparable to that using a PZT. In addition, the feasible adoption of the simplest interpolation enhances the potential of the proposed method when the most sophisticated methods of interpolation are adopted. Despite the reduced cost of the proposed technique, which makes it assessable to a greater number of users, its adoption when only one fringe can be acquired, such as in the case of pulsed laser application, or when it is necessary to monitor an object under movement, justifies its application.
4. Conclusions The digital phase-shifting approach was able to monitor the deformations of a cantilever beam as well as the traditional method using the PZT as the phase-shift generator. The proposed technique was able to reproduce the deformation of the cantilever beam in the intermediary region between the clamping point and its free end, presenting sensitivity, repeatability and robustness, which justify its use as an alternative to PZT in traditional applications, and mostly when the PZT cannot be used.
Acknowledgements This study was supported by the Physiology Department of Universitat de València, the Federal University of Lavras, and the Federal Institute of Minas Gerais, Campus Bambuí, and was also financed partially by CNPq (302805/2012-5), Capes, FINEp, and FAPEMIG (CAG - PPM-00075-15).
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Roberto A. Braga Jr. graduated with a Master's degree in Electrical Engineering from UFMG, Brazil, and has a Doctoral degree in Agricultural Engineering UNICAMP, Brazil. He has been an Associate Professor at UFLA, Brazil, since 1996. He teaches electricity and automation and researches optical instrumentation. His sabbatical leaves include BIOSS Scotland (2005 and 2008), senior stage in BIOSS, and JHI Scotland in 2011. His publications include more than 40 papers, the majority in international journals; he has some patents, books chapters, and books published.
Thiago Paiva Almeida is a technician in informatics (core technical education) since 2005, BA in Information Systems UNIPAC (2009), Specialist in Computer Networks from ESAB (2013), and is currently a Master's student in Systems Engineering and Automation at UFLA (Federal University of Lavras). He currently teaches courses in IT and technical drawing at the Institution of Higher Education CEMES and is a temporary teacher at the Federal Institute of Minas Gerais (Bom Sucesso), where he teaches technical courses in computer areas.
Rolando J. González-Peña received his Master of Science degree in Optics from the High Technology Institute of Havana in 1998, and his PhD in Technical Science in 2001. His current research is focused on digital speckle interferometry and dynamic speckle lasers. Currently, he is Professor at the Physiology Department in the Faculty of Medicine and Odontology at the University of Valencia.
Igor V. A. Santos is an undergraduate student of Control Engineering and Automation at UFLA Federal University of Lavras. He has received a scholarship for research from FAPEMIG in the area of shearography.
Marlon Marcon graduated with a Master's degree in Systems Engineering from the Federal University of Lavras (2009). He has been a Professor at the Federal University of Technology - Paraná. He graduated with a Bachelor of Computer Science from the State University of West Paraná (2006). He has experience in Computer Science, with an emphasis on Computer Vision, Natural Language Processing (NLP), Machine Learning and Pattern Recognition, mainly in the following areas: construction of mathematical models based on image analysis, lexicography, computational linguistics and Support Vector Machines (SVM).
Moisés Batista Martins is an undergraduate student of Control Engineering and Automation at UFLA Federal University of Lavras. He is a technician in Mechanics at the Municipal Technical School of Sete Lagoas. He has taken a training course of Industrial Training in Machining Mechanical SeteLagoas/MG, and experience as a Maintenance Mechanic at the Ambev and Iveco companies.
Ricardo R. Magalhaes received his Master of Science degree in Mechatronics (2008) and PhD in Industrial Engineering (2011), both from the University of Bahia. He is a Professor at the University of Lavras. He has experience in automotive companies with an emphasis on product development. His current research is focused on measurement techniques, the finite element method, and stress analysis.