## Abstract

Time-averaged fringe patterns in vibration testing of MEMS (microelectromechanical systems) are unaffected by carrier displacements. They are additive superimposition type moirés. These features and Hilbert transform vulnerability to additive trend are utilized for visualization of centers of dark Bessel fringes. Two frames with shifted carrier are subtracted for background and noise correction. Two normalized images of this pattern are calculated with slightly different bias levels and subtracted. The method does not require precise phase shifting between two frames, cosinusoidal carrier and linear recording. It enables detecting light power variations and phase shifting nonuniformities. Synthetic and experimental results corroborate the robustness of the method.

©2013 Optical Society of America

## 1. Introduction

Optical whole-field measurement methods with simultaneous acquisition of experimental data and their parallel processing are well suited to evaluate both macro and microscale objects. One of the most important mechanical tests is vibration testing. It provides material properties in macro- and microscale, validation of designs and simulations, and information for the manufacturing process optimization. Among optical full-field methods time-averaged and stroboscopic techniques are commonly used for that purpose [1–4].

Time-averaged approach with comparison to other measurement techniques demonstrates the experimental setup simplicity and independence of the vibration frequency value. In the case of a sinusoidal vibration the interferogram modulation changes are described by the zero-order Bessel function J_{0} of the first kind. This function encodes the vibration amplitude information in its argument. To find the fringe pattern modulation envelope several multi- and single-frame automatic fringe pattern analysis (AFPA) techniques were proposed, for example temporal phase shifting (TPS) [4–6], Fourier transform (FT) [7], spatial carrier phase shifting (SCPS) [8], continuous wavelet transform (CWT) [9], and the 2D Hilbert transform aided by the bidimensional empirical mode decomposition [10].

Note that references [4–10] concern the studies of MEMS devices showing great potential of the time-average technique in testing MEMS components operating at high resonance frequencies. Fast growing applications of MEMS and their elements, including actuators and sensors, introduce pioneering requirements on their design and testing to ensure product quality and reliability (see, for example [11–13], and references therein).

Once the pattern modulation envelope is determined, the vibration amplitude information has to be extracted. Most comprehensive approach uses heterodyning with additional phase modulation (of the same frequency as the vibrating object) in one arm of the interferometer [2,14]. In this way the Bessel fringes can be analyzed by temporal phase shifting (TPS). Controlled phase shifting of the modulation envelope enables determination of the vibration phase. Heterodyning approach complicates, however, the experimental setup hardware considerably. In some cases of analyzing harmonic vibration modes, the vibration amplitude sign change localizations can be readily predicted. The prime goal, therefore, is to find the amplitude distribution over the object under test. Tracing dark Bessel fringe centers representing equidistant contours of equal amplitude serves the purpose. Interpolating the data between the centers provides a 3D map of vibration amplitude. In the case of considerable vibration amplitudes, however, high order Bessel fringes become very dark and their tracing might become quite difficult or just impossible.

For highly contrasted Bessel fringe extrema (minima) determination we propose a novel method based on the Hilbert transform (HT) processing of two Bessel fringe patterns with phase shifted carriers (projected gratings in incoherent vibration profilometry [15–17] and two-beam interference fringes [4–10] in coherent projection methods). Processing the difference of two frames yields much better results than using a single frame because the information we want to access is located where the modulating Bessel function reaches 0. Background and common low frequency noise removal by subtracting two π shifted images was proposed several years ago to Fourier transform profilometry to extend the measurable slope of height variation [18]. Our algorithm for precise detection of the centers of dark Bessel fringes starts with subtraction of two phase shifted frames as well. We exploit two unique properties of Bessel fringes: (a) they are stationary against the projected carrier displacement, and (b) their intensity distribution corresponds to the additive type moiré [19–22]. The latter feature means that the pattern has a constant average intensity level (as opposed to the most commonly encountered multiplicative type moiré [19–22]) accompanied by the amplitude modulated carrier (AM in communication techniques). The modulation envelope (macrostructure) corresponds to the moiré pattern modulating the carrier (microstructure). Because of the cosine variation of the moiré term (the case of linear summation of two cosine patterns) the phase of the other one, the microstructure, changes intrinsically by π at the crossover points. In time-averaged patterns the carrier phase jumps are connected with Bessel function minima (actually with the modulus of Bessel function, as we do not detect negative intensity).

We propose to visualize the crossover points by normalizing Bessel fringe patterns using the Hilbert transform [10, 23–25]. Before applying HT the input pattern should be background and noise corrected. This is implemented by subtracting two carrier phase-shifted Besselograms. Next the pattern normalization process is conducted twice with slightly different uniform background levels and two normalization results are mutually subtracted. Very narrow and bright contour lines of the crossover points on a dark background enable highly contrasted visualization and tracing of the centers of dark Bessel fringes.

The principle of our method, i.e., subtracting two normalized fringe patterns makes it very robust from the experimental point of view. Firstly, phase shifting of two input frames does not require a precise phase displacement value, although the brightest output result is obtained for the phase shift of π [26,27]. Secondly, the requirements regarding linearity of the recording process and using sinusoidal carrier (projected pattern) do not have to be strictly obeyed as in the case of interferogram phase calculations using the Hilbert transform [28].

Additional interesting features of the proposed method relate to the detection of possible light source power variations and nonuniform phase displacements between the two subtracted time-averaged frames. The second error, the so-called tilt-shift error, can be caused by tilting or twisting of (a) the PZT system during translation of the interferometer reference mirror [29–32] in two-beam interference projection systems, or (b) the projected grating in incoherent projection techniques. Both errors cause blurring of otherwise sharp bright contours. One of the errors, i.e., light source power variations can be removed by decomposing the result of subtracting two phase-shifted frames and excluding the residue (uneven background illumination). For image decomposition we use the FABEMD algorithm [22,25].

The method simulation models and applications to real data processing of time-averaged two-beam interferograms are given to corroborate the principles.

## 2. Method and algorithm description

Let us describe the principle of our method on the example of the interference generated sinusoidal intensity distribution carrier projected on the object specular reflection surface (interferometric studies of vibrations of MEMS/MOEMS silicon elements). For a harmonic sinusoidal vibration, the intensity distribution of a two-beam time-averaged interferogram can be expressed as [4–10]

_{stat}(x,y) is the contrast of interference fringes (nonvibrating object), φ

_{vib}(x,y) = (2π/λ)OPD(x,y), where OPD(x,y) is the optical path difference between the beams, and λ is the light wavelength. J

_{0}is the zero-order Bessel function of the first kind, and a

_{o}(x,y) is the vibration amplitude distribution encoded in the amplitude modulation of the cosine term. In the case of vibration profilometry OPD(x,y) is usually a linearly varying function, e.g., along the x direction, resulting in straight, vertical, sinusoidal carrier fringes projected on the object surface under test. The intensity distribution described by Eq. (1) corresponds to additive moiré [9,22]. The J

_{0}(x,y) function is the moiré term carrying the measurand information, and the second cosine term describes the carrier.

The approach to resolve the carrier to monitor the modulation function is rather arduous; moreover, the contrast of modulation bands in additive-type moiré is very low. The simplest and most common solution to increase the visibility of time-averaged fringes is to apply nonlinear detection [20, 33–36]. In result, an isolated term in the intensity distribution, similarly to the case of multiplicative moiré, is generated. More elegant solution to visualize additive-type moiré fringes is to calculate the modulation envelope using multi- and single-frame automatic fringe pattern analysis techniques [4–10]. Once the modulation envelope is found a quantitative evaluation of the vibration amplitude distribution follows.

Another approach of tracing dark Bessel fringes providing equidistant contours of vibration amplitude and interpolating the data between the contours belongs to the group of intensity demodulation methods [37,38]. These methods are simpler, from the experimental setup point of view, than the phase demodulation techniques. In the case of large vibration amplitudes high order Bessel fringes become very dark and their tracing is difficult. This paper deals with a novel method for highly contrasted dark Bessel fringe center visualization.

Figure 1(a) shows, as an example, a time-averaged two-beam interference pattern obtained for a circular silicon membrane vibrating at 659 kHz [9,10]. Figure 1(b) highlights its magnified local region with characteristic additive-type moiré intensity distribution including inherent phase shifts of π of the carrier (note the carrier in the form of circular two-beam interference fringes; the silicon active membrane surface departed from flatness [5, 11]). The crossover points lie along the center of the additive-type moiré fringes where the modulation envelope assumes its zero values. Accessing the information on the crossover points is aided by minimizing the noise and possible background variations in the time-averaged patterns. Very effective solution is to subtract two time-averaged frames, Fig. 1(a), with mutually phase shifted carriers. Figure 1(c) shows the result for the phase shift of π in the same local region as in Fig. 1(b). Significant improvement of the pattern quality used as the input to our dark fringe center contouring algorithm, based on the Hilbert transform, can be readily apprehended.

The exact phase step value of π as proposed for Fourier transform profilometry to increase the object measurable slope [18], is not a stringent requirement for us. We do not need phase calculations [18] but the pattern modulation distribution. It is computed basing on the analytic signal approach as the square root from the sum of squares of the pattern real and imaginary parts. The latter one is generated by the Hilbert spiral transform [39,40], denoted below as HS [10,25]. In such a case the carrier exact phase displacement between the two frames is not required. Detailed mathematical description of this very useful and robust feature can be found in references [26,27] and will be not repeated here. The calculated modulation distribution output is the brightest for the phase shift of π, it reduces with the phase step departure from this value. Please note that although the papers [26,27] concern the full-field optical coherence tomography (FF-OCT), the issue of the image modulation determination is in common with our technique.

We would like to note, additionally, that studies presented in [26,27] concerned the case of sinusoidal carrier. The influence of the phase shift value between frames encountered in the case of possible occurrence of system nonlinearities, i.e., (a) sinusoidal carrier nonlinear recording and (b) the use of nonsinusoidal carrier (e.g., binary Ronchi-type grating) will be discussed below. Their presentation constitutes a novelty as far as the Hilbert transform determination of the fringe pattern modulation distribution and interferogram normalization are concerned. To the best of authors’ knowledge, up to now only the influence of fringe pattern nonlinearities in phase distribution calculations has been treated [28].

Our processing path proceeds as described in flow chart depicted in Fig. 2. As a result of the algorithm we obtain highly contrasted dark Bessel fringe contour lines (*dif_n*) computed as a difference of two normalized fringe patterns with slightly different bias level.

To compute Hilbert transform correctly one needs to clear out the bias level (the uneven background illumination term) of the analyzed signal. For this purpose we use fast adaptive bidimensional empirical mode decomposition (FABEMD) algorithm proposed in [41]. It is adaptive and data-driven approach. Unlike in the Fourier or wavelet methods, no predefined decomposition basis is used; it is rather derived from the signal itself. It decomposes an image, in so-called sifting process, into a set of bidimensional intrinsic mode functions (BIMFs), representing image features at various spatial scales and quasi-monotonic residue which we consider as a background term. In the FABEMD the envelope estimation required for image decomposition is performed by a nonlinear order-statistics-based filtering followed by a smoothing operation. It significantly shortens computation time and augments the extracted BIMF quality. Recently our group has successfully employed the FABEMD method for studying the moiré phenomenon [22, 42] and low quality fringe pattern adaptive and automatic enhancement [25,43]. For more details regarding the FABEMD method please refer to the aforementioned literature and references therein.

## 3. Numerical evaluation studies

For the sake of simplicity we start our simulation works with the input pattern of the form

_{stat}(x,y) are usually slowly varying functions, we do not enter them into the simulated input pattern. In simulations 512x512 pixels images were tested; computation time obtained for one image was around 0.15 seconds on medium class PC.

#### 3.1 Sinusoidal carrier projection and linear detection

Figure 3 presents processing results of the “starting case” corresponding to:

- 1. frame1 = bessel * cos(x); bessel = J
_{0}(x,y), constant background distribution, noise free input pattern; - 2. frame2 = bessel * cos(x + π) + noise + DC; bessel – as above, added noise with SNR = 2, DC denotes the background intensity difference between time-averaged frames frame1 and frame2;
- 3.
*dif*= frame1 – frame 2; - 4.
*modu1*and*norm1*= HS(*dif*); - 5.
*modu2*and*norm2*= HS (*dif*+ bias * dif_amplitude), where “bias” denotes normalized difference between constant bias levels used when calculating two normalized patterns*norm1*and*norm2*by the amplitude value of*dif*equal to 2 (as it is the difference of two modulated cosines). - 6.
*dif_m = modu2 – modu1*; - 7.
*dif_n = norm2 – norm1*; it is the desired contour pattern of the centers of dark Bessel fringes.

The conclusions following from Fig. 3 are as follows.

- 1. In the case of the input pattern
*dif*without DC and noise we obtain a perfect reconstruction (c) of the modulation function J_{0}(x,y) and a perfect normalized image (d). Note the error in the modulation distribution*modu2*containing residual parasitic carrier fringes, and the error in the normalized image*norm2*(presence of finite width, very narrow bright bands centered along the centers of dark modulation bands). These intentionally introduced “errors” arise because of the nonzero value of parameter “bias” entered when computing images*modu2*and*norm2*. This is the basis of our method – upon subtracting*norm1*from*norm2*we get highly contrasted delineation of the extrema of dark Bessel fringes. Very narrow contours in*dif_n*are generated for DC = 0 (i.e., both input time-averaged frames frame1 and frame2 are devoid of DC or represent exactly the same bias intensity level which cancels upon the frame subtraction) and small values of the parameter “bias”. This conclusion results from several computer simulations conducted for increasing values of DC and “bias”; they are not reproduced here for the sake of paper reasonable length. Contour lines blur quite fast with increasing DC and “bias” values. - 2. The denser (higher spatial frequency) is the projected carrier the better (smoother) are the contour lines (the effect of “denser sampling”).
- 4. Obtained contour pattern of the centers of dark Bessel fringes, Fig. 3(h), may contain negative intensity values (as a result of subtracting two patterns normalized with slightly different DC). As we do not consider negative intensities the modulus operation can be applied setting contrast level to 1. Cross section of dark Bessel fringe contours (Fig. 3(h)) is shown in Fig. 4. Its high contrast can be readily noticed (strong peaks; minimum intensity value is equal to zero).

The similarity between images *norm2* and *dif_m* seen in Fig. 3 occurs in the case of both time-averaged frames free of noise. This similarity conclusion is no longer valid for real experimental situations; the superiority of the two-frame approach will be evidenced in the experimental part of this paper.

The case with noise is shown in Fig. 5. Actually the results presented in Fig. 5 can be interpreted as the ones corresponding to processing a single frame contaminated with noise. It can be clearly seen how important is noise reduction when the pattern processing is conducted using the Hilbert transform. Please note that the simulations with the presence of additive noise in both frames does not provide representative results since the noise cancels to a great extent upon subtraction. In the case of real experiments with silicon reflective surfaces (with a common low frequency noise present) we will show both single and two frame processing results. The advantage of the latter one becomes evident (see experimental part below).

#### 3.2 Detection and correction of uneven background error caused by light source power variations between the frames

We will supplement the simulation results presented in Fig. 3 by assuming a nonuniform change of DC level between the two time-averaged frames (corresponding to light source power changes in time). For simplicity, we assume the DC level difference between two frames to be simulated by linear function, i.e., DC = sin(x), other parameters assumed are the same as in Fig. 3.

Characteristic brightness changes in *dif_m* and blurring of the contour lines in *dif_n* along assumed direction of DC variation are readily seen in Figs. 6(a) and 6(b). Negative DC values on the left hand side (uncorrected background intensity sine function attains zero value at the image center, see Fig. 6(c)) show stronger influence on the contour line sharpness than the positive ones (right hand side). Accordingly, we may say that our algorithm can detect possible bias changes between the two frames being compared.

This error can be easily corrected employing the FABEMD algorithm for the frame difference image *dif* decomposition followed by neglecting the residual part (high-pass filtering to remove background variation stored in the decomposition residue) [10,22,24,25,43]. Figures 6(d)-6(f) show the results after FABEMD correction applied to the simulation data of Figs. 6(a)-6(c). The RMS value computed using Fig. 3(h) as a reference dropped from 0.17 to 0.02 after bias correction.

#### 3.3 Sinusoidal carrier projection and nonlinear detection

The effect of nonlinear detection of a sinusoidal intensity distribution carrier projected on the object surface under test was simulated by representing the intensity distribution of two recorded time-averaged frames by

It follows from Fig. 7 that quality deterioration of contour lines (their blurring) increases for stronger nonlinearities. Nevertheless, for very small nonlinearities provided by CCD detectors present on the market we can conclude that our algorithm can be considered as quite robust to nonlinear recording of time-averaged patterns. This property follows from the final step of our algorithm, i.e., subtraction of two normalized patterns with slightly differentiated uniform “bias” values. Errors present in both normalized patterns (due to unsatisfied linearity requirement for HS) seem to cancel out during the subtraction process.

The above conclusion was drawn from the comparison of Figs. 1 and 3 corresponding to the case of two time-averaged frames with a carrier shifted by π. It follows from our simulations that the departure from the phase shift value of π influences the sharpness of dark Bessel fringe contour lines quite slowly. Additionally, similarly to the case of linear detection, the best results are achieved for DC = 0 and very small values of the parameter “bias”.

#### 3.4 Nonsinusoidal carrier projection and linear detection

The case of simulating highly nonsinusoidal carrier projection (e.g., binary amplitude Ronchi type grating with aspect ratio equal to 0.5) with linear detection is presented below. Figure 8 shows our simulation results for the same algorithm parameters as in Fig. 3, i.e., carrier phase displacement between two time-averaged frames equal to π, DC = 0, noise free, “bias” = 0.025.

Despite the erroneous modulation distribution calculation, Fig. 8(c), the final result provides highly contrasted dark fringe center lines, Fig. 8(f), with very low RMS value equal to 0.01 (computed with Fig. 3(h) as a reference). This fact indicates that our algorithm works equally well for sinusoidal and nonsinusoidal projected carriers.

#### 3.5 Nonsinusoidal carrier projection and nonlinear detection

This is the case when two previously studied nonlinearities appear simultaneously. In consequence the effects of both nonsinusoidal carrier projection and nonlinear detection add producing slightly broadened contour lines. This effect increases with increasing the degree of recording nonlinearity. With small nonlinearities offered by CCD matrices our method can be considered as quite robust to both types of nonlinearities.

#### 3.6 Detecting nonuniform phase displacement between time-averaged frames

The so-called tilt shift error corresponding to nonuniform phase shift between two time-averaged frames (caused by tilting or twisting of the PZT system during translation of the interferometer reference mirror [29–32] or by tilting or twisting incoherently projected pattern) can be detected by our algorithm. Figure 9 shows simulation results of this property with the frame2 cosinusoidal carrier period changed by 1% with comparison to frame1.

Note the changes in intermediate modulation and normalization maps contributing to final *dif_n*. Unsymmetrical blurring in *dif_n* is readily observed (producing relatively high RMS value equal to 0.19). It looks similar to the one introduced by linear change of DC term between the frames, see Fig. 6. There is a considerable difference, however. In Fig. 6 blurring is accompanied by strong decrease in intensity level over the blur area (large Hilbert transform error). In Fig. 9 blurring is of the same intensity (contrast) level as the rest of the image area due to different origin of the error. The tilt-shift error cannot be removed as the DC error by using the empirical mode decomposition method.

## 4. Experimental data processing

Our experimental work begins with single frame processing of the two-beam time-averaged interferogram obtained in vibration testing of a circular active silicon micro-membrane (1 mm diameter) with resonance vibration frequency equal to 659 kHz [5,6]. Every experimental image examined in this section had 768x576 pixel dimensions and calculation time under 0.25 seconds on medium class PC (if the FABEMD filtering was used calculations lasted around 1 second). Results are outlined in Fig. 10. Single frame (frame1 shown in Fig. 10(a)) was high-pass filtered (neglecting the image residual part) using FABEMD algorithm [22,25,42,43] to remove the bias intensity term. Denoising by means of excluding the first BIMF is not possible due to presence of the information carrying high spatial frequency carrier in BIMF1. Output of our algorithm, Fig. 10(d), in terms of noisy contour lines is not satisfactory. To augment the quality of dark Bessel fringes tracking we use the above described two-frame processing approach.

We have at our disposal five π/2 carrier phase shifted frames, (frame1 presented in Fig. 10(a) and frame2 to frame5 shown in Fig. 11) originally recorded for the five frame modulation distribution calculation scheme [5,6]. We take into account three mutually π shifted frame pairs, e.g. pairs (1,3), (2,4) and (3,5), as this phase step value is most suitable for our dark Bessel fringes visualization technique (see previous sections). We proceed with subtracting frames within the pairs to create three common noise and bias free input patterns referred to as *dif(1,3), dif(2,4), dif(3,5).* Processing results obtained using the proposed Hilbert transform based algorithm are presented in Fig. 12, Fig. 13 and Fig. 14, respectively.

It is worth to notice that the best outcome was obtained for *dif(2,4)* processing, i.e., the modulation distribution *modu(2,4)* is free of parasitic fringes, *dif_m(2,4)* is of the same average intensity in the membrane part and the contour lines *dif_n(2,4)* are slim, sharp and bright over the whole test region. It means that frame2 and frame4 have the same DC term which was canceled out in subtraction - thus the Hilbert transform performed very well on the bias free *dif_n(2,4)* pattern. Processing results of *dif(1,3)* and *dif(3,5)* display uneven background error caused by light source power variations between recorded frames. The bias intensity term varies between the frames and does not cancel out upon subtraction. This error can be easily detected by our approach (at the image processing level) as blurring of the contour lines (*dif_n(1,3), dif_n(3,5)*) and uneven averaged intensity level in *dif_m(1,3)* and *dif_m(3,5)*, see Fig. 12 and Fig. 14. The blur of contour lines appears where the DC term of *dif* departs from zero (and is heavier for positive values) due to the Hilbert transform vulnerability to uneven bias term (see the numerical simulation part of the paper). Moreover, the light source power variation error influences modulation distributions *modu(1,3)* and *modu(3,5)* introducing parasitic fringes readily observable in Fig. 12(b) and Fig. 14(b). It is not the case in properly recorded frame2 and frame4 processing, Fig. 13(b), where parasitic fringes are almost indiscernible.

The result of correcting the effect of illumination variation between the frames is illustrated in Fig. 15. The FABEMD algorithm was used to eliminate uneven bias term. Note a smooth modulation distribution, Fig. 15(b), uniform averaged intensity level in *dif_m*, Fig. 15(c), and bright sharp contour lines in *dif_n,* Fig. 15(d).

For the sake of our algorithm further potential and robustness corroboration we present processing results for three selected time-averaged patterns. First two patterns, Fig. 16(a) and Fig. 17(a), were obtained in vibration testing of a circular silicon micro-membrane vibrating at 930 kHz and 365 kHz, respectively [5]. First Besselogram corresponds to a complex resonance mode with small bright spots (nodes) characteristic to multiple irrational vibration frequency excitation. In this case the joint characteristic function becomes the product of the separate characteristic functions for each separate time function [1, 44]. Nodes are encountered only at the intersection points of the component Besselogram nodes. The second one shows high vibration amplitude regions represented by large number of Bessel fringes in four fringe “islands”. Both analyzed time-averaged interferograms were preprocessed using the FABEMD bias correction technique and satisfactory results were obtained, see Fig. 16 and Fig. 17. They corroborate that our algorithm performs well when applied to complex shape Bessel patterns with a large number of fringes. The final experimental result presented concerns testing of a rectangular silicon micro-membrane (1mm x 1mm) vibrating at 172 kHz. Results outlined in Fig. 18 were produced without FABEMD correction, this case illustrates a proper frame acquisition process. Despite parasitic fringe-free modulation distribution, Fig. 18(b), the contour line quality drop can be observed in the upper center part of Fig. 18(c). It is not connected with DC variation but with local modulation departure from the Bessel shape function, see Fig. 18(b). It can be attributed to a membrane locally different performance. It is worth to note that our algorithm is very fast - with efficient implementation of the FABEMD algorithm for *dif* preprocessing our method completes calculations under one second.

## 5. Conclusions

The method for highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry has been proposed. It uses Hilbert transform processing of the image obtained by subtracting two time-averaged frames with phase shifted carriers to remove the common background and noise. This filtered pattern is then two times normalized by its modulation distribution calculated using the Hilbert spiral transform for the case of the zero bias level and slightly increased DC. The Hilbert transform errors (encountered in the latter case) found in the carrier phase jump area, localized at the Bessel fringe minima, enable highly contrasted visualization of the contour lines of dark Bessel fringes. It is obtained by the subtraction of two normalized patterns. The method proposed is very fast (computation time below 0.25 second, with the FABEMD filtering - around 1 second) and robust - it does not require precise phase stepping between two time-averaged frames, strictly cosinusoidal carrier and linear recording. Moreover the presented algorithm exhibits two additional features, e.g., detecting possible light source power variations and phase change nonuniformities (tilt-shift error) between the two recorded time-averaged frames. The solution for correcting the first error using the fast and adaptive empirical mode decomposition method has been proposed and implemented. Results of processing real and simulated patterns corroborate the potential of the presented technique.

The method has been developed and tested for processing time-averaged out-of-plane vibration patterns of MEMS piezoelectric micromachined devices. It would be interesting to apply it to process other fringe patterns with the above mentioned intensity distribution characteristics, i.e., additive type moiré with constant average intensity level and stationary against carrier displacement. The method can be directly extended to the studies of the influence of grating profile on the Besselogram intensity distributions when testing in-plane vibrations [45]. Ragulskis and associates showed [45] that in the case of the grating opening ratio (defined as the ratio of slit width to grating period) not equal to 0.5 the centers of dark Bessel fringes shift from the positions common to the use of sinusoidal and Ronchi gratings. Our processing scheme is expected to detect dark fringe centers with high accuracy. This topic, however, deserves separate treatment and we do not include it in this work dealing with out-of-plane vibrations. Other applications of our method concern the interferograms and moirégrams generated in the self-imaging phenomenon based lateral shear interferometry, i.e., Talbot interferometry [46–50] and the three-beam Ronchi test [51]. The additive type moiré carries an information on the second derivative of the phase function under investigation. It is unaffected by lateral displacement of carrier fringes with first derivative information. This topic, however, is out of scope of this study.

## Acknowledgments

This work was supported, in part, by Ministry of Science and Higher Education budget funds for science in the years 2012-2015 as a research project in the “Diamond Grant” programme and partly by statutory funds. The experimental work described in [5], from which some data were taken, was co-supported by the European Union (EU) Project OCMMM (a part of the experiments under this project was performed at Laboratoire d’Optique P.M. Duffieux, Université de Franche-Comté, Besancon, France). Work of MT was partially supported by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme.

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