Time average vibration fringe analysis using Hilbert transformation

Time average vibration fringe analysisusing Hilbert transformation

Upputuri Paul Kumar, Nandigana Krishna Mohan,* and Mahendra Prasad KothiyalApplied Optics Laboratory, Department of Physics, Indian Institute of Technology Madras, Chennai-600036, India

*Corresponding author: [email protected]

Received 7 June 2010; revised 9 September 2010; accepted 14 September 2010;posted 14 September 2010 (Doc. ID 129715); published 14 October 2010

Quantitative phase information from a single interferogram can be obtained using the Hilbert transform(HT). We have applied the HT method for quantitative evaluation of Bessel fringes obtained in timeaverage TV holography. The method requires only one fringe pattern for the extraction of vibration am-plitude and reduces the complexity in quantifying the data experienced in the time average referencebias modulation method, which uses multiple fringe frames. The technique is demonstrated for the mea-surement of out-of-plane vibration amplitude on a small scale specimen using a time average microscopicTV holography system. © 2010 Optical Society of AmericaOCIS codes: 120.6165, 100.2650.

1. Introduction

Knowledge of resonance frequencies and modeshapes of engineering structures subjected to harmo-nic excitation is of interest in industrial applications.Vibration testing and evaluation provides informa-tion on the material properties at the microlevel, va-lidation of the design and simulation, prevention offatigue failure, detection of noise generating parts,and information for optimization of the manufac-turing process. TV holography, or digital/electronicspeckle pattern interferometry, is a well-establishedoptical interferometric technique for the measure-ment deformation under static as well as a dynamicload on rough surfaces [1,2]. Time average TV holo-graphy is generally used as a method for vibrationamplitude measurement [3–5]. The method gener-ates speckle correlation time averaged J0 fringesthat can be used for full-field qualitative visualiza-tion of mode shapes at resonant frequencies of anobject under harmonic excitation. It is well-suitedfor qualitative fringe analysis in real time. In orderto map the amplitudes of vibration, quantitative eva-luation of the time averaged fringe pattern is de-

sired. Interferometric techniques use temporal orspatial phase modulation for phase evaluation [1,2].For quantitative vibration fringe analysis, the stro-boscopic illumination method [6] where there is con-ventional phase shifting can be applied, and for thetime average method [3–5], where there is the refer-ence phase modulation method, must produce phaseshifted frames. Both methods use multiple framesand require a complicated and expensive optical set-up for producing phase shifted frames. Analysisusing a single time average frame obviously is anattractive proposition.

In this paper, we explore the use of the Hilberttransform (HT) method to produce a phase shiftedframe for phase analysis of a time average fringe pat-tern. This procedure requires a single fringe pattern,avoiding the complexity in the experimentally ob-tained phase shifted time average fringe patterns.The phase error introduced due to the assumed J0function as a cosine function along with the errordue to the limited number of fringe cycles availablein the frame is corrected by generating a lookup ta-ble. The method is demonstrated for the measure-ment of vibration amplitude at the fundamentalfrequency of a piezoelectric transducer (PZT) cantile-ver beam and a circular diaphragm. We have also

0003-6935/10/305777-10$15.00/0© 2010 Optical Society of America

20 October 2010 / Vol. 49, No. 30 / APPLIED OPTICS 5777

compared the results obtained with the HT and biasmodulation methods.

2. Theory

A. Visualization of the Time Average Speckle Fringes

The intensity distribution of a time average frameobtained by sinusoidally exiting the object with a fre-quency much higher than the video frame rate can beexpressed as [1–5]

Iavgð0Þðx; yÞ ¼ Ioðx; yÞð1þ Vðx; yÞ cosðϕðx; yÞÞJ0ðΩðx; yÞÞÞ; ð1Þ

where Ioðx; yÞ is the bias intensity, Vðx; yÞ is the vis-ibility, ϕðx; yÞ is the random speckle phase,J0ðΩðx; yÞÞ is the zero order Bessel function of thefirst kind, and Ωðx; yÞ is the fringe locus function gi-ven by Ω ¼ ð4π=λÞA, where A is the amplitude of vi-bration of the object. The fringes represent contourvalues of Ω. Equation (1) results in poor contrastfringes due to the bias fluctuations. Several methodsto improve the contrast of the time average fringeshave been developed, such as (a) the four-framemethod [3–5], (b) the subtraction of the initial state

from the excited state of the object [7,8], and (c) theautomatic refreshing reference frame method [5,7,8].The procedures (a) and (c) give similar contrast, but(c) requires only two frames. It requires a π phaseshifted frame with respect to the frame given byEq. (1). The intensity distribution of the π phaseshifted frame can be expressed as

IavgðπÞðx; yÞ ¼ Ioðx; yÞð1− Vðx; yÞ cosðϕðx; yÞÞJ0ðΩðx; yÞÞÞ: ð2Þ

A contrast enhanced time average fringe pattern canbe obtained by using the following equation:

P ¼ jIavgð0Þ − IavgðπÞj ¼ j2VIo cosðϕÞJ0ðΩÞj: ð3Þ

We have found that a further improvement in con-trast is achieved by eliminating the term Io as

R ¼��Iavgð0Þ − IavgðπÞIavgð0Þ þ IavgðπÞ

��¼ jV cosðϕÞJ0ðΩÞj: ð4Þ

Fringes obtained as per Eqs. (3) and (4) are shown inFig. 5.

Fig. 1. (a) Line scan of a simulated closed fringe pattern, (b) π=2 phase shifted signal from 1(a), and (c) HT signal of the pattern shown inFig. 1(a).

5778 APPLIED OPTICS / Vol. 49, No. 30 / 20 October 2010

Fig. 2. (a) Simulated J0ðΩÞ fringes, (b) fringe after applying HT to Fig. 2(a), and (c) line scan profile of the patterns shown inFigs. 2(a) and 2(b).

Fig. 3. (a) Wrapped phase map calculated using the patterns shown in Figs. 2(a) and 2(b), and (b) line scan profile.


B. Analysis of Time Average Fringe Pattern Using HilbertTransformation

The parameter of interest in Eqs. (3) and (4) is Ω,which has information on the amplitude of vibration.

In the conventional cosine fringes, the phase (orphase difference) information is obtained most accu-rately by using the phase shifting technique with thehelp of several phase calculation algorithms. Severalphase shifted intensity frames are experimentallyobtained for this purpose. This approach has alsobeen applied to time average fringes by assumingthe Bessel fringes as cosine fringes and using oneof the phase calculating algorithms [3–5]. The errorintroduced as a result of this approximation is thencorrected with the help of lookup tables. To producephase shifted Bessel fringes, bias phase modulationprocedures have been used [3–5]. This requires addi-tional optical and other hardware making the proce-dure involved. We have investigated the possibility ofusing the Hilbert transformation with Eq. (4) to pro-duce a phase shifted frame [9–11]. The application ofthe HT in a space domain, as required here, is notstraightforward. Several procedures have been in-vestigated [12–14]. However, here, we have used a1D analysis combined with a phase mask.

C. Hilbert Transformation

In signal theory, the concept of the analytic signalwas introduced by Denis Gabor [15]. According toit, with each real wave function uðxÞ, we may associ-ate a complex wave function ψðxÞ ¼ uðxÞ þ ivðxÞ. Theimaginary part vðxÞ can be obtained by the HT of thereal signal. The HT thus generates an analytical sig-nal from a real signal. The real part is the originalsignal and the imaginary part is the HT of the origi-nal signal, which is phase shifted by π=2. The HT ofuðxÞ is defined by [15]

vðxÞ ¼ HifuðxÞg ¼ 1π

Z∞

−∞

uðx0Þx0 − x

dx0: ð5Þ

In fact, it is obtained from uðxÞ by convolution withð−πxÞ−1 expressed by

Fig. 4. (a) Calibration curve (lookup table) between the input Ω and calculated Ω00, and (b) phase error εv plot.

Fig. 5. (a) Simulated J0 fringe pattern, (b) fringe after applyingHT to (a), (c) wrapped phase map with the π shift, (d) mask gen-erated by identifying the parts where π shift is happening, and (e)wrapped phase map after removing the sign ambiguity.


HifuðxÞg ¼ −1πx ⊗ uðxÞ; ð6Þ

where ⊗ represents convolution. The Hilbert trans-form is equivalent to the filtering, in which theamplitudes of the spectral components are left un-changed, except that their phases are altered byπ=2, positively or negatively, according to the signof x. The HT of the signal f ðφðxÞÞ ¼ cosðφðxÞÞ givessinðφðxÞÞ. This helps us determine the phase of theinterference signal, as given by

Ω00 ¼ arctan�Hiff ðϕðxÞÞg

f ðϕðxÞ�: ð7Þ

The value of Ω00 differs from the correct argument Ωof the J0 because of the difference between the cosineand J0 functions and can be expressed as

Ω00 ¼ Ωþ εv; ð8Þ

where εv is the total error due to the differencebetween the J0 and cosine functions, as well as thelimits of the HT function.

The effectiveness of the HT procedure in the pre-sent application is determined using simulationresults. The applications of the HT to produce aphase shifted fringe pattern can be tricky in practice.If the fringe pattern does not have closed fringes (forexample, Bessel fringes, as in Fig. 2), the fringe sys-tem can be decomposed into an array of line scansalong the x axis stacked together along the y axis. Ap-plying the HT to each one of them will produce a π=2

phase shifted frame, as in Fig. 2(c). Equation (7) canthen be applied to determine Ω.

In the presence of closed fringes, however, a phaseshift introduced in an interferometer will result inthe fringes opening up or closing together, dependingon the change in the path difference. Using line scansand applying HT in this case does not work. Figure 1shows this problem. Figure 1(a) shows the simulatedline scan of a closed fringe system, and Fig. 1(b)shows the scan profile of the corresponding π=2 phaseshifted fringe system as would be obtained in an in-terferometer. The HT generated signal from the scanin Fig. 1(a) is shown in Fig. 1(c). Clearly, Figs. 1(a)and 1(c) are not identical.

As explained later, this produces a π shift in partsof the phase maps obtained using Eq. (7). In order toexplain how the HT has been applied in the presentapplication, we have carried out simulations fortime average cantilever beam fringes (no closedfringes), as well as for time average fringes from aedge clamped diaphragm, which are essentiallycircular.

Fig. 6. (Color online) Schematic of the time average microscopic TV holographic system: SF, spatial filter; BS1, beam splitter; CL, col-limating lens; BS2, cube beam splitter; NDF, neutral density filter; M, mirror; A1, amplifier; A2, amplifier for object excitation; PZTM,piezoelectric transducer mirror; FG, function generator; DAQ, digital to analog converter.

Fig. 7. Fringes obtained with a vibrating PZT cantilever at reso-nance frequency (5:72 kHz) using Eqs. (3) and (4), respectively.


3. Simulation Results

A. Open Fringe Pattern

A simulated zero order Bessel fringe pattern for a vi-brating cantilever beam shown in Fig. 2(a) is gener-ated from Eq. (4) taking V and cosðϕÞ as 1, and it isused to test the method. The image has about sevenfringes, which correspond to Ω ¼ 50 rad. The patternhas the brightest fringe at one end, which shows theposition of zero displacement. The visibility of thehigher order fringes is relatively poor. The Hilbert

transformation (MATLAB function “hilbert”) is ap-plied to the pattern shown in Fig. 2(a), and the trans-formed pattern is shown in Fig. 2(b). The line scanprofile of the images shown in Figs. 2(a) and 2(b)is plotted in Fig. 2(c), which clearly shows a phaseshift between the profiles.

The patterns shown in Figs. 2(a) and 2(b) are thenused to calculate the phase (Ω00) using the Eq. (7). Thecalculated phase map Ω00 is wrapped between −π toþπ and is shown in Fig. 3(a). The corresponding linescan profile is shown in Fig. 3(b). It is unwrapped to

Fig. 8. PZT cantilever beam vibrating at fundamental frequency 5:72 kHz: (a) time average fringes, (b) central line scan profile, (c) fil-tered fringe pattern, (d) fringe after applying HT to (c), and (e) central line scan profile of the patterns shown in Figs. 8(c) and 8(d).


get a continuous phase curve. This phase is obviouslyassociated with an error. Figure 4(a) shows Ω00plotted against Ω, the input parameter used in simu-lating the Bessel fringes shown in Fig. 4(a). Thephase error εv is shown in Fig. 4(b). Figure 4(a) canbe used as a calibration curve to generate a lookuptable for correction of Ω00 values.

B. Closed Fringe Pattern

Figure 5(a) shows a simulated J0 fringe pattern cor-responding to an edge clamped diaphragm. The Hil-bert transformation is applied on this pattern usingthe line scans, as explained earlier. The HT signalthus generated from the pattern [Fig. 5(a)] is shownin Fig. 5(b). The original and HT signals are thenused to calculate the corresponding phase usingEq. (7). The wrapped phase map thus calculated isshown in Fig. 5(c). There is a phase shift of π inthe lower part, below the line passing through thecenter. The reason for this π shift was discussed inSubsection 2.C. In the present case of the circularfringe pattern, it is a simple matter to identify the

parts of the phase map where sign ambiguity is pre-sent. A sign map mask as shown in Fig. 5(d) can begenerated in which the dark portions represent apositive sign and the bright region represents a ne-gative sign. It is then used to remove the π shift inFig. 5(c), and the sign ambiguity removed phase isshown in Fig. 5(e). The phase map in Fig. 5(e) canbe unwrapped and corrected for the error usingthe lookup table (Fig. 4). Keeping inmind our presentapplication, we have carried out a simulation for thecircular pattern. However, we have also carried out asimulation for more complex closed fringes and foundit is feasible to generate filter mask as mentionedabove to remove π phase shifts.

4. Experimental Arrangement

We have used a microscopic TV holography system toobtain time average fringes. The schematic of thesystem is shown in Fig. 6. The narrow beam froma 50 mW laser beam from a diode pumped solid state532 nm CW Nd:YAG laser (Compass 315M) is di-vided into two beams using a beam splitter (BS1).

Fig. 9. (Color online) (a) Wrapped phase map obtained using the HT method, (b) 3D view of the shape made, (c) wrapped phasemap obtained using bias phase modulation method, and (d) Profiles A and B are the central line scan profiles of the phase maps shownin Figs. 9(a) and 9(c) after unwrapping and correcting the error.


Fig. 10. (Color online) Thin circular aluminum membrane vibrating at the fundamental frequency of 4 KHz: (a) fringe pattern, (b)wrapped phase map obtained using the HT method, (c) 3D view of the mode shape, (d) wrapped phase map obtained using bias phasemodulation method, and (e) central line scan profiles (A and B) of the phase maps shown in Figs. 10(b) and 10(d) after unwrapping andcorrecting phase. A shift is introduced between Profiles A and B for clarity.


One beam is expanded using a spatial filtering setup(SF) and collimated with a 150 mm focal length col-limating lens (CL) to act as an object beam to illu-minate the object via a mirror M1, a piezoelectrictransducer mirror, or “PZTM” (STr 25/150/6 PZTfrom Piezo-Mechanik), and a cube beam splitter(BS2). The PZTM is driven by an amplifier, A1,(LE150 from Piezo-Mechanik), which is interfacedto a PC with a DAQ card (NI6036E). The scatteredobject beam from the specimen enters the micro-scopic imaging system via the same cube beam split-ter (BS2). The microscopic imaging system consists ofa long working distance microscope with extendedzoom range and a Sony 2=300 CCD camera (XC-ST70CE). The CCD is interfaced to a PC with anNI1409 frame grabber card. The reference beam alsoexpanded using an SF and collimated with the sup-port of a 150 mm focal length CL. The referencebeam enters the microscopic imaging system via thesame cube beam splitter (BS2). The scattered objectwave and the smooth reference wave are combinedcoherently onto the CCD plane. A neutral density fil-ter (NDF) in the reference beam allows us to controlthe intensity ratio between the object and referencewave. In the present arrangement, the collimated il-lumination and the observation beams are in-lineand, hence, the sensitivity vector is perpendicularto the test object. The function generator (FG) is ac-tivated for harmonic excitation of the object. ThePZTM in the setup is to shift the phase of the scat-tered object beam by 180° for obtaining the contrastreversed time average image, which is then sub-tracted from the preceding time average image toremove the bias intensity in order to enhance thefringe contrast, as in Eq. (3). Programs based on Lab-VIEW have been developed for storing the framesfor real-time visualization and storing the framesfor fringe analysis.

5. Experimental Results

A PZT cantilever beam is excited sinusoidally at dif-ferent frequencies by connecting to an amplifier (A2)and a FG. A high contrast real-time visualization ofthe vibration mode shapes at resonant frequencies,at a video rate of 25 frames per second is made pos-sible using software developed in LabVIEW to im-plement Eq. (3). The resonance frequencies and theirmode shapes on the PZT cantilever beam areobserved. It is to be noted from the fringe patternsthat the bright regions on the subtracted frameare the positions of zero displacement. Figures 7(a)and 7(b) show the fringe patterns obtained usingEqs. (3) and (4), respectively. It is important to notethat Eq. (4) gives fringes with significantly improvedcontrast over that obtained with Eq. (3).

The fundamental mode of the PZT cantilever beamat the 5:72 kHz resonant frequency is shown inFig. 8(a). Figure 8(b) shows the line scan profile alongthe central x axis of the fringe pattern shown inFig. 8(a). To reduce the influence of the noise onthe calculated phase, it is removed prior to perform-

ing the HT by using an average filtering with a 3 × 3window. Figures 8(c) and 8(d) show the filtered pat-tern and the corresponding imaginary part of the HTgenerated fringe pattern. Line scans from Figs. 8(c)and 8(d) are shown in Fig. 8(e). A phase shift betweenthe two signals can be clearly seen. The patternsshown in Figs. 8(c) and 8(d) are used to calculatethe phase. The wrapped phase map generated usingEq. (7) associated with an error as explained inSection 3.A is shown in Fig. 9(a). The phase mapis then unwrapped using a multigrid method [16]to get a continuous amplitude profile, which is thencorrected using a lookup table. The 3D view of thefundamental mode is shown in Fig. 9(b). Figure 9(c)shows the corresponding wrapped phase map ob-tained using the multiple frame bias modulationmethod [5]. The central line scan profiles of the am-plitude obtained using the HTmethod (Profile A) andbias modulation method (Profile B) are shown inFig. 9(d). Profiles A and B are deliberately separatedfor clarity. There is a very close agreement betweenthe results obtained using the two methods.

Similar analysis has been carried out on a circularaluminum diaphragm of diameter 5 mm and thick-ness 10 μm. The results on the sample are shownin Fig. 10.

6. Conclusions

We have demonstrated the use of Hilbert transformfor the analysis of time average vibration fringes.The method uses only one frame containing fringesfor quantifying the fringe data. This is particularlysignificant in the case of time average vibrationfringes, where getting multiple frames is a difficultprocess. The results obtained are in good agreementwith a bias phase modulation technique, which usesmultiple frames for phase analysis.

This work is supported by the Defense Researchand Development Organization (DRDO), Govern-ment of India.

References1. P. K. Rastogi, Digital Speckle Pattern Interferometry and

Related Techniques (Wiley, 2001).2. W. Osten, Ed.,Optical Inspection of Microsystems (CRC, 2007).3. R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibra-

tion pattern using electro-optic holography,” Proc. SPIE 1162,456–468 (1989).

4. S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis ofphase measurement errors in electro-optic holography,” Opt.Rev. 8, 26–31 (2001).

5. U. Paul Kumar, Y. Kalyani, N. Krishna Mohan, and M. P.Kothiyal, “Time average TV holography for vibration fringeanalysis,” Appl. Opt. 48, 3094–3101 (2009).

6. L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F.Voessing, “Stroboscopic digital speckle pattern interferometryfor vibration analysis of microsystems,” Opt. Lasers Eng. 47,252–258 (2009).

7. L. X. Yang and A. L. Bhangaonka, “Investigation of naturalfrequencies under free-free conditions on objects by digital ho-lographic speckle pattern interferometry,” ISEM, Charlotte,USA, 2–4 June, 2003.


8. B. Bhaduri, M. P. Kothiyal, and N. KrishnaMohan, “Vibrationmode shape visualization with dual function DSPI system,”Proc. SPIE 6292, 629217 (2006).

9. V. D. Madjarova and H. Kadono, “Dynamic electronic specklepattern interferometry (DESPI) phase analyses with tempor-al Hilbert transform,” Opt. Express 11, 617–623 (2003).

10. F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phasemeasurement improvement in temporal spackle pattern inter-ferometry using empirical mode decomposition,” Opt. Com-mun. 275, 38–41 (2007).

11. X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis oftextures,” Pattern Recogn. 42, 718–734 (2009).

12. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demo-dulation of two-dimensional fringe patterns. I. General back-

ground of the spiral phase quadrature transform,” J. Opt. Soc.Am. A 18, 1862–1870 (2001).

13. K. G. Larkin, “Natural demodulation of two-dimensionalfringe patterns. II. Stationary phase analysis of the spiralphase quadrature transform,” J. Opt. Soc. Am. A 18, 1871–1881 (2001).

14. W. Lohmann, E. Tepichin, and J. G. Ramirez, “Optical im-plementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).

15. M. Born and E. Wolf, Principles of Optics: ElectromagneticTheory of Propagation, Interference and Diffraction of Light,7th extended ed. (Cambridge, , 1999).

16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional PhaseUnwrapping: Theory, Algorithms and Software (Wiley,1998).


Documents

Time average vibration fringe analysis using Hilbert transformation