Strain contouring with Gabor filters: filter bank design

Strain contouring with Gabor filters: filter bank design

Wang Jun and Anand Asundi

A novel fringe processing method is proposed to segment whole-field strain distributions from interfero-metric deformation patterns by use of Gabor filters. This novel strategy is specifically proposed forstrain measurement with a Gabor filter used as a set of wavelets. To increase computational speed aswell as for selection of contour intervals, judicious design of the filter bank, based on the fringe patternand the requirements of the user, is crucial in this methodology. A filter design strategy is developedand, based on the proposed filter design scheme, properly designed filter banks are generated and appliedfor strain contouring in low-strain and strain concentration regions. This scheme allows one to measureengineering strains within regions of interest and hence provides the design engineer great flexibility ofmonitoring, testing, or analysis. © 2002 Optical Society of America

OCIS codes: 070.6110, 120.4120, 120.2650, 100.7410.

1. Introduction

Moire techniques are a powerful assortment of meth-ods for measurement of deformation.1–3 However,although a fringe pattern representing deformationor distortion is readily obtained, expert interpreta-tion is still necessary for conversion of these fringesinto the desired strain information. For accuratemapping of the deformation field, which will thuspermit numerical differentiation, various fringe pro-cessing algorithms, notably the Fourier transform4,5

and phase shifting,6 have been proposed and devel-oped. Nevertheless, numerical differentiation isstill error prone. Optical7, digital,8,9 and optodigi-tal10 methods are also used.

One of the optical approaches to measurement ofdeformation, moire of moire, uses a copy of the moirepattern that is shifted and superposed upon the pat-tern. The resultant fringes contour the derivative ofdisplacement in the direction of the shift. Althougha dense moire pattern is required, the strain �deriv-ative of displacement� contour interval also dependson the shift, which might be large if sufficient con-tours are to be exhibited. An alternative approachproposed by Asundi and Cheng11 involves superpos-ing upon the moire pattern a grating of known pitch.

W. Jun and A. Asundi �[email protected]� are with the Schoolof Mechanical and Production Engineering, Nanyang Technologi-cal University, Singapore 639798.

Received 1 April 2002; revised manuscript received 19 August2002.

0003-6935�02�347229-08$15.00�0© 2002 Optical Society of America

The reason for using this method is that the strain atany point depends on the fringe spacing and hence, ifthe fringe spacing can be obtained, the strain can bedetermined. A grating of known pitch is used toidentify points on the fringe pattern with the samespacing. Strains at these points can then be readilydetermined. However, strains at only specific pointscan be determined, and there is need for many grat-ings if the whole image needs to be segmented; digitalor computer-aided moire12 can circumvent this prob-lem, but identifying contours is not an easy task.One method for automating this process was recentlyreported13,14: engineering strains are contoured bymatching of local fringes and a set of channels ofGabor filters. This method allows one to obtain di-rectly full-field engineering strains from one inter-ferogram without the need for phase unwrapping andnumerical differentiation.

In this method, the design of a suitable filter bankis crucial. The choice of filter is a compromise be-tween contour resolution and computation speed.In this research paper we address this specific pointand related issues.

2. Principles

A. Gabor Filter Method for Strain Contouring

A two-dimensional Gabor function is a harmonicfunction �i.e., a sinusoidal grating� that is modulatinga Gaussian; it can be expressed as

h� x, y� � g� x�, y��exp� j2��Ux � Vy��, (1)

where �x�, y�� x cos � y sin �, x sin � y cos ��are the rotated spatial coordinates and �U, V� are the

1 December 2002 � Vol. 41, No. 34 � APPLIED OPTICS 7229

center frequency components, which specify the filterin the spatial domain. g�x, y� is a Gaussian function,denoted

g� x, y� �1

2��x�yexp�

12 � x2

�x2 �

y2

�y2�� , (2)

where ��x, �y� represent the standard deviation of theGaussian and specify the width of the Gabor filter inthe spatial domain. The Fourier transform showsthat the frequency response function of the Gaborfilter is also a Gaussian, centered at frequency �U, V�with standard deviation ��u�, �v��. This responsefunction can be expressed as

H�u, v� � exp�0.5��u� � F�u�

�2

� � v�

�v��2�� , (3)

where �u�, v�� u cos � v sin �, u sin � v cos ��are the rotated coordinates in the frequency domain,F � �U2 V2 is the radial center frequency, and��u�, �v�� 1�2��x, 1�2��y� is bandwidth of the filteras defined at the half-peak of the frequency responsein the u� and v� directions.

Because the envelope is a Gaussian, Gabor filtersare local in both the spatial and the frequency do-mains. These filters can be thought of as bandpassfilters with narrow bandwidths that are tuned to aspecific frequency � fx, fy�. Convolution of a fringepattern and a Gabor filter gives a response valueR�x, y� at point �x, y�. This value indicates thecorrelation between the filter and the local fringefrequency about the point. A maximum responseindicates that the fringe frequency matches the Ga-bor filter frequency; i.e., the fringe frequency com-ponents at the point are equal to tuned parametersof the filter. In this way, fringe frequencies arediscretely sampled and hence derivatives of the dis-

placement can be determined. The ratio of fringefrequency to the reference grating frequency, f�f,can be shown to represent engineering strains;i.e.,

εx � fx�U��f, εy � fy

�V��f �xy � fy�U��f � fx

�V��f,

(4)

where fx�U� and fy

�U� are fringe frequencies for the Ufield pattern in the x and y directions, respectively.Similarly, fx

�V� and fy�V� are the frequencies in the x

and y directions for the V field, and f is the frequencyof the reference grating. Equations �4� show thatfringe frequencies are proportional to engineeringstrains. Thus the idea behind the method is to get afringe frequency with Gabor filters and hence to de-termine engineering strains. For whole-field con-touring it is vital to design a filter bank thatcomprises a set of suitable channels tuned to theavailable range of fringe frequencies.

B. Multichannel Filter Bank Design

The procedure for designing a suitable filter bank ispresented below. Based on a priori knowledge of thecenter frequency �carrier frequency� of the fringe pat-tern, one determines the location of the center of thebank. The center frequency is obtained by applica-tion of a peak-finding algorithm to the positive half-plane in the Fourier domain. The center frequencyalso assists in determining frequency separation f,which is defined as the frequency step between twoadjacent channels.

In the filter bank, frequency steps from the centerare integer multiples of the frequency separation, i.e.,�Um, Vn� � ��1 � f, �2 � f ��, where �1, �2 ��N, . . . , 1, 0, 1, . . . , N� and m, n � �0, 2N� arealso integers, which identify the row and the column,

Fig. 1. �a� Moire fringes, showing the contours of displacement in the vertical �load� direction. �b� The filter bank to be used issuperimposed onto the frequency spectrum.

7230 APPLIED OPTICS � Vol. 41, No. 34 � 1 December 2002

respectively, of each filter index. Thus, in terms ofthe center frequency, the spatial frequencies of filterscan be

� fx, fy� � �U � Um, V � Vn�. (5)

This gives a total of �2N 1�2 channels.There are several ways to select a frequency sepa-

ration. Let

f1 � 1�2��, (6)

where

� �2B � 12B � 1

�

�F,

� � �ln 2�2 is constant, F is the radial center fre-quency as defined in Eq. �3�, and B � 1.0 octave is the

bandwidth of the Gabor filter, which is defined at thehalf-peak of the frequency response in the Fourierdomain. The frequency response for step f1 equalse0.5. These frequency steps form an octave-basedfilter bank with no overlap.15

Alternatively, the frequency separation can be cho-sen as

f2 � 1��8�2�2, (7)

where � � 1�F. The associated response at this stepinterval is e0.25. Thus an overlapped filter bank isformed.16 As f1 � f2, the bank that uses theformer frequency separation �Eq. �6�� is called loosefilters and the latter bank �Eq. �7�� is referred to astight filters.

Furthermore, N f � F; i.e., the fringe frequency isrestricted to the positive half of the Fourier plane andpermits the determination of the number of channels�N�. If the value of N is based on the Nyquist crite-rion, N f � F�2, then frequency separation f1 re-sults in 25 equispaced filter bank channels, whereasfor frequency separation f2 the bank has 49 chan-nels.

Furthermore, note that the frequency separationcontrols the contour interval �measurement sensitiv-ity� and that the number of channels �N� determinesthe computational efficiency �computation time�.Thus, to obtain a desired resolution with acceptablecomputational efficiency, we propose using a mixedscheme. In the research reported in this paper thebasis for filter bank design is that regions with lowstrain need finer sampling intervals than regionswith large strain. Such a design allows one to havefiner strain contours in small deformation regionsand coarser contours in high strain regions. That is,one set of Gabor filters with small frequency separa-tion is used to sample fringe frequency about thecarrier frequency; another set with larger separation,to sample fringe frequency away the carrier. Addi-tionally, such a design allows one segment high-strain regions from the background based on the factthat strain concentration regions have higher fringefrequencies than the carrier frequency. The follow-

Fig. 2. Derivatives of displacement along the �a� horizontal and�b� vertical directions. Contour interval, 1.08 � 104.

Fig. 3. Mixed filter bank superimposed upon the frequency spec-trum of the moire pattern of Fig. 1�a�.


ing demonstrations will help to clarify this state-ment.

3. Experimental Results and Discussion

Consider the moire interferometric fringe patternshown in Fig. 1�a�. The pattern depicts contours ofdisplacement in the vertical �load� direction of onequarter of an isotropic tensile specimen with a hole.The frequency spectrum �Fig. 1�b�� assists in iden-tifying the center frequency and the frequencyspread of the fringe pattern; thus a suitable bankthat comprises a set of channels of Gabor filters canbe selected. This bank is centered at the centerfrequency, and the spacing between adjacent chan-nels is a function of this central frequency as well.

Using a peak-finding algorithm, we found the cen-ter frequency to be F � 0.9 lines�mm. Substituting

center frequency F into Eq. �6� yields the frequencyseparation

f1 �1

2��

12��2B � 1

2B � 1�

�F� � 0.26, (8)

where f1 is measured in lines per millimeter and B �1.0 octave is the filter bandwidth. Furthermore, sub-stituting center frequency F into Eq. �7� gives fre-quency separation f2 in lines per millimeter:

f2 �1

�8�2�2�

F

�8�2� 0.10. (9)

Both of these frequency scales, f1 and f2, are usedbelow.

First, an equispaced filter bank, shown in Fig. 1�b�,

Fig. 4. Derivatives of displacement along the �a� horizontal and�b� vertical directions. Finer contour interval, 4.17 � 105. Fig. 5. Moire-of-moire fringes representing the derivatives for �a�

the horizontal �x� direction and �b� the vertical �y� direction.


is formed with a uniform frequency separation f1between adjacent channels. Each cross representsone channel with tuned parameters fx and fy. Intotal, 42 channels are used. Convolution of thesechannels with the moire pattern �Fig. 1�a�� yields aset of response patterns. At each point, the channelthat has the maximum response is determined; i.e.,the channel tuned to the local fringe frequency at thatpoint is identified from the 42 response patterns.

Thus the fringe frequency is assigned according tothe tuned channel. Subtracting the carrier fre-quency from the resultant frequency components fxand fy and dividing by the reference grating fre-quency, f � 2400 lines�mm, gives the derivatives ofthe displacement, �v��x and �v��y, as shown in Figs.2�a� and 2�b�. The latter figure represents the nor-mal strain in the y �vertical� direction, as the fringepattern contours the displacement component in the

Fig. 6. Distribution of the derivative for the x direction along a diagonal that results from the equispacing filter design �Looser� and themixed design �Mixed�, compared with the moire-of-moire �MOM� fringes.

Fig. 7. Distributions of displacement derivatives in the y direction along a peripheral curve from the equispacing bank �Loose� and themixed bank �Mixed�, compared with the moire-of-moire �MOM� fringes.


vertical direction. The contour interval is f�f �0.26�2400 � 1.08 � 104.

Next, an example of the mixed filter design is dem-onstrated. As in Fig. 3, this design uses two fre-quency scales, f1 � 0.26 and f2 � 0.10 lines�mm.Each cross represents one channel of the mixed filterbank. There are 49 channels, nearly the same num-ber as in the previous example. Channel separa-tions are smaller close to the center frequency andlarger away from the center. Thus the strain reso-lution is higher in low-strain regions corresponding tofringe frequencies about the central frequency andlower in high-strain areas. The resultant deriva-tives are shown in Fig. 4�a� for the x direction and inFig. 4�b� for the y direction. A greater number ofcontours in the low-strain region than in Fig. 2 can beseen. In this case a finer sampling is implementedover the small-strain regions, i.e., close to the uniformcarrier fringes. The finer contour interval was f�f � 0.10�2400 � 4.17 � 105. The contour intervalin the high-strain region near the hole is the same asin the previous example.

Additionally, the moire-of-moire interferometricpatterns are generated, to illustrate the displacement

derivative in the x and the y directions and are shownin Figs. 5�a� and 5�b�, respectively. These patternshave the same contour interval, 1.08 � 104, as forthe equispacing filter design. Good correlation is ob-served. For quantitative comparison, the results ob-tained with an equispaced filter bank �Fig. 1�b�� andwith a mixed filter bank �Fig. 3� are compared withthe result obtained with moire-of-moire patterns.Figure 6 shows the distribution of the derivative inthe x direction, �v��x, along a diagonal; Fig. 7 showsthe distribution of the derivative in the y direction,�v��y, along an arc that is concentric with the hole.In these figures the solid curves represent the resultsobtained with the mixed filter bank and the dottedcurves represent distributions obtained with theequispaced filter bank. The moire-of-moire contoursalong the same arc are indicated by squares.Whereas the moire-of-moire fringes contour the de-rivatives only at certain points, the proposed methodprovides whole-field, quantitative contours of the dis-placement derivatives. Also, the mixed filter designprovide more-accurate results in the small-strain re-gion away from the hole.

Fig. 8. Gabor filter bank �a� for measuring the derivative �b� forthe x direction within the strain concentration region.

Fig. 9. Gabor filter bank �a� for measuring normal strain �b� in they direction within the strain concentration region.


To measure strain within high-strain regions weconducted another experiment, with the mixed filterdesign. From the moire-of-moire fringes the regionsof high strain can be determined. Based on thisknowledge, the part of the Fourier spectrum that cor-responds to these fringes is determined, as shown in

Fig. 8�a�. With the designed bank of Gabor filters thederivative of displacement in this region is obtained, asshown in Fig. 8�b�. The same procedure is imple-mented for the high-strain region in the y direction.The specially designed Gabor filters are shown in Fig.9�a� as a superimposition of the frequencies of Gabor

Fig. 10. Distribution of the derivative for the x direction along a diagonal within the strain concentration region that results from themixed filter bank in Fig. 8�a� �Mixed-Higher Strain� compared with the moire-of-moire �MOM� contours.

Fig. 11. Distribution of the derivative for the y direction along a peripheral curve within the strain concentration region that results fromthe mixed filter bank in Fig. 9�a� �Mixed-Higher Strain� compared with the moire-of-moire �MOM� contours.


filters onto the fringe frequencies in this region. Theresultant distribution of strain in the y direction isshown in Fig. 9�b�. A segmented strain concentrationregion can be seen, which can be used directly by thedesign engineer for monitoring, testing, or analysis.

To establish the validity of the proposed filter de-sign method we plot the distribution of the displace-ment derivative in the x direction, �V��x, along adiagonal in Fig. 10 as well as the distribution of thederivative in the y direction, �V��y, along a concen-tric arc in Fig. 11. The distributions along thesecurves are compared with the contours from themoire-of-moire interferometric pattern. Goodagreement can be seen within the segmented strainconcentration region. It confirms our filter designconcept in the application of the Gabor filter methodfor determining derivatives of displacement.

4. Conclusion

A novel method has been proposed with which straindistribution can be segmented based on matchingfringe frequency with a set of Gabor filters. In thismethod the filter bank design is crucial. The pro-posed strategy allows strains to be measured withineither high-strain regions or low-strain regions.Whereas an equispaced filter bank gives strain con-tours with a constant contour interval that is suitablein some situations, a mixed filter bank provides theflexibility of being able to select different contour in-tervals and thus to enhance the speed of calculation.Thus, depending on the requirement of the user, con-tour intervals can be varied over the region of inves-tigation. The experimental results confirm theconcept proposed for strain contouring with Gaborfilters.

The support of the Ministry of Education and Nan-yang Technological University, Singapore, throughgrant MLC 1�97 is acknowledged. Also, the supportof the Center for Graphics and Imaging Technology isappreciated.

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Strain contouring with Gabor filters: filter bank design

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