Application of S-transform profilometry in eliminating nonlinearity in fringe pattern

Application of S-transform profilometry in eliminatingnonlinearity in fringe pattern

Min Zhong, Wenjing Chen,* and Mohua JiangOpto-Electronics Department, Sichuan University, Chengdu 610064, China

*Corresponding author: [email protected]

Received 25 July 2011; revised 24 October 2011; accepted 25 October 2011;posted 25 October 2011 (Doc. ID 151726); published 7 February 2012

The S-transform, conceptually viewed as an extension of the short-time Fourier transform and thewavelet transform, features a time-frequency representation known for its local spectral propertyand multiresolution strategy. It has been introduced in optical three-dimensional shape measurementbased on a fringe projection technique recently. In this paper, an application of S-transform for demo-dulating fringe patterns affected by the nonlinearity has been discussed. Two methods based on theS-transform, called S-transform ridge method and S-transform filtering method, are used to eliminatethe phase errors caused by nonlinear factors of the projector and CCD camera. The theoretical repre-sentations of S-transform ridge method and S-transform filtering method in fringe analysis are given.The computer simulation and experiment are carried out to verify our research. Compared with the re-constructions of the windowed Fourier transform, the wavelet transform and the S-transform ridgemeth-od, the S-transform filtering method gives a better result under considering the influence of the fringenonlinearity. © 2012 Optical Society of AmericaOCIS codes: 120.6650, 070.4790, 070.2590, 190.2620.

1. Introduction

Optical 3D measurement profilometry based on thestructured light illumination is widely used in var-ious kinds of research fields, including biomedicine,industry inspection, dynamical process analysis, andmachine vision, because of the characteristics of non-contact, whole field analysis, and high speed. Manyoptical 3D measurement techniques have been pro-posed and studied extensively in recent years, suchas Fourier transform profilometry, phase measure-ment profilometry, and modulation measurementprofilometry [1–3]. A large number of researchachievements have been obtained.

Fourier transform profilometry was proposed byTakeda [3] in 1983, in which the 3D shape informa-tion of a testing object modulated in a deformedfringe pattern can be calculated by Fourier trans-form, filtering and inverse Fourier transform in spa-tial domain or frequency domain. As a reversible

linear transform, Fourier transform is used to decom-pose a recorded signal into its frequency components,but it does not tell us when/where the pointed fre-quency component appears. In addition, Fouriertransform provides an excellent frequency resolutionbut no spatial localization ability. In order to over-come the disadvantage of Fourier transform, theshort-time (or windowed) Fourier transform (STFT)[4] and wavelet transform [5] are introduced in fringeanalysis. The windowed Fourier transform (WFT)profilometry suffers from the shortcoming that thestationary window brings the fixed spatial and fre-quency resolution in the spatial frequency domain.The application of the dilating windows [6] improvesthe method, but it is difficult to estimate the sizesof the optimized windows in practices [7,8]. Thewavelet transform was introduced by Morlet at thebeginning of 1985 to analyze the signal by the multi-resolution strategy. The wavelet transform profilo-metry, subsequently, uses the continuous wavelettechnique to measure 3D shapes from deformedfringe patterns because of its local analysis capabil-ity and multiresolution strategy. However, wavelet

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10 February 2012 / Vol. 51, No. 5 / APPLIED OPTICS 577

transform profilometry uses the scale factor insteadof frequency concept.

S-transform, proposed in 1996 by R. G. Stockwell[9], can be regarded as an extension of the WFT andthe continuous wavelet transform (CWT). The formu-lation of the S-transform (ST) is similar to that of thewindowed/short-time Fourier transform (WFT/STFT) except that the width of the window varieswith the frequency, which enables the multiresolu-tion strategy. Moreover, it has been proved thatthe ST can be considered as a special case of theCWT. The S-transform formula can be derived fromthe WFT and the CWT, respectively. One of the im-portant characteristics of S-transform is that thetime averaging of S-transform coefficients of a signalproduces the Fourier transform spectra of the signal.After being presented, S-transform has been appliedin many fields, including medical imaging [10], elec-trical and mechanical engineering [11,12], and geo-physics [13,14] and fringe analysis. Serhat Özder[15] and Ali Dursun [16] proposed an optical phasedistribution evaluation by the S-transform and fin-ished the phase recovery from interference fringes.But up to now, the research on the application ofS-transform in fringe analysis focuses on theS-transform ridge method, which is similar to thecontinuous wavelet transform ridge analysis techni-que (CWTR). Whereas, the research ignores the dis-cussion based on the S-transform filtering method,which is similar to the WFT/STFT. In fact, wecan obtain the correct fundamental spectra fromS-transform coefficients by filtering operation evenif the global spectra of the fringe have some seriousfrequency overlapping.

In the real application of the optical 3D measure-ment profilometry, the deformed fringe patternsinclude the nonlinear components caused by the non-linearity from both CCD cameras and projectors.Two methods based on the S-transform, calledS-transform ridge method and S-transform filteringmethod, are used to eliminate the phase errorscaused by nonlinearity. The theoretical representa-tions of the two methods based on S-transform infringe analysis are given. The computer simulationand the experiment have been carried out to comparethe reconstructions based on S-transform with thosebased on Fourier transform and wavelet transform.

The organization of the paper is as follows: InSection 2, we give the principle of three-dimensionalmeasurement technology based on structured lightillumination. Section 3 elaborates a review of the the-ory of the S-transform and gives strict expressions ofthe S-transform ridge method and the S-transformfiltering method applied in analyzing fringe patternswith nonlinearity. In Section 4, computer simula-tions are carried out to compare the results obtainedfrom the FT, wavelet transform, and S-transformridge method and S-transform filtering method, re-spectively. While in Section 5, an experiment is ap-plied to verify the advantages of S-transform. Lastbut not least, the conclusion is made in Section 6.

2. Principle of Optical 3D Measurement Based on theStructured Light Illumination

The general geometry of the 3D measurement setupbased on triangulation theory is shown in Fig. 1. Theoptical axis P1P2 of the projector lens crosses that ofthe camera lens I1I2 at point O on the referenceplane R. L0 is the distance between I2 and O, andd depicts the distance between P2 and I2. The planeR serves as a reference plane. A and C express thepoints on R, while D expresses a tested point onthe measured object and h is its height. The gratingG, whose image is projected onto the measured objectsurface, has its lines perpendicular to the plane of thefigure.

If a sinusoidal grating image is projected onto themeasured object surface, the deformed fringe patterncaptured by CCD is expressed as

g�x; y� � a�x; y� � b�x; y� cos�2πf 0 � φ�x; y��: (1)

On reference plane R�h�x; y� � 0�, the referencefringe is written as

g0�x; y� � a�x; y� � b�x; y� cos�2πf 0 � φ0�x; y��; (2)

where a�x; y�, b�x; y� are, respectively, the back-ground intensity and the fringe contrast. f 0 is thecarrier frequency of the observed grating image,and φ�x; y� is the phase modulation resulting fromthe object plane. While φ0�x; y� is the original phasemodulation on reference plane (R�h�x; y� � 0�). Thus,Δφ�x; y� � φ�x; y� − φ0�x; y� represents the core phasedistribution caused by the height variation of theobject.

Analyzing similar triangles in Fig. 1 and consider-ing the condition of L0 ≫ h�x; y� in the practicemeasurement, the relationship between the heightvariation of the object and the modulation phaseΔφ�x; y� can be approximately expressed as

Δφ�x; y� ≈ 2πf 0dL0

h�x; y�: (3)

Fig. 1. Scheme of the geometry.

578 APPLIED OPTICS / Vol. 51, No. 5 / 10 February 2012

3. Principle of S-transform

A. 1D S-transform

Given h�t� is a 1D signal, its 1D S-transform is de-fined as [9]

S�τ; f � �Z �∞

−∞

h�t� jf j��2π

p exp�−f 2�τ − t�

2

�exp�−i2πf t�dt;

(4)

where jf j��2π

p exp �− f 2�τ−t�2 � is a Gaussian window func-

tion adjusted by a frequency variable f . Both τ andt are the time variables, τ is the translation param-eter representing the center of Gaussian windows.

Applying the Fourier transform definition, the STcan also be written as follows [9]:

S�τ;f ��Z

∞

−∞

H�α� f �exp�−2π2α2f 2

�exp�i2πατ�dα; (5)

where α is a frequency variable, H�α� f � is theFourier transform of h�t�e−j2πf t. Therefore, the S-transform can also be regarded as the inverse Four-ier transform of the product ofH�α� f �multiplied bya low-pass Gaussian filter adjusted by the frequencyparameter f. S�τ0; f � denotes a local spectrum at anyposition τ0, in which f is a frequency variable, whileS�τ; f 0� is a voice section with a fixed frequency f 0, inwhich τ is a position variable. An important propertyof S-transform is that the averaging of the local spec-trum of S-transform over the τ direction producesthe Fourier spectrum, that is

R�∞

−∞S�τ; f �dτ � H�f �.

Figure 2 shows the relationship scheme betweenS-transform and Fourier transform. Figure 2(a) is a1D fringe added some noises, whose Fourier spec-trum and S-transform spectrum are shown inFig. 2(b) (only the positive frequency) and Fig. 2(c),respectively. The time average of the S-transformspectrum along τ direction, indicated by the redarrow, is shown in Fig. 2(d) (only the positive fre-quency). Comparing Fig. 2(b) with Fig. 2(d), wecan clearly see that they are totally the same witheach other, which verifies the above-mentioned prop-erty of S-transform.

B. Description of a Deformed Fringe PatternAffected by Nonlinearity

The perfect deformed fringe pattern captured byCCD camera is expressed as Eq. (1). However, inmany practical applications, nonlinear terms alwaysexist because of the inherent nonlinearities of theprojector and the CCD in the experimental setup.The deformed fringe pattern should be expressed as

g0�x; y� � �a�x; y� � b�x; y� cos�2πf 0x� φ�x; y��γ; (6)

where γ is a value used to express the nonlinear ef-fect, ranging from 1.8 to 2.5 for majority projectors[17]. In a practical experimental system used tocapture deformed fringe patterns, taking nonlinear

behaviors of the CCD camera and the projector intoaccount, γ can choose a numbermore than 2.5. Usual-ly, the deformed fringe pattern with nonlinearity canbe approximately expressed by the following non-linear polynomial function with the first four terms[18,19]:

g0 � g� k1g� k2g2 � k3g3; (7)

where g � g�x; y�, k1, k2, k3 are the weight coeffi-cients. Substituting Eq. (1) into Eq. (7) and simplify-ing the deformed grating image, we obtain

g0 � C0 � C1fexp�i�2πf 0x� φ�� exp�−i�2πf 0x� φ��g� C2fexp�i2�2πf 0x� φ�� exp�−i2�2πf 0x� φ��g� C3fexp�i3�2πf 0x� φ�� exp�−i3�2πf 0x� φ��g;

(8)

where

C0 � a� k1a� k2

�a2 � 1

2b2�� k3

�a3 � 3

2ab2

�;

C1 � b2

�1� k1 � 2k2a� 3k3a2 � 3

2k3b2

�;

C2 � 14b2�k2 � 3k3a�;

C3 � 18k3b3: (9)

C. Phase Demodulates of Deformed Fringe Patterns withNonlinearity Based on S-transform

1. Filtering Method Based on S-transformAccording to Eq. (5), S-transform can be calculated bythe fast Fourier transform formula. The 1D Fouriertransform spectra of Eq. (8) (along the x direction)can be expressed as (omitting the variable y)

G0�α� � C00�α� � C0

1�α − f 0� � C0�1 �α� f 0�

� C02�α − 2f 0� � C0�

2 �α� 2f 0�� C0

3�α − 3f 0� � C0�3 �α� 3f 0�; (10)

where G0�α� is the 1D Fourier transform of g0�x; y�,C0

k�α − kf 0� (k � 1, 2, 3) are the Fourier spectra ofC1 exp�ik�2πf 0x� φ�� (k � 1, 2, 3). C0�

k �α� kf 0�(k � 1, 2, 3) are the complex conjugates of C0

k�α −

kf 0� (k � 1, 2, 3).When complex testing objects are measured, the

big scope of the height variation may cause thefrequency overlapping between the fundamentalcomponents and the zero frequency components orhigh-order harmonious frequency components. Theperfect fundamental frequency components are not


available. Under this situation, the surface shape ofthe testing object usually cannot be reconstructed byFourier transform profilometry correctly.

According to the formula (4), S-transform coeffi-cients can be regarded as the spectra of the localfringe extracted by Gaussian window functions withdifferent centers and sizes. The multiresolution S-transform output, an M ×N matrix called S-matrix,is a complex matrix, whose columns pertain to timeand rows to frequency. Each column is a representa-tion of “local spectrum” for that point in time. Unlikethe Fourier spectrum representing the global spectra

of the fringe and presenting some serious frequencyoverlapping under the circumstance of the existenceof nonlinearity or complex testing objects, the zerofrequency components and high-order harmoniousfrequency components of S spectrum are separatedfrom each other even if it is presented in the abovecondition. So the local spectra based on S-transformare much simpler than that of the global Fourierspectra of the deformed fringe pattern, and it willbring conveniences for us to filter out the local fun-damental spectra. After local filtering operation, theaveraging of the local fundamental spectra over

Fig. 2. (Color online) Relationship between S-transform spectrum and Fourier transform spectrum. (a) 1D fringe pattern; (b) Fouriertransform spectrum; (c) S-transform spectrum; (d) product by superimposing S spectrum along the red arrow; (e) 2D flattop filter.


the position variable τ direction can correctly formthe whole fundamental spectra and effectively elim-inate the impact of high-order harmonics.

According to the formula (5), S-transform coeffi-cient can be calculated by the fast Fourier transformalgorithm. Supposing an appropriate 2D filter is cho-sen to filter out local fundamental spectra along the τdirection, and then these local fundamental spectraare added to yield

Z∞

−∞

S�τ; f �dτ �Z

∞

−∞

S�τ; f �F�τ; f ridτ jfwhigh

fwlow�dτ

�Z

∞

−∞

Z∞

−∞

C01�α� f − f 0� exp

�−2π2α2f 2

�exp�i2πατ�dαF�τ; f ridτ jfwhigh

fwlow�dτ

�Z

∞

−∞

C01�α� f − f 0� exp

�−2π2α2f 2

��Z∞

−∞

exp�i2πατ�F�τ; f ridτ jfwhigh

fwlow�dτ

�dα

�Z

∞

−∞

C01�α� f − f 0� exp

�−2π2α2f 2

�δα;0dα

� C01�f − f 0�; (11)

where F�τ; f ridτ jfwhigh

fwlow�, using f ridτ as the center, is the lo-

cal filter at the position of τ. fwlow and fwhigh are thedown-half-width and up-half-width of the filter, re-spectively. Both α and f are frequency parameters.C0

1�α − f 0� is the rebuilt fundamental spectra. Fromthe point of view of Fourier transform theory, the in-tensity of each frequency component expresses thecontribution of the harmoniouswavewith the pointedfrequency in forming the signal. The bigger the inten-sity of the frequency is, the bigger the contribution ofthe harmonious wave to the signal is. At position τ0,the peak of S�τ0; f � is called a ridge value. The curveformed by all ridge values along the τ direction iscalled a ridge line. In the fringe pattern, the local fun-damental components have the biggest contributionto forming the local fringe, while the other harmo-niouswaves have smaller contribution. In order to ex-tract local fundamental components, a simple flattopfilter can be used to eliminate the components farfrom the ridge position. The filter is expressed as

wei �

8>>><>>>:

0.5f1� cos��π�f − f rid�τ� � fwlow∕3�∕�2fwlow∕3��g f rid�τ� − fwlow ≤ f < f rid�τ� − fwlow∕3

1 f rid�τ� − fwlow∕3 ≤ f ≤ f rid�τ� � fwhigh∕3

0.5f1� cos��π�f − f rid�τ� − fwhigh∕3�∕�2fwhigh∕3��g f rid�τ� � fwhigh∕3 < f ≤ f rid�τ� � fwhigh

0 else

; (12)

where f rid�τ� is the “ridge” frequency at the position τ,which controls the center of the filter at each positionτ. f rid�τ� is found out by searching the maximum of thelocal S spectra. fwhigh is used to decide the up-half-width of the filter and fwlow decides the down-half-width of the filter. Both of them are decided by findingthe positions of the first minimum at the two side ofthe ridge line. The designed 2D flattop filter is shownin Fig. 2(e).

2. The “Ridge” Method Based on S-transformS-transform can also be regarded as the extension ofwavelet transform. The deformed fringe pattern withnonlinearities, as expressed by Eq. (8) and Eq. (9),can be demodulated by the S-transform ridge meth-od, which is similar to the CWT profilometry. InEq. (9), a�x; y� and b�x; y� are low varying functions;therefore, it is accepted to regard that a�x; y� �a�τ; y�, b�x; y� � b�τ; y�, and phase value φ is ex-pressed as the Tailor expression with first approxi-mation at position τ:

φ�x� � φ�τ� � φ0�x��x − τ�: (13)

Substituting both Eqs. (8) and (13) into (4), thensimplifying the result, we obtain the S-transformcoefficients

S0�τ; f � � S0�τ; f � � S1�τ; f � � S�1�τ; f � � S2�τ; f �

� S�2�τ; f � � S3�τ; f � � S�

3�τ; f �; (14)


where

S0�τ;f ��C0exp�−2πf τ�exp�−2π2�;

S1�τ;f ��12C1expfi�2π�f�f 0�τ

�φ�τ��gexp�−2π2f 2

�−f�f 0�

φ0�τ�2π

�2�;

S�1�τ;f ��

12C1expf−i�2π�f�f 0�τ

�φ�τ��gexp�−2π2f 2

�−f −f 0−

φ0�τ�2π

�2�;

S2�τ;f ��12C2expfi�2π�−f�2f 0�τ

�2φ�τ��gexp�−2π2f 2

�−f�2

�f 0�

φ0�τ�2π

��2�;

S�2�τ;f ��

12C2expf−i�2π�f�2f 0�τ


�−f −2

�f 0�

φ0�τ�2π

��2�;

S3�τ;f ��12C3expfi�2π�−f�3f 0�τ


�−f�3

�f 0�

φ0�τ�2π

��2�;

S�3�τ;f ��

12C3expf−i�2π�f�3f 0�τ


�−f −3

�f 0�

φ0�τ�2π

��2�: (15)

Because f > 0 in S-transform, S0�τ; f � have a maxi-mum when f τ1 � f 0 � φ0�τ�

2π and two local maximawhen f τ2 � 2�f 0 � φ0�τ�

2π � and f τ3 � 3�f 0 � φ0�τ�2π �. How-

ever, when f τ1 � f 0 � φ0�τ�2π , f τ2 � 2�f 0 � φ0�τ�

2π �, andf τ3 � 3�f 0 � φ0�τ�

2π �,

exp�−2π2f 2τ1

�−f τ1 − f 0 −

φ0�τ�2π

�2�

� exp�−2π2f 2τ2

�−f τ2 − 2

�f 0 −

φ0�τ�2π

��2�…

� exp�−2π2f 2τ3

�−f τ3 − 3

�f 0 −

φ0�τ�2π

��2�

� exp�−8π2� ≈ 0.

Besides exp�−2π2� < 10−8 ≈ 0. Therefore, in Eq. (15),S0 ≈ S�

1 ≈ S�2 ≈ S�

3 ≈ 0. That is, S-transform coeffi-cients of g0�x� can be expressed as

S0�τ; f � � S1�τ; f � � S2�τ; f � � S3�τ; f �

� 12C1 expfi�2π�−f � f 0�τ� φ�τ��g

× exp�−2π2f 2

�−f � f 0 �

φ0�τ�2π

�2�

� 12C2 expfi�2π�−f � 2f 0�τ� 2φ�τ��g

× exp�−2π2f 2

�−f � 2

�f 0 �

φ0�τ�2π

��2�

� 12C3 expfi�2π�−f � 3f 0�τ� 3φ�τ��g

× exp�−2π2f 2

�−f � 3

�f 0 �

φ0�τ�2π

��2�: (16)

When f τ1 � f 0 � φ0�τ�2π , f τ2 � 2�f 0 � φ0�τ�

2π �, and f τ3 �3�f 0 � φ0�τ�

2π �, the three local maxima of S-transformcoefficients are decided by factors C1, C2, and C3. Be-cause C1 > C2 > C3, it is easy to find out f τ1 at eachposition τ by searching S-transform coefficients. Thephase value of the point τ can be calculated by

φS�τ; f τ1� � arctan�imag�S�τ; f τ1��real�S�τ; f τ1��

�

� 2π�−f � f 0�τ� φ�τ�; (17)

where imag�S�τ; f τ1�� and real�S�τ; f τ1�� are imaginaryand real parts of the S-transform, respectively.φS�τ; f τ1� is wrapped in the range of �−π;�π� due toarctangent. Employing phase unwrapping algori-thm [20], the continuous phase distribution can beobtained:

φ�τ� � Ounwrap�ϕS�τ; f τ1� � 2π�f τ1 − f 0�τ�: (18)

Ounwrap denotes the unwrapping operator. The abovephase calculation procedure is carried out to treatwith each row of the fringe pattern, and then thephase distribution φ�x; y� can be obtained.

4. Computer Simulations

In order to verify the two methods based onS-ransform in analyzing fringe patterns with thenonlinearity, computer simulations are carried out.For showing the advantages of the S-transform, wecompare the reconstructions of the two methodsbased on S-transform with that of Fourier transformand the wavelet transform. In our simulation, thesize of the fringe image is 256 pix × 256 pix, the car-rier frequency is f 0 � 1∕�12 pix�, and the nonlinearparameter is γ � 2.9. The system parameters, dand L0, are 190 and 540 cm, respectively. Some noiseis also added on to match the actual situation.

The simulated reference fringe pattern and thedeformed fringe pattern are expressed as

Ire � �a� b cos�2πf 0x��γ � n�x; y�; (19)


Ide � �a� b cos�2πf 0�x� tan θ · Z�x; y��γ� n�x; y�; (20)

where a � b � 0.5, tan θ � d∕L0, n�x; y� is the noisedistribution, and the signal-to-noise rate is SNR �22 dB. The simulated is Z�x; y� � 7.35peaks�x; y�,as shown in Fig. 3(a). The peaks function is providedby Matlab function base, whose expression is

Peaks�x; y� � 3 × �1 − x�2 × exp�−�x2� − �y� 1�2�

− 10 ×�x5− x3 − y5

�× exp�−x2 − y2�

−13× exp�−�x� 1�2 − y2�: (21)

The simulated deformed fringe pattern is shown inFig. 3(b), in which the 64th row of deformed fringe

Fig. 4. Results based on FTPmethod. (a) Spectrum of the 64th row of the fringe pattern; (b) reconstruction surface; (c) error distribution.

Fig. 3. Simulation: (a) simulated object; (b) deformed fringepattern (white line stands for the 64th row).

Fig. 5. Results based on wavelet transform. (a) Reconstructionsurface; (b) error distribution.


pattern is marked by a white line for the nextanalysis.

Because the object has sharp height variations, be-sides, the nonlinearity and the bigger noise are alsoadded, in the global Fourier transform spectra, thefundamental spectra overlap the other components.For example, the spectrum of the 64th line of thefringe is shown in Fig. 4(a) (only the positivefrequency). The fundamental spectrum overlapsthe high-order harmonious spectra caused by nonli-nearity and noise. The reconstruction by Fouriertransform profilometry is not satisfied, as shownin Fig. 4(b), and the error distribution is shown inFig. 4(c).

In addition, wavelet transform profilometry is alsoused to analyze the above deformed fringe pattern.The complex Morlet wavelet is used as the mother

wavelet due to its good local property in both thespatial domain and the frequency domain. The recon-struction surface and error distribution are shownin Fig. 5(a) and Fig. 5(b), respectively. The standarderror is 0.57mm.

As we mentioned in the above section, theS-transform is an extension of the wavelet trans-form and the WFT with the width of the windowvarying with a frequency factor jf j. We use the“ridge” method based on S-transform and the filter-ing method based on S-transform to demodulate thesame deformed fringe pattern. The 2D S-transformcoefficient distribution of a row of the fringe (the64th row) is shown in Fig. 6. For clarity, the Sspectrum is displayed in both gray image and 3Dimage, which are shown in Fig. 6(a) and Fig. 6(b),respectively.

Fig. 6. (Color online) Computer simulation based on spectrum of S filtering method. (a) Gray display of S spectrum of the 64th row of thefringe; (b) 3D display of S spectrum of the 64th row of the fringe; (c) spectrum obtained by adding the local S spectrum after filteringoperation; (d) reconstruction surface; (e) error distribution.


Theoretically, there are three ridge lines when con-sidering the influence of the nonlinear effect, as ex-pressed by Eq. (16), but the third one is not obviousbecause the third-order nonlinear coefficients aresmall. Besides the existence of the noise, there aresome stripes vertical to the ridge line. It is causedby the energy dispersion of S-transform [21]. Usingthe flattop filter mentioned before, we can filterout the local S-transform spectrum around the ridge,and then superimpose the obtained spectra along thered arrow to yield the fundamental spectrum shownin Fig. 6(c) (only the positive frequency). When eachrow of the deformed fringe pattern is demodulated byS-transform filtering method, the whole reconstruc-tion surface of the simulated object and the error areobtained, as shown in Fig. 6(d) and Fig. 6(e), respec-tively. The standard error is 0.22mm.

The reconstruction surface can also be obtained byS-transform ridge method. Even though the fringe isaffected by both the nonlinear terms and the noisedisturbance, it is not hard to find out the ridge linecorresponding to the fundamental spectra. By ex-tracting the phase distribution of the ridge line,the reconstruction surface can be obtained, as isshown in Fig. 7(a). The error distribution is shownin Fig. 7(b) and the standard error is 0.41mm.

5. Experiment

A primary experiment is carried out to verify ourmethods. Figure 8 is the schematic diagram of theexperimental device, and the system parametersare d � 30 cm, L0 � 100 cm. In the experiment, a si-nusoidal fringe pattern generated by a computer isprojected onto the reference plane by a projector(EPSON EMP-280). A digital camera (EOS503) re-cords the reference fringe pattern. The fringe patterncaptured by the camera is cut to be an image with1024 by 1024 pixels, as shown in Fig. 9(a). For clarity,the 512th row of the fringe pattern is displayed inFig. 9(b), which exhibits that the fringe is not sinu-soidal distribution any more because of nonlinear ef-fects. When a testing object, resembling a bowl withthe maximum height 25.08 cm, is upended on thereference plane, the deformed fringe pattern is cap-tured by the camera. The image is cut to the samesize as the reference fringe, as shown in Fig. 9(c).The 512th row of the deformed fringe pattern is dis-played in Fig. 9(d). Fourier transform, wavelet trans-form, and S-transform are used to demodulate thedeformed fringe pattern. Because nonlinear effectsare obvious, Fourier transform fails to give accu-rate measurement results. Take the 512th row ofdeformed fringe pattern as an example to analyze,shown in Fig. 9(e) (only the positive frequency). Itis a global Fourier spectrum existing severe fre-quency overlapping with a flattop filter (dotted line),which can generally filter out the useful fundamentalfrequency components. The result of this operation isshown in Fig. 9(f). Figure 9(g) is the reconstruction byFourier transform. The reconstruction surfaces bywavelet transform profilometry and S-transform

ridge method are shown in Fig. 9(h) and Fig. 9(i),respectively. The reconstructed maximum heightis 24.219mm by wavelet transform method and24.525mm by S-transform ridge method. However,because S spectra have an advantage of offering se-parated harmonious spectra, using the S-transformfiltering method, not only the fundamental frequencycomponents can be obtained, but also the correct fun-damental spectra can be rebuilt. The 512th row of thedeformed fringe pattern is still used as an example todemonstrate its characteristics. The rebuilt funda-mental spectrum is shown in Fig. 9(j) (only thepositive frequency). Making a comparison between

Fig. 7. Computer simulation based on S-transform ridge method.(a) Reconstruction surface; (b) error distribution.

Fig. 8. Schematic of experimental equipment.


Fig. 9(f) and Fig. 9(j), even though a certain funda-mental frequency component can be obtain by Four-ier transform, high-frequency components still exist,which can only be reduced instead of eliminated per-

fectly through the operation of filtering. While usingthe method of S-transform we can get correct funda-mental spectra, unlike the fundamental spectra ob-tained from Fourier transform method, whose edge

Fig. 9. Experiment: (a) reference fringe; (b) 512th row of the reference fringe pattern; (c) deformed fringe pattern; (d) 512th row of thedeformed fringe pattern; (e) spectrum of the 512th row of the fringe pattern; (f) fundamental spectrum of the 512th row of the fringe patternby Fourier transform method; (g) reconstruction by Fourier transform method; (h) reconstruction by wavelet transform method; (i) recon-struction by S-transform ridge method; (j) fundamental spectrum of the 512th row of the fringe pattern by S-transform filtering method;(k) reconstruction by S-transform filtering method.


is smooth rather than rough. The reconstructed max-imum height by S-transform filtering method is25.091mm, as shown in Fig. 9(k). It would be veryclose to the maximum height of the actual bowl.We can see that the detailed distribution on the rimof the bowl by S-transform filtering method is betterthan those of the other methods, because the correctfundamental spectra can be rebuilt from the local Sspectra even though there exists severe frequencyoverlapping.

6. Conclusion

The ST is not only similar to the WFT/STFT exceptthe width of the window varying with the frequency,but also has been proved to be a special case of theCWT. This paper discussed both the filtering methodand the ridge method based on the S-transform anddeduced their strict expressions in the demodulationof the fringe pattern with nonlinear terms. The the-oretical representations of these twomethods used inthe fringe analysis are given. The computer simula-tion and experiment are carried out to verify that S-transform can provide us satisfied results when thedeformed fringe pattern includes nonlinear terms.

The authors wish to acknowledge the support bythe National Natural Science Foundation of China(NSFC) under grants 60838002 and 60677028

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Application of S-transform profilometry in eliminating nonlinearity in fringe pattern

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