ARTICLE IN PRESS

Optics and Lasers in Engineering 48 (2010) 141–148

Contents lists available at ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Comparison of Fourier transform, windowed Fourier transform, and wavelettransform methods for phase extraction from a single fringe pattern in fringeprojection profilometry

Lei Huang a,�, Qian Kemao b, Bing Pan a, Anand Krishna Asundi a

a School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singaporeb School of Computer Engineering, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e i n f o

Available online 13 May 2009

Keywords:

Optical profilometry

Fringe pattern processing

Phase extraction

Fourier transform

Windowed Fourier transform

Wavelet transform

66/$ - see front matter & 2009 Elsevier Ltd. A

016/j.optlaseng.2009.04.003

esponding author.

ail address: huanglei0114@gmail.com (L. Hua

a b s t r a c t

Fringe projection profilometry is widely used for three-dimensional (3-D) surface shape measurement

using phase-shifting (PS) methods with multiple images or transform methods with single projected

fringe pattern. In this paper, phase extraction methods from a single fringe pattern using different

transform methods are compared using both simulations and experiments. The principles of Fourier

transform (FT), windowed Fourier transform (WFT) and wavelet transform (WT) methods for fringe

pattern processing are introduced. Implementation of 1-D and 2-D algorithms and phase compensation

are discussed. Noisy and non-sinusoidal waveforms are involved into this comparison. The merits and

limitations of each of these processing methods are indicated.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Three-dimensional (3-D) surface profiles can be obtained byusing non-contact optical profilometry techniques, such as laserscanning [1], stereovision [2], and structured-light-based method[3,4]. The whole-field capability coupled with high speed and highaccuracy of the fringe projection methods has resulted in variedresearch and industrial applications. Phase retrieval is generally acrucial process in fringe projection profilometry. The well-knownphase-measuring profilometry (PMP) uses multiple fringe patternsalong with the phase-shifting algorithm to obtain 3-D profiles withhigh accuracy [4,5]. However, for dynamic objects measurement,single projected patterns are processed using transform methods,such as the Fourier transform (FT) [3,6,7], windowed Fouriertransform (WFT) [8–10], wavelet transform (WT) methods [11–14].

In this paper, the FT, WFT, and WT are compared for differentfringe patterns characteristics. Both the 1-D FT [3] and 2-D FT[6,15] methods are tested, so are the 1-D WT and 2-D WTalgorithms. For WFT, the windowed Fourier filtering (WFF) andwindowed Fourier ridges (WFR), in both 1-D and 2-D areexamined. Since the phase extraction from one fringe pattern isusually influenced by noise and non-sinusoidal waveform, thecomparisons of the different algorithms under noisy condition ornon-sinusoidal waveform condition are compared.

The contents of the paper are organized as follows. In Section 2,a brief introduction to principles of the transform methods for

ll rights reserved.

ng).

fringe processing is given. In Section 3, some annotationsfor implementation of these eight algorithms are explained(Section 3.1), and the phase compensation in ridge-basedalgorithms are discussed (Section 3.2). Section 4 shows thecomparison results and discussions by simulations under noisycondition (Section 4.1) and non-sinusoidal waveform condition(Section 4.2). The experimental results are demonstrated inSection 5. Finally, Section 6 concludes this paper.

2. Principles of Fourier transform, windowed Fourier transform,and wavelet transform methods in fringe pattern processing

In this section, transform-based methods for fringe patternanalysis are introduced as a background and preparation for thecomparison and discussion.

A fringe pattern, f(x,y) captured by a CCD camera can beexpressed as

f ðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos½jðx; yÞ�, (1)

where A(x, y), B(x, y), and j(x, y) are, respectively, the backgroundintensity, the fringe modulation, and the phase to be measured atthe location of (x, y) in a fringe pattern.

2.1. Fourier transform method [3,6,7]

The principle of Fourier transform is to extract the funda-mental frequency component of the fringe pattern in the 1- or 2-Dfrequency domain. An inverse transform of the filtered frequencydomain signal then provides the modulo 2p phase of the fringe

ARTICLE IN PRESS

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148142

pattern. For example, for a 2-D fringe pattern as described byEq. (1), after the Fourier transform, filtering in the frequencydomain, and inverse Fourier transform, the filtered exponentialfringe pattern f̄ (x,y) can be obtained as

f̄ ðx; yÞ ¼ F�1fFff ðx; yÞgg ¼ aðx; yÞ expðj�jðx; yÞÞ, (2)

where functions of F and F�1 represent forward and inverseFourier transform, respectively. The over-bar denotes the filteringprocess, after which the fundamental component can be ex-tracted, and a(x, y) is the amplitude of the filtered exponentialfringe pattern. From the exponential fringe pattern f̄ (x,y), themodulo 2p phase j(x, y) can be retrieved as

jðx; yÞ ¼ tan�1 Imf̄ ðx; yÞ

Ref̄ ðx; yÞ

" #. (3)

2.2. Windowed Fourier transform method [8,10]

Without loss of generality, the 2-D WFT and 2-D inverse WFTcan be expressed as [10]

Sf ðu;v;x;ZÞ ¼Z 1�1

Z 1�1

f ðx; yÞgðx� u; y� vÞ exp ð�jxx� jZyÞdx dy,

(4)

f ðx; yÞ ¼1

4p2

Z 1�1

Z 1�1

Z 1�1

Z 1�1

Sf ðu;v; x;ZÞgðx� u; y� vÞ

� exp ðjxxþ jZyÞdxdZdu dv, (5)

where the imaginary unit j ¼ffiffiffiffiffiffiffi�1p

and the window function g(x, y)is a normalized Gaussian function given as

gðx; yÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffipsxsy

p exp �x2

2s2x

�y2

2s2y

!, (6)

where sx and sy are the standard deviations (SD) of the Gaussianfunction in x and y directions. In the windowed Fourier filteringalgorithm, the phase j and filtered fringe pattern can be expressed as

jðx; yÞ ¼ tan�1 Imf̄ ðx; yÞ

Ref̄ ðx; yÞ

" #, (7)

f̄ ðx; yÞ ¼1

4p2

Z 1�1

Z 1�1

Z 1�1

Z 1�1

Sf ðu;v; x;ZÞgðx� u; y� vÞ

� expðjxxþ jZyÞdxdZdu dv. (8)

The fundamental frequency component is extracted from thefiltered spectrum by adjusting x and Z

Sf ðu;v;x;ZÞ ¼Sf ðu;v; x;ZÞ if jSf ðu;v; x;ZÞjXthr;

0 if jSf ðu;v; x;ZÞjpthr:

((9)

where thr is a preset threshold to suppress noise.For the windowed Fourier ridges algorithm, local frequencies

ox and oy, and phase j can be expressed as

½oxðu;vÞ;oyðu;vÞ� ¼ arg maxx;ZjSf ðu;v; x;ZÞj, (10)

jðu;vÞ ¼ tan�1 ImSf ½u;v;oxðu;vÞ;oyðu;vÞ�

ReSf ½u;v;oxðu;vÞ;oyðu;vÞ�

� �þoxðu;vÞuþoyðu;vÞv. (11)

2.3. Wavelet transform method [12–14,16]

As a non-constant signal processing method, wavelet trans-form has been introduced into many signal processing fields,including phase retrieval from fringe pattern. In WT, the window

size is changed according to the frequency. For 1-D WT, the phasej can be retrieved as [13]

jðbÞ ¼ tan�1 ImWf ðar;b; bÞ

ReWf ðar;b; bÞ

� �, (12)

where ar,b is the value of scale dilation parameter a relating to theridge point at position b; 1-D wavelet transform of f(x), Wf(a,b),and 1-D Morlet wavelet c(x) can be expressed as [16]

Wf ða; bÞ ¼1ffiffiffiap

Z 1�1

f ðxÞcn x� b

a

� �dx, (13)

cðxÞ ¼1

p1=4

2pg

� �1=2

exp �ð2p=gÞ2x2

2þ j2px

" #,

with g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2= ln 2

q. (14)

Furthermore, for 2-D WT the phase j can be calculated by [14]

jðbÞ ¼ tan�1 ImWf ðb; ar;b; yr;bÞ

ReWf ðb; ar;b; yr;bÞ � f 0ðp=mÞ expð�p2=mÞ

� �, (15)

where a40 is a scale dilation parameter corresponding to thewidth of the wavelet, y is a rotation angle, ar,b and yr,b are thevalues of a and y relating to the ridge point of Wf at position b,while b is a 2-D translation parameter corresponding to theposition of the wavelet. The wavelet transform of f(x) and themodified wavelet can be expressed as

Wf ðb; a; yÞ ¼ a�n

Z 1�1

f ðxÞcn½a�1r�yðx� bÞ�d2x, (16)

cðxÞ ¼ expð�mjxj2 þ j2pxÞ, (17)

where x ¼ (x, y) comprises the 2-D coordinates of the individualpixel in the fringe pattern; r�y is the conventional 2�2 rotationoperator matrix, and m is a window-modifying parameter.

3. Annotations of algorithms and phase compensation

Considering the flexibility of these methods in practicalapplications, some annotations for the implementation of thedifferent algorithms and phase compensation of ridge-basedmethods are given to enhance the reproducibility of the resultsin this paper.

3.1. Annotations for implementations of algorithms

(1)

The implementation of 2-D FT is conventional [6] and 2-DHanning window is chosen as the frequency response of theband pass filter [15,17,18]. Four algorithms of WFT (1-D WFF,1-D WFR, 2-D WFF, and 2-D WFR) are implemented accordingto the Kemao’s method [8,10] and the window size is set as(6sx+1)� (6sy+1) to achieve an unbiased estimation for phasedistribution [19]. The 1-D WT implemented here is the sameas in Refs. [12,13,16] and uses an 1-D complex Morlet wavelet,while the 2-D WT is implemented according to approachproposed by Wang [14] with the parameter m ¼ 2.

(2)

The DC component in a fringe pattern is suppressed by simplysubtracting the average intensity of the fringe pattern toreduce the influence of the DC component.

(3)

Reliability-guided phase unwrapping algorithm is used in theexperimental comparison in Section 5. The reliability for WFF[23] is the amplitude of the filtered fringe pattern f̄(x,y) in Eq.(8), while those for WFR [20] and WT [16] are the calculatedridge maps.

(4)

The pixels around image boundaries are not involved in ourcomparison because of the boundary effect of these methods.

ARTICLE IN PRESS

Fig. 11-D

com

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148 143

The tested objects are usually suggested to be settled in thecenter of the image in practical measurements.

3.2. Phase compensation in ridge-based algorithms

In these eight algorithms, there are four ridge-based algo-rithms: 1-D WFR, 1-D WT, 2-D WFR, and 2-D WT. It should benoted that the extracted phase by using these ridge-basedalgorithms needs phase compensation to reduce the theoreticalphase error [19]. The phase results of WFR should be compensatedby adding a compensatory phase jc, and jc can be expressed as

jc ¼ �1

2arctan s2

x

@2j@x2

!�

1

2arctan s2

y

@2j@y2

!. (18)

In the implementation of phase compensation for WFR, thesame Gaussian window used in WFR in Eq. (6) is suggested tosmooth the extracted local frequencies by spatial convolution, andthen the second partial derivatives of the phase with respect to x

and y can be easily calculated.The phase compensation in WT uses a Gaussian window with

an average size of the windows used in WT to smooth theextracted phase for 1-D WT or local frequencies for 2-D WT,respectively. Then, central difference is applied twice to theextracted phase for 1-D WT and once to local frequencies for 2-DWT to get the second partial derivatives of the phase. Finally, thecompensatory phase jc in WT can be calculated by Eq. (18).

Simulation examples are shown in Fig. 1 to indicate the needfor phase compensation in ridge-based algorithms. The true phasein the simulation is given as 3.5� peaks(x, y) shown in Fig. 2. Thepeaks is a built-in MATLABs function expressed as

peaksðx; yÞ ¼ 3� ð1� xÞ2 � expð�ðx2Þ � ðyþ 1Þ2Þ

� 10� ðx=5� x3 � y5Þ � expð�x2 � y2Þ

� 1=3� expð�ðxþ 1Þ2 � y2Þ. (19)

. Phase compensation in ridge-based algorithms. (a) Phase error by 1-D WFR befor

WT before compensation; (d) phase error by 1-D WT after compensation; (e) p

pensation; (g) phase error by 2-D WT before compensation; and (h) phase error b

The size of fringe patterns is 256�256 pixels. This true phasewill be used again in the next section for the simulationcomparison. Throughout the paper, all the results from theridge-based algorithms have been compensated.

4. Comparison by simulation

In this section, effect of noise and non-sinusoidal waveformwill be simulated to compare the performances of the eightalgorithms.

4.1. Noise

Noise in fringe patterns is unavoidable in practical mea-surements. To compare the sensitivity to noise of these eightalgorithms, normally distributed noise with a mean value of zeroand different standard deviations is added onto fringe patterns.The reference and deformed fringe patterns can be expressed as

Iref ðx; yÞ ¼ 255 0:5þ 0:3 cos2px

p

� �� �þ nðx; yÞ, (20)

Idef ðx; yÞ ¼ 255 0:5þ 0:3 cos2px

pþ 3:5� peaksðx; yÞ

� �� �þ nðx; yÞ, (21)

where the standard deviation of n(x, y) ranges from 0 to 10 withan interval of 1. The noisy fringe pattern with standard deviationof 3 and 10 are shown in Fig. 3(a) and (b), respectively. We alsoinclude four-step phase-shifting (4-step PS) technique as areference. The results are shown in Fig. 3(c). The comparisonresults in Fig. 3(c) show that the four-step phase-shifting methodis sensitive to noise for it is a pixel-wise fringe pattern processingmethod, and all the 1-D local methods (1-D WFF, 1-D WFR, and1-D WT) and the global methods (1-D and 2-D FT) are moresensitive to noise than 2-D local methods (2-D WFF, 2-D WFR, and

e compensation; (b) phase error by 1-D WFR after compensation; (c) phase error by

hase error by 2-D WFR before compensation; (f) phase error by 2-D WFR after

y 2-D WT after compensation.

ARTICLE IN PRESS

Fig. 2. True phase and deformed fringe pattern. (a) Phase using in simulation: 3.5� peaks(x, y) and (b) simulated deformed fringe patterns.

Fig. 1. (Continued)

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148144

2-D WT). If parameters are appropriately selected, 2-D WFF, 2-DWFR and 2-D WT can retrieve the phase from a more noisy fringepattern with relatively small phase error. This is because the 1-Dalgorithms calculate the phase distribution line by line and are,thus, more vulnerable to noise than 2-D methods. The 2-D FTextracts the fundamental component with a suitable window infrequency domain. Being a global method, the fundamentalcomponent usually has wide frequency band. Noise in thisfrequency band cannot be suppressed in the 2-D FT method,which ultimately affects the phase precision. Therefore, 2-D FTcould also be sensitive to noise. In addition, the 2-D localalgorithms retrieve the phase information from 2-D local datawith different noise reduction strategies: for a point in the spatialdomain, the 2-D WFF uses a threshold for the local spectrum todifferentiate the noise from the signal, while 2-D WFR and 2-DWT retrieve the phase from the most reliable point (i.e., the ridge)of the local spectrum. Thus, the 2-D local algorithms are usuallymore tolerant to noise than the 1-D local algorithms.

Noise in practical structured-light-based 3-D optical profilo-metric fringe patterns is usually not severe. Considering the speed,1-D local algorithms are much faster than the 2-D localalgorithms. Also, the 2-D FT, which involves one forward andone inverse Fourier transforms, is much faster than the 2-D localalgorithms. Thus, the 2-D local algorithms are not suitable forreal-time processing, which is their major limitation, and hence2-D FT, 1-D methods, and phase-shifting method are still widelyapplied in practice.

4.2. Non-sinusoidal waveform condition

Digital light projectors have been widely used in opticalprofilometry nowadays. Non-sinusoidal waveform may be dueto the gamma parameter [21] of the digital projector or the non-linear response function of the whole system [22,23]. In addition,for transform-based methods, the DC component and the second

ARTICLE IN PRESS

Fig. 4. Standard deviation of phase error vs. Ratio of a2/a1.

Fig. 3. Comparison results under noisy condition. (a) Fringe pattern with noise of SD ¼ 3; (b) fringe pattern with noise of SD ¼ 10; and (c) standard deviation of phase error

vs. standard deviation of additive noise.

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148 145

harmonic hinder the extraction of the fundamental frequencycomponent in the frequency domain and are thus considered inthis simulation. The reference and deformed fringe patterns arehence simulated as

Iref ðx; yÞ ¼ 255 a0 þ a1 cos2px

p

� �þ a2 cos 2

2px

p

� �� �� �, (22)

Idef ðx; yÞ ¼ 255 a0 þ a1 cos2px

pþ 3:5� peaksðx; yÞ

� ��

þa2 cos 22px

pþ 3:5� peaksðx; yÞ

� �� ��, (23)

where a0 ¼ 0.5, a1 ¼ 0.4, and the ratio of a2/a1 changes from 0% to25% with a step size of 5%.

From the simulation results in Fig. 4, it can be seen that theridge-based methods (1-D WFR, 1-D WT, 2-D WFR, and 2-D WT)are less sensitive to the higher harmonics than methods based onforward and inverse transforms (1-D FT, 1-D WFF, 2-D FT, and 2-DWFF). In the forward and inverse transforms methods, higherharmonics extend into the fundamental frequency component,thus affecting phase retrieval. In the ridge-based methods, theridge will not be severely affected by the higher harmonics, sincethe ridge has the highest amplitude of the spectrum and is usuallyquite far away from the harmonics. Therefore, these ridge-basedmethods are less sensitive to the higher harmonic components.

5. Experimental comparison

The results of a scanning mechanical stylus could possibly beconsidered as the standard, although the mechanical stylusdirectly gives the height rather than phase. Therefore, errors fromthe phase-height mapping and the calibration of camera will beinvolved. We propose an approach for constructing a phasestandard for continuous phase distributions.

5.1. Experimental phase standard

It is well known that in the phase-shifting technique, the phaseerror due to non-sinusoidal waveforms can be reduced by using alarge number of phase-shifting steps, and in addition, the noise

can also be partially suppressed [5]. Furthermore, the 2-D WFF hasbeen shown theoretically and experimentally to be very effectivein removing noise in exponential fringe patterns with continuousphase maps with a SNR gain of around 20 [19]. It is thusreasonable to construct an experimental phase standard using theN-step phase-shifting approach (N should be large enough toreduce the error from non-sinusoidal waveform), followed by the2-D WFF for denoising. To verify this ‘N-step PS+2-D WFF’approach, a pair of fringe patterns are simulated as shown inFig. 5(b) with the phase, Fig. 5(a), similar to the experimentalphase distribution in respect of the fringe period, noise level, andsecond harmonic wave. In simulation, the true phase jt(x, y) over256�256 pixels is set as

jtðx; yÞ ¼ �ðx� 30Þ2=1000. (24)

The fringe patterns can be written as

Iref ðx; yÞ ¼ 255 a0 þ a1 cos2px

p

� �þ a2 cos 2

2px

p

� �� �� �þ nðx; yÞ, (25)

ARTICLE IN PRESS

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148146

Idef ðx; yÞ ¼ 255 a0 þ a1 cos2px

pþjtðx; yÞ

� ��

þa2 cos 22px

pþjtðx; yÞ

� �� ��þ nðx; yÞ, (26)

where a0 ¼ 0.5, a1 ¼ 0.3, a2 ¼ 0.045, p ¼ 8 and the standarddeviation of noise n(x, y) is 2. A phase jps(x, y) can be extractedfrom N phase-shifted reference and deformed fringe patterns byusing conventional N-step phase-shifting algorithm [4], and thenconverted to an exponential fringe pattern fexp(x, y) as

f expðx; yÞ ¼ exp½j�jpsðx; yÞ�. (27)

Since in this simulation only the second harmonic is added inthe fringe patterns, N ¼ 4 phase shifts is sufficient [5]. The phasealong the solid red line in Fig. 5(b) are shown in Fig. 5(c) toindicate efficiency of ‘N-step PS+2-D WFF’ approach. Phase errorsby using only the phase-shifting technique and the ‘N-step PS+2-DWFF’ approach are shown in Fig. 5(d) and (e), respectively,indicating that the 2-D WFF performs well in filtering exponentialfringe patterns. Furthermore, phase errors from the differentalgorithms are compared to either the true phase or the result ofphase standard ‘N-step PS+2-D WFF’. The results in Fig. 6 show the

Fig. 5. Simulated true phase and phase errors. (a) True phase in simulation; (b) simulate

in (b) by using ‘N-step phase shifting+2-D WFF’ approach; (d) phase error by using ph

mean and 3� standard deviation of phase error for thesealgorithms.

Form Fig. 6, it is seen that for each algorithm, the error meansbased on these two different standards are quite close, so are thestandard deviations of the errors. It suggests that the phase resultsfrom ‘N-step PS+2-D WFF’ approach can be used as a phasestandard for our experimental comparison.

5.2. Experimental results

For the experimental demonstration, a model is first analyzedusing the 16-step phase-shifting method (i.e., N ¼ 16). Thereference and deformed fringe patterns in two experiments areshown in Fig. 7(a). Due to the boundary effect, only the centralpart, as shown inside the blue solid rectangle in Fig. 7(b), isprocessed. The phase calculated by a 16-step phase-shiftingalgorithm and further denoised by the 2-D WFF are shown inFig. 7(b) and (c), respectively, and the denoised phase isconsidered as the phase standard for experimental comparison.The reduced noise in central area is shown in Fig. 7(d).

In the first experiment, there are no observable higherharmonics in the fringe patterns, as shown in the left of Fig.

d reference and deformed fringe patterns; (c) phase distribution of the solid red line

ase shifting; and (e) phase error after 2-D WFF.

ARTICLE IN PRESS

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148 147

7(a), while in the second experiment, some harmonics, especiallya strong second harmonic is seen (right of Fig. 7(a)). The phaseerrors of these algorithms under both conditions are shown inFig. 8. The significant influence of harmonics can be found bycomparing 1-D or 2-D WFF results in Fig. 8. Fig. 9 shows theretrieved phase by the 2-D WFF under both conditions and theircorresponding phase errors. For fringe patterns with no higher-order harmonics, there is no overlap in the spectrum and the WFFperforms well for phase retreival. However, the non-sinusoidalwaveform situation, there are obvious overlaps. Consequently, theWFF results would be affected by the higher harmonics.

The experimental results mirror the results from the simula-tion. First, 2-D local methods are generally more tolerant to noise

Fig. 7. Experimental result and noise reduction result. (a) Reference and deformed frin

shifting method; (c) phase result after noise reduction by using 2-D WFF; and (d) redu

Fig. 6. Means and 3� SDs of phase error based on the true phase or the result of

‘N-step phase shifting+2-D WFF’ as the phase standard.

than the 1-D methods. Second, local methods are usually lesssensitive to noise than global methods. Third, ridge-basedmethods are more tolerant to higher-order harmonics as seenwith a non-sinusoidal waveform than methods based on forwardand inverse transforms. Finally, 2-D WFR and 2-D WT seem to bethe most robust methods in these eight algorithms undercontinuous phase condition, but they are more time-consumingthan the other six algorithms. In addition, the essence of 2-D WFRand 2-D WT is almost the same. The only main difference betweenthem is that 2-D WFR uses a fixed window size while 2-D WTchanges the window size according to the frequencies. Never-theless, when processing the fringe patterns with carrier fringe,the window size in the 2-D WT usually changes within a certain

ge patterns under two sorts of conditions; (b) raw phase data from 16-step phase-

ced noise in central area.

Fig. 8. Means and SDs of phase errors under conditions of fringe pattern with no

observable harmonics and with some harmonics.

ARTICLE IN PRESS

Fig. 9. Retrieved phase and phase error by 2-D WFF under conditions of fringe pattern with no observable harmonics and with some harmonics. (a) Phase by 2-D WFF with

almost no harmonics; (b) phase error by 2-D WFF with almost no harmonics; (c) phase by 2-D WFF with some harmonics; and (d) phase error by 2-D WFF with some

harmonics.

L. Huang et al. / Optics and Lasers in Engineering 48 (2010) 141–148148

range which generally covers the selected window size in the 2-DWFR. Consequently the WFR and WT give very similar results.

6. Conclusions

In fringe projection profilometry several one-frame fringepattern processing methods are evaluated and compared in thispaper. Effects of noise and the higher harmonics in a non-sinusoidal waveform are considered in the comparison using bothsimulation and actual experiments. Among the transform-basedmethods, the 2-D WFR and 2-D WT seem to be the most robustalgorithms, but they are time-consuming and not suitable for real-time processing. All transform-based algorithms perform poorlyaround discontinuities in a phase map, which needs to be furtherinvestigated. In summary, algorithms should be chosen properlyaccording to the requirement of an experiment to foster strengthsand circumvent weaknesses of these algorithms.

Acknowledgements

The authors want to show gratitude to Prof. Xianyu Su inSichuan University, China, for many helpful discussions, and thereviewers for their excellent comments and suggestions.

References

[1] Zeng L, Matsumoto H, Kawachi K. Two-directional scanning method forreducing the shadow effects in laser triangulation. Measurement Science andTechnology 1997;8:262–6.

[2] Anderson BL. Stereovision: beyond disparity computations. Trends inCognitive Sciences 1998;2:214–22.

[3] Takeda M, Mutoh K. Fourier transform profilometry for the automaticmeasurement of 3-D object shapes. Applied Optics 1983;22:3977–82.

[4] Srinivasan V, Liu HC, Halioua M. Automated phase-measuring profilometry of3-D diffuse objects. Applied Optics 1984;23:3105–8.

[5] Surrel Y. Design of algorithms for phase measurements by the use of phasestepping. Applied Optics 1996;35:51–60.

[6] Bone DJ, Bachor HA, Sandeman RJ. Fringe-pattern analysis using a 2-D Fouriertransform. Applied Optics 1986;25:1653–60.

[7] Su X, Chen W. Fourier transform profilometry: a review. Optics and Lasers inEngineering 2001;35:263–84.

[8] Kemao Q. Windowed Fourier transform for fringe pattern analysis. AppliedOptics 2004;43:2695–702.

[9] Zhong J, Weng J. Dilating Gabor transform for the fringe analysis of 3-D shapemeasurement. Optical Engineering 2004;43:895–9.

[10] Kemao Q. Two-dimensional windowed Fourier transform for fringe patternanalysis: Principles, applications and implementations. Optics and Lasers inEngineering 2007;45:304–17.

[11] Sandoz P. Wavelet transform as a processing tool in white-light interfero-metry. Optics Letters 1997;22:1065–7.

[12] Zhong J, Weng J. Spatial carrier-fringe pattern analysis by means of wavelettransform: Wavelet transform profilometry. Applied Optics 2004;43:4993–8.

[13] Zhong J, Weng J. Phase retrieval of optical fringe patterns from the ridge of awavelet transform. Optics Letters 2005;30:2560–2.

[14] Wang Z, Ma H. Advanced continuous wavelet transform algorithm fordigital interferogram analysis and processing. Optical Engineering 2006;45:045601.

[15] Zhang Q, Su X. High-speed optical measurement for the drumhead vibration.Optics Express 2005;13:3110–6.

[16] Li S, Chen W, Su X. Reliability-guided phase unwrapping in wavelet-transformprofilometry. Applied Optics 2008;47:3369–77.

[17] Lin J-F, Su X. Two-dimensional Fourier transform profilometry for theautomatic measurement of three-dimensional object shapes. Optical En-gineering 1995;34:3297–302.

[18] Zhang Q, Su X, Cao Y, et al. Optical 3-D shape and deformation measurementof rotating blades using stroboscopic structured illumination. OpticalEngineering 2005;44:113601–7.

[19] Kemao Q, Wang H, Gao W. Windowed Fourier transform for fringe patternanalysis: theoretical analyses. Applied Optics 2008;47:5408–19.

[20] Kemao Q, Gao W, Wang H. Windowed Fourier-filtered and quality-guidedphase-unwrapping algorithm. Applied Optics 2008;47:5420–8.

[21] Hongwei G, Haitao H, Mingyi C. Gamma correction for digital fringeprojection profilometry. Applied Optics 2004;43:2906–14.

[22] Zhang S, Yau S-T. Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector. AppliedOptics 2007;46:36–43.

[23] Pan B, Kemao Q, Huang L, et al. Phase error analysis and compensation fornonsinusoidal waveforms in phase-shifting digital fringe projection profilo-metry. Optics Letters 2009;34:416–8.