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Optics and Lasers in Engineering 43 (2005) 645–658

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Fourier fringe processing using a regressiveFourier-transform technique

J. Vanherzeele�, Patrick Guillaume, Steve Vanlanduit

Department of Mechanical Engineering, Vrije Universiteit Brussel, Acoustics and Vibrations Research

Group, Pleinlaan 2, 1050 Brussels, Belgium

Received 13 May 2004; accepted 9 September 2004

Available online 28 October 2004

Abstract

Since the introduction of a fourier fringe algorithm by Takeda, it has been possible to

determine the phase of a particular light source impinging on an object from one sole image.

This has led to applications in many whole field optical measurement techniques such as ESPI,

holography, profilometry and so on. However, the basic processing technique, in case of the

2D-Fourier transform, is subject to a major drawback. Because this technique supposes

periodicity in a fringe image, the so-called leakage effects occur. This gives rise to non-negligible

errors, which can be resolved by using a regressive Fourier transformation technique. In the

method introduced in this article, the fringe signal is represented by a model using sines and

cosines where the frequency is not fixed (which is the case for classical FFT-techniques). The

coefficients of those sines and cosines together with the frequency components are then

estimated locally by means of a frequency domain system identification technique. This allows

the fringe pattern to be unwrapped without any distortion. This method will be applied in

particular to Fourier-transform profilometry (determines object geometry using shifts of

projected fringes) although it can be used in any of the techniques mentioned above. Moreover,

it will be shown that the proposed method can deal with other distortions that occur in practice

such as over-modulation and varying fringe visibility. The proposed technique will be validated

on both simulations and on a profile measurement of a pipe section.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Fourier-transform profilometry; Regressive Fourier-transform technique

see front matter r 2004 Elsevier Ltd. All rights reserved.

.optlaseng.2004.09.010

nding author. Tel.: +322 629 28 07; fax: +322 629 28 65.

dress: joris.vanherzeele@vub.ac.be (J. Vanherzeele).

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J. Vanherzeele et al. / Optics and Lasers in Engineering 43 (2005) 645–658646

1. Introduction

In industrial environments tactile measurement systems have long been used toobtain the geometry of separate components as well as whole structures themselves.However, these systems suffer some important disadvantages such as relativeslowness and cost price. Recently, however, with the ever ongoing progress in opticaltechniques some interesting alternatives have been developed. Progress in opticaltechniques were possible due to the improvements in computer hardware whichallowed automatic process of images within a reasonable time span. This gave way tothe possibility of taking a whole-field image on which one projected a series of(sinusoidal) fringes, as was proposed in the optical society [1,2]. In particular, theFourier-transform profilometry (FTP) method—introduced by Takeda [3,4]—hasthe potential to compete with classical tactile systems. The FTP method uses a2D-Fourier transform to estimate the phase of a projected grating (a brief review onthe theory of the FTP will be given in Section 2). This allows an immediateextraction of the structure’s geometry from one single image.Since the introduction of Fourier-transform profilometry different improvements

have been suggested over the last decades (a recent literature review is given in [5]).First of all techniques were proposed to eliminate the background illumination byusing two [6] or more [7] images from phase-shifted gratings. Also, the FTPtechnique was extended to frequency-multiplexed gratings [8].Although the FTP technique has evolved into a fairly robust approach it still

lacks the ability to tackle all possible geometries. This is inherently tied to the useof a Fourier transform where the fringe and height distribution are assumed tobe periodic within the image window, if not distortions (leakage) will be the result(the effect will be illustrated in Section 4). A common approach to this problem isusing windows [9] to reduce the effect, which, however, results in a diminishingfrequency resolution. Another drawback is its sensitivity to over-modulation andlow fringe visibility. In this article an FTP method, based on a local estimation ofthe phase and frequency of the grating will be presented. First, the FTP theorywill be briefly resumed and afterwards the proposed regressive Fourier approachwill be explained in Section 3. Via computer simulations the strength of the proposedregressive technique toward the aforementioned shortcomings will be illustratedin Section 4. In Section 5 the profile of a pipe is measured using both a classicalFFT and the proposed regressive FFT. Finally, conclusions will be drawn inSection 6.

2. Fourier-transform profilometry

Assume that a sinusoidal grating with frequency f X in the x-direction is projectedon an object with a height distribution given by hðx; yÞ: If an image is acquired(for instance with a CCD camera) the intensity can be written as

iðx; yÞ ¼ rðx; yÞð1þ cosð2pf X ðx þ hðx; yÞ tan yÞÞÞ (1)

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with rðx; yÞ the reflectance of the object. For simplicities sake both projection andobservation are made by an optical system whose optical axis lie in a plane normal toy-axis which form an angle y in the x-axis direction. Therefore the phase modulationfor the carrier frequency can be written as follows:

fðx; yÞ ¼ 2pf X hðx; yÞ tan y: (2)

This equation automatically produces the desired height distribution. The phasefðx; yÞ is estimated by means of a 2D-Fourier transform. To perform that task, acomplex signal—the so-called analytical continuation iðx; yÞ of iðx; yÞ [10]—isgenerated as follows:

�

One computes the 2D-Fourier transform

Iðk; lÞ ¼XM�1

m¼0

XN�1

n¼0

iðxm; ynÞ exp �2p{mk

Mþ

nl

N

� �� �: (3)

�

Thereafter all frequency components above the Nyquist frequency are put to zero:Iðk; lÞ ¼ 0 for k ¼ M=2; . . . ;M and l ¼ N=2; . . . ;N) and I ¼ I elsewhere.

�

Finally the inverse 2D-Fourier transform is applied

iðxm; ynÞ ¼1

MN

XM�1

k¼0

XN�1

l¼0

Iðk; lÞ exp 2p{k

Mþ

l

N

� �� �: (4)

Extracting the phase of the image, iðx; yÞ is achieved by simply taking the complexangle of iðx; yÞ : fðx; yÞ ¼ arctanIðiðx; yÞÞ=Rðiðx; yÞÞ with R and I the real andimaginary part of a complex number. However, the phase fðx; yÞ is still wrapped inthe ½�p=2;p=2� interval. As this was not the main objective of this paper, one of theexisting 2D phase unwrapping methods—a simple sequential integration method[11]—was used (an overview of unwrapping methods is given in [12,13]).Although different authors have shown independently that the 2D Fourier

transform method works well on various examples, there is an important restrictionon the applicability: the image iðx; yÞ should be periodic (values on the left and rightside of the image borders should be equal). If this is not the case leakage will occur(the energy of a spectral line at frequency k will spread out over its neighboringfrequency lines). This means that in order to avoid distortions in the reconstructedimage with classical 2D Fourier transform methods a few ‘rules’ have to berespected.

�

The carrier frequency f X should be chosen in such a way that the image iðx; yÞcontains an integer number of fringes.

�

The height distribution hðx; yÞ itself should be periodic in the time window (this isfor example satisfied when the object under investigation falls completely withinthe image because then the height is zero near the border of the image).

In the next section, a method is proposed which does not exhibit this leakageeffect. Instead of applying a Fourier transform on the full image, the image is divided

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J. Vanherzeele et al. / Optics and Lasers in Engineering 43 (2005) 645–658648

into small partitions. For each of these partitions the fringe frequency is estimatedusing a regressive Fourier-transform technique. It will be shown that besides thefact that the proposed technique is not affected by leakage, other effects that takeplace in practice, leading to distortions using a 2D classic Fourier method, will beresolved using the regressive technique. Therefore, the technique is much moregenerally applicable.

3. Local phase and frequency estimation using a regressive Fourier-transform

technique

3.1. Overview of the method

Both phase and frequency can be estimated either globally, which is the case for atraditional 2D Fourier-transform technique or locally. In this paper the latterapproach was opted for. The method can be divided into the following steps:

�

Define a sliding window of a certain length M0: Apply it to N M image iðx; yÞ insuch a fashion that it is perpendicular to the projected fringes. This implies thatN � M0 (or M � M0) sub-images are taken depending on how the fringes areprojected.

�

Then one has to assume that inside one of those particular sub-images only onesingle frequency component is present. This implies that the window length mustbe chosen within certain ranges with respect to the carrier frequency. A largecarrier frequency will imply a small window and vice versa.

�

Estimate phase and frequency for every consecutive position of the sliding windowand repeat this for every column (row respectively).

�

Unwrap the obtained phase f (in this article a fairly straight forward sequentialintegration method was used [11]).

The most important step is of course the determination of the phase and frequencyof each independent sine. This is a non-linear problem which can be solved in anumber of ways. Many of them are quite time-consuming especially for the purposeof phase extraction from images (many of these techniques were introduced and arestill used in telecommunication). In the next section, a so-called regressive Fourier-transform technique based on a transfer function approach in the frequency domainwill be introduced which is quite robust towards varying distortions.

3.2. Sine estimation using the regressive FFT

Assume that the fringe projection is in the x-direction and the lth row of the kthsub-image ik;l ¼ iðxk; :::; xkþM0

; ylÞ can be modelled by a single sine with frequencyf k;l and phase fk;l

ik;lðxÞ ¼ Ak;l expðið2pf k;lx þ fk;lÞÞ þ Ak;l expð�ið2pf k;lx þ fk;lÞÞ: (5)

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One can simplify this equation by taking the phase together with the sine amplitude:Ak;l expðfk;lÞ ¼ ak;l and do the same for the complex conjugate. Now let us look atthe first term in the equation. By representing expð2pf k;lÞ with lk;l that term can bewritten as: alx

k;l : In order not to burden the reader with countless indices, the k; lsubscripts will be left out in the following reasoning as they do not contribute tomaths. The z-transform of that particular sequence is

X ðzÞ ¼XM�1

n¼0

alnz�n ¼ a1� lMz�M

1� lz�1: (6)

The discrete Fourier transform of aln is obtained by evaluating Eq. (6) on the unitcircle zk ¼ expði2pk=MÞ

X ½k� ¼ X ½zk� ¼ a1� lM

1� lz�1k

: (7)

This form can now be applied to the complete sequence in Eq. (5)

X ½k� ¼~a

1� lz�1k

þ~a

1� lz�1k

; (8)

where ~a ¼ að1� lMÞ:

Eq. (8) is nothing else than the pole/residue representation of the discrete Fouriertransform X ½k�: This clearly proves that it is possible to model sinusoids using atransfer function model in the frequency domain. Therefore, it is possible by meansof a simple least-squares approach (or for more reliable results a maximumlikelihood (ML) approach [14]) to retrieve values for the poles l and the residues ~a:The sinusoid frequencies can then be obtained from the poles and similarly thesinusoid amplitudes can be calculated from the residues with following formula:a ¼ ~a=ð1� lM

Þ: This result also illustrates the fact that it is possible to compensatethe residue estimates ~a for the leakage error. Indeed from the estimated poles l andresidues ~a an estimate of a has been derived. The final step consists of merelycalculating the angle of a which provides the desired phase.This shows that the technique will be quite robust dealing with distortions of varying

nature. However the computational load is still quite heavy, because for every window,first of all least-squares has to be performed to generate initial values for the MLequations which in turn have to be iterated at least about 10–20 times. This process hasto be repeated ðM � M0ÞN times to evaluate the entire image. This is something that isbeing looked into and already some promising results have been achieved.

4. Computer simulations

In this part different N M-sized images simulating an object with two differentheight distributions are presented

(1)

hðx; yÞ ¼ peaksðN;MÞ;N ¼ M ; (9)

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(2)

Fig.

cosð

hðx; yÞ ¼ cosð2pf xxÞ sinð2pf yyÞ: (10)

The former is interesting in a way that the classical 2D-Fourier transform will failalmost completely (results are shown at the end of this section). (Fig. 1(a) gives anexample.) The latter allows simulations of varying distortions which can occur inpractice (Fig. 1(b) gives an example.).

�

A carrier frequency f X which does not coincide with a frequency line of the FFTgrid, which simply means that the number of fringes in the image is not an integer.

�

The height distribution hðx; yÞ is not periodic in the time window (i.e. f x or f y arenot integers).

�

A variation in the fringe visibility is present. � Over-modulation by taking a modulation index [10] b ¼ 2pf X jhj tanðyÞ which istoo high. When b is too high the modulation spectrum comes across the DC-lineand is thereby mirrored back around DC, creating distortions. From theexpression above, one can easily see that this will be a problem for either anangle y or a height variation which is too great.

(b)

(a)

1. (a) Simulated height distribution hðx; yÞ ¼ peaks ðNÞ: (b) Simulated height distribution hðx; yÞ ¼2pf xxÞ sinð2pf yyÞ with f x and f y ¼ 3:

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Fig. 2 gives an example of a projected fringe image with a carrier frequencyf X ¼ 10:5 with no over-modulation and a periodic height distribution. Thefrequency spectrum clearly illustrates the presence of leakage. The result for theheight distribution will be some substantial distortions with the classical approachwhile the regressive technique does not suffer these anomalies (Fig. 3).Leakage due to a non-periodic height distribution is of even greater importance.

The most important reason is a practical one. When measuring large objects oneusually has to rely on taking separate sub-images (Fig. 4). This almost guaranteesthat the height will not be the same at the borders, which means a discontinuity andin turn leads to leakage. On top of that these distortions are not confined to theimage edges but are spread out over the entire projection. The latter is especiallyvisible in Fig. 5 where f x ¼ 2:5; f y ¼ 2:5:When the fringe visibility is no longer constant a low-frequency component is

added to the modulation spectrum (Fig. 6). This can be remedied by fitting a high-pass filter to the spectrum, in other words by eliminating all low-frequencycomponents in the spectrum. However, this will inevitably mean that some of theinformation held in the modulation spectrum will also be cut off, resulting in anunderestimation of the height and in some cases—such as in this article—where thecarrier frequency f X is fairly low by a complete miscalculation of the distributionusing the classical 2D FFT. When using the regressive Fourier technique no suchdistortions are visible (Fig. 7).Finally, the effect of over-modulation can be taken into account. Fig. 8(a) shows

an image with carrier frequency f X ¼ 10 and f x ¼ 3; f y ¼ 3 and modulation indexb ¼ 1:6: Fig. 8(b) shows how the spectrum is indeed mirrored back around DCwhich will lead to substantial distortions for the classical approach. The regressivetechnique gives far better results (Fig. 9).Now a comparison can be made in how the two techniques tackle the image in

Fig. 1(a). This is a height distribution built up by function PEAKS in Matlab. It is afunction of two variables, obtained by translating and scaling Gaussian distribu-tions. The image was then cut off after 8 lines in the x-direction to create a distinct

(a)

0 5 10 15 20 25 3035

40

45

50

55

60

65

Frequency

Am

plitu

de (

dB)

(b)

Fig. 2. (a) Intensity image, f X ¼ 10:5 and b ¼ 0:4; (b) frequency spectrum of the intensity image in (a).

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0 10 20 30 40 50 60 70-6

-4

-2

0

2

4

6

8x 10-3

heig

ht in

m

y-coordinate(c)

0 10 20 30 40 50 60 70-8

-6

-4

-2

0

2

4

6

8x 10-3

y-coordinate

heig

ht in

m

(d)

(a) (b)

Fig. 3. (a) Height demodulated from Fig. 2 using a classical 2D FFT, f x ¼ 3 and f y ¼ 3; (b) regressiveFourier technique, (c) slice of the image in (a) at x ¼ 35; (d) slice of the image in (b) at x ¼ 35: The -dotted

lines represent the exact height distribution.

J. Vanherzeele et al. / Optics and Lasers in Engineering 43 (2005) 645–658652

difference in intensity between left and right border. The results are shown in Fig. 10.The classical 2D approach gets no where near the exact distribution whilst in theregressive solution the profile is clearly visible.

5. Experimental results

In this section the regressive technique will be put to a practical test. The case athand is a profile measurement of a household pvc cylindrical pipe (diameterd ¼ 31:5 cm). Even though this geometry is a simple one the classical 2D FFTcalculation will still be subject to errors (leakage effect), simply because this practicalcase will be a combination of all prior effects mentioned in Section 4.The fringes were projected onto the tube using a NEC VT540 LCD projector

(1600 1200 pixels and 1000 ansi brightness) at a stand-off distance of 157 cm. Theimages were captured using a Fujifilm Finepix s602Z CCD camera whose lens waspositioned at a distance of 8 cm from the projector lens (same floor height). Thesinusoidal fringes were generated in Matlab.

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0 50 100 150 200 250 300-0.015

-0.01

-0.005

0

0.005

0.01

0.015

x-coordinate

heig

ht in

m

(c)

0 50 100 150 200 250 300-0.015

-0.01

-0.005

0

0.005

0.01

0.015

x-coordinate

heig

ht in

m

(d)

(a) (b)

Fig. 4. (a) Height demodulated from a projection with f X ¼ 10; b ¼ 0:4 and sin. height distribution with

f x ¼ 1:5 and f y ¼ 1:5 using a classical 2D FFT, (b) regressive Fourier technique, (c) slice of the image in

(a) at y ¼ 35; (d) slice of the image in (b) at y ¼ 35: The -dotted lines represent the exact height

distribution.

0 10 20 30 40 50 60 70-0.015

-0.01

-0.005

0

0.005

0.01

0.015

x-coordinate

heig

ht in

m

(a)

0 10 20 30 40 50 60 70-0.015

-0.01

-0.005

0

0.005

0.01

0.015

x-coordinate

heig

ht in

m

(b)

Fig. 5. (a) Slice of intensity image at y ¼ 70 with f X ¼ 20 and f x ¼ 2:5; f y ¼ 2:5 and b ¼ 0:4 using a

classical 2D FFT, (b) slice of intensity image at y ¼ 70 with f X ¼ 20 and f x ¼ 2:5; f y ¼ 2:5 and b ¼ 0:4using a regressive Fourier transform.

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The 2D frequency spectrum depicts leakage in the x-direction which of course wasto be expected, but what seems more surprising is the fact that the same effect takesplace in the y-direction (Fig. 11(b)). Normally there should be absolutely nocomponents above DC for y but because the fringes were not projected exactly

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0 10 20 30 40 50 60 70-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

x-coordinate

heig

ht in

m

(c)

0 10 20 30 40 50 60 70-0.015

-0.01

-0.005

0

0.005

0.01

0.015

x-coordinate

heig

ht in

m

(d)

(a) (b)

Fig. 7. (a) Height demodulated from Fig. 6 using a classical 2D FFT, (b) regressive Fourier technique, (c)

slice of the image in (a) at y ¼ 35; (d) slice of the image in (b) at y ¼ 35: The -dotted lines represent the

exact height distribution.

(a)

0 5 10 15 20 25 30 35-60-50-40-30-20-10

010203040

Frequency

Am

plitu

de (

dB)

(b)

Fig. 6. (a) Intensity image with sin. varying visibility (border to center: 0.2–1); f X ¼ 10 and b ¼ 0:4; (b)frequency spectrum of the intensity image in (a).

J. Vanherzeele et al. / Optics and Lasers in Engineering 43 (2005) 645–658654

parallel to the pipe axis a low-frequency component is generated which results inmassive leakage in the y-direction. However, in Fig. 11(a) one can clearly notice avariation in fringe visibility which is also responsible for triggering low-frequencycomponents.

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0 10 20 30 40 50 60 70-0.025-0.02

-0.015-0.01

-0.0050

0.0050.01

0.0150.02

0.025

x-coordinate

heig

ht in

m

(c)

0 10 20 30 40 50 60 70-0.02

-0.015-0.01

-0.0050

0.0050.01

0.0150.02

x-coordinate

heig

ht in

m

(d)

(a) (b)

Fig. 9. (a) Height demodulated from Fig. 8 using a classical 2D FFT, (b) regressive Fourier technique, (c)

slice of the image in (a) at y ¼ 70; (d) slice of the image in (b) at y ¼ 70: The -dotted lines represent the

exact height distribution.

(a)

0 10 20 30 40 50 60-300

-250

-200

-150

-100

-50

0

50

100

Frequency

Am

plitu

de in

dB

(b)

Fig. 8. (a) Intensity image with over-modulation index b ¼ 1:6; f X ¼ 10; (b) frequency spectrum of the

intensity image in (a).

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The profile obtained with the classical approach is given in Fig. 12(a). The leakageon the edges of the image is clearly visible. In the result of the 2D FFT technique(Fig. 12) the distortions due to the leakage effect (both in x- and y-direction) are

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Fig. 11. (a) Intensity image of the fringes on the test pipe, (b) 2D frequency spectrum of image (a).

(a) (b)

Fig. 10. (a) Height demodulated from Fig. 1(a) using a classical 2D FFT, (b) using a regressive Fourier

transform.

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clearly visible. The regressive Fourier approach provides a profile which is free ofany distortions at the borders.

6. Conclusions

In this paper it has been shown that due to leakage, distortions occur while using aclassical 2D Fourier transform. This led to distortions in the different predictedheight distributions. In theory this can be avoided by taking images where theobject borders are completely entailed in the image (height is zero at the edges ofthe image). However, in practical examples this is not always a possibility, as wasshown here, where the pipe was too big to fit into one single take. A solution wasfound in a regressive Fourier approach—based on a transfer function model ofthe measurement spectra—which did not exhibit any of the distortions present in a

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Fig. 12. (a) Demodulated profile using a classical 2D FFT, (b) using a regressive Fourier transform.

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classical FFT solution. This was shown on both simulations and a practicalmeasurement on a pvc pipe.

Acknowledgements

This research has been sponsored by the Flemish Institute for the Improvement ofthe Scientific and Technological Research in Industry (IWT), the Fund for ScientificResearch—Flanders (FWO) Belgium. The authors would also like to acknowledgethe Flemish government (GOA-Optimech) and the research council of the VrijeUniversiteit Brussel (OZR) for their funding.

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