Optics and Lasers in Engineering 49 (2011) 1118–1123

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Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

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

A demodulation method for the circular carrier interferogram usingphase stitching

Bo Li a, Lei Chen a,n, Jiang Bian b, Yan Li a

a School of Electronic Engineering and Optoelectronic Technology, Nanjing University of Science and Technology, Nanjing 210094, Chinab The Chinese Academy of Sciences, The Institute of Optics and Electronics, Chengdu 610209, China

a r t i c l e i n f o

Article history:

Received 8 February 2011

Received in revised form

3 May 2011

Accepted 22 May 2011Available online 8 June 2011

Keywords:

Optical testing

Phase retrieval

Circular carrier

Fourier transform

Phase stitching

66/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlaseng.2011.05.013

esponding author.

ail address: chenleiy@126.com (L. Chen).

a b s t r a c t

The four phases stitching algorithm is proposed to demodulate the circular carrier interferogram, which

can eliminate the sign ambiguity and avoid serious local errors in the Fourier transform method. A pair

of orthogonal low-pass filters is used to obtain four demodulated phases with local errors concentrated

in different areas, then their reliable parts are chosen and combined to get the phase without sign

ambiguity by a stitching strategy, which makes a significant error suppression. Furthermore, due to the

stitching strategy the sign flip location just needs to be detected approximately, therefore the detection

can be done using an automatic procedure. The algorithm is validated by the numerical simulations,

where the calculation precisions are better than l/50 with suitable carrier. Besides, an actual

interferogram is analyzed and the result is in good accordance with the Zygo phase-shifting

interferometer.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

To demodulate phase (or wavefront) from single interfero-gram, the Fourier transform (FT) method [1] is usually the bestchoice where a linear carrier greater than the maximum wave-front slope is required in the interferogram, or equivalently, thefringes are not closed. However, the highest carrier frequency isalways restricted by the resolution of detectors. Sometimes, forexample, if the test wavefront contains large aberrations or ishighly aspheric, the required linear carrier is hard to satisfy, and aclosed fringe interferogram is needed to demodulate [2].

Currently there are many methods to process closed fringes. Theregularized phase-tracking (RPT) method [3] is an optimizationmethod with robustness, which avoids phase unwrapping. But itoften takes long time for its nonlinear optimization calculation, andthe initial condition must be chosen carefully. The Hilbert transformmethod [4] solves the isotropic Hilbert transform and orientation oflocal fringes, where the later is hard to be calculated accurately. Thecoordinate transform method [5] converts closed fringes in therectangular coordinate to open fringes in the polar coordinate, whilethe calculation errors may result from the non-integral coordinateand various fringes frequency after transform. The FT method is alsoused to process the closed fringes similar as the case of linear carrierinterferogram [6–11], but the result contains the sign ambiguity, so

ll rights reserved.

a correct operation is necessary, which is guided by a pair of phaseshift interferograms [6], the smoothness condition of phase vari-ety[8], the orientation of local fringes[9] and so on.

Some researchers [10,11] focused on a special kind of closedfringes with the shape of approximate concentric circles, asshown in Fig. 1(a), which are often generated under two condi-tions: (1) The measured phase contains large spherical aberrationor other rotational symmetric aberrations and the circular carrieris introduced by defocusing to ensure the phase monotonicityfrom the center to the edge. (2) The measured phase itself isnearly circular such as the measurement of the fiber connectorend face, where the circular fringes are not introduced by thecarrier. Nevertheless, in this paper we still take it as the circularcarrier interferogram for the description simplicity, because thesame mathematical presentation and demodulation methodare used.

There is only one extreme in the phase included in the circularcarrier interferogram, so the sign flip resulting from the FTmethod locates along a line approximately, which means the signcan be corrected by just reversing the phase on one side of thesign flip line [10]. Using this method, however, there will besignificant phase errors around the sign flip location, which willbecome more serious if the flip locations are not detectedaccurately.

In this paper we propose an improved method to process thecircular carrier interferogram. A pair of orthogonal low-pass filtersare used to get four phases with sign ambiguity, then the reliableparts with continue sign are stitched to the complete retrieved

Fig. 1. Phases retrieving. (a) Interferogram. (b) Frequency spectrum after low-pass filtering in y direction. (c,d) Phase maps with sign flip in y and x direction. (e) Retrieved

phase by sign correction. (f) Error distribution of (e) where the brightest pixel represents a value of 0.43l.

B. Li et al. / Optics and Lasers in Engineering 49 (2011) 1118–1123 1119

phase where the area of large errors is avoided. Furthermore, thedetection of sign flip locations has a loose tolerance so that it canbe finished automatically with an approximate result.

2. Theory and algorithm

2.1. FT method to demodulate circular carrier interferogram

The intensity of an interferogram can be written as

Iðx,yÞ ¼ aþbcos½kWðx,yÞ�, ð1Þ

where a, b are the background and modulation of the interfero-gram, k¼2p/l, and W(x, y) is the test wavefront. By increasing thedefocusing, a circular carrier can be introduced to the wavefront,and the interferogram shown in Fig. 1(a) is given by

Iðx,yÞ ¼ aþbcos jðx,yÞ� �

¼ aþb

2exp ijðx,yÞ

� �þ

b

2exp �ijðx,yÞ

� �, ð2Þ

where the phase j(x,y)¼k[D(x2þy2)�W(x,y)] and D is the defo-

cusing coefficient. If the spectrum of Eq. (1) is low-pass filtered inone side of the vertical direction as shown in Fig. 1(b), its inverseFourier transform can be written as

ZReðx,yÞþ iZImðx,yÞ

¼

b2 cos ijðx,yÞ

� �þ i b

2 sin ijðx,yÞ� �

if @j@y Z0,

b2 cos ijðx,yÞ

� ��i b

2 sin ijðx,yÞ� �

otherwise,

8<: ð3Þ

and the retrieved phase with sign ambiguity is given by

jv ¼ tan�1 ZImðx,yÞ

ZReðx,yÞ

� �: ð4Þ

The phase jv shown in Fig. 1(c) equals to the original phase orconjugate phase, with the same magnitude but inverse sign in thearea of positive or negative local spatial frequency, respectively. Ifthe monotonicity condition

@ðDr2�WÞ

@r40 ð5Þ

is satisfied (r2¼x2þy2), there will be only one extreme in the

phase j so the sign of @j=@y changes only once in the whole zone.That means there is a sign flip in jv along a horizontal linethrough the image center, so the phase can be corrected by simplyreversing the sign in the �y half-plane [10]. That is

jeðx,yÞ ¼jvðx,yÞ, yZ0,

jeðx,yÞ ¼ �jvðx,yÞ, yo0:

(ð6Þ

Then the estimated phase je shown in Fig. 1(e) is obtained andcan be unwrapped to reconstruct the original phase.

Unfortunately, as shown in Fig. 1(f), there is a serious localerror in the retrieved phase around the area of phase flip, orequivalently, the area of zero spatial frequency, which is intro-duced by the Gibbs phenomenon.

In addition, there is a potential problem that sometimes thelocation of sign flip is not at y¼0 and the accurate position is hardto estimate. The inaccurate flip detection will result in severalrows of the phase being reversed wrongly, so the direct flipstrategy need to be improved.

2.2. Phases stitching method to improve the result

To solve the above problem, another low-pass filter in thehorizontal direction is used to obtain the wrapped phase ju withsign flip in the x direction as shown in Fig. 1(d). Similar to Eq. (6),je can be given by

jeðx,yÞ ¼jhðx,yÞ, xZ0,

jeðx,yÞ ¼ �jhðx,yÞ, xo0,

(ð7Þ

where the demodulated phase je distorts at the position x¼0.Note that je given by Eqs. (6) and (7) have different error

areas, so an improved result is available by combining them.In other words, je is estimated by stitching four parts fromphases 7jv and 7ju as

jwe ðx,yÞ ¼jw

v ðx,yÞ, fy�y0Zx�x0g \ fy�y0Z�xþx0g,

jwe ðx,yÞ ¼�jw

v ðx,yÞ, fy�y0ox�x0g \ fy�y0o�xþx0g,

jwe ðx,yÞ ¼jw

u ðx,yÞ, fy�y04x�x0g \ fy�y0o�xþx0g,

jwe ðx,yÞ ¼�jw

u ðx,yÞ, fy�y0ox�x0g \ fy�y04�xþx0g,

8>>>><>>>>:

ð8Þ

Fig. 2. Four phases stitching procedure.

Fig. 3. Searching the stitching center. (a,b) Distribution of the fringe orientation

angles by Eqs. (6) and (7). (c) Binary image using edge detection to (a). (d) Two

lines from the binary images using Hough transform.

Table 1The coefficients of Kingslake polynomial.

Coefficients A B C D E F

Value 0.5 1 2 20 0.5 0.5

B. Li et al. / Optics and Lasers in Engineering 49 (2011) 1118–11231120

in which the point (x0, y0) is the stitching center (here x0¼y0¼0).The chosen parts of the phases for stitching are shown in Fig. 2where it can be seen that the areas around x¼0 and y¼0 thatcontain phase distortions are both avoided so the phase demo-dulation error decreases obviously. We call this method the fourphases stitching (FPS) method.

If the sign flip location is not at (0, 0), the stitching center hasto be changed correspondingly. However, unlike the case ofSection 2.1, FPS has a much looser tolerance of the sign fliplocations. Because in Section 2.1 the location detecting error of epixels will distort the demodulated phase in the rows from y¼0to e, while in FPS the same location error just affects a very smallregion around the stitching center with the area of about e2. It isanother merit of FPS method, due to which the stitching centercan be approximately estimated by an automatic strategydescribed as follows.

Since the phase flips where the local spatial frequency is zero,the flip locations can be estimated through the fringes orientationangle given by [4]

Yðx,yÞ ¼ tan�1 @Iðx,yÞ

@y

@Iðx,yÞ

@x

� ��ð9aÞ

or

Y0ðx,yÞ ¼ tan�1 @Iðx,yÞ

@x

@Iðx,yÞ

@y

� �:

�ð9bÞ

As shown in Fig. 3(a) and (b), there is a jump from �p to p inthe orientation angles Y and Y0 at the locations @I=@x¼ 0 and@I=@x¼ 0, respectively. This inherent jump can be searched by thegeneral edge detection technique, such as the Roberts operator[13] and the result is represented by a binary image. InFig. 3(c) this binary image takes the jump location in Fig. 3(a) asits foreground points, while some of them deviate from the ideallocations since estimating the orientation angle by Eq. (9) is notrobust. Nevertheless, with the circular carrier, these points can beassumed to distribute along a horizontal line approximately. Sothe standard Hough transform [13], which is a statistical methodwith good resistance to the disturbance of invalid foregroundpoints, can be used to recognize the horizontal straight line. In thesame way, the vertical edge in Fig. 3(b) also can be detected, and

the two straight lines are shown in Fig. 3(d) where the intersec-tion point is recognized as the stitching center.

Now the procedure of FPS method can be presented as follows:

Step. 1: The spectrum of interferogram is filtered by a pair oforthogonal low-pass filter, and then the filtered spectrums areinverse transformed so that the phases ju and jv with signambiguity can be obtained by Eq. (4).Step. 2: Eqs. (9a) and (9b) are used to estimate the fringesorientation angles, from which two sign flip lines are searchedby the edge and line detection techniques, and the stitchingcenter (x0, y0) is decided by their interception point.Step. 3: Eq. (8) is used to stitch 7ju and 7jv to a phasewithout sign ambiguity.Step. 4: The unwrapping operation is used to retrieve theoriginal phase.

3. Discussion

In this section we will validate the assumption in the abovesection that the phase flip locations distributes approximatelyalong straight line in FPS algorithm.

A wavefront containing the primary aberrations can be repre-sented by the Kingslake polynomial [12]

Wðx,yÞ ¼ Aðx2þy2Þ2þByðx2þy2ÞþCðx2�y2ÞþDðx2þy2ÞþExþFyþG, ð10Þ

where A�G are the coefficients of spherical aberration, coma,astigmatism, defocusing, tilt and the constant, respectively. Thecoefficients A�F are set with the values in Table 1.

B. Li et al. / Optics and Lasers in Engineering 49 (2011) 1118–1123 1121

With above setting the monotonicity condition of Eq. (5) issatisfied evidently, which makes W with only one extreme, so thesign flip locations in ju and jv must distribute along the twocurves @W=@x¼ 0 and @W=@y¼ 0, respectively.

Firstly, using Eq. (10) the sign flip curve of ju is given by

4Axðx2þy2Þþ2Bxyþ2Cxþ2DxþE¼ 0, ð11Þ

whose nonlinearity can be estimated by analyzing its deviationfrom the best fit line. The coefficients of Eq. (11) are given inTable 1, and the numerical calculation is carried out in a256�256 image. The best fit line of Eq. (11) is x¼�0:022, whosemaximum deviation from the curve is less than 1 pixel so thenonlinearity can be omitted.

Secondly, the sign flip curve of jv is given by

4Ayðx2þy2ÞþBðx2þ3y2Þ�2Cyþ2DyþF ¼ 0, ð12Þ

and its best fit line is y¼�0:012. The deviations between themare less than 3 pixels, and the deviations of 60% points are lessthan 1 pixel. It is a small value compared to the image size, so thesign flip distribution is reasonable to be regarded as a lineapproximately.

4. Simulation and experiment

Numerical simulations and experiments are carried out tovalidate the precision of the algorithm.

The simulated interferograms are divided into four groups thathave three interferograms, respectively:

Fig. 4. Simulated interferograms. (a) Circular phase. (b

In Group One, the measured phases are circular with the PVvalue of 10l. The first interferogram is ideal without any noiseand the illumination is uniform, while in the other two interfer-ograms the illumination is assumed as the Gauss function withthe amplitude around the margin about the half of the center, andthe low and heavy random noise with 7% and 15% amplitudes ofthe intensity signal are added into them. The third interferogramis shown in Fig. 4(a).

In Group Two, the measured phases contain the sphericalaberration of 3l. The circular carriers of these three interfero-grams are 12, 16 and 18l, where the background and noise aresame as the second one of Group One (low noise). The firstinterferogram of this group is shown in Fig. 4(b).

In Group Three and Group Four, the measured phases containthe astigmatism and the coma, respectively with both 2l PVvalue, where the first interferograms of each group are shown inFig. 4(c) and (d). The noise is set similarly to Group Two, while thecarriers of the interferograms are 6, 12 and 18l.

The interferograms are processed by the FPS method and theresults are evaluated by the root mean square (RMS) value of thedeviation between the demodulated phase and the ideal one,however, several pixels around the edge of aperture are excludedconsidering the boundary effect of the FT method. The stitchingcenters searched by the algorithm are illustrated with square flags inFig. 4, where the center searching strategy performs reliably underthe noisy condition. In addition, the phase around the fringes centercan be improved by a spatial or frequency domain filtering since thephase here has a very low spatial frequency.

The demodulated phases of the interferograms in Fig. 4 areshown in Fig. 5, where in Fig. 5(b–d) the carriers are removed

) Spherical aberration. (c) Astigmatism. (d) Coma.

Fig. 5. Demodulated phases of Fig. 4(a–d).

Fig. 6. Experiment. (a) Interferogram captured by Zygo GPI interferometer. (b) Phase obtained by PSI. (c) Phase demodulated by FPS method. (d) Deviation between (b) and

(c) with RMS of 0.022l.

B. Li et al. / Optics and Lasers in Engineering 49 (2011) 1118–11231122

from the phases by polynomial fitting since they are introducedby defocusing. The errors of all simulated interferograms aregiven out in Table 2.

From Group One we can see that the non-uniform illuminationand low noise will not affect the precision evidently, but the error

tends to increase with heavy noise. Although the noise can besuppressed by reducing the pass-band of the filter, more phaseinformation of high frequency will be lost. If these two errors arebalanced suitably, however, the demodulation precision betterthan 0.03l can be obtained.

Table 2

The demodulation error of the simulated interferograms (unit: l).

Group One Two Three Four

Interferogram 1 2 3 1 2 3 1 2 3 1 2 3

Noise None Low Heavy Low Low Low Low Low Low Low Low Low

Carrier 10 10 10 12 16 18 6 12 18 6 12 18

Error RMS 0.013 0.016 0.029 0.014 0.013 0.013 0.024 0.018 0.017 0.026 0.019 0.019

B. Li et al. / Optics and Lasers in Engineering 49 (2011) 1118–1123 1123

In the Group Two, the precision remains almost unchangedwith different carriers as long as the monotonicity condition issatisfied. The reason is that the spherical aberration is rotationalsymmetrical, where FPS method will obtain the best resultbecause it stitches the parts with equal area from four phases.

In Group Three and Group Four, the error becomes large whenthe carrier is low, since the astigmatism and the coma are notrotational symmetrical. But the precision will be better than l/50if carrier is suitable.

In the experiment, a 223�223 interferogram shown inFig. 6(a) is captured by the ZYGO GPI XP interferometer and theresult by phase shifting interferometry (PSI) is shown in Fig. 6(b),where the test object is a double convex lens with 80 mmaperture and 100 mm focus length. Here the circular carrier isalso removed by polynomial fitting.

The FPS method is used to process the interferogram and thestitching center searched by the algorithm is also illustrated withthe rectangle flag, where the location detection is accurate. Thedemodulated phase is shown in Fig. 6(c) whose PV value is 1.401lwith 95% aperture while in Fig. 6(b) the value is 1.438l, and theRMS values of these two phases are both 0.274l. In addition, thedeviation between these two is shown in Fig. 6(d) with the RMS of0.022l. Therefore it can be concluded that the two results by FPSand PSI accord well.

5. Conclusion

We propose the FPS method to demodulate the circular carrierinterferogram, in which the phase stitching strategy has twomerits as follows:

(1)

The sign ambiguity is eliminated and the area of unreliableresult is avoided, so the precision is improved. And theresidual error concentrated in a small region around thefringes center can be suppressed by spatial or frequencydomain filtering.

(2)

It gives a loose tolerance for sign flip location, which leads toan automatic detection, and the potential errors introducedby the inaccurate sign flip detection is evaded.

This method can be used to process the closed fringes inter-ferogram whose phase contains only one extreme, thus a propercircular carrier should be introduced to ensure the monotonicity,or the phase itself is nearly circular. A good demodulationprecision will be obtained for an approximate rotational symme-trical phase, and a more complex stitching may improve theresult in other situations, which will be researched in the future.

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