1966 OPTICS LETTERS / Vol. 31, No. 13 / July 1, 2006

Fast blind extraction of arbitrary unknown phaseshifts by an iterative tangent approach ingeneralized phase-shifting interferometry

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. ShenDepartment of Optics, Shandong University, Jinan, 250100, China

Received March 15, 2006; accepted April 13, 2006; posted April 13, 2006 (Doc. ID 68986)

A novel fast convergent algorithm to extract arbitrary unknown phase shifts in generalized phase-shiftinginterferometry (PSI) is proposed and verified by a series of computer simulations. In this algorithm an errorfunction is introduced and then the unknown phase shifts are found by an iterative tangent approach. Incombination with the statistical method, this algorithm can give the most exact results in the fewest itera-tion steps. It can be used for generalized PSI of arbitrary frames for both smooth and diffusing objects andcan usually reach the exact phase shifts with only four or five iterations for three- or four-frame PSI. © 2006Optical Society of America

OCIS codes: 090.2880, 100.3010, 100.2000, 120.5050.

Phase-shifting interferometry (PSI) is a powerful toolfor wavefront reconstruction in a variety ofapplications.1–3 Standard PSI requires a special con-stant phase shift, 2� /N with integer N�3. This re-quirement, however, is often difficult to meet exactlyin reality. To remove this inconvenience, some alter-native approaches such as generalized data reductionand a least-squares algorithm have beendeveloped.4–15 Usually these methods have substan-tial computation loads, and some of them need many(as many as 15) interferograms,10 additional opticaldevices,11 or additional measurement of object andreference wave intensities.12,13 Moreover, often theleast-squares method cannot yield exact results ofthe phase shifts.14 In this Letter we introduce an al-gorithm that can calculate the unknown phase shiftsin any generalized PSI of N�3 with arbitrary phaseshifts. It seems, to the best of our knowledge, thefastest and most exact method of this kind until now.We first explain its properties and then confirm themby computer simulations.

If we denote the complex object field in the record-ing plane O�x ,y�=Ao�x ,y�exp�i�o�x ,y��, the constantamplitude and phase of a plane reference wave in thejth step are Ar and �j, respectively, and the intensitydistribution of the jth interferogram is

Ij�x,y� = Ao2�x,y� + Ar

2 + 2Ao�x,y�Ar cos��o�x,y� − �j�,

j = 1,2, . . . , N. �1�

Let us take three-frame PSI as an example and as-sume that �1=0, �2=�1, and �3=�1+�2, where �jstands for the reference phase difference between thejth and the �j+1�st frames. From Eq. (1) � j=1,2,3�we can find the expression15

AoAr exp�i�o� = A1 + iA2, �2�

where A1 denotes the real part and A2 the imaginary

part, and

0146-9592/06/131966-3/$15.00 ©

A1 =1

4 sin��2/2�� cos��1/2�

sin���1 + �2�/2��I1 − I3�

−cos���1 + �2�/2�

sin��1/2��I1 − I2�� , �3�

A2 =1

4 sin��2/2�� sin��1/2�

sin���1 + �2�/2��I1 − I3�

−sin���1 + �2�/2�

sin��1/2��I1 − I2�� . �4�

It is easy to see that

AoAr = �A12 + A2

2�1/2, �5�

cos �o = A1�A12 + A2

2�−1/2,

sin �o = A2�A12 + A2

2�−1/2. �6�

From Eq. (6) we can determine �o in its principal re-gion of −� to �. Furthermore, we have

�Ao + Ar�2 = I1 − 2A1 + 2�A12 + A2

2�1/2, �7�

�Ar − Ao�2 = �I1 − 2A1 − 2�A12 + A2

2�1/2�. �8�

Since both Ar and Ao are positive, with Ar larger thanthe maximum of Ao�x ,y�, in practice to guarantee cor-rect recording we get

Ar + Ao = �I1 − 2A1 + 2�A12 + A2

2�1/2�1/2, �9�

Ar − Ao = �I1 − 2A1 − 2�A12 + A2

2�1/2�1/2. �10�

Equations (5), (9), and (10) yield

Ao = �1/2���I1 − 2A1 + 2�A12 + A2

2�1/2�1/2

− �I1 − 2A1 − 2�A 2 + A 2�1/2�1/2� , �11�

1 2

2006 Optical Society of America

July 1, 2006 / Vol. 31, No. 13 / OPTICS LETTERS 1967

Ar = �1/�2M�� � �I1 − 2A1 + 2�A12 + A2

2�1/2�1/2

+ �I1 − 2A1 − 2�A12 + A2

2�1/2�1/2. �12�

In Eq. (12) and what follows, the sign � indicates thesummation over all the pixels in the correspondingframe and M denotes the number of pixels in oneframe. The sign of absolute value in Eqs. (8), (10),and (11) is not necessary for the exact phase shiftsbut is required sometimes to ensure reasonable Aoand Ar during iterations if the initial values of thephase shifts used deviate significantly from their realvalues.

Now we may introduce an error function

� = �j=1

3

� �Ij�x,y� − Ao2�x,y� + Ar

2

+ 2Ao�x,y�Ar cos��o�x,y� − �j��, �13�

where �1=0, �2=�1, and �3=�1+�2. From Eqs. (3), (4),(6), (11), and (12) we can see that all the Ao�x ,y�, Ar,and �o�x ,y� in Eq. (13) can be expressed with thephase steps �1 and �2 and the intensities of threeframes. Therefore � is a function of variables �1 and�2 for a known set of I1, I2, and I3, and the actual �1and �2 will yield a minimum �. Based on this idea, wecan find the correct �1 and �2 by varying them to geta minimum �. In practice, we may first give �1 and �2two initial values, �10 and �20, randomly from 0 to �,and calculate the error function with them and thegiven I1, I2, and I3. Then we fix �2 as �20 and change�1 to find its improved value �11, yielding smaller �.The next step is to search for the new value of �2, �21for fixed �11. Now the first iteration cycle is com-pleted. Similarly, using �11 and �21 as initial values,a new value �12 and then �22 can be obtained in suc-cession.

Here we propose a novel and unique method withwhich to find new �1 or �2 by a process that may becalled an iterative tangent approach as shown in Fig.1 for the kth iteration. In this cycle the initial valuesare �1,k−1 and �2,k−1. If �2,k−1 is fixed, � is a function of�1 as shown by the curve in this figure. Now we drawa tangent line of the curve at point A correspondingto the error function at ��1,k−1 ,�2,k−1�, and denote thecoordinate of the intersection point of the tangentwith the �1 axis as �1k. Obviously

Fig. 1. Variation of the error function with �1 for a fixed �2

and the tangent approach.

�1k = �1,k−1 − �A/�, �14�

where � is the partial derivative of � with respect to�1 at point A, which can be calculated numerically.After �1k is found, we fix �1k and change �2 to find thenew �2k in the same way. This process will go on untilthe difference between �1k and �1,k−1 and that be-tween �2k and �2,k−1 �k=1,2,3, . . . � are all less than apreset small amount.

As we shall see in following simulations, thismethod can give a fast convergence of �1 and �2 andusually yields their exact values in a few iterationsteps. The reason is that, as calculations have shown,in the vicinity of real �1 and �2 the variation of curve� with �1 or �2 is approximately linear, and thereforethe tangent can lead to a fast approach to the exactvalues of �1 and �2. In addition, it is not necessaryhere to solve a group of equations, as is needed in theleast-squares method (LSM).14

When N3, we can first calculate �1 and �2 withthe first three interferograms and then find �3 ,�4 , . . .one by one. For example, for a four-frame PSI, the in-tensity of the fourth interferogram is

I4�x,y� = Ao2�x,y� + Ar

2 + 2Ao�x,y�Ar

cos��o�x,y� − �1 − �2 − �3�. �15�

When all the Ao�x ,y�, Ar, and �o�x ,y� are obtainedand �1 and �2 are found from I1, I2, and I3 as men-tioned above, we can introduce a new error function:

�� = � �I4�x,y� − Ao2�x,y� + Ar

2 + 2Ao�x,y�Ar

cos��o�x,y� − �1 − �2 − �3��. �16�

Evidently �� is now only a function of �3. By the samemeans we can differentiate �� with respect to �3 toget a new �31, and repeat this process if necessary.Hence a one-frame increase needs only a new errorfunction consisting of only one variable, which is easyto calculate.

A series of computer simulations was carried out toverify the validity of this algorithm. In all thesesimulations the recording geometry and system pa-rameters are the same as we reported before,13,15 sothey are ignored here. Both the smooth and the dif-fusing spherical surfaces with different radii R areutilized as test objects. Different interferograms arecomputer generated for different randomly presetphase shifts. For comparison, we also made some cal-culations in the same conditions, using other meth-ods. Here for brevity we give only part of our results.

Table 1 shows the phase-shift extraction results fora three-frame PSI with both our algorithm and theLSM for an object of diffusing surface R=0.2 m. Herethe preset phase shifts are �1=1.5977 and �2=2.6438 (all in radius), and the initial values that wetook arbitrarily in calculations are �1=�2=1.0000.From this table we can see that our algorithm canfind the exact �1 and �2 in six iterations. The LSM,however, cannot give exact values of �1 and �2 afteras many as 40 iterations; actually its results fluctu-

ate about the exact values of �1 and �2 even after

15

51

1968 OPTICS LETTERS / Vol. 31, No. 13 / July 1, 2006

many iteration steps, and the exact values can neverbe reached. Similar calculations with different ran-domly chosen initial values of �1 and �2 and withsmooth surfaces have led to similar results.

To further increase the speed of convergence of ouralgorithm, we can combine it with the statisticalmethod for the choice of initial values of �1 and �2, aswe proposed before.15 To show the effectiveness ofthis idea and the feasibility of our approach for gen-eralized PSI, in Table 2 we give the calculation re-sults for a four-frame PSI with arbitrary phase shifts�1, �2, and �3 (Table 2, line a) by using our newmethod here and the one reported in our previous

15

Table 1. Comparison of Phase-Shift ExtractionResults of Our Method and the LSMa for a

Three-Frame PSI

Phase Shifts

Our Methodb LSM

Algorithms �1 �2 �1 �2

Preset values 1.5977 2.6438 1.5977 2.6438Initial values 1.0000 1.0000 1.0000 1.00001st iteration 1.5777 1.4981 0.6284 2.46692nd iteration 2.2035 2.1499 0.6794 2.74373rd iteration 1.6561 2.5912 0.8406 2.74824th iteration 1.6194 2.6435 0.9938 2.72175th iteration 1.5976 2.6438 1.1214 2.69936th iteration 1.5977 2.6438 1.2241 2.6835

10th iteration 1.4599 2.655720th iteration 1.5876 2.645530th iteration 1.5979 2.644940th iteration 1.5988 2.6448

aRef. 14.bA cross means no iteration.

Table 2. Comparison of Phase-Shift ExtractionResults of Our Method and the Direct Searcha for a

Four-Frame PSI

Our Algorithm Direct Search

Lineb �1 �2 �3 Lineb �1 �2 �3

a 1.3963 1.8326 1.6581 a 1.3963 1.8326 1.658b 1.3976 1.8354 1.6625 b 1.3976 1.8354 1.6621 1.3933 1.8354 4 1.3936 1.8354

2 1.3955 1.8329 7 1.3966 1.8324

3 1.3962 1.8326 9 1.3964 1.8326

4 1.3963 — 10 — 1.8325

5 — — 1.6581 11 1.3963 —

12 — 1.8326

16 — — 1.65820 — — 1.658

aRef. 15.bIn this table, row a gives preset values of �1, �2 and �3; b gives

the values calculated by statistics.15 The numbers 1–20 are the it-eration steps. A horizontal line means no change from the formervalue, and an means no iteration.

paper, referred to as the direct search method in

Table 2 for a smooth surface, R=0.2 m. These resultshave revealed clearly that our statistical approachcan considerably reduce the number of iterations,and our new method can yield much faster conver-gence than the direct search technique. In fact, thevalues of �1, �2, and �3 obtained from the statisticalmethod in this table (line b) have already been neartheir exact values. Using them as initial values, weneed only four steps to retrieve the exact values of �1and �2, and the exact value of �3 can be further foundwith only one more step with our new method (totalof five iterations). Even with the same approximateinitial values of �1, �2, and �3, however, 20 iterationsare required in the direct search method. The diffus-ing surfaces under the same conditions were alsotested, and the results were similar.

In summary, we have proposed a new, fast conver-gent algorithm, mainly by use of an iterative tangentapproach, to extract arbitrary unknown phase shiftsin generalized PSI and verified its effectiveness by aseries of computer simulations. This method, com-bined with our statistical approach, is the simplestand fastest method compared with least-squares andother methods in this field. It needs no additionalmeasurement and can be used for both smooth anddiffusing objects for any PSI of three or more frames.And it can yield the exact phase shifts in very few it-eration steps, say, four steps for a three-frame PSIand five steps for a four-frame PSI. After the actualphase shifts are extracted, the complex object fieldO�x ,y� can be retrieved exactly.15

This work is supported by the National NaturalScience Foundation of China (grant 64077005). L. Z.Cai’s e-mail address is lzcai@sdu.edu.cn.

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