Abstract

This paper presents a novel algorithm for phase extraction based on the computation of the Euclidean distance from a point to an ellipse. The idea consists in extracting the intensities from a data row or column in three interferograms to form points of intensity and then fitting them to an ellipse by the method of least squares. The Euclidean distance for each intensity point is computed to find a parametric phase whose value is associated to the object phase. The main advantage of the present method is to avoid the use of tangent function, reducing the error in the desired phase computation.

© 2015 Optical Society of America

1. Introduction

Phase-shifting interferometry (PSI) is a method capable of obtaining the desired phase by resolving a system of equations formed by three or more interferograms changed in known phase-steps equally spaced between any two consecutive fringe patterns [1–4]. Attempts to bring these important and needed conditions in the experimental world have required the careful and exhaustive calibration of a phase shifter, demanding high cost, much effort and hard work, among others [1–6]. Because of the exceptional characteristics of PSI such as being non-invasive, non-contact, providing full field observation, and its high accurate given in fractions of the wavelength, has been successfully employed in many fields of science and technology, such as Physics, Astronomy, Biology, Chemistry, Medicine, Mechanics, Metrology, Microscopy, Holography, among many others. It has been employed to measure the refractive index in transparent media including gasses, liquids, solids and plasma, and therefore, to measure viscosity, density, permittivity, diffusion, temperature, homogeneity, purity, etc.; also PSI has been used to measure angular or linear displacement or very short distances and therefore estimates thickness, rugosity, topography, vibration modes in materials, and many others measurements [7, 8].

Bruning et al [1] proposed in 1974 the use of a piezoelectric transducer (PZT) as a phase shifter to create a phase-step and to carry out the PSI method for first time. After him many other experimental techniques for achieving a phase shifter were proposed, including polarizers [9], gratings [10, 11], spatial light modulators [12], by the lateral displacement of the light source [13], and recently with amplitude filters [14–16] by applying the theory of amplitude modulation out-of-phase in quadrature (QAM) [17] and non-quadrature NQAM [18]. Nevertheless, despite all the great efforts to develop a synchronous detection of interferograms as required by the PSI method, the desired conditions are not met exactly, due to the inhomogeneity in the phase shifter materials, and other characteristics such as their non-linear response, hysteresis, and their dependence on the temperature, and of course by fabrication defects, as well as many other conditions outside of the phase shifters that form part of experimental setup such as mechanical vibrations, atmospheric turbulence, temperature gradients, instability of the light source, non-linearly in the detector, and defects in the used optical components. The intrinsic characteristics of the phase shifter and the innate fluctuations in the experimental setup introduce undesirable variations not only in the phase-step but also in the background and modulation light of the interferograms. For this reason, several studies to estimate the uncertainty in the measurement of object phase have been widely developed by Schreiber [19], Hariharan [20], Schmit and Creath [21], and many others researchers [22–30].

A first alternative to improving the drawbacks in PSI was to propose algorithms capable of demodulating phase from interferograms with unequal phase-steps, that is asynchronous detection of interferograms, named as generalized phase-shifting interferometry (GPSI). Nevertheless, these steps must still be known, implying calibration of the phase shifter. Then in order to avoid this hard task, a second alternative emerged to estimate the unknown phase-steps in the interferograms introduced generally with a miscalibrated phase shifter, named as self-calibrating phase-shifting algorithms. In 1982 Morgan [31], and in 1984 Grievenkamp [32] presented a proposal based on the method of least squares, and after they many interesting proposals [33–46] that combined GPSI with self-calibrating methods as classified by Patit et al [47] were presented based in concepts such as Fourier transform [33], statistical [34, 44, 45], elliptical curve fitting [35, 36, 48, 49], iterative [37], spatio-temporal [41], and optimization [43]. The merit of these methods is in how calculate in the most of cases the phase-step from the interferograms and after calculating the object phase by applying some PSA based on PSI or GPSI. Assuming a synchronous detection, the simplest case was proposed by Carré [50] in 1966, and an uncertainty analysis so as a wide state of art was presented by Kemao et al [51], Novak and Miks [52], Hack [53], Rastogi and Hack [54], and in the references cited therein. The first proposal for self-calibrating with asynchronous detection was resolved by the use of iterative methods [55–58] demanding long computer calculation time and with processing only a few frames or with many but with low resolution. Non-iterative methods were also proposed reducing considerably the computing time allowing many frames of high resolution and with better accurate in the phase demodulation. Recently, Vargas et al [59, 60] introduced a novel idea for phase extraction based on principal component analysis (PCA), and afterwards its generalization [61] in order to remove the drawbacks that occur when there is less than a fringe. Juarez-Salazar et al [62] proposed a method for normalizing interferogram using the method of least squares for estimating the phase-steps and obtaining the object phase. Other novel approaches have used the frequency space of the interferograms to deduce several PSA methods with special characteristics. In 1990 Freischlad, and Koliopoulos [63] presented by first time this idea, followed by Larkin and Oreb [64], Schmit and Creath [21], and finally this method was generalized by Servin et al [65, 66], Mosiño et al [67, 68], and Tellez et al [69]. Basically, the Fourier transform was used as a principal tool to develop this approach. However, other integral transforms have been used such as the wavelet transform [70], S-transform [71], and Z-transform used by Surrel in 1996 [72] to introduce the idea of characteristic polynomial.

All the methods above mentioned including PSI, GPSI, and self-calibrating can be classified under the phase-shifting algorithms, which have at least two common characteristics such as the spatially uniform or homogenous phase shift (phase-step), and the computation of phase via the tangent function. As far as we know, only a few algorithms different from PSA have been proposed for instance Zeng et al [73] in 2103 pointed out that the phase shift can be an unknown tilt plane instead of a non-tilt plane (phase-step) with a different tilt in each interferogram. A little later, Juarez-Salazar et al [74] in 2014 proposed the phase shift as a spatial function of any shape, different to a phase-step as in the PSA method and tilt planes as proposed by Zeng. Both approaches use the method of least squares to carry out their proposal, and although these results are important, these are obtained in an approximated way. A little earlier that these works, Rivera-Ortega et al [14–16] presented with an exact mathematical model a novel method for phase extraction named phase-visibility modulating interferometry (PVMI), that resolved not only the case of phase shift considered as inhomogeneous and arbitrary spatial functions but also the case of visibility modulated in a way inhomogeneous and arbitrary. The algorithm of PVMI is non-iterative, and does not have restrictions or special considerations, therefore it is very fast and very easy to implement in the computer, and its experimental implementation is also very easy and does not need calibration. The idea of PVMI is built on the novel method for modulating an optical field in its phase and amplitude based QAM and NQAM [17, 18]. One of the biggest advantages of PVMI is the ability to be implemented in applications in real time, as it was reported in [16].

In this paper we propose a new method for phase extraction that does not use the tangent function as in the well-known PSA methods, based on the simple idea of Euclidean distance from a point to an ellipse. This proposal considers three interferograms changed in unknown phase-steps within the range (0,π)with the same background and the same contrast varying slightly. In the first stage, two consecutive subtractions are done to obtain two secondary fringe patterns with which the background is eliminated. In the second stage ordered pairs are formed from data in a row or column of these secondary fringe patterns named intensity points, which are fitted to an ellipse by applying the method of least squares. It is important to note that the method used in this paper for fitting disperse data to ellipses is an alternative way from the method introduced by Bookstein [48] used by Farrel and Player in self-calibrating GPSI [35, 36] and by Albertazzi et al [49]. Finally, we obtain the wrapped phase map by computing the Euclidean distance from each intensity point to the fitted ellipse, and it is compared with a PSA method as deduced by Zeng [73].

2. Theory and simulation

Let’s consider three interferograms of the form,

I0(x,y)=a(x,y)+b(x,y)cosϕ(x,y),
I1(x,y)=a(x,y)+b(x,y)cos[ϕ(x,y)+α1],
I2(x,y)=a(x,y)+b(x,y)cos[ϕ(x,y)+α2],
where a and b represent the background and modulation light expressed in units of irradiance, α1 and α2 are unknown phase-steps, and ϕ represents the phase of some object under test expressed in radians. Omitting coordinates from Eq. (1) a can be eliminated to obtain two secondary fringe patterns
I0I1=2bsin12α1sin(ϕ+12α1),
I1I2=2bsin12(α2α1)sin(ϕ+12(α1+α2)),
and after several operations ϕ also can be eliminated,
sin2(α22)=(I0I1)24b2sin2(α1/2)+(I1I2)24b2sin2((α2α1)/2)(I0I1)(I1I2)cos(α2/2)2b2sin(α1/2)sin((α2α1)/2).
Typically, the interferograms are captured with a CCD camera, so the variables take discrete values, such as x=jΔx=xj and y=iΔy=yi, where Δx and Δy are the pitches of the CCD camera, and therefore the notations K(x,y)=K(xj,yi)=Ki,j, with K=I,a,b,ϕ are assumed. Thus, the interferograms in Eq. (1) become matrixes of m×n, with i=1m and j=1n.

A graphical representation of the interferograms described in Eq. (1) is shown in Fig. 1(1). These interferograms were obtained considering the phase-steps of α1=π/3 rad, and α2=2π/3 rad with a Gaussian noise only for α1 of σ1=π/20 rad in standard deviation, and the phase object, background, and contrast of the form

ϕ(x,y)=103(x2sinx+y2),
a(x,y)=a0exp((xxa)2σax2(yya)2σay2),
b(x,y)=b0exp((xxb)2σbx2(yyb)2σby2),
where the constant values: a0=7, xa=3.6, ya=3.5, σax=27, σay=26; and b0=3.5, xb=2.9, yb=3.9, σbx=25, σbx=23 were chosen for simplicity. The form of ϕ, a and b were chosen taking into account a typical experimental situation. Figure 1(2) shows the object phase, background, and contrast maps used in the interferograms. The images in Fig. 1 were codified in rainbow color maps with levels of 8 bits and 200×200 in resolution, m=n=200 and evaluated in the range defined by x(π,π) and y(π,π) with Δx=Δy=2π/m.

 

Fig. 1 (1) Interferograms I0,I1,I2 with α1=π/3 rad, α2=2π/3rad; (a2) phase function ϕ rad; (b2) background a; and (c2) modulation light b

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Fig. 2 (1) Secondary fringe patterns, (a2) orange line depicts the fitted ellipse from the red dots extracted of (1) for i=100 as indicated with dashed red lines, and (b2) shows b100 and B100 for a comparison.

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The present method can extract the object phase by rows or columns, in this paper; this method is developed by rows from the secondary fringe patterns given in Eq. (2). For the i-th row, the ordinated pairs (pi,j,qi,j) are formed, named intensity points, where p=I0I1 and q=I1I2, and are fitted to an ellipse by applying the method of least squares. This is due to the secondary fringe patterns have the form of the parametric equations of an ellipse, where ϕi,j is an unknown phase function that plays the role of the phase parameter, since for each value of ϕi,j two values of intensity (pi,j,qi,j) are obtained, while 2bsinα1/2 and 2bsin((α2α1)/2) play the role of amplitudes, and α2/2 is the difference of phase between the two equations. As known, the amplitudes in the parametric equations of an ellipse are constant values, its parametric phase varies linearly, and the phase difference between the parametric signals is also a constant. An important difference with respect to Eq. (2) is that the amplitudes are not constants, since depend on the phase-steps and the modulation intensity varying point to point as a spatial function. For this reason, a graph of these points would not obey an ellipse but rather would be seen as disperse points [48]. Nevertheless, as known, in an experiment the superposition of two beams with homogeneous amplitudes can get bi to be a smooth enough function. Then, if bi varies smoothly the intensity points may seem an ellipse, and fitting them to this conic curve can cause bi to be approximated to a constant value. We demonstrate that this fact is a very important step in the success of our method, and under certain conditions of noise in the modulation light and phase-steps it could be more accurate than the methods that use the tangent function for phase extraction, such as the phase-shifting algorithms.

Let us consider Bi as a constant that substitutes to bi, and βi,1 and βi,2 that substitute the phase-steps α1 and α2 in Eq. (3), we can write in a simpler way

εi,j(ci,0,ci,1,ci,2)=pi,j2ci,0ci,1pi,jqi,jci,2qi,j2,
where the constants ci,k, k=0,1,2, depend on Bi, βi,1 and βi,2 of the form
ci,2=sin2(βi,1/2)sin2[(βi,2βi,1)/2],
ci,1=2sin(βi,1/2)cos(βi,2/2)sin[(βi,2βi,1)/2],
ci,0=4Bi2sin2(βi,1/2)sin2(βi,2/2).
The local error εi,j associated at the i-th row and j-th column in Eq. (5) depends on ci,k emerged by the approximation used mainly in bi. The form how this error is described in Eq. (5) can be interpreted as the approximation from measured data pi,j2 for some function of the form ci,0+ci,1pi,jqi,j+ci,2qi,j2. A global error for the i-th row taking in account all the values of j=1n to determine ci,k by the method of least squares can be described by
εi(ci,0,ci,1,ci,2)=j=1n(pi,j2ci,0ci,1pi,jqi,jci,2qi,j2)2,
where εi is the sum of the quadratic errors in the i-th row. The minimization of εi can be achieved by equaling to zero the first partial derivative of εi with respect to ci,k and by resolving the formed system of equations. One can get to prove
ci=(iTi)1iTpi,
where ()T and ()1 denote the transpose and inverse of a matrix, ci is a column vector of 3×1, i is a matrix of n×3, and pi is a column vector of n×1, given by
ci=(ci,0ci,1ci,2);i=(1pi,1qi,1qi,121pi,2qi,2qi,221pi,nqi,nqi,n2);pi=(pi,12pi,22pi,n2),
with ci computed from Eq. (8), Bi, βi,1 and βi,2 can be obtained from Eq. (6), having
tan(βi,12)=4ci,2ci,122+ci,1,
cos(βi,22)=ci,12ci,2,
Bi=24ci,2+ci,12ci,0ci,2(1+ci,1ci,2).
Then the parametric equations of the ellipse that represent the best fit to the experimental data in the i-th row can be written by choosing a linear function for the parameter φj=2πj/nπ

Pi,j=2Bisin(12βi,1)sin(φj+12βi,1),
Qi,j=2Bisin(12(βi,2βi,1))sin(φj+12(βi,1+βi,2)),

Figure 2(1) shows the secondary fringe patterns, and a dashed red line at the i=100 row indicates the set of data taken into account to form the intensity points (p100,j,q100,j). These data correspond to y=0 in Eq. (4). In Fig. 2(a2) these points are plotted with red dots, and are fitted by applying the procedure explained above, founding B100=3.32, β100,1=1.0422 rad and β100,2=2.0889 rad, which generate the ellipse plotted with an orange line. In this numerical example b100,j is approximated to a horizontal line of height B100=3.32, and in Fig. 2(b2) these curves are plotted together to see their comparison. The errors in the calculation of the phase-steps are of 0.0049=α1β100,1 rad and 0.0055=α2β100,2 rad.

In order to retrieve ϕi,j, we propose calculating the Euclidean distance from a point (pi,j,qi,j) to the fitted ellipse given in its parametric form in Eq. (11). Then, the distance from an experimental point (pi,j,qi,j) to any point of the adjusted ellipse is given in terms of φ,

li,j(φ)=(pi,jPi(φ))2+(qi,jQi(φ))2.
Because the Euclidean distance from a point to an ellipse is defined as the shortest distance, the value of the parameter φ over the adjusted ellipse that complies this condition can be found by computing the minimum of l(φ), that is, equaling to zero its first derivative, and resolving for φ. Thus, using Eq. (11) the following expression can be obtained,
0=si,jcos(φ+βi,12)+ui,jcos(φ+βi,1+βi,22)visin(2φ+βi,1)wisin(2φ+βi,1+βi,2),
where si,j=pi,jsin(βi,1/2), ui,j=qi,jsin[(βi,2βi,1)/2], vi=Bisin2(βi,1/2), and wi,j=Bisin2[(βi,2βi,1)/2] are coefficients known.

Resolving for φ in Eq. (13) is not a trivial task, in fact, as far as we know an analytic solution does not exist. In this paper, we find φ by resolving numerically an equations system formed with Eq. (13) and the trigonometric identity cos2φ+sin2φ=1, whose solution gives 4 roots, from which, the complexes are discriminated, while the real are substituted in Eq. (12) and the minimum value is chosen as a desired solution, named φi,j. Then this phase is assigned to the wrapped phase in this point, ϕwi,j=φi,j, and in this way the object phase at the point (i,j) is found. This procedure is repeated for j=1n and for i=1m to obtain the wrapped phase in each point in the map. We named this method the Euclidean Distance (ED) method.

Figures 3(a)-3(c) show the wrapped phase: theoretical ϕwT, computed by the ED method ϕwED, and by a typical formula that uses the tangent function such as in phase-shift algorithms [73] ϕwPSA, respectively. From Eq. (1) this formula is obtained

 

Fig. 3 Wrapped phase, measurement 342 of 1000 with Gaussian noise of σ1=π/20 in α1, (a) theoretical, computed by (b) the ED, and (c) the PSA method, and (d) data over the row i=100.

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tanϕ=I0(cosβ1cosβ2)I1(1cosβ2)+I2(1cosβ1)I0(sinβ1sinβ2)+I1sinβ2I2sinβ1.

Although actually, in this numerical simulation, these wrapped phases represent the 342-th measurement of r=1000. To show with better clarity the comparison of accuracy between the ED and PSA methods: Fig. 4(a) shows with red dots the 1000 measurements of ϕwED,100 for j=84, from which the mean is computed and depicted with a continuous red line having a value of ϕ¯wED,100=0.9878 rad, and in this point ϕwT,100=0.950772 rad is plotted with a continuous green line. The bias percent with respect to a phase period of 2π is of εED,100=0.0059023 computed by means of εED,100=|ϕ¯wED,100ϕwT,100|/2π, while the standard deviation is of σED,100=0.220225 rad, which is indicated with two dashed purple lines. The subscripts ED and PSA in ϕ, σ, and ε indicate that the ED and PSA methods were applied.

 

Fig. 4 Wrapped phase retrieved by the ED method for i=100 to estimate its accuracy (a) 1000 measurements of ϕwED,100 at j=84, (b) bias percent, (c) standard deviation, (d) average phase, (e) zoom to see the bias and standard deviation in each point j

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If this computation is repeated for each point j a variation curve for εED,100 is obtained as shown in Fig. 4(b), similarly for σED,100 as shown in Fig. 4(c), and similarly for ϕ¯wED,100 as shown in Fig. 4(d). To see in better detail, Fig. 4(e) shows a sample of Fig. 4(d) from j=80 to j=120 to illustrate the bias and standard deviation in the ED method that is indicated with the bar size in the graph. Observing Fig. 4(b) and 4(c), the bias percent and the standard deviation is different for each point j as waited. A way to quantify the accuracy in this row, the average of εED,100 and σED,100 are computed, giving ε¯ED,100=0.00883035 and σ¯ED,100=0.147919 rad, which are plotted with red and blue continuous lines, respectively.

If the calculations for the ED method above explained are repeated for the PSA method using the same interferograms, similar plots could be obtained, then for a comparison of both methods: Fig. 5(a1) shows with red color the ED method and with blue color the PSA method; εED,100 and εPSA,100 are shown with dots and are joined with dashed lines, while their means ε¯ED,100=0.00883035 and ε¯PSA,100=0.00988584 are depicted with continuous lines, and with the same color and lines notations Fig. 5(a2) shows σED,100, σPSA,100 and their means σ¯ED,100=0.147919 rad and σ¯PSA,100=0.175776 rad. As it can be noted, the ED method is better than the PSA method in both bias percent and standard deviation for this particular case where Gaussian noise was only introduced to α1 of σ1=π/20=0.157 rad in standard deviation while the other parameters were kept constant.

 

Fig. 5 Computation of the accuracy (bias and standard deviation) of the ED and PSA methods for i=100 in the cases: Gaussian noise of (a) σ1=π/20 rad in α1, (b) σ2=π/20 rad in α2, (c) σb=0.1%b0 in b constant (σbx) (d) without noise but with σbx=8. Each point in the graphs represents the average of 1000 measurements.

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We performed an additional analysis in order to show the properties of the ED and PSA methods for the cases where only α2, or b are introduced with Gaussian noise of σ2=π/20 rad or of σb=0.1%b0, respectively, and when only b is modulated spatially by σbx=8, as defined in Eq. (4c). The result of this analysis is shown in Figs. 5(b)-5(d) for εED,100 and σED,100 with the same color and line notations as Fig. 5(a). As it can be noted, the ED method is better than the PSA method in both bias percent and standard deviation for these particular cases of levels of Gaussian noise in α2, or b, or spatial variations of b. Nevertheless, in order to have better description of εED,100 and σED,100 for several variations of noise introduced in α1, or α2, or b and for several values of σbx, the analysis was repeated iteratively and the results of the means ε¯ED,100 and σ¯ED,100 were plotted in Fig. 6(a)-6(d), respectively. As noted in Fig. 6, when the parameters are introduced without noise and without variations, both methods obtain the same results. However when the level noise introduced is between zero and a threshold value, which is different for each case (see Figs. 6(a)-6(d)), the ED method is more accurate than the PSA method, and when this threshold noise is rebased, the situation is reversed the ED method is less accurate than the PSA method. Note that each point in the graphs of Fig. 6 represents the average of 200 data, such as is indicated for a case particular of noise in Fig. 5, and each one of these 200 points was obtained by averaging 1000 measurements of phase, as shown in Fig. 4 for j=84 in the numerical simulation.

 

Fig. 6 Accuracy estimation in the ED and PSA methods for i=100 and different Gaussian noise levels in: (a) α1, (b) α2, and b in the cases (c) σb with σbx, and (d) σbx. (1) bias percent, and (2) standard deviation. Each point depicted in the graphs represents the average of 200 measurements (see Fig. 5), and each one of these 200 represents the average of 1000 measurements (see Fig. 4(a)).

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3. Experimental results

The presented method also was applied to interferograms captured experimentally as shown in Fig. 7(a1), which are codified in grayscale levels of 8 bits and 221×241 in resolution, and have known phase-steps α1=1.35 rad and α2=2.7 rad for comparison with the phase-steps β1 and β2 computed with the presented method. The secondary fringe patterns are computed with Eq. (2) and are graphed in Figs. 7(b1)-7(b2). The red dots in Fig. 7(a2) are formed with the intensities extracted of the 110-th row indicated with red dashed lines in Figs. 7(b1)-7(b2). The fitted ellipse obtained by applying the method of least squares is depicted with orange lines in Fig. 7(a2), which gives β1=1.18 rad and β2=2.77, being comparable with the experimentally introduced phase-steps. Then by applying the ED method ϕwED is obtained as shown in Fig. 7(a3), and by applying the typical PSA method by using Eq. (14) ϕwPSI is obtained as shown in Fig. 7(b3).

 

Fig. 7 (a1) Experimental interferograms, (b1-b2) secondary fringe patterns, (a2) intensity points and fitted ellipse for i=110, (3) wrapped phase computed by the ED and PSA methods.

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4. Conclusions and remarks

We have presented a novel method for phase extraction named the Euclidean Distance Method that does not use the tangent function such as the PSA method. The ED method was based on the geometrical concept of Euclidean distance from a point to an ellipse, involving two important tasks: the first task was to fit an ellipse to disperse data by applying the method of least squares; the second task was to compute the Euclidean distance from a point to the fitted ellipse by solving numerically the system of equations formed by the trigonometric identity cos2φ+sin2φ=1 and Eq. (13). Although a detailed analysis to characterize the time of computation was not carried out, we realized it was of several minutes, which we think that might be reduced greatly if an analytic solution were implemented. On the other hand, we demonstrated that the ED method is more accurate than the PSA method for low levels of noise, and less accurate than the PSA method for high levels of noise. In particular, because was approximated to a constant, the ED method had little tolerance to its spatial variations, as observed in Fig. 6(d) the error increments as the function b has high variations, and furthermore the algorithm could become unstable. We think that this drawback may be avoided in two ways: first by dividing each row in intervals instead of taking all its elements, and second by approximating b to a polynomial instead of a constant value. These approaches can obtain possibly higher noise thresholds than the thresholds shown in Fig. 6. On the other hand, we think that, the noise levels where the ED method computed the object phase with better accurate than the PSA method, it is possible to be gotten experimentally, which is an important advantage in the applications where the accuracy is need. It is open the possibility to improve the ED method by fitting any conic curve or in general any parametric curve in two or more dimensions.

Acknowledgments

F. A. Lara-Cortes appreciates the scholarship from Consejo Nacional de Ciencia y Tecnología (México) under grant 242776. This work was partially supported by Consejo Nacional de Ciencia y Tecnología (México) under grant 166742 and by Vicerrectoría de Investigación y Estudios de Posgrado of Benemérita Universidad Autónoma de Puebla under grant MEFC-EXC15-G. Authors thank N. Keranen for her advice on wording.

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14. U. Rivera-Ortega, C. Meneses-Fabian, and G. Rodriguez-Zurita, “Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation,” Opt. Express 21(15), 17421–17434 (2013). [CrossRef]   [PubMed]  

15. U. Rivera-Ortega, C. Meneses-Fabian, G. Rodriguez-Zurita, and C. Robledo-Sanchez, “Phase-visibility modulating interferometry by binary non-quadrature amplitude modulation with neutral density filters,” Opt. Lasers Eng. 55, 226–231 (2014). [CrossRef]  

16. C. Meneses-Fabian, U. Rivera-Ortega, and G. Rodriguez-Zurita, “One-shot phase-visibility modulating interferometry by on-off non-quadrature amplitude modulation,” Opt. Lasers Eng. 58, 33–38 (2014). [CrossRef]  

17. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. 36(13), 2417–2419 (2011). [CrossRef]   [PubMed]  

18. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng. 50(7), 905–909 (2012). [CrossRef]  

19. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).

20. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef]   [PubMed]  

21. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef]   [PubMed]  

22. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef]   [PubMed]  

23. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express 17(11), 8789–8794 (2009). [CrossRef]   [PubMed]  

24. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef]   [PubMed]  

25. Y. Zhu and T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40(25), 4540–4546 (2001). [CrossRef]   [PubMed]  

26. J. van Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30(19), 2718–2729 (1991). [CrossRef]   [PubMed]  

27. B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8(2), 147–153 (1997). [CrossRef]  

28. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7(4), 537–541 (1990). [CrossRef]  

29. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36(10), 2070–2075 (1997). [CrossRef]   [PubMed]  

30. K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, “Noise and information in interferometric cross correlators,” Appl. Opt. 36(17), 3948–3958 (1997). [CrossRef]   [PubMed]  

31. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef]   [PubMed]  

32. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).

33. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991). [CrossRef]  

34. H. Kadono and S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16(12), 883–885 (1991). [CrossRef]   [PubMed]  

35. C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992). [CrossRef]  

36. C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994). [CrossRef]  

37. G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994). [CrossRef]   [PubMed]  

38. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997). [CrossRef]  

39. G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997). [CrossRef]  

40. A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999). [CrossRef]  

41. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]   [PubMed]  

42. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9(5), 236–253 (2001). [CrossRef]   [PubMed]  

43. B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).

44. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003). [CrossRef]   [PubMed]  

45. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef]   [PubMed]  

46. A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 29(12), 1381–1383 (2004). [CrossRef]   [PubMed]  

47. A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5), 475–490 (2005). [CrossRef]  

48. F. L. Bookstein, “Fitting conic sections to scattered data,” Comput. Graphics Image Process. 9(1), 56–71 (1979). [CrossRef]  

49. A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

50. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966). [CrossRef]  

51. Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000). [CrossRef]  

52. J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008). [CrossRef]  

53. E. Hack, “Measurement uncertainty of Carre´-type phase-stepping algorithms,” Opt. Lasers Eng. 50(8), 1023–1025 (2012). [CrossRef]  

54. P. Rastogi and E. Hack, Phase estimation in optical interferometry, (CRC Press 2014)

55. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef]   [PubMed]  

56. I. B. Kong and S. W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). [CrossRef]  

57. J. L. Marroquin, M. Servin, and R. Rodriguez Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23(4), 238–240 (1998). [CrossRef]   [PubMed]  

58. K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9(5), 236–253 (2001). [CrossRef]   [PubMed]  

59. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011). [CrossRef]   [PubMed]  

60. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011). [CrossRef]   [PubMed]  

61. J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013). [CrossRef]  

62. R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013). [CrossRef]  

63. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]  

64. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]  

65. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009). [CrossRef]   [PubMed]  

66. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef]   [PubMed]  

67. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef]   [PubMed]  

68. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express 18(24), 24405–24411 (2010). [CrossRef]   [PubMed]  

69. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010). [CrossRef]   [PubMed]  

70. K. Qian, S. H. Soon, and A. Asundi, “Phase-shifting windowed Fourier ridges for determination of phase derivatives,” Opt. Lett. 28(18), 1657–1659 (2003). [CrossRef]   [PubMed]  

71. B. Zielinski and K. Patorski, “Application of the S-transform to the phase-shift extraction in phase shifting interferometry,” Proc. SPIE 7746, 17th Slovak-Czech-Polish Optical conference on wave and quantum aspects of contemporary optics, 77460J (2010) [CrossRef]  

72. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef]   [PubMed]  

73. F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013). [CrossRef]   [PubMed]  

74. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014). [CrossRef]   [PubMed]  

References

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    [Crossref] [PubMed]
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  24. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983).
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  25. Y. Zhu and T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40(25), 4540–4546 (2001).
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  26. J. van Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30(19), 2718–2729 (1991).
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  27. B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8(2), 147–153 (1997).
    [Crossref]
  28. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7(4), 537–541 (1990).
    [Crossref]
  29. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36(10), 2070–2075 (1997).
    [Crossref] [PubMed]
  30. K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, “Noise and information in interferometric cross correlators,” Appl. Opt. 36(17), 3948–3958 (1997).
    [Crossref] [PubMed]
  31. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
    [Crossref] [PubMed]
  32. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
  33. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
    [Crossref]
  34. H. Kadono and S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16(12), 883–885 (1991).
    [Crossref] [PubMed]
  35. C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
    [Crossref]
  36. C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
    [Crossref]
  37. G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994).
    [Crossref] [PubMed]
  38. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
    [Crossref]
  39. G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
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  40. A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999).
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  41. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000).
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  42. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9(5), 236–253 (2001).
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  43. B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).
  44. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
    [Crossref] [PubMed]
  45. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
    [Crossref] [PubMed]
  46. A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 29(12), 1381–1383 (2004).
    [Crossref] [PubMed]
  47. A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5), 475–490 (2005).
    [Crossref]
  48. F. L. Bookstein, “Fitting conic sections to scattered data,” Comput. Graphics Image Process. 9(1), 56–71 (1979).
    [Crossref]
  49. A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).
  50. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
    [Crossref]
  51. Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000).
    [Crossref]
  52. J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008).
    [Crossref]
  53. E. Hack, “Measurement uncertainty of Carre´-type phase-stepping algorithms,” Opt. Lasers Eng. 50(8), 1023–1025 (2012).
    [Crossref]
  54. P. Rastogi and E. Hack, Phase estimation in optical interferometry, (CRC Press 2014)
  55. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [Crossref] [PubMed]
  56. I. B. Kong and S. W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
    [Crossref]
  57. J. L. Marroquin, M. Servin, and R. Rodriguez Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23(4), 238–240 (1998).
    [Crossref] [PubMed]
  58. K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9(5), 236–253 (2001).
    [Crossref] [PubMed]
  59. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
    [Crossref] [PubMed]
  60. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
    [Crossref] [PubMed]
  61. J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
    [Crossref]
  62. R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
    [Crossref]
  63. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [Crossref]
  64. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
    [Crossref]
  65. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
    [Crossref] [PubMed]
  66. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [Crossref] [PubMed]
  67. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009).
    [Crossref] [PubMed]
  68. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express 18(24), 24405–24411 (2010).
    [Crossref] [PubMed]
  69. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010).
    [Crossref] [PubMed]
  70. K. Qian, S. H. Soon, and A. Asundi, “Phase-shifting windowed Fourier ridges for determination of phase derivatives,” Opt. Lett. 28(18), 1657–1659 (2003).
    [Crossref] [PubMed]
  71. B. Zielinski and K. Patorski, “Application of the S-transform to the phase-shift extraction in phase shifting interferometry,” Proc. SPIE 7746, 17th Slovak-Czech-Polish Optical conference on wave and quantum aspects of contemporary optics, 77460J (2010)
    [Crossref]
  72. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref] [PubMed]
  73. F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013).
    [Crossref] [PubMed]
  74. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
    [Crossref] [PubMed]

2014 (4)

U. Rivera-Ortega, C. Meneses-Fabian, G. Rodriguez-Zurita, and C. Robledo-Sanchez, “Phase-visibility modulating interferometry by binary non-quadrature amplitude modulation with neutral density filters,” Opt. Lasers Eng. 55, 226–231 (2014).
[Crossref]

C. Meneses-Fabian, U. Rivera-Ortega, and G. Rodriguez-Zurita, “One-shot phase-visibility modulating interferometry by on-off non-quadrature amplitude modulation,” Opt. Lasers Eng. 58, 33–38 (2014).
[Crossref]

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
[Crossref] [PubMed]

2013 (7)

F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013).
[Crossref] [PubMed]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).

C. Meneses-Fabian, R. Kantun-Montiel, G.-P. Lemus-Alonso, and U. Rivera-Ortega, “Double aperture common-path phase-shifting interferometry by translating a ruling at the input plane,” Opt. Lett. 38(11), 1850–1852 (2013).
[Crossref] [PubMed]

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21(14), 17228–17233 (2013).
[Crossref] [PubMed]

U. Rivera-Ortega, C. Meneses-Fabian, and G. Rodriguez-Zurita, “Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation,” Opt. Express 21(15), 17421–17434 (2013).
[Crossref] [PubMed]

2012 (2)

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng. 50(7), 905–909 (2012).
[Crossref]

E. Hack, “Measurement uncertainty of Carre´-type phase-stepping algorithms,” Opt. Lasers Eng. 50(8), 1023–1025 (2012).
[Crossref]

2011 (3)

2010 (2)

2009 (5)

2008 (2)

T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt. 47(21), 3784–3788 (2008).
[Crossref] [PubMed]

J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008).
[Crossref]

2005 (1)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3–5), 475–490 (2005).
[Crossref]

2004 (3)

2003 (2)

2001 (3)

2000 (2)

X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000).
[Crossref] [PubMed]

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000).
[Crossref]

1999 (1)

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999).
[Crossref]

1998 (2)

C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Opt. Laser Technol. 30(8), 545–550 (1998).
[Crossref]

J. L. Marroquin, M. Servin, and R. Rodriguez Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23(4), 238–240 (1998).
[Crossref] [PubMed]

1997 (5)

1996 (1)

1995 (3)

1994 (2)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994).
[Crossref] [PubMed]

1992 (2)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
[Crossref]

1991 (3)

1990 (2)

1987 (1)

1985 (1)

1984 (1)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).

1983 (1)

1982 (1)

1979 (1)

F. L. Bookstein, “Fitting conic sections to scattered data,” Comput. Graphics Image Process. 9(1), 56–71 (1979).
[Crossref]

1974 (1)

1967 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Albertazzi, A.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Arévalo Aguilar, L. M.

Arevalo-Aguilar, L.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Asundi, A.

Basinger, S. A.

Belenguer, T.

Benedet, M. E.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Bookstein, F. L.

F. L. Bookstein, “Fitting conic sections to scattered data,” Comput. Graphics Image Process. 9(1), 56–71 (1979).
[Crossref]

Brady, D. J.

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

Burow, R.

Cai, L. Z.

Carazo, J. M.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Chen, X.

Cheng, Y.-Y.

Creath, K.

Cywiak, M.

Doblado, D. M.

Dragostinov, T.

G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
[Crossref]

Eiju, T.

Elssner, K. E.

Estrada, J. C.

Fangjun, S.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000).
[Crossref]

Fantin, A. V.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Farrant, D. I.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Fermin-Granados, W.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Fisher, R. B.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999).
[Crossref]

Fitzgibbon, A.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999).
[Crossref]

Frankena, H. J.

Freischlad, K.

Gallagher, J. E.

Gemma, T.

Gramaglia, M.

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).

Grzanna, J.

Gu, H.

Guerrero-Sanchez, F.

Guerrero-Sánchez, F.

Gutiérrez-García, J. C.

Gutiérrez-García, T. A.

Hack, E.

E. Hack, “Measurement uncertainty of Carre´-type phase-stepping algorithms,” Opt. Lasers Eng. 50(8), 1023–1025 (2012).
[Crossref]

Han, B.

Han, G.-S.

Hariharan, P.

Hernández, D. M.

Herriott, D. R.

Hibino, K.

Hill, K. B.

Hioki, R.

Ixba-Santos, V.

Jin, G.

Juarez-Salazar, R.

Kadono, H.

Kantun-Montiel, R.

Kemao, Q.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000).
[Crossref]

Kiire, T.

Kim, S. W.

I. B. Kong and S. W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kim, S.-W.

Koliopoulos, C. L.

Kong, I. B.

I. B. Kong and S. W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Lai, G.

Larkin, K.

Larkin, K. G.

Lemus-Alonso, G.-P.

Liu, Q.

Macías-Preza, J. M.

Maia, A. F.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Malacara-Doblado, D.

Marroquin, J. L.

Meneses-Fabian, C.

U. Rivera-Ortega, C. Meneses-Fabian, G. Rodriguez-Zurita, and C. Robledo-Sanchez, “Phase-visibility modulating interferometry by binary non-quadrature amplitude modulation with neutral density filters,” Opt. Lasers Eng. 55, 226–231 (2014).
[Crossref]

C. Meneses-Fabian, U. Rivera-Ortega, and G. Rodriguez-Zurita, “One-shot phase-visibility modulating interferometry by on-off non-quadrature amplitude modulation,” Opt. Lasers Eng. 58, 33–38 (2014).
[Crossref]

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21(14), 17228–17233 (2013).
[Crossref] [PubMed]

U. Rivera-Ortega, C. Meneses-Fabian, and G. Rodriguez-Zurita, “Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation,” Opt. Express 21(15), 17421–17434 (2013).
[Crossref] [PubMed]

C. Meneses-Fabian, R. Kantun-Montiel, G.-P. Lemus-Alonso, and U. Rivera-Ortega, “Double aperture common-path phase-shifting interferometry by translating a ruling at the input plane,” Opt. Lett. 38(11), 1850–1852 (2013).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng. 50(7), 905–909 (2012).
[Crossref]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. 36(13), 2417–2419 (2011).
[Crossref] [PubMed]

Merkel, K.

Miks, A.

J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008).
[Crossref]

Morgan, C. J.

Mosiño, J. F.

Nakadate, S.

Novak, P.

J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008).
[Crossref]

Novák, J.

J. Novák, P. Novak, and A. Miks, “Multi-step Phase-shifting Algorithms Insensitive to Linear Phase Shift Errors,” Opt. Commun. 281(21), 5302–5309 (2008).
[Crossref]

Oreb, B. F.

Patil, A.

Pilu, M.

A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qian, K.

Quan, C.

C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Opt. Laser Technol. 30(8), 545–550 (1998).
[Crossref]

Quiroga, J. A.

Rangel-Huerta, A.

Raphael, B.

B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).

A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 29(12), 1381–1383 (2004).
[Crossref] [PubMed]

Rastogi, P.

Rivera-Ortega, U.

C. Meneses-Fabian, U. Rivera-Ortega, and G. Rodriguez-Zurita, “One-shot phase-visibility modulating interferometry by on-off non-quadrature amplitude modulation,” Opt. Lasers Eng. 58, 33–38 (2014).
[Crossref]

U. Rivera-Ortega, C. Meneses-Fabian, G. Rodriguez-Zurita, and C. Robledo-Sanchez, “Phase-visibility modulating interferometry by binary non-quadrature amplitude modulation with neutral density filters,” Opt. Lasers Eng. 55, 226–231 (2014).
[Crossref]

C. Meneses-Fabian, R. Kantun-Montiel, G.-P. Lemus-Alonso, and U. Rivera-Ortega, “Double aperture common-path phase-shifting interferometry by translating a ruling at the input plane,” Opt. Lett. 38(11), 1850–1852 (2013).
[Crossref] [PubMed]

U. Rivera-Ortega, C. Meneses-Fabian, and G. Rodriguez-Zurita, “Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation,” Opt. Express 21(15), 17421–17434 (2013).
[Crossref] [PubMed]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng. 50(7), 905–909 (2012).
[Crossref]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. 36(13), 2417–2419 (2011).
[Crossref] [PubMed]

Robledo-Sanchez, C.

Robledo-Sánchez, C.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, W. Fermin-Granados, and L. Arevalo-Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least square method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Rodriguez Vera, R.

Rodriguez-Zurita, G.

U. Rivera-Ortega, C. Meneses-Fabian, G. Rodriguez-Zurita, and C. Robledo-Sanchez, “Phase-visibility modulating interferometry by binary non-quadrature amplitude modulation with neutral density filters,” Opt. Lasers Eng. 55, 226–231 (2014).
[Crossref]

C. Meneses-Fabian, U. Rivera-Ortega, and G. Rodriguez-Zurita, “One-shot phase-visibility modulating interferometry by on-off non-quadrature amplitude modulation,” Opt. Lasers Eng. 58, 33–38 (2014).
[Crossref]

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21(14), 17228–17233 (2013).
[Crossref] [PubMed]

U. Rivera-Ortega, C. Meneses-Fabian, and G. Rodriguez-Zurita, “Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation,” Opt. Express 21(15), 17421–17434 (2013).
[Crossref] [PubMed]

Rosenfeld, D. P.

Schmit, J.

Schwider, J.

Servin, M.

Shang, H.

C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Opt. Laser Technol. 30(8), 545–550 (1998).
[Crossref]

Shibuya, M.

Smith, I. F. C.

B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).

Smorenburg, C.

Soon, S. H.

Sorzano, C. O. S.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Spolaczyk, R.

Stack, R. A.

Stoilov, G.

G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
[Crossref]

Surrel, Y.

Susuki, T.

Tan, Q.

Tay, C.

C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Opt. Laser Technol. 30(8), 545–550 (1998).
[Crossref]

Téllez-Quiñones, A.

Toyooka, S.

van Wingerden, J.

Vargas, J.

Viotti, M.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Wang, Z.

White, A. D.

Willemann, D. P.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized N-dimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Wyant, J. C.

Xiaoping, W.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11(8), 1220–1223 (2000).
[Crossref]

Yang, X. L.

Yatagai, T.

Yeazell, J. A.

Zeng, F.

Zhao, B.

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36(10), 2070–2075 (1997).
[Crossref] [PubMed]

B. Zhao, “A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,” Meas. Sci. Technol. 8(2), 147–153 (1997).
[Crossref]

Zhu, Y.

Appl. Math. Comput. (1)

B. Raphael and I. F. C. Smith, “A direct stochastic algorithm for global search,” Appl. Math. Comput. 146(2–3), 729–758 (2013).

Appl. Opt. (14)

G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994).
[Crossref] [PubMed]

X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000).
[Crossref] [PubMed]

A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010).
[Crossref] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[Crossref] [PubMed]

Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985).
[Crossref] [PubMed]

T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt. 47(21), 3784–3788 (2008).
[Crossref] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 (1) Interferograms I 0 , I 1 , I 2 with α 1 =π/3 rad, α 2 = 2π /3 rad; (a2) phase function ϕ rad; (b2) background a ; and (c2) modulation light b
Fig. 2
Fig. 2 (1) Secondary fringe patterns, (a2) orange line depicts the fitted ellipse from the red dots extracted of (1) for i=100 as indicated with dashed red lines, and (b2) shows b 100 and B 100 for a comparison.
Fig. 3
Fig. 3 Wrapped phase, measurement 342 of 1000 with Gaussian noise of σ 1 =π/ 20 in α 1 , (a) theoretical, computed by (b) the ED, and (c) the PSA method, and (d) data over the row i=100 .
Fig. 4
Fig. 4 Wrapped phase retrieved by the ED method for i=100 to estimate its accuracy (a) 1000 measurements of ϕ wED,100 at j=84 , (b) bias percent, (c) standard deviation, (d) average phase, (e) zoom to see the bias and standard deviation in each point j
Fig. 5
Fig. 5 Computation of the accuracy (bias and standard deviation) of the ED and PSA methods for i=100 in the cases: Gaussian noise of (a) σ 1 =π/ 20 rad in α 1 , (b) σ 2 =π/ 20 rad in α 2 , (c) σ b =0.1% b 0 in b constant ( σ bx ) (d) without noise but with σ bx =8 . Each point in the graphs represents the average of 1000 measurements.
Fig. 6
Fig. 6 Accuracy estimation in the ED and PSA methods for i=100 and different Gaussian noise levels in: (a) α 1 , (b) α 2 , and b in the cases (c) σ b with σ bx , and (d) σ bx . (1) bias percent, and (2) standard deviation. Each point depicted in the graphs represents the average of 200 measurements (see Fig. 5), and each one of these 200 represents the average of 1000 measurements (see Fig. 4(a)).
Fig. 7
Fig. 7 (a1) Experimental interferograms, (b1-b2) secondary fringe patterns, (a2) intensity points and fitted ellipse for i=110 , (3) wrapped phase computed by the ED and PSA methods.

Equations (24)

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I 0 ( x,y )=a( x,y )+b( x,y )cosϕ( x,y ),
I 1 ( x,y )=a( x,y )+b( x,y )cos[ ϕ( x,y )+ α 1 ],
I 2 ( x,y )=a( x,y )+b( x,y )cos[ ϕ( x,y )+ α 2 ],
I 0 I 1 =2bsin 1 2 α 1 sin( ϕ+ 1 2 α 1 ),
I 1 I 2 =2bsin 1 2 ( α 2 α 1 )sin( ϕ+ 1 2 ( α 1 + α 2 ) ),
sin 2 ( α 2 2 )= ( I 0 I 1 ) 2 4 b 2 sin 2 ( α 1 /2 ) + ( I 1 I 2 ) 2 4 b 2 sin 2 ( ( α 2 α 1 ) /2 ) ( I 0 I 1 )( I 1 I 2 )cos( α 2 /2 ) 2 b 2 sin( α 1 /2 )sin( ( α 2 α 1 ) /2 ) .
ϕ( x,y )= 10 3 ( x 2 sinx+ y 2 ),
a( x,y )= a 0 exp( ( x x a ) 2 σ ax 2 ( y y a ) 2 σ ay 2 ),
b( x,y )= b 0 exp( ( x x b ) 2 σ bx 2 ( y y b ) 2 σ by 2 ),
ε i,j ( c i,0 , c i,1 , c i,2 )= p i,j 2 c i,0 c i,1 p i,j q i,j c i,2 q i,j 2 ,
c i,2 = sin 2 ( β i,1 /2 ) sin 2 [ ( β i,2 β i,1 ) /2 ] ,
c i,1 =2 sin( β i,1 /2 )cos( β i,2 /2 ) sin[ ( β i,2 β i,1 ) /2 ] ,
c i,0 =4 B i 2 sin 2 ( β i,1 /2 ) sin 2 ( β i,2 /2 ).
ε i ( c i,0 , c i,1 , c i,2 )= j=1 n ( p i,j 2 c i,0 c i,1 p i,j q i,j c i,2 q i,j 2 ) 2 ,
c i = ( i T i ) 1 i T p i ,
c i =( c i,0 c i,1 c i,2 ); i =( 1 p i,1 q i,1 q i,1 2 1 p i,2 q i,2 q i,2 2 1 p i,n q i,n q i,n 2 ); p i =( p i,1 2 p i,2 2 p i,n 2 ),
tan( β i,1 2 )= 4 c i,2 c i,1 2 2+ c i,1 ,
cos( β i,2 2 )= c i,1 2 c i,2 ,
B i = 2 4 c i,2 + c i,1 2 c i,0 c i,2 ( 1+ c i,1 c i,2 ) .
P i,j =2 B i sin( 1 2 β i,1 )sin( φ j + 1 2 β i,1 ),
Q i,j =2 B i sin( 1 2 ( β i,2 β i,1 ) )sin( φ j + 1 2 ( β i,1 + β i,2 ) ),
l i,j ( φ )= ( p i,j P i ( φ ) ) 2 + ( q i,j Q i ( φ ) ) 2 .
0= s i,j cos( φ+ β i,1 2 )+ u i,j cos( φ+ β i,1 + β i,2 2 ) v i sin( 2φ+ β i,1 ) w i sin( 2φ+ β i,1 + β i,2 ),
tanϕ= I 0 ( cos β 1 cos β 2 ) I 1 ( 1cos β 2 )+ I 2 ( 1cos β 1 ) I 0 ( sin β 1 sin β 2 )+ I 1 sin β 2 I 2 sin β 1 .

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