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
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  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 , gratings [10, 11], spatial light modulators , by the lateral displacement of the light source , and recently with amplitude filters [14–16] by applying the theory of amplitude modulation out-of-phase in quadrature (QAM)  and non-quadrature NQAM . 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 , Hariharan , Schmit and Creath , 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 , and in 1984 Grievenkamp  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  were presented based in concepts such as Fourier transform , statistical [34, 44, 45], elliptical curve fitting [35, 36, 48, 49], iterative , spatio-temporal , and optimization . 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  in 1966, and an uncertainty analysis so as a wide state of art was presented by Kemao et al , Novak and Miks , Hack , Rastogi and Hack , 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  in order to remove the drawbacks that occur when there is less than a fringe. Juarez-Salazar et al  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  presented by first time this idea, followed by Larkin and Oreb , Schmit and Creath , and finally this method was generalized by Servin et al [65, 66], Mosiño et al [67, 68], and Tellez et al . 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 , S-transform , and Z-transform used by Surrel in 1996  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  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  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 .
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 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  used by Farrel and Player in self-calibrating GPSI [35, 36] and by Albertazzi et al . 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 .
2. Theory and simulation
Let’s consider three interferograms of the form,Eq. (1) can be eliminated to obtain two secondary fringe patternsEq. (1) become matrixes of , with and .
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 rad, and rad with a Gaussian noise only for of rad in standard deviation, and the phase object, background, and contrast of the formFigure 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 in resolution, and evaluated in the range defined by and with .
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 -th row, the ordinated pairs are formed, named intensity points, where and , 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 is an unknown phase function that plays the role of the phase parameter, since for each value of two values of intensity are obtained, while and play the role of amplitudes, and 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 . Nevertheless, as known, in an experiment the superposition of two beams with homogeneous amplitudes can get to be a smooth enough function. Then, if varies smoothly the intensity points may seem an ellipse, and fitting them to this conic curve can cause 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 as a constant that substitutes to , and and that substitute the phase-steps and in Eq. (3), we can write in a simpler wayEq. (5) depends on emerged by the approximation used mainly in . The form how this error is described in Eq. (5) can be interpreted as the approximation from measured data for some function of the form . A global error for the -th row taking in account all the values of to determine by the method of least squares can be described byEq. (8), , and can be obtained from Eq. (6), having
Figure 2(1) shows the secondary fringe patterns, and a dashed red line at the row indicates the set of data taken into account to form the intensity points . These data correspond to in Eq. (4). In Fig. 2(a2) these points are plotted with red dots, and are fitted by applying the procedure explained above, founding , rad and rad, which generate the ellipse plotted with an orange line. In this numerical example is approximated to a horizontal line of height , 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 rad and rad.
In order to retrieve , we propose calculating the Euclidean distance from a point to the fitted ellipse given in its parametric form in Eq. (11). Then, the distance from an experimental point to any point of the adjusted ellipse is given in terms of ,Eq. (11) the following expression can be obtained,
Resolving for 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 by resolving numerically an equations system formed with Eq. (13) and the trigonometric identity , 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 . Then this phase is assigned to the wrapped phase in this point, , and in this way the object phase at the point is found. This procedure is repeated for and for 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 , computed by the ED method , and by a typical formula that uses the tangent function such as in phase-shift algorithms  , respectively. From Eq. (1) this formula is obtained
Although actually, in this numerical simulation, these wrapped phases represent the 342-th measurement of . 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 for , from which the mean is computed and depicted with a continuous red line having a value of rad, and in this point rad is plotted with a continuous green line. The bias percent with respect to a phase period of is of computed by means of , while the standard deviation is of rad, which is indicated with two dashed purple lines. The subscripts ED and PSA in , , and indicate that the ED and PSA methods were applied.
If this computation is repeated for each point a variation curve for is obtained as shown in Fig. 4(b), similarly for as shown in Fig. 4(c), and similarly for as shown in Fig. 4(d). To see in better detail, Fig. 4(e) shows a sample of Fig. 4(d) from to 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 as waited. A way to quantify the accuracy in this row, the average of and are computed, giving and 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; and are shown with dots and are joined with dashed lines, while their means and are depicted with continuous lines, and with the same color and lines notations Fig. 5(a2) shows , and their means rad and 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 of rad in standard deviation while the other parameters were kept constant.
We performed an additional analysis in order to show the properties of the ED and PSA methods for the cases where only , or b are introduced with Gaussian noise of rad or of , respectively, and when only b is modulated spatially by , as defined in Eq. (4c). The result of this analysis is shown in Figs. 5(b)-5(d) for and 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 , or , or spatial variations of . Nevertheless, in order to have better description of and for several variations of noise introduced in , or , or and for several values of , the analysis was repeated iteratively and the results of the means and 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 in the numerical simulation.
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 in resolution, and have known phase-steps rad and rad for comparison with the phase-steps and 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 rad and , being comparable with the experimentally introduced phase-steps. Then by applying the ED method is obtained as shown in Fig. 7(a3), and by applying the typical PSA method by using Eq. (14) is obtained as shown in Fig. 7(b3).
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 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 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.
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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