Phase-shifting is one of the most useful methods of phase recovery in digital interferometry in the estimation of small displacements, but miscalibration errors of the phase shifters are very common. In practice, the main problem associated with such errors is related to the response of the phase shifter devices, since they are dependent on mechanical and/or electrical parts. In this work, a novel technique to detect and measure calibration errors in phase-shifting interferometry, when an unexpected phase shift arises, is proposed. The described method uses the Radon transform, first as an automatic-calibrating technique, and then as a profile measuring procedure when analyzing a specific zone of an interferogram. After, once maximum and minimum value parameters have been registered, these can be used to measure calibration errors. Synthetic and real interferograms are included in the testing, which has thrown good approximations for both cases, notwithstanding the interferogram fringe distribution or its phase-shifting steps. Tests have shown that this algorithm is able to measure the deviations of the steps in phase-shifting interferometry. The developed algorithm can also be used as an alternative in the calibration of phase shifter devices.
© 2017 Optical Society of America
Phase-Shifting Interferometry (PSI) is widely used in phase extraction [1,2], due to its fastness and accuracy [3–6]. PSI is a non-invasive technique that can be used to measure displacement of surfaces associated with physical variables, such as temperature, vibrations; and also for failure detection in reflective profiles, such as biological cells, optical elements, etc [7,8].
Phase-Shifting Algorithms (PSA) published in recent years, generally estimate wrapped phase distribution from phase displaced interferograms with known or unknown step lengths. PSA methods with supposedly known steps can carry errors in the measured phase due to mechanical vibrations, air turbulences, calibration errors and other disturbances [1,9].
Determination of background intensity and modulation amplitude applying a temporal method is demonstrated in , after this process, extraction of the phase is possible by applying the arccosine operation; however, this method needs time to process background intensity and modulation amplitude of the interferogram. A more recent work  proposes the analysis of independent components that can also be used to extract the phase and the background with the arctangent function. This method decomposes the interferograms with unknown steps by an independent component analysis technique, but in this case, the process is iterative.
In  a two-dimensional continuous wavelet transform is used to obtain the phase of a surface applying a two-step algorithm with arbitrary length phase-shifts; however, this technique requires the filtering of the background in the interferogram. Other techniques use the combination of principal components and the least-square methodology in order to extract random phase-shifting interferograms , or the spatiotemporal hybrid method , but this last process needs a set of frames to reduce possible phase errors. Artificial intelligence has been used to extract the phase utilizing PSI, and specific techniques such as neural networks ; nonetheless, the training of the network demands computational time.
Different techniques use mathematical analysis to obtain expressions with the purpose of estimating the true phase. Mosiño et al.  proposed an analytical quadrature filter and Ayubi et al.  found a general expression for PSA considering random steps. This last work contemplates error analysis and minimization of noise ; nevertheless, the last algorithms do not consider miscalibration errors of the introduced phase steps for the phase extraction. In order to alleviate drawback, this work advances a novel method that detects errors when using four-step algorithms with π/2 step length.
One disadvantage of iterative algorithms [10,14] or those that need to estimate the background , is the computational time required to achieve good results. Other methods only work properly with the real values of the shift displacements [9,13] or with a high number of interferograms that help to reduce errors in the presence of miscalibration  or mechanical perturbations. In this work, the use of the Radon transform is proposed in order to estimate any error in phase step values, due to the problems mentioned before. The use of the Radon transform is fast and robust against noise, which allows its use with any number of interferograms according to a chosen PSA procedure. If any error in the phase shift is detected, one can chose between acquiring new error-free interferograms to use an N-bucked family [15–19] or to use algorithms for any phase-shifting values [9,13].
The method here presented, Phase-Shifting Interferometry with Radon Transform (PSI-RT) can mean a solution for both, open fringe and closed fringe interference patterns; however, it is important to highlight that the analysis of closed-fringe patterns implies the analysis of small regions with nearly open fringes inside the interferogram field. In the following section, the principle of the PSI is explained as well as the concepts of the Radon transform used in the proposed PSI-RT method; then, the simulation and the experimental results are presented, and finally the conclusions are declared.
This section explains the theory of phase shifting interferometry, the Radon transform and the method that has been developed to detect errors when introducing phase shifts of π/2 radians.
2.1 Phase-shifting interferometry
Several physical variables can be measured by using optical phase recovering techniques of an object under test; phase-shifting interferometry is one of the most popular methods used to find the phase distribution (wrapped phase format), due to its accuracy [20,21]. PSI records in a digital format a succession of interferograms introducing step phase changes in one beam of the interferometer . Intensity images of interferograms, in the spatial coordinates (x, y) can be described through the next equation :22]. The number of k steps depends on the algorithm. The most popular algorithm utilizes four steps, but there are other algorithms that can consist of 2, 3, 5, 7 steps, or more . In case of a linear phase error εk, which may occur due to a different increment in the voltage applied to the piezoelectric , the Eq. (1) can be rewritten as:
In order to obtain the phase with any PSA, it is necessary to follow a number of general steps to fulfill the phase extraction task: first, it is necessary to record the shifted k interferograms; then, to filter the information in the interferograms (pre-processing stage); next, to apply basic operations between interferogram images (addition and/or subtraction, and division); after that, to use the inverse tangent function, to obtain the wrapped phase map. After these PSA steps, there are some unwrapped phase algorithms that can be used to reconstruct the phase that has been measured, and make it into a continuous form . In general, all the PSA methods that had been published can be expressed as a quotient of combinations of all registered interferograms , i.e.Eq. (3), with only four interferograms (k = 4), b = c = ± 1, and α = π/2, and applying the inverse tangent function, one of the most popular PSA is obtained, to then find the wrapped phase φw  as follows:
2.2 Radon transform
The Radon transform has many different applications in the most diverse fields of physics and engineering. It has been used to determine image profiles , and it acts as a projection of the gray levels of images at an angle θ with respect to the abscissa axis, as shown in Fig. 1; this means that, the Radon transform can be seen as a distribution of pixels of an image that has been scanned in a direction along the position of the line ρ. This transform has been utilized as a profilometry tool in diverse fields, such as astronomy, crystallography, electron microscopy, geophysics, optics, and materials science, among others, with the purpose of finding patterns . The following equation describes the Radon transform :
2.3 Error detection in phase-shifting
The novel method PSI-RT was devised to find an angle of projection for the Radon transform, with the purpose of obtaining a single sinusoidal waveform of the fringe patterns (interferograms) characterized by a similar amplitude and frequency. With this waveform, it is possible to calculate the deviations in phase-shifting steps. If an interferogram is studied directly with a single row or a single column, an estimation of the deviations cannot be carried out, because the graph obtained may show a nonsinusoidal form. In concordance with what has been said, the need to locate an angle arises; this angle can be calculated by applying the Radon transform.
In order to find the adequate angle θA of the Radon transform, it is necessary to follow the procedure here outlined: first, four images of intensity with α = π/2 [Eq. (1)] are recorded; then, the Radon transform is applied to one of the interferograms (i.e. I1) with all the range of θ values, this is θ = [0°-180°]; next, the Radon's image obtained by the 180 projections is processed using a thresholding procedure; this process consists of finding the dominant presence of white level in a region of the resulting image. At this angle, the Radon transform gives information of those regions of an interferogram where it behaves in a sinusoidal form. This process can be shown as a flow diagram in Fig. 2(a). Figure 2(b) shows an example for θA selection: an image of the Radon transform from 0°-180°, then the thresholding and the searching process are utilized; finally, a sinusoidal profile is obtained for θA.
Once the value for θA is found through the use of the thresholding procedure, the next step is to rotate the four interferograms at the angle that has been previously calculated. In order to simplify the analysis of the interferograms, the images are cropped to obtain rectangular regions that contains nearly straight fringes, then the four interferograms are transformed through the use of Radon at the 0° angle; and the results of the transforms are normalized from 0 to 1. Additionally, a half sinusoidal cycle (0, π) is extracted from each of the Radon transforms. The purpose of this is to find the errors in the phase-shifting steps, they are identified by means of the behavior of the four waveforms. The normalized Radon transform, obtained from the four interferograms, is expressed in Eq. (6),Fig. 3. If the four interferograms do not have a deviation εk, the behavior of the waveforms is showed in Fig. 3(a); the opposite case is presented in the Fig. 3(b), where an error ε3 can be observed in .
The errors εk are detected through the deviations found in five points of ρ around [-π, π] for all ; these are:
- 1. ρ = 0, when the waveform has its maximum value (PmaxI1) and the waveform has its minimum value (PminI3);
- 2. ρ = π/2, the waveform has its maximum value (PmaxI4) and has its minimum value (PminI2);
- 3. when ρ = -π/2 it is the opposite of point 2, has its maximum value (PmaxI2) and has its minimum value (PminI4);
- 4. finally, the last two points with similar answers are ρ = π and ρ = -π, take place where has its maximum value (PmaxI3) and has its minimum value (PminI1).
Once the points have been found, it is possible to determine whether there is an error in the phase steps by using the following equations:
The error ε1 is not considered, due to the fact that the first interferogram works as a reference point; if the values of ε2, ε3 and ε4 are identical among themselves, but different from zero, then it means that no miscalibration has occurred.
Summarizing, the complete algorithm PSI-RT error detector here presented is graphically explained in the flow diagram of Fig. 4, and it contains the following steps:
- 1. Interferogram capture;
- 2. θA selection (automatic calibration);
- 3. rotation and cropping process;
- 4. normalization of the Radon transform;
- 5. selection of the parameters and,
- 6. estimation of the errors in phase-shifting steps.
The capturing process consists of acquiring k interferograms Ik(x,y) with π/2 phase-shifts among images (in this work k = 4), through of a CCD (charge-coupled device) and stored in a computer; the second step is θA selection, it has previously been explained. The next step of the method is to rotate Ik(x,y) at an angle θA and to crop it in a rectangular region; the objective of turning the interferograms is to obtain an image with horizontal fringes, in order to apply the Radon transform at 0°. The criteria for the image cropping process was the selection of the maximum square region inside the rotated Ik(x,y), as it is observed in the image that was added to the flow diagram of Fig. 4.
Once all interferograms have been processed, the Radon transforms are obtained at θ = 0° for the cropped interferograms; then, they are normalized; subsequently, a half cycle of the normalized Radon transforms are selected. The sinusoidal waveforms are analyzed, in order to find the maximum (Pmax) and minimum (Pmin) points for each. Finally, errors (ε1, ε2, ε3 and ε4) are obtained with Eqs. (7). The stated process allows the finding of deviations in the phase-shift steps. A factor that influences the accuracy of the PSI-RT method is the resolution of the interferogram image. If the interferogram resolution is reduced, then the calculus of the deviations may present inaccuracies due to what was just mentioned before.
A computer simulation applied to three different sets of Synthetic Interferograms (SI), is used to verify the effectiveness of PSI-RT method that has been proposed, based on the Radon transform as a profile measurement, in order to retrieve the phase errors steps in the interferograms. In order to test the robustness of this method, different characteristics of their features are included in these SI: an interferogram with horizontal fringes and static period as shown in Fig. 5(a), whose equation is ; a second interferogram with curved fringes and varying period as seen in Fig. 5(b); and a third synthetic interferogram was programed , where tilt fringes and varying period is observed in Fig. 5(c). With A = B = 127.5 gray levels, C1 = 0.005, C2 = 0.00001, C3 = 0.00003 and the interferograms sizes of all the SI are 1024x1024 pixels.
The θA selection process explained in the previous section was used with the three SI, in order to obtain the three respective angles. In a range of 0 to 180°, the Radon transform is applied at the I1 of each SI, as shown in Fig. 6(a). After the thresholding process is utilized to obtain the adequate angle, such angle is signaled by the dominant presence of white level; this procedure is showed in Fig. 6(b). Depending on the straight fringes content in a given direction of each one of the interferograms, the white regions obtained after the thresholding process show a reduced range of θ values; as a consequence, the adequate angle is the average of the θ values. Applying the process that has been mentioned before, the results obtained for θA in SI1, SI2 and SI3 were 90, 83 and 140°, respectively.
In the simulation, four phase-shifted interferograms were generated for each SI; phase-shifted α value was π/2 and in the three SI, the following errors: ε2 = 0.3 rad, ε3 = 0.2 rad and ε4 = 0.15 rad were programmed. The three errors are included for each set of SI; deviations are proposed with these values, according to those that can occur in experimental setups.
All interferograms were rotated in the angle that has been selected through the thresholding process previously mentioned. Figure 7 shows the cycle of sinusoidal waveform used, in order to find maximum and minimum values. Applying the Eqs. (7), for SI1, the amount of deviation in phase steps is ε2 = 0.2903 rad, ε3 = 0.18 rad and ε4 = 0.1394 rad; for SI2, the amount of deviation in phase steps is ε2 = 0.2742 rad, ε3 = 0.1985 rad and ε4 = 0.1241 rad; and for SI3, the amount of deviation in phase steps is ε2 = 0.2837 rad, ε3 = 0.1985 rad and ε4 = 0.1332 rad. The difference against the real amount of deviations is a result of the resolution of the interferogram.
The simulation results show the effectiveness of the PSI-RT method. For SI1, SI2 and SI3 the deviations found are very close to those that have been here proposed. The estimated ε2 implies that the interferogram 2 has a different phase-shifting step from that of π/2; a similar case occurs with interferograms 3 and 4. Deviations may estimate an erroneous wrapped phase, which might result in a different phase retrieval to the real one.
4. Experimental results
The proposed novel PSI-RT method was also tested using Real Interferograms (RI) with a fringe pattern that corresponded to a flat profile [Fig. 8(a)]. A set of four frames was obtained using a Michelson interferometer. The size of interferograms is 1280x1024 pixels. Experimental setup consists of a He-Ne laser (632.8 nm), the phase-shifting steps (π/2) were produced using a Thorlabs DRV120 piezoelectric transducer (PZT), and the digital information was acquired by a Thorlabs DCU224 monochrome CCD.
Figure 8(b) shows the Radon transform of the real interferogram, and then its thresholding [Fig. 8(c)], this image shows the θA used to rotate all the interferograms. The Radon transform is applied to the interferogram. Figure 8(b) shows a region with a sinusoidal profile; this area can be found between angles 80° and 90°; this means that the tilt of the interferogram can be found in that region; after a black and white image is obtained [Fig. 8(c)] through the thresholding of the Radon transform; this is done in order to find the dominant presence of white regions; this result highlights the section of interest. The thresholding image is processed following the steps explained in the 2.3 subsection, giving a result of θA = 83°.
According to the simulation results, the deviations are very close to the real ones for the experimental results. Figure 9 shows that the maximum and minimum points do not occur at 0, π or π/2; this means that there are deviations that can cause errors in the estimated phase; therefore, deviations in the PSI-RT process of the RI may be due to a malfunctioning of the PZT used or to external perturbations added to the system.
A low-pass filtering process is necessary to obtain improved results with real interferograms; this happens because of the natural noise mainly caused by speckle, low CCD resolution and/or poor quality, low quality in mirrors or lenses, etc.
The PSI-RT can be used to measure step deviations in phase-shifting, and it was developed by applying the Radon transform. This new approach is demonstrated in simulated and real data with PSA method of π/2 phase steps. The procedure was able to successfully detect phase-shifts deviations over the ideal value proven its viability. The process can be extended to any number of phase steps or step size. To this end the error equations [Eqs. (7)] must be modified according to the specific number of phase-shifting steps to be used.
The Radon transform is useful to find a sinusoidal waveform in a fringe pattern, even in presence of a tilt or curvature in a fringe pattern or an interferogram. The PSI-RT algorithm makes it possible to calculate phase-shifting deviations that can result from PSI system miscalibrations or errors resulting from piezoelectric malfunctions. Processing time is shorter than that of methods that use N frames, because it utilizes a single vector in the fringe patterns.
The proposed algorithm can be useful in the calibration of the PZT transducer to generate equally spaced phase-shifted interferograms or more directly to find the wrapped phase by using PSA procedures with arbitrary phase-steps.
Programa para el Desarrollo Profesional Docente, para el Tipo Superior (PRODEP) (UPA-013); Fondos Mixtos (FOMIX) (AGS-2012-C01-198467).
The authors would like also to acknowledge to Mr. Eric Hidalgo, Ph. D. for performing the revision of style.
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