Measurement of step surface contour based on variable sampling phase shift interference phase extraction algorithm based on selective sampling


 Variable frequency phase shift interferometry is widely applied in optical precision measurement, with the accuracy of phase extraction’s direct impact on that of phase shift interferometry. In the variable-frequency phase-shift interferometry, the commonly used phase-shifting devices are prone to phase shift errors, because the ordinary equal-step phase extraction algorithm, which can be merely used to measure simple and smooth surface, influences the accuracy of phase extraction resulting in measuring error, and causes inefficiency led by the long time the iterative process lasts for when applied in complex stepped surfaces measurement. As a sort of step-by-step phase-shifting phase extraction algorithm based on selective sampling is used to measure the step surface contour, the interference image is firstly sampled at equal intervals to reduce the iterative calculation, and in view of the fact that the phase calibration of the test system is not required in this algorithm, the measured phase is given by using the alternating iterative method despite the unknown phase and unknown phase shift amount. The phase extraction accuracy and iteration time among traditional iterative algorithm, four-step phase shift algorithm and the variable phase shift phase interpolation algorithm based on selective sampling are compared in the simulation and experiment. It is shown that the variable frequency phase shifting interference phase extraction algorithm based on selective sampling has shorter operation time, less error and higher accuracy than traditional iterative algorithm in measuring complex step surface.


Introduction
The interferometric method to measure the high-accuracy surface shape was first carried out in the 1960s. With the rapid development of interferometry technology, phase-shifted interferometry (phaseshifting interferometry, PSI), the primary presentative of interferometry technology, becomes one of the main methods in modern measurement of optics surface shape. From the early phase-shift algorithm including the Hariharan algorithm and the Schwider algorithm to the current frequency-conversion phase-shift algorithm including the iterative algorithm and the principal element analysis method, we can come to a conclusion that the research of the frequency conversion phase shift algorithm has become the trend of modern phase-shift interferometry research [1]. Although the principal element analysis (PCA) and Advanced iterative algorithm (AIA) can be used to recover the phase information from a large number of phase-shifted interferogram efficiently [2][3][4][5], a series of error sources and the complexity of measured surface do exist in the actual measuring process [6,7]. And the traditional phase-shift interference algorithm exposes some drawbacks, such as the effect of noise and phase shift error on phase extraction precision of stochastic phase-shift algorithm [8], the influence of the background intensity and modulation system on phase-shift algorithm, and slow rate of operation as well as low precision of the result [3,9,10].
In order to solve the above drawbacks, Yang Mu, Hou Lizhou and others proposed a stochastic phase-shifted phase solution method based on least squares iteration, whereas running computing time will be increased obviously and phase extraction accuracy will mostly be lowered when interference fringe graphs with lots of pixels are available to be dealt with in the iterative operation period [3,11,12]. In this paper, the iterative algorithm is used to solve the phase, and it is improved and deeply researched, and a high efficient and accurate phase extraction algorithm is proposed, which differs from the traditional method in the study of phase extraction with random sampling to simplify operation and improve accuracy [13][14][15][16]. In the measurement process, the algorithm selectively samples the original interference fringe image, which ensures that all pixel points of the iterative operation are valid, thus lessening the amount of data [17][18][19]. Besides, in the case of unknown phase shifts and phases, the algorithm can determine the phase shift and phase simultaneously through constant iterative computing [15], and achieve higher accuracy in measuring complex step surface profile due to the insensitivity to the phase shift error.The reliability of the algorithm is verified by simulation experiment and experimental data [18,19].

basic principle of frequency conversion phase shift interference
In the frequency conversion phase shift interferometry, the assumption is that the background light is ! ( , ), ! ( , ) is the modulation, ℎ( , ) is the length of the interference cavity, and the light intensity expression of the t-step phase shift is: (1) represents the test wavelength value corresponding to the t-step phase shift, the ! is the wavelength tunable semiconductor laser starting output wavelength (that is, the center wavelength), is the wavelength phase shifting step, (1) can be approximated to： It can be seen from equation (2) that in the variable-frequency phase-shift interferometry, the phase shift of the phase shift in the t-th step is determined by the joint length and the wavelength step. In the measurement, in order to make the phase shift amount a certain value under different interference cavity lengths, phase calibration must be performed [20].

Frequency-shifted phase-shift interferometric phase extraction algorithm based on selection sampling
In phase-shifted interferometry, if the phase shift δ . can be determined, then the K-phase-shifted interference image can be expressed as: In ( According to the principle of least squares, the residual function is obtained by using the formula (4): To solve the optimal estimate of ! ( , )， / ( , )， 0 ( , ), we need to make the square sum of (5) and take the minimum value. Further acquisition of the phase value To be measured: The true phase shift amount during the phase shift can be expressed as the nominal phase shift plus the phase shift error value. The first iteration can make the initial phase shift error zero.
,a r c t a n ,
Substituting the phase shift amount and the phase shift error into the formula, the residual function is as follows: The phase shift error value in the formula is an unknown quantity [21][22][23]. Similarly, the phase shift error value can be solved by the least squares method. A partial derivative is obtained for each phase shift error amount . , and the result is zero. The phase shift error value . can be obtained by solving the equation. At this time, the phase shift amount δ A . = δ . + . is updated to perform the next iteration operation. The threshold condition T is set according to the phase shift error value . , and the iteration is terminated when the . change amount is less than T. At this point, an accurate phase shift amount can be obtained [21,23].

Simulation experiment based on selective sampling
Use initial cap for first word in title or for proper nouns. Use lowercase following colon. Title should not begin with an article or contain the words "first," "new" or "novel." This paper simulates four interference fringe patterns with a pixel resolution of 450pix×450pix. The contrast of a region in the interference image is reduced, and the low reflectance sample region in the actual measurement is simulated; and the salt and pepper noise with a density of 0.002 and the Gaussian noise with a standard deviation of 15 are added to the entire image to simulate the interference image noise. The interference diagram is shown in Figure. 1.
The original image is first sampled every d pixel points. Then the sampled points are brought into the iterative algorithm, the four-step phase shift algorithm and the improved algorithm for iterative calculation [25]. The images after equal interval sampling are filtered out at low quality points, and a reasonable threshold T is selected to divide the four interferences. In the fringe image, the point where the maximum value of the gray value of the pixel at the same position is smaller than the threshold T is filtered, and the pixel whose image value is always low in the image [26][27][28]. The threshold T=80 is determined according to the pixel value of the dark interference fringe. At this time, the filtering ratio is 20%, indicating that most of the sampling pixels still participate in the iterative operation after filtering, and the relationship between the sampling interval and the running time and the sampling interval and the error value are as shown in Figure.

Experimental system construction
The optical path diagram of the measurement system we use is shown in Figure.  consists of a projector, an area array camera, a lens, a moving displacement platform, and a computer. A CCD camera is used to take a picture above the reference plane to obtain the surface contour of the object. Information and sync to the PC side. The image resolution of the camera is 768 × 576 pixels, and the actual size of the object corresponding to the width of the captured image is 100 mm. By adjusting the distance between the lens and the projector, the scattered light field projected by the projector is projected through the lens and projected onto the object to be tested in parallel, and the projection angle of incidence is 30° [25]. The measured surface shape of the experiment uses a complex measured surface whose front end is a stepped surface with a smooth curved back end, as shown in Figure.5. In this way, we compare the variable frequency phase shift phase extraction algorithm based on selective sampling.
Whether it is more accurate when measuring the step surface, the experiment includes the following steps: (1) The phase-shifted interferograms with different phase shifts are obtained from the measured surface, and both have a random error with a mean of 0 and a variance of 0.01; (2) Solving the measured phase by using an iterative algorithm, a four-step phase shift algorithm, and a variable frequency phase shift phase extraction algorithm based on selective sampling; (3) Unwrapping the package phase to obtain the measured surface for experimental analysis.

Interferogram with phase shift amount of arbitrary value
Let the phase shift step be an arbitrary value. The phase shift amount and the theoretical phase shift amount obtained by the above iterative method are as shown in Table 1, and the number of iterations is 65 times. From the above results, the simulated wave front diagram of Figure.6 (a) and the phase diagram of the envelope of Figure 6 (b) are obtained. It is found that the phase shift magnitude obtained by the iterative method is compared with the theoretical phase shift magnitude, and the error is 10^(-5) .The order of magnitude indicates that the iterative algorithm is used to solve the phase shift with high precision, which can reduce the measurement error introduced by the step error during interferometry [8,15].

Iterative algorithm with phase shift of π/2
The iterative algorithm is used to solve the measured surface, and the phase shift and theoretical phase shift are as shown in Table 2. The number of iterations is 37. The shape of the surface to be measured obtained after unwrapping and fitting is shown in Figure   7(a) and Figure 7(b) is the surface cross section of the surface to be measured.

Four-step phase shift algorithm with phase shift of π/2
Using a four-step phase shift algorithm (four steps of phase shift, each phase shift is π/2), four phase shifting interferograms with phase shift steps of π/2 rad are acquired, and the four-step phase shift algorithm is used to test The phase solution formula φ = arctan I  Table 3, and the number of iterations is 37 times. The shape of the surface to be measured obtained after unwrapping and fitting is shown in Figure   8(a), and Figure 8(b) is the surface cross section of the surface to be measured. is taken as the initial phase shift amount into the algorithm, and then the original image is sampled every other pixel point, and then the sampled points are brought into The algorithm performs an iterative operation, and finally the profile of the measured surface obtained by unwrapping and fitting and the cross-sectional view of the measured surface shape are as shown in Figure 9(a) and Figure 9(b).
(a)The measured shape surface solved by the variable frequency phase shift phase extraction algorithm based on the selected sampling (b)Sectional view of the measured shape solved by the variable frequency phase shift phase extraction algorithm based on the selected sampling sampled phase-shift phase-shifting algorithm to solve the shape of the wrapped phase, the above three algorithms can obtain the higher-precision measurement pattern. In the surface step of the measured surface, the iterative algorithm and the four-step phase shift algorithm are fitted to obtain a larger surface error value, and the measurement accuracy is lower, but when using the variable phase shift phase extraction algorithm based on selective sampling It can be seen from Figure 9 (a) and (b) that the surface error level obtained after fitting is kept at a relatively low level, and the maximum iteration error value is only about 0.1°, which has a high Measurement accuracy, and running time is shorter than the other two algorithms.

Conclusions
Phase-shifted interferometry has been widely used in 3D scanning modeling, image restoration, and accurate measurements. In actual measurements, the accuracy of phase result has a great impact on that of subsequent phase to be worked out. and measurement experiments, we come to a conclusion that the algorithm proposed in this paper has a more scope application, not only to the simple and smooth surfaces, but also to complex step surfaces. When the algorithm mentioned before is used to measure complex step surfaces, a high accuracy and short running time can be reached. As a result, such algorithm can be used in actual measurements. Fig. 1 Four simulated interference fringe images.         Table 1 Phase shift by interaction method and the theoretical phase shift (arbitrary phase shift). Table 2 Phase shift by interaction method and the theoretical phase shift (arbitrary phase shift). Table 3 Phase shift by interation method and the theoretical phase shift(arbitrary phase shift). Figure 1 Four simulated interference fringe images Comparison of root mean square error of three algorithms for equal interval sampling Test system light path diagram   Results measured by a variable-phase phase shift phase extraction algorithm based on selected samples. (a)The measured shape surface solved by the variable frequency phase shift phase extraction algorithm based on the selected sampling (b)Sectional view of the measured shape solved by the variable frequency phase shift phase extraction algorithm based on the selected sampling