CN111006611A - Rapid high-precision phase extraction method based on two-step nonlinear phase shifting - Google Patents

Rapid high-precision phase extraction method based on two-step nonlinear phase shifting Download PDF

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CN111006611A
CN111006611A CN201911330172.4A CN201911330172A CN111006611A CN 111006611 A CN111006611 A CN 111006611A CN 201911330172 A CN201911330172 A CN 201911330172A CN 111006611 A CN111006611 A CN 111006611A
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姚勇
傅艳萍
吴乾超
刘楚彦
淦亚苹
杨彦甫
田佳峻
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Abstract

The invention is suitable for the field of signal processing technology improvement, and provides a rapid high-precision phase extraction method based on two-step nonlinear phase shift, which comprises the following steps: s1, fitting an oblique ellipse equation by using the two interferograms; s2, solving the coefficient of the oblique elliptical equation by using a least square method; s3, solving the parameters of the oblique ellipse according to the solved oblique ellipse equation coefficients; s4, converting the oblique ellipse into a standard ellipse equation according to the obtained oblique ellipse parameters; and S5, extracting phase information through a standard ellipse equation. The process of converting the oblique ellipse into the standard ellipse corrects the phase shift error caused by the instability of the light source, the air disturbance and the instability of the phase shifter, and improves the precision of the algorithm. Direct current components do not need to be filtered in advance, phase information is directly extracted from the interference pattern by a non-iterative method, the operation speed of the algorithm is improved, and rapid measurement can be realized.

Description

Rapid high-precision phase extraction method based on two-step nonlinear phase shifting
Technical Field
The invention belongs to the field of signal processing technology improvement, and particularly relates to a rapid high-precision phase extraction method based on two-step nonlinear phase shifting.
Background
The magnitude of the precision zero device is in the micron or even nanometer level, various errors inevitably exist in the production and processing of the precision zero device, and the functions of the zero device are affected by some tiny deformations, so the detection of the micro-morphology of the zero device is very important. The detection of topography can be divided into contact and non-contact measurements. Contact measurement uses mechanical probes, contact of the probes with the object can damage the object, and the probes also need to be replaced frequently, increasing cost. The non-contact measurement has the advantages of high efficiency, good stability and capability of avoiding secondary damage to zero devices, and is widely concerned by people. To obtain phase information, a good phase extraction algorithm is crucial. The classical phase extraction algorithm needs to know a determined phase shift value, however, in the phase shift, the precision of the classical phase extraction algorithm is reduced sharply due to phase shift errors caused by unstable light sources, air disturbance and unstable phase shifters. To improve the phase shifting accuracy and overcome the effects of translational errors (caused by phase shifter misalignment), a large number of phase extraction algorithms have emerged, including Principal Component Analysis (PCA), lissajous figure-to-ellipse fitting (LEF), smith's orthogonal and ellipse fitting (GS & LEF), smith's orthogonal and least squares iteration (GS & LSI), principal component analysis and ellipse fitting (PCA & LEF), ellipse fitting and least squares iteration (LEF & LSI), and so on. However, the tilt phase shift error caused by the nonlinearity and the non-uniformity of the phase shifter is not overcome, and the accuracy of the algorithm is affected.
In order to overcome the skew phase shift error, researchers have proposed a number of phase extraction algorithms. In 2000, Chen et al used Taylor expansion to compensate for random tilt-shift errors, which only dealt with small amplitude tilt-shifts. In 2008, xu et al propose to extract an oblique phase shift value by using an improved least squares iterative algorithm, and then to calculate phase information by using the phase shift value, and the algorithm needs to be iterated, resulting in long running time of the algorithm. In order to expand the application range, Liu et al propose to utilize a three-step iterative algorithm, which can extract a large-amplitude tilt phase shift, however, the algorithm is time-consuming and is not beneficial to dynamic measurement. Recently, WIELGUS extracted the tilt-shift value from the two images with non-linear error minimization to obtain faster speed, however, this algorithm requires filtering out the background light intensity in advance, which introduces new errors into the algorithm. Moreover, the algorithm is only effective for linear tilt phase shift, and for nonlinear tilt phase shift, the algorithm cannot effectively extract phase information.
Although the proposed phase-shifting interferometric phase extraction algorithm can extract the phase under the condition of oblique phase shifting, the methods have respective defects, and cannot meet the requirement of accurately extracting the phase from an oblique phase-shifting interferogram at high speed and high precision.
Disclosure of Invention
The invention aims to provide a rapid high-precision phase extraction method based on two-step nonlinear phase shift, and aims to solve the problems of random nonlinear phase shift errors, background light intensity, modulation amplitude disturbance and the like caused by nonlinearity and nonuniformity of a phase shifter.
The invention is realized in such a way that a rapid high-precision phase extraction method based on two-step nonlinear phase shifting comprises the following steps:
s1, fitting an oblique ellipse equation by using the two interferograms;
s2, solving the coefficient of the oblique elliptical equation by using a least square method;
s3, solving the parameters of the oblique ellipse according to the solved oblique ellipse equation coefficients;
s4, converting the oblique ellipse into a standard ellipse equation according to the obtained oblique ellipse parameters;
and S5, extracting phase information through a standard ellipse equation.
The further technical scheme of the invention is as follows: in the step S1, the two interferograms are represented as background light intensity, modulation amplitude disturbance and random nonlinear phase shift error
Figure BDA0002329358430000031
Wherein, a1(x,y),a2(x, y) represents background light intensity, b1(x,y),b2(x, y) represents the modulation amplitude, and δ (x, y) represents the amount of phase shift between the two interferograms.
The further technical scheme of the invention is as follows: sin in said step S12φ+cos2Phi 1 is an oblique elliptic equation
Figure BDA0002329358430000032
I1Representing a first phase-shifting interferogram, I2A second interference correlation plot is shown.
The further technical scheme of the invention is as follows: the step S2 further includes the following steps:
s21, diagonal ellipse equation:
Figure BDA0002329358430000033
simplified to obtain a and I1 2+b·I1·I2+c·I2 2+d·I1+f·I2+g＝0；
S22, obtaining coefficients of oblique ellipse equation by least square method
Figure BDA0002329358430000034
Figure BDA0002329358430000035
Wherein a represents I1B represents I1,I2Coefficient of product term, c represents I2D represents I1F represents I2G represents a constant term coefficient.
The further technical scheme of the invention is as follows: the step S3 further includes the following steps:
s31, obtaining an equation according to the relation between the ellipse and the oblique ellipse
Figure BDA0002329358430000041
S31, obtaining ellipse parameters from the standard ellipse relation equation through the oblique ellipse according to the oblique ellipse coefficient obtained in the step S22
Figure BDA0002329358430000042
Figure BDA0002329358430000043
Figure BDA0002329358430000044
Wherein, axRepresents the major semi-axis of the ellipse, ayRepresenting the minor semi-axis of the ellipse, theta being the angle of rotation, x0，y0Is the amount of translation.
The further technical scheme of the invention is as follows: in the step S4, the oblique ellipse is converted into a standard ellipse equation by using the ellipse parameters obtained in the step S32
Figure BDA0002329358430000045
The further technical scheme of the invention is as follows: in the step S5, according to the obtained standard ellipse equation, using the equation
Figure BDA0002329358430000051
Obtaining phase information, wherein r ═ ay/axAnd the ratio of the major semi-axis to the minor semi-axis of the ellipse is shown, X represents a first corrected phase-shifting interferogram, and Y represents a second corrected phase-shifting interferogram.
The invention has the beneficial effects that: the process of converting the oblique ellipse into the standard ellipse corrects the phase shift error caused by the instability of the light source, the air disturbance and the instability of the phase shifter, and improves the precision of the algorithm. Direct current components do not need to be filtered in advance, phase information is directly extracted from the interference pattern by a non-iterative method, the operation speed of the algorithm is improved, and rapid measurement can be realized.
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Fig. 1 is a schematic flow chart of a two-step nonlinear phase shift-based fast high-precision phase extraction method according to an embodiment of the present invention.
FIG. 2 is a graph of the phase extraction results of a computer simulation in which a perturbation is present, provided by an embodiment of the present invention;
fig. 3 is a graph showing the phase extraction result of the number of interference fringes smaller than 1 in the computer simulation provided by the embodiment of the present invention.
Fig. 4 is a graph of phase extraction results under different noises simulated by a computer according to an embodiment of the present invention.
Detailed Description
As shown in fig. 1, the two-step nonlinear phase shift-based fast high-precision phase extraction method provided by the present invention is detailed as follows:
and step S0, collecting two interference patterns, wherein the light source adopts a collimated parallel LED light source, a beam of light is divided into two beams of light with a certain shearing amount and the same direction by using a uniaxial plane crystal, random phase shifting is carried out by using a 1/4 wave plate and an analyzer, and two phase-shifting interference patterns are collected by using a CCD industrial camera.
Step S1, fitting an oblique ellipse equation by using the two interferograms; a tilted ellipse is fitted. In the presence of background light intensity and modulation amplitude perturbations, and in the presence of random non-linear phase shift errors, the two interferograms can be represented as:
Figure BDA0002329358430000061
a1(x,y),a2(x, y) represents background light intensity, b1(x,y),b2(x, y) represents the modulation amplitude. δ (x, y) represents the amount of phase shift between the two interferograms.
In the following derivation, we omit (x, y), perform the reduction operation on equation (1),
Figure BDA0002329358430000062
the following results were obtained:
Figure BDA0002329358430000063
because of sin2φ+cos2φ＝1，
Figure BDA0002329358430000064
Step S2, solving the coefficient of the oblique elliptical equation by using a least square method; observe equation (4), denote I by a1Coefficient of quadratic term of (a), expressed as I by b1,I2Coefficient of product term, denoted as I by c2Coefficient of quadratic term of (d) representing I1Coefficient of first order term of (1), expressed as I by f2The coefficient of the first order term in g is used to represent the coefficient of the constant term, we can get:
a·I1 2+b·I1·I2+c·I2 2+d·I1+f·I2+g＝0 (5)
wherein the content of the first and second substances,
Figure BDA0002329358430000071
Figure BDA0002329358430000072
these parameters can be obtained using a least squares method. Oblique ellipse equation:
Figure BDA0002329358430000073
simplified to obtain a and I12+b·I1·I2+c·I22+d·I1+f·I2+ g ═ 0; obtaining coefficients of an oblique elliptic equation by a least square method
Figure BDA0002329358430000074
Wherein a represents I1B represents I2C represents I1,I2Coefficient of product term, d represents I1F represents I2G represents a constant term coefficient.
Let M be [ a, b, c, d, f, g ═ g]，
Figure BDA0002329358430000077
The object is to
Figure BDA0002329358430000075
Wherein the content of the first and second substances,
Figure BDA0002329358430000076
constructing lagrange functions
L(M,λ)＝MTNNTM-λ(MTHM-1) (7)
Has a derivative of zero to obtain
Figure BDA0002329358430000081
Let S be NNTThen SM ═ λ HM, M of 6 possible alternatives can be obtained by solving the generalized eigenvector. Using MTThe condition that HM is 1 is used for screening qualified M.
Next, the screening conditions will be described. Assuming the resulting eigenvalue and eigenvector pairs { λ }i,viFor the elliptic equation a.I }1 2+b·I1·I2+c·I2 2+d·I1+f·I2The discriminant when + g is 0 is:
Figure BDA0002329358430000082
when Λ ≠ 0, and ac-b2>An ellipse at 0.
Step S3, solving the parameters of the oblique ellipse according to the solved oblique ellipse equation coefficients; a standard ellipse parametric equation is shown below
Figure BDA0002329358430000083
Wherein a isxRepresents the major semi-axis of the ellipse, ayRepresenting the minor semi-axis of the ellipse.
The oblique ellipse is derived from the rotation and translation of a standard ellipse, and first, considering only the rotation, the new ellipse after rotation can be represented by:
Figure BDA0002329358430000084
wherein, x ', y' represent the new ellipse parameters after rotation, and theta represents the rotation angle.
Second, considering only translation, the new ellipse after translation can be represented by:
Figure BDA0002329358430000085
where x ", y" represents the new ellipse parameters after translation, x0，y0Is the amount of translation.
Therefore, the equation of the standard ellipse after rotation and translation can be expressed as:
Figure BDA0002329358430000091
wherein x and y represent oblique ellipses after rotation and translation.
The standard ellipse X, Y is expressed by X, Y after the rotation translation, and the derivation process of converting the oblique ellipse into the standard ellipse is as follows:
Figure BDA0002329358430000092
substituting equation (14) into equation
Figure BDA0002329358430000093
Equation (15) can be obtained
Figure BDA0002329358430000094
The key of converting the oblique ellipse into the standard ellipse is to solve the ellipse parameter ax，ay，θ，x0，y0；
Reducing equation (15) to the form of equation (5), and using x of equation (15) as I in equation (5)1Instead, y of formula (15) is represented by I in formula (5)2Instead. Comparing equation (5) with equation (15), the coefficients a, b, c, d, f, g of equation (5) can be expressed as:
Figure BDA0002329358430000101
Figure BDA0002329358430000102
Figure BDA0002329358430000103
by observing the formula (16), d, f, g are represented by a, b, c
Figure BDA0002329358430000104
By using this equation, x can be found0,y0,ax,ay(ii) a From a to I1 2+b·I1·I2+c·I2 2+d·I1+f·I2The equation of + g ═ 0 can be written to express the tilt angle θ, so we can obtain the parametric expression of the ellipse as follows:
Figure BDA0002329358430000105
Figure BDA0002329358430000106
Figure BDA0002329358430000107
step S4, converting the oblique ellipse into a standard ellipse equation according to the obtained oblique ellipse parameters; using these ellipse parameters, the oblique ellipse is obtained from the standard ellipse rotation and translation, and we find the rotation angle θ and the translation x0,y0Then, x in the formula (14) is represented by I1Is represented by I2Thus, using equation (14), we can convert the oblique ellipse to a standard ellipse, of the form:
Figure BDA0002329358430000115
in step S5, phase information is extracted by a standard ellipse equation.
The standard elliptic equation parameters are of the form:
Figure BDA0002329358430000111
phi is the phase information we want to solve.
Figure BDA0002329358430000112
Wherein the content of the first and second substances,
Figure BDA0002329358430000113
therefore, by using the standard elliptic equation, the phase information can be accurately extracted
Figure BDA0002329358430000114
In order to verify the effectiveness of the method, a computer is used to perform simulation on various conditions.
Case 1: in phase-shifting interferometry, there are source instabilities, external environmental perturbations, which can cause phase-shifting errors. The parameters of the two interferograms are thus set as follows:
background light intensity: a is1(x,y)＝1+0.1·exp(-0.02(x2+y2))，a2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
Modulation amplitude: b1(x,y)＝1+0.1·exp(-0.02(x2+y2))，b2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
The object phase: phi (x, y) 5 pi (x)2+y2)
The phase shift difference between the two interferograms is: delta is 0.75+0.05 x2+0.15·y
The initial phase of 512 × 512 pixels is generated through numerical simulation, as shown in fig. 2(a), fig. 2(b) is one of two random interferograms, fig. 2(c) is a phase shift amount with a nonlinear error between the two interferograms, and fig. 2(d) is object surface phase information obtained by extracting a phase through the method provided by the invention, calculating through phase unwrapping operation, and performing tilt adjustment.
The phase extraction time is: 0.046847 s.
Case 2: in phase-shifting interferometry, the interference fringes tend to be less than 1. However, in many existing multiphase extraction algorithms, the number of interference fringes is required to be more than 1 to effectively extract the phase. To verify the accuracy of the invention in this case, we simulated the case where the number of interference fringes is less than 1.
Background light intensity: a is1(x,y)＝1+0.1·exp(-0.02(x2+y2))，a2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
Modulation amplitude: b1(x,y)＝1+0.1·exp(-0.02(x2+y2))，b2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
The object phase: phi (x, y) 0.6 pi (x)2+y2)
The phase shift difference between the two interferograms is: delta is 0.75+0.05 x2+0.15·y
The initial phase of 512 × 512 pixels is generated by numerical simulation, as shown in fig. 3(a), fig. 3(b) is one of two random interferograms, fig. 3(c) is a phase shift amount of a nonlinear error between the two interferograms, and fig. 3(d) is object surface phase information obtained by extracting a phase by the method provided by the invention, calculating through phase unwrapping operation, and performing tilt adjustment.
The phase extraction time is: 0.044079s
Case 3: in phase-shifting interferometry, noise is inevitable. To verify the noise immunity of the present invention, we performed simulation experiments at different signal-to-noise ratios (SNRs).
Background light intensity: a is1(x,y)＝1+0.1·exp(-0.02(x2+y2))，a2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
Modulation amplitude: b1(x,y)＝1+0.1·exp(-0.02(x2+y2))，b2(x,y)＝0.9+0.21·exp(-0.02(x2+y2))
The object phase: phi (x, y) 5 pi (x)2+y2)
The phase shift difference between the two interferograms is: delta is 0.75+0.05 x2+0.15·y
In addition, white gaussian noise with different signal-to-noise ratios is added. SNR is: 13dB, 15dB,17dB,20dB,30 dB.
The numerical simulation generated 512 x 512 pixels of initial phase, as shown in fig. 4 (a). Fig. 4(b) -4(f) show the phase extraction results of the object phase information calculated by extracting the phase by the method provided by the invention and performing phase unwrapping operation, which correspond to the signal-to-noise ratios of 13dB, 15dB,17dB,20dB and 30dB, respectively.
The phase extraction time is shown in table one:
watch 1
SNR(dB) 13 15 17 20 30
Time(s) 0.053104 0.043537 0.045660 0.044094 0.043574
Case 1, case 2 and case 3 illustrate that the method can extract the phase information of the interferogram rapidly and precisely under different conditions, and has strong robustness.
Through the process of converting the oblique ellipse into the standard ellipse, the phase shift error caused by instability of a light source, air disturbance and instability of a phase shifter is corrected, and the accuracy of the algorithm is improved. Direct current components do not need to be filtered in advance, phase information is directly extracted from the interference pattern by a non-iterative method, the operation speed of the algorithm is improved, and rapid measurement can be realized.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (7)

1. A rapid high-precision phase extraction method based on two-step nonlinear phase shifting is characterized by comprising the following steps of:
s1, fitting an oblique ellipse equation by using the two interferograms;
s2, solving the coefficient of the oblique elliptical equation by using a least square method;
s3, solving the parameters of the oblique ellipse according to the solved oblique ellipse equation coefficients;
s4, converting the oblique ellipse into a standard ellipse equation according to the obtained oblique ellipse parameters;
and S5, extracting phase information through a standard ellipse equation.
2. The two-step nonlinear phase shift-based fast high-precision phase extraction method according to claim 1, wherein the two interferograms in step S1 are represented as background light intensity, modulation amplitude disturbance and random nonlinear phase shift error
Figure FDA0002329358420000011
Wherein, a1(x,y),a2(x, y) represents background light intensity, b1(x,y),b2(x, y) represents the modulation amplitude, and δ (x, y) represents the amount of phase shift between the two interferograms.
3. The fast high-precision phase extraction method based on two-step non-linear phase shifting according to claim 2, characterized in that in step S1 sin2φ+cos2Phi 1 is an oblique elliptic equation
Figure FDA0002329358420000012
Wherein, I1Representing a first phase-shifting interferogram, I2A second interference correlation plot is shown.
4. The fast high-precision phase extraction method based on two-step nonlinear phase shift according to claim 3, characterized in that the step S2 further comprises the following steps:
s21, diagonal ellipse equation:
Figure FDA0002329358420000021
simplified to obtain a and I1 2+b·I1·I2+c·I2 2+d·I1+f·I2+g＝0；
S22, obtaining coefficients of oblique ellipse equation by least square method
Figure FDA0002329358420000022
Figure FDA0002329358420000023
Figure FDA0002329358420000024
Wherein a represents I1B represents I1,I2Coefficient of product term, c represents I2D represents I1F represents I2G represents a constant term coefficient.
5. The fast high-precision phase extraction method based on two-step nonlinear phase shifting according to claim 4, wherein the step S3 further comprises the following steps:
s31, obtaining an equation according to the relation between the ellipse and the oblique ellipse
Figure FDA0002329358420000025
S31, obtaining ellipse parameters from the standard ellipse relation equation through the oblique ellipse according to the oblique ellipse coefficient obtained in the step S22
Figure FDA0002329358420000031
Figure FDA0002329358420000032
Figure FDA0002329358420000033
Wherein, axRepresents the major semi-axis of the ellipse, ayRepresenting the minor semi-axis of the ellipse, theta being the angle of rotation, x0，y0Is the amount of translation.
6. The two-step nonlinear phase shift-based fast high-precision phase extraction method according to claim 5, wherein the step S4 is implemented by transforming the oblique ellipse into a standard ellipse equation using the ellipse parameters obtained in step S32
Figure FDA0002329358420000034
7. The two-step nonlinear phase shift-based fast high-precision phase extraction method according to claim 6, wherein the equation is utilized according to the obtained standard ellipse equation in the step S5
Figure FDA0002329358420000035
Obtaining phase information, wherein r ═ ay/axAnd the ratio of the major semi-axis to the minor semi-axis of the ellipse is shown, X represents a first corrected phase-shifting interferogram, and Y represents a second corrected phase-shifting interferogram.
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