CN111238398B - Phase shift error detection method based on probability distribution function - Google Patents

Phase shift error detection method based on probability distribution function Download PDF

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CN111238398B
CN111238398B CN202010136180.1A CN202010136180A CN111238398B CN 111238398 B CN111238398 B CN 111238398B CN 202010136180 A CN202010136180 A CN 202010136180A CN 111238398 B CN111238398 B CN 111238398B
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phase shift
phase
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probability distribution
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CN111238398A (en
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刘元坤
于馨
薛俊鹏
张启灿
陈文静
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Sichuan University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G01B11/24—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures
    • G01B11/2441—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures using interferometry
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01J—MEASUREMENT OF INTENSITY, VELOCITY, SPECTRAL CONTENT, POLARISATION, PHASE OR PULSE CHARACTERISTICS OF INFRARED, VISIBLE OR ULTRAVIOLET LIGHT; COLORIMETRY; RADIATION PYROMETRY
    • G01J9/00—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength
    • G01J9/02—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01J—MEASUREMENT OF INTENSITY, VELOCITY, SPECTRAL CONTENT, POLARISATION, PHASE OR PULSE CHARACTERISTICS OF INFRARED, VISIBLE OR ULTRAVIOLET LIGHT; COLORIMETRY; RADIATION PYROMETRY
    • G01J9/00—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength
    • G01J9/02—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods
    • G01J2009/0211—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods for measuring coherence

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Abstract

The invention discloses a phase shift error detection method based on a probability distribution function, which is used for acquiring actual truncated phase distribution and calculating a corresponding PDF curve; and performing correlation operation on the PDF curve and a pre-established simulation curve database to find a simulation curve with the maximum correlation degree with the PDF curve in the simulation curve database, wherein the phase shift error of the simulation curve with the maximum correlation degree is used as the detected phase shift error. The phase shift error detected by the method is very close to the actual phase shift error, and the detection error can be further reduced by shortening the sampling interval when the analog curve database is established.

Description

Phase shift error detection method based on probability distribution function
Technical Field
The invention relates to the field of phase shift interferometry, in particular to a phase shift error detection method based on a probability distribution function.
Background
Phase-shift interferometry is a high-precision optical phase measurement technique, and is one of the important means for detecting the surface type of an optical surface. The current general methods for realizing phase shift mainly include a mechanical method and a wavelength tuning method, and the mechanical method generally adopts means of pushing piezoelectric ceramics (PZT), rotating polarization devices (wave plates and polarizing plates), moving diffraction gratings or tilting plates and the like to realize phase modulation. Of which piezoelectric ceramics (PZT) has been pushed to produce phase shifts for the most widespread applications. The wavelength tuning mode realizes phase modulation by changing the frequency of a light source, and the light source adopts a wavelength tunable laser. Both of the two phase shifting methods have certain phase shifting errors, including initial phase shifting error, phase shifting error caused by hardware aging and the like, and the phase shifting error is considered to be one of main errors affecting the measurement accuracy. Therefore, to realize high-precision three-dimensional surface shape measurement, calibration of precise phase shift amount is required.
Disclosure of Invention
The invention aims to solve the problem of phase shift error existing in phase shift interferometry measurement phase shift in the prior art, and provides a phase shift error detection method based on a probability distribution function.
In order to achieve the above purpose, the invention provides the following technical scheme:
a phase shift error detection method based on a probability distribution function is characterized in that N-1 phase shift errors are generated by N-step phase shift interferometry, N is greater than or equal to 3, and the phase shift error detection method comprises the following steps:
step one, acquiring actual truncated phase distribution, and calculating a corresponding PDF curve;
and step two, carrying out correlation operation on the PDF curve and a pre-established simulation curve database to find a simulation curve with the maximum correlation degree with the PDF curve in the simulation curve database, wherein the phase shift error of the simulation curve is used as the detected phase shift error.
Preferably, the method for establishing the simulation curve database comprises the following steps:
and simulating a series of PDF curves under different phase shift errors in a set interval, and establishing a simulation curve database.
Preferably, the first step includes:
acquiring N phase shift pictures, describing light intensity of pixels in the phase shift pictures, and measuring a phase shift amount under an ideal condition according to N steps of phase shift interferometry to obtain a truncated phase with a phase shift error;
and obtaining the corresponding PDF curve according to the truncated phase with the phase shift error.
Preferably, the light intensity I of the pixel in the nth phase-shift picturen(x,y)：
Figure GDA0002464930130000021
Wherein A (x, y) and B (x, y) represent background light intensity and modulation, respectivelyThe degree of the magnetic field is measured,
Figure GDA0002464930130000022
is the initial phase, δ, to be measurednIs a phase shift, and δ0＝0。
Preferably, the process of detecting phase shift errors of N-step phase shift interferometry is decomposed into detection
Figure GDA0002464930130000023
Phase shift error of three-step phase shift interferometry.
Preferably, the phase shift error detection method of the three-step phase shift interferometry includes:
s1, acquiring three phase shift pictures, describing the light intensity of pixels in the phase shift pictures, and measuring the phase shift amount under an ideal condition according to three-step phase shift interferometry to obtain a truncated phase with an error;
obtaining a corresponding three-step phase shift interferometry PDF curve according to the truncated phase with the error;
s2, carrying out correlation operation on the three-step phase shift interferometry PDF curve and a three-step phase shift interferometry simulation curve database established in advance, finding out a simulation curve with the maximum correlation degree with the three-step phase shift interferometry PDF curve in the three-step phase shift interferometry simulation curve database, and taking the phase shift error of the simulation curve as the detected phase shift error.
Preferably, the three-step phase-shift interferometry simulation curve database is established by: simulating to generate a series of different delta in a set interval1And delta2PDF curve F for the case0(m, n) where m and n are delta1、Δδ2The number of (2).
Preferably, in step S1, the light intensity I of any pixel of the nth phase-shifted picturen(x,y)：
Figure GDA0002464930130000031
When the three-step phase shift is the full-period three-step phase shift, the phase shift amounts are 0 and delta1＝2π/3、δ 24 pi/3, resulting in a truncated phase with phase shift error
Figure GDA0002464930130000032
Figure GDA0002464930130000033
The PDF curve F:
Figure GDA0002464930130000034
wherein, M represents the number of sampling points, and M is 0,1, 2.
Preferably, in step S2, the correlation operation calculates the correlation coefficient R (m, n) by using the following formula:
Figure GDA0002464930130000041
wherein F is the PDF curve, F0(m, n) is a simulation curve; where m, n are the numbers of phase shift errors.
Preferably, the simulated curve with the maximum correlation is a simulated curve in which the correlation coefficient R (m, n) is maximized.
Compared with the prior art, the invention has the beneficial effects that:
the invention discloses a phase shift error detection method based on a probability distribution function, which is used for acquiring actual truncated phase distribution and calculating a corresponding PDF curve; and performing correlation operation on the PDF curve and a pre-established simulation curve database to find a simulation curve with the maximum correlation degree with the PDF curve in the simulation curve database, wherein the phase shift error of the simulation curve with the maximum correlation degree is used as the detected phase shift error. The phase shift error detected by the method is very close to the actual phase shift error, and the detection error can be further reduced by shortening the sampling interval when the analog curve database is established. The invention also decomposes N-step phase shift measurement into three-step phase shift measurement and then carries out phase shift error detection, only needs to establish a simulation curve database with two phase shift errors, reduces the difficulty of establishing the simulation curve database and reduces the volume of the simulation curve database.
Description of the drawings:
fig. 1 shows the truncated phase and phase error of the present embodiment.
Fig. 2 shows the truncated phase (a) and PDF curves (b) for the ideal case and for the four phase shift errors.
Fig. 3 is a schematic diagram illustrating a flow of the calculation process of the embodiment.
Fig. 4 is a schematic diagram of a process for detecting phase shift errors based on probability distribution.
Detailed Description
The present invention will be described in further detail with reference to test examples and specific embodiments. It should be understood that the scope of the above-described subject matter is not limited to the following examples, and any techniques implemented based on the disclosure of the present invention are within the scope of the present invention.
In the phase shift technique, N phase shift pictures need to be obtained for calculating the phase value, where the light intensity of any pixel in the nth fringe pattern can be represented as:
Figure GDA0002464930130000051
wherein A (x, y) and B (x, y) respectively represent background light intensity and modulation degree,
Figure GDA0002464930130000052
is the initial phase, δ, to be measurednIs a phase shift, and δ 00. When deltanWhen known, only A (x, y), B (x, y) and
Figure GDA0002464930130000053
three unknowns, and therefore at least three fringe patterns are required.
The N-step phase-shift interferometry has N-1 phase-shift errors, and when N is greater than 3, the process of detecting the phase-shift error of the N-step phase-shift interferometryDecomposition into detection
Figure GDA0002464930130000054
Phase shift errors of the three-step phase shift interferometry, for example, the five-step phase shift interferometry has 4 phase shift errors, and the first two phase shift errors and the second two phase shift errors are obtained by detecting the phase shift errors of the three-step phase shift interferometry according to the method; for another example, the six-step phase shift interferometry has 5 phase shift errors, and the first two phase shift errors and the middle phase shift error are obtained by respectively detecting the phase shift errors of the three-step phase shift interferometry according to the method and selecting one of the phase shift errors and the last two phase shift errors.
This example discusses the case where N is 3, and the method mentioned applies equally to N>3, in the case of the first embodiment. In a full-period three-step phase-shift interferometry, the amount of phase shift is ideally: 0. delta1＝2π/3、δ 24 pi/3, there are often two phase shift error amounts in practical measurements: delta delta1And delta2. The existence of the phase shift error introduces a periodic phase error, and greatly influences the precision of three-dimensional surface shape measurement. Therefore, accurate determination of the phase shift amount is the core of phase shift interferometry, and how to calibrate the phase shift amount quickly and accurately has important significance for ensuring the measurement precision. The method for calibrating the accurate phase shift amount by using a statistical method has good noise suppression capability, can realize rapid and real-time phase shift amount calibration, eliminates periodic phase errors and improves the measurement precision. Equation (1) can be reduced to:
Figure GDA0002464930130000061
in the formula (2), the first and second groups,
Figure GDA0002464930130000062
and
Figure GDA0002464930130000063
Figure GDA0002464930130000064
in place of A (x, y), B (x, y) and
Figure GDA0002464930130000065
become three new unknowns [ Guo]。
Figure GDA0002464930130000066
Can be expressed as:
Figure GDA0002464930130000067
for convenience, we will omit (x, y) hereafter and calculate I from equation (2)1(x,y)-I0(x, y) and I2(x,y)-I0(x, y) can give:
Figure GDA0002464930130000068
get solution of c1，c2：
Figure GDA0002464930130000069
And
Figure GDA00024649301300000610
if the exact phase shift value is known, the exact phase value can be calculated by substituting equations (5) and (6) into equation (3). In fact, in the interferometric technique based on phase shift, the motion error of the mechanical structure is one of the sources of the phase shift error, and the actual phase shift amount can be considered as the preset phase shift amount plus the error term. I.e. deltan＇＝δn+Δδn
In this embodiment, a full-period three-step phase shift is taken as an example, but the algorithm is also applicable to three-step phase shifts with any phase shift amount and phase shifts with more than 3 steps, that is, the phase shift amount is 0, delta1＝2π/3，δ 24 pi/3, at which time the phase is cut offThe bit calculation formula can be further simplified as:
Figure GDA0002464930130000071
without loss of generality, it is assumed that two shifts introduce two phase shift errors, i.e., Δ δ1And delta2The truncated phase and phase error curves for several phase shift errors are shown in fig. 1, where fig. 1(a) is the ideal phase and fig. 1(b) - (e) are the delta, respectively1＝π/19、Δδ2＝π/17，Δδ1＝π/19、Δδ2＝-π/17，Δδ1＝-π/19、Δδ2＝π/17，Δδ1＝-π/19、Δδ2Truncated phase obtained at-pi/17, fig. 1(f) - (i) phase error introduced for the corresponding phase shift error: (a) ideally truncating the phase. (b) With phase shift error delta1Pi/19 and delta2Pi/17 truncated phase. (c) With phase shift error delta1Pi/19 and delta2A truncated phase of-pi/17. (d) With phase shift error delta1-pi/19 and Δ δ2Pi/17 truncated phase. (e) With phase shift error delta1-pi/19 and Δ δ2A truncated phase of-pi/17. (f) And (i) is the phase error corresponding to each phase shift error in (b) - (e).
It can be seen that the phase error caused by the phase shift error is periodic. And the phase error varies with the phase shift error, not only in amplitude, but also in overall motion. Therefore, the phase error can only be eliminated by first obtaining an accurate amount of phase shift. The invention adopts a statistical method to search for the accurate phase shift quantity.
1. Probability distribution function of truncated phase
Probability Distribution Function (PDF) is a common statistical method. We take PDF as the objective function of the truncated phase, which can be expressed as
Figure GDA0002464930130000081
Wherein M represents the number of sampling points, M is 0,1,2, and M-1, and the larger M is, the more accurate the result is. In this document, M is 64. The meaning is as follows: truncated phase to be obtained from three-step phase-shifted stripes
Figure GDA0002464930130000082
Figure GDA0002464930130000083
Dividing the phase value into 64 phase value regions, namely 64 sampling points, wherein the value range of each region is [2 pi m/64-pi, 2 pi (m + 1)/64-pi), counting the number of pixel points of which the truncation phase falls in each phase value region, and the ratio of the number of the pixel points to the total number of the pixel points is the probability of the sampling point. And drawing probability values corresponding to 64 sampling points respectively, namely the PDF curves corresponding to the three-step phase shift stripes.
Theoretically, when there is no phase shift error in the measurement system, the probability of each phase value occurring is the same, and the PDF curve is a uniform curve. If a phase shift error exists, the phase error value changes the probability of the original precise phase value, that is, the PDF curve changes. FIG. 2(a) shows the truncated phase for several phase shift errors, with red representing the ideal case, i.e., Δ δ1＝Δδ2The other four colors represent four different Δ δ values, 01And delta2The case of a combination. Fig. 2(b) shows a PDF curve corresponding to the phase shift error. It is obvious that the amplitude of the PDF curve and the peak position will vary with the phase shift error. When delta1＝Δδ2When the value is 0, the PDF curve is a straight line, and the probability of each sampling point is 1/M. For example, when M is 64, f (M) is 0.0156.
2. Phase shift error calculation method based on probability distribution function
Because the PDF curves correspond to the phase shift error conditions one by one, the invention provides a method for calculating the phase shift error based on the characteristic. The three-step phase shift of the full period is selected, the phase shift is set to be 0, 2 pi/3 and 4 pi/3, but the algorithm is also applicable to the three-step phase shift of any phase shift amount and the phase shift of more than 3 steps, and the calculation process is shown in FIG. 3. Therefore, the actual phase shift error finding step in the present invention is shown in fig. 4:
the computer is in [ - π/10, π/10]Simulation within interval produces a series of different delta1And delta2PDF curve F for the case0(m, n), wherein m, n is Δ δ1And delta2The number of (2). FIG. 4(a) simulates Δ δ1And delta2All fall within the interval [ - π/10, π/10]PDF curve of time, sampling interval is pi/200, due to delta1And delta2The sign of (a) will influence the position of the peak of the PDF curve, and for ease of understanding, only Δ δ is shown1>0,|Δδ2|<|Δδ1Part of l.
Acquiring actual truncated phase distribution, and calculating a corresponding PDF curve F; FIG. 4(b) shows the PDF curve of the fringe pattern to be measured, where Δ δ is simulated1＝π/19，Δδ2-pi/17, i.e. (0.165, -0.185).
Through correlation operation, finding a simulation curve cluster F0The curve with the maximum correlation degree with the actual curve is obtained, and the correlation coefficient calculation formula is as follows:
Figure GDA0002464930130000091
when the correlation coefficient R takes the maximum value, the phase shift error corresponding to the simulation curve is the phase shift error of the curve to be measured, namely the phase shift error existing in the actual measurement system. The correlation coefficient results of the curve to be measured and the simulated curve are shown in fig. 4(c), and the maximum value of the correlation coefficient falls at the position of (0.157, -0.189), so that the detection result is Δ δ1＝0.157，Δδ2The detection error is (-0.008, -0.004), almost close to the correct value (-0.189) (note: the detection error is related to the sampling interval, when the simulation error case is one of the curve clusters, the detection error is 0).

Claims (8)

1. A phase shift error detection method based on a probability distribution function is characterized in that N-1 phase shift errors are generated by N-step phase shift interferometry, N is greater than or equal to 3, and the phase shift error detection method comprises the following steps:
step one, acquiring actual truncated phase distribution, and calculating a corresponding probability distribution function curve;
the probability distribution function curve F:
Figure FDA0002974786230000011
wherein, M represents the number of sampling points, M is 0,1, 2.
Figure FDA0002974786230000012
To truncate the phase;
simulating a probability distribution function curve under the condition of generating a series of different phase shift errors in a set interval, and establishing a simulation curve database;
performing correlation operation on the probability distribution function curve and a simulation curve database established in advance, finding out a simulation curve with the maximum correlation degree with the probability distribution function curve in the simulation curve database, and taking the phase shift error of the simulation curve as the detected phase shift error; wherein the correlation degree calculates a correlation coefficient R (m, n) using the following formula:
Figure FDA0002974786230000013
wherein F is the probability distribution function curve, F0(m, n) is a simulation curve; where m, n are the numbers of phase shift errors.
2. The method for detecting phase shift error based on probability distribution function of claim 1, wherein the step one comprises:
acquiring N phase shift pictures, describing light intensity of pixels in the phase shift pictures, and measuring a phase shift amount under an ideal condition according to N steps of phase shift interferometry to obtain a truncated phase with a phase shift error;
and obtaining the corresponding probability distribution function curve according to the truncated phase with the phase shift error.
3. The method as claimed in claim 1, wherein the light intensity I of the pixels in the n-th phase-shifted picture is detected by using the probability distribution functionn(x,y)：
Figure FDA0002974786230000021
Wherein N is 0,1,.., N-1; a (x, y), B (x, y) respectively represent the background light intensity and the modulation degree,
Figure FDA0002974786230000022
is the initial phase, δ, to be measurednIs a phase shift, and δ0＝0。
4. A method as claimed in any one of claims 1 to 3, characterized in that the process of detecting phase shift errors of N-step phase shift interferometry is decomposed into detection
Figure FDA0002974786230000023
Phase shift error of three-step phase shift interferometry.
5. The method of claim 4, wherein the method of detecting phase shift error of three-step phase shift interferometry comprises:
s1, acquiring three phase shift pictures, describing the light intensity of pixels in the phase shift pictures, and measuring the phase shift amount under an ideal condition according to three-step phase shift interferometry to obtain a truncated phase with an error;
obtaining a corresponding three-step phase shift interferometry probability distribution function curve according to the truncated phase with the error;
s2, carrying out correlation operation on the three-step phase shift interferometry probability distribution function curve and a three-step phase shift interferometry simulation curve database established in advance, finding out a simulation curve with the maximum correlation degree with the three-step phase shift interferometry probability distribution function curve in the three-step phase shift interferometry simulation curve database, and taking the phase shift error of the simulation curve as the detected phase shift error.
6. The method of claim 5, wherein the three-step phase-shift interferometry simulation curve database is established by: simulating a series of different phase shift errors delta in a set interval1And phase shift error delta2Simulation curve F for the case0(m, n) where m and n are phase shift errors delta1Phase shift error delta2The number of (2).
7. The method as claimed in claim 6, wherein the intensity I of any pixel in the nth phase-shifted picture is S1n(x,y)：
Figure FDA0002974786230000031
Wherein n is 0,1, 2;
when the three-step phase shift is the full-period three-step phase shift, the phase shift amounts are 0 and delta1＝2π/3、δ24 pi/3, resulting in a truncated phase with phase shift error
Figure FDA0002974786230000032
Figure FDA0002974786230000033
8. A method for detecting phase shift error based on probability distribution function as claimed in claim 1, wherein the analog curve with the maximum correlation is the analog curve with the maximum correlation coefficient R (m, n).
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