CN108648239B - Calibration method of phase-height mapping system based on sectional polynomial fitting - Google Patents
Calibration method of phase-height mapping system based on sectional polynomial fitting Download PDFInfo
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Abstract
The invention discloses a calibration method of a phase-height mapping system based on sectional polynomial fitting, namely sectional calibration of the phase-height mapping system is carried out by adopting polynomial fitting, according to the method provided by the scheme, a measuring range is divided into intervals, and polynomial fitting calibration is carried out by adopting different fitting orders in different height ranges, so that the relative error of each position in the measuring range is minimum, the three-dimensional reconstruction precision under the sectional polynomial fitting calibration in the scheme is higher than that under the polynomial fitting calibration of 1-6 orders, and the process is simplified.
Description
Technical Field
The invention relates to phase measurement profilometry, in particular to a calibration method of a phase-height mapping system based on sectional polynomial fitting.
Background
In the phase measurement profilometry, grating phase change is caused due to the effect of the height of an object to be measured on sinusoidal grating modulation, and the height information of the three-dimensional profile of the object to be measured is hidden in the phase change. Therefore, to measure the height information of the object to be measured, it is critical to establish a corresponding relationship between the height and the phase change, especially a mathematical relationship between the height and the phase, i.e. a phase-height mapping mathematical model. The existing phase-height mapping mathematical model is mainly divided into two types: an explicit phase-height mapping model and an implicit phase-height mapping model. The explicit phase-height mapping model needs to accurately measure specific parameters of a projector-camera system structure in advance in the calculation process, such as a horizontal distance between a Digital Light Processing (DLP) device and a Charge Coupled Device (CCD), a vertical distance between the DLP and a reference plane, an included angle between a DLP optical axis and a CCD optical axis, and the like; in contrast, the implicit phase-height mapping model does not need to accurately measure specific parameter values of the projector-camera system structure in the calculation process, and only needs to calibrate the corresponding relation between the phase and the height through a calibration object.
The polynomial fitting is a process of finding out a polynomial function which is the best match of a group of experimental data with hidden relations according to the principle of square error and minimum. The polynomial fitting calibration method has the advantages of simple calibration process, high calibration accuracy, avoidance of projector calibration and the like, and is widely researched and applied, the existing research mainly focuses on the influence of polynomial fitting orders on the calibration accuracy, no relevant research is carried out on the influence of fitting data quantity on the calibration accuracy within a certain height range, and the data quantity has important influence on the process complexity, the result accuracy and the like of the polynomial fitting calibration method.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a sectional polynomial fitting calibration method by dividing different height ranges, so as to simplify the calibration process and improve the result precision.
The invention provides a calibration method of a phase-height mapping system based on sectional polynomial fitting, which is characterized by comprising the following steps of:
1) obtaining a plurality of height values and phase differences of corresponding positions of an object to be measured, and respectively performing multi-order polynomial fitting on each group of height values and phase differences to sequentially obtain fitting parameters of each order; carrying out deviation and amplification processing on the fitting parameters of each order, and storing the fitting parameters as image data;
2) performing three-dimensional reconstruction and detection on the standard correction plate by using the fitting parameters of each order in the step 1) to obtain fitting calibration parameters of each order, and simultaneously calculating the fitting measurement height value of each bulge in the standard correction plate under each order and the absolute error between the fitting measurement height value and a standard value, wherein the fitting order with the minimum absolute error is taken as the first order;
3) extracting a group of phase difference values of equal difference intervals in a range to be calibrated, performing polynomial fitting calibration of the first order to obtain calibration parameters of each interval and performing precision verification on the calibration parameters, wherein the interval with the minimum average relative error is a first fitting interval;
4) respectively carrying out height detection under each order of fitting calibration parameters in the step 2) at theoretical heights of the object to be detected with equal difference, and calculating relative errors between actual detection values and theoretical height values;
5) dividing the height of the object to be measured into a plurality of intervals according to the relative error obtained in the step 4), wherein the minimum relative error occurs when each interval is fitted by a polynomial of a specific order;
6) fitting each interval divided in the step 5) by the first-order polynomial through the calibration parameters in the step 3), performing three-dimensional reconstruction and detection, and judging the best fitting order of each interval according to the detection values;
7) calculating system calibration parameters of each interval according to the optimal fitting order in the step 6), carrying out three-dimensional detection and reconstruction, and further carrying out sectional polynomial fitting calibration on the object to be tested.
Preferably, the formula of the polynomial fitting in step 1) is:
H(x,y)＝anΔφn(x,y)+an-1Δφn-1(x,y)+…+a1Δφ(x,y)+a0；
where n is the fitting order, H (x, y) is the fitting height value of the (x, y) position, Δ φ (x, y) is the phase difference of the (x, y) position, an,an-1,…,a0Fitting coefficients for the polynomial.
Preferably, the center of the correction plate in step 2) has 16 protrusions, including rectangular protrusions and circular protrusions; and the fitting calibration parameters of each order are respectively subjected to 30 times of height reconstruction and detection repeatability tests.
Preferably, the fitting of the multi-order polynomial in the step 1) is a fitting of a 1-6 order polynomial, and the range of the plurality of height values is-250 to +250 μm; the first order in step 2) is order 2.
Preferably, the range to be calibrated in the step 3) is between-250 and +250 μm, and the group of equal difference intervals is 10 μm, 20 μm, 30 μm, 40 μm, 50 μm, 60 μm, 70 μm, 80 μm, 90 μm and 100 μm; the first fitting interval is 90 μm, the minimum average relative error is δ is 1.06%, and the system calibration time t is 33.884 s.
Preferably, the theoretical height described in step 4) is 10 μm, 20 μm, 30 μm, …, 500 μm.
Preferably, the plurality of intervals in the step 5) are 0-90 μm, 100-190 μm, 190-480 μm and 480-500 μm.
Preferably, the best fit order of each interval in step 6) is: fitting by 4-order polynomial at 0-90 μm, fitting by 3-order polynomial at 100-190 μm, fitting by 1-order polynomial at 190-480 μm, and fitting by 4-order polynomial at 480-500 μm.
The invention has the beneficial effects that: the calibration method provided by the invention is to adopt polynomial fitting to perform sectional calibration of a phase-height mapping system, according to the method provided by the scheme, the range of the measuring range is divided into intervals, and different fitting orders are adopted to perform polynomial fitting calibration in different height ranges, so that the relative error of each position in the measuring range is minimum, namely the three-dimensional reconstruction precision under the sectional polynomial fitting calibration in the scheme is higher than that under the 1-6 th-order polynomial fitting calibration, and the process is simplified.
Drawings
FIG. 1 is a graph showing the comparative analysis of the fitting measurement values and the standard values in the order of 1 to 6;
FIG. 2 is a graph of absolute error versus standard value for fit measurements at 1-6 orders;
FIG. 3 is a standard deviation of fit measurements at 1-6 orders;
FIG. 4 is a graph of the average relative error under different fitting interval calibrations;
FIG. 5 is a system calibration time under calibration of different fitting intervals;
FIG. 6 is a comparative analysis of relative errors under 1-6 order fitting;
FIG. 7 is a graph of fitted order change with minimal relative error;
FIG. 8 is a relative error analysis for a segmented fit and for each order of fit;
FIG. 9 is a plot of the average relative error under the segmented fit and the various order fits.
Detailed Description
The present invention is further described in detail below with reference to the attached drawings so that those skilled in the art can implement the invention by referring to the description text.
It will be understood that terms such as "having," "including," and "comprising," as used herein, do not preclude the presence or addition of one or more other elements or groups thereof.
The invention is further illustrated by the following specific examples. However, the specific details of the embodiments are merely for explaining the present invention and should not be construed as limiting the general technical solution of the present invention.
Step one, in this embodiment, each height value of the phase difference at the corresponding position is obtained by means of an external high-precision mobile device, and then a polynomial fitting parameter between the phase difference and the height is obtained by a polynomial fitting method, wherein a formula of the polynomial fitting is as follows:
H(x,y)＝anΔφn(x,y)+an-1Δφn-1(x,y)+…+a1Δφ(x,y)+a0；
where n is the fitting order, H (x, y) is the fitting height value of the (x, y) position, Δ φ (x, y) is the phase difference of the (x, y) position, an,an-1,…,a0Fitting coefficients for the polynomial.
Setting the height of the reference surface as 0, the downward direction as negative (-) and the upward direction as positive (+), driving the optical structure (composed of a projector and a camera) to move up and down at equal intervals (the interval is 50 μm) by a high-precision servo motor, and obtaining the phase difference of the positions with the height (unit: μm) equal to-250, -200, … and 200,250 relative to the reference surface as delta phi-250、Δφ200、…、Δφ250、Δφ200For a total of 11 sets of data pairs. Respectively carrying out 1-6-order polynomial fitting on 11 groups of data pairs through a polynomial fitting formula, wherein the pixel polynomial fitting parameters are mutually independent due to relative independence between pixels, and finally processing (offset 100 and method 100 times) each-order fitting parameter and storing as image data (PNG format), taking 2-order fitting as an example, the height reconstruction formula is as follows:
H(x,y)＝p2(x,y)×Δφ2(x,y)+p1(x,y)Δφ(x,y)+p0(x,y)；
wherein p is2(x,y)，p1(x,y)，p0(x, y) are constant term coefficients, first term coefficients and second term coefficients of the 2-order fitting, respectively; h (x, y) is the three-dimensional reconstruction height of the (x, y) position, delta phi (x, y) is the phase difference of the (x, y) position, and the rest orders can be analogized.
And step two, performing three-dimensional reconstruction and detection on the standard correction plate by using the obtained 1-6 order fitting parameters to obtain each order fitting calibration parameter, and simultaneously calculating the fitting measurement height value of each bulge in the standard correction plate under each order and the absolute error of the fitting measurement height value and a standard value, wherein the fitting order with the minimum absolute error is taken as the first order.
Specifically, the standard correction plate is verified by a third party, 16 protrusions (circular protrusions 1-8 and rectangular protrusions 9-16) are arranged in the middle of the correction plate, and corresponding height values are listed in table 1. And (3) performing three-dimensional reconstruction on the standard correction plate by using the 1-6 th-order polynomial fitting parameters in the step one, and detecting the height values of the No. 1-16 bulges in the 1-6 orders, which are listed in Table 1.
TABLE 1 Standard and measured values (units: μm) of the Standard calibration plate
| |
|
No. 3 | |
|
|
No. 7 | |
|
| Standard value | 100.8 | 101.1 | 101.3 | 101.8 | 101.5 | 101.5 | 101.9 | 101.9 |
| 1 st order | 96.098 | 99.764 | 99.709 | 100.6 | 100.95 | 101.86 | 101.38 | 102.14 |
| 2 order | 97.846 | 100.46 | 100.28 | 101.5 | 101.56 | 102.59 | 102.1 | 102.63 |
| 3 order | 96.543 | 97.704 | 98.243 | 100.99 | 99.816 | 100.79 | 100.51 | 100.13 |
| 4 th order | 98.553 | 96.402 | 97.268 | 98.585 | 97.402 | 98.313 | 97.591 | 100.11 |
| 5 th order | 117.78 | 123.21 | 160.05 | 108.62 | 140.01 | 163.83 | 128.23 | 132.26 |
| 6 th order | 504.33 | 340.07 | 600.83 | 343.46 | 346.54 | 314 | 392.55 | 344.11 |
| |
|
|
|
|
|
|
|
|
| Standard value | 100.7 | 100.7 | 100.7 | 101.2 | 101.2 | 100.8 | 101.6 | 101.9 |
| 1 st order | 101.15 | 101.89 | 102.12 | 101.69 | 102.73 | 103.22 | 102.63 | 103.36 |
| 2 order | 101.9 | 102.65 | 102.53 | 102.35 | 103.27 | 103.82 | 103.06 | 104.02 |
| 3 order | 100.51 | 100.74 | 100.91 | 100.75 | 101.96 | 102.55 | 101.93 | 102.47 |
| 4 th order | 100.92 | 99.506 | 95.702 | 99.053 | 100.6 | 101.49 | 99.752 | 100.34 |
| 5 th order | 139.93 | 141.75 | 136.95 | 128.88 | 153.75 | 125.57 | 125.57 | 129.34 |
| 6 th order | 518.46 | 430.12 | 508.57 | 630.38 | 402.41 | 486.93 | 486.93 | 488.84 |
FIG. 1 is a graph showing the comparison between the measured values of 1 to 6 orders and the standard value for the third party certification, and FIG. 2 is a graph showing the comparison between the measured values of 1 to 6 orders and the absolute error of the standard value for the third party certification. As can be seen from FIGS. 1 and 2, the absolute error Delta is less than 5 μm (the height value is about 100 μm) in the order of 1-4; 5. the absolute errors of the 6 th order fit are around 40 μm and 300 μm, respectively, and greatly deviate from the detection height. Therefore, the 1-4 order fitting is more suitable for polynomial fitting calibration, and in the 1-4 order fitting, the absolute error 4 mu m < delta < 5 mu m obtained by the 1, 3 and 4 order fitting is less than the absolute error delta; the absolute error Delta < 3 mu m obtained by the 2 nd order fitting, and the average relative error thereof
Therefore, in the 1-4 order fitting, the 2-order fitting can obtain smaller absolute error, and the three-dimensional reconstruction precision is higher.
In order to verify the stability of polynomial fitting calibration under 1-6 orders, the parameters under 1-6 orders of fitting calibration are respectively subjected to 30 times of height reconstruction and detection repeatability tests, and the standard deviation (fig. 3) and the average standard deviation (recorded in table 2) of the measured values under each order are obtained.
TABLE 2 mean standard deviation
As shown in fig. 3 and table 2, the standard deviation σ at order 1 to 4 is less than 1; the standard deviation is more than 0.5 and less than 2.5 under 5 orders; standard deviation 2.5 < at 6 th orderSigma is less than 9.5. In Table 2, the average standard deviations in the order of 1 to 2
Mean standard deviation at 3-4 orders
Mean standard deviation in order 5-6
The smaller the standard deviation, the better the stability, from which it can be seen that: the multiple ounce fitting calibration stability under the order of 1-2 is the best; 3 to 4 order stability order; the stability is the worst in 5-6 orders. Through polynomial fitting calibration and three-dimensional height reconstruction under the 1-6 order, the fact that higher reconstruction accuracy (the absolute error delta is less than 5 mu m) can be obtained when the fitting order is 1-4 is found. Further analysis and study were conducted on the 1-4 order polynomial fitting calibration coefficients, and it was found that when the fitting order is 2, the reconstruction accuracy (absolute error: Δ ═ 1.36 μm; relative error: δ ═ 1.34%) and stability (average standard deviation:
)。
and step three, extracting a group of phase difference values of equal difference intervals in the range to be calibrated, performing polynomial fitting calibration of the first order to obtain calibration parameters of each interval and performing precision verification on the calibration parameters, wherein the interval with the minimum average relative error is the first fitting interval.
Specifically, phase difference values at height (depth) intervals of 10 μm, 20 μm, 30 μm, …, and 100 μm are extracted, respectively, and corresponding polynomial fitting calibration (the fitting order is 2) is performed, resulting in 10 sets of calibration parameters. The calibration range is-250 μm to 250 μm, there are 51 pairs of fitting data (phase difference + height) when the height (depth) interval is 10 μm, and so on, and there are 6 pairs of fitting data when the height (depth) interval is 100 μm. Respectively carrying out precision verification on 10 pairs of calibration parameters, respectively detecting at theoretical heights of 20 mu m, 40 mu m, 60 mu m, … and 500 mu m, and respectivelyThe average relative error for each set of measurements was calculated, and as shown in FIG. 4, a smaller average relative error was obtained when the fitting interval was 90 μm
Fig. 5 shows the system calibration time at different fitting intervals, and as the intervals increase, the time required for system calibration decreases, and the whole system calibration tends to decrease. It can be seen that, in the range of 500 μm, the polynomial fitting order is 2, and when the fitting interval is 90 μm, a smaller average relative error can be obtained
And has higher system calibration speed (calibration time)t＝33.884s)。
Respectively carrying out height detection under each order of fitting calibration parameters at theoretical heights of the object to be detected with equal difference, and calculating relative errors between actual detection values and theoretical height values;
dividing the height of the object to be measured into a plurality of intervals according to the obtained relative error, wherein the minimum relative error occurs when each interval is fit by a polynomial of a specific order;
sixthly, fitting each divided interval by a polynomial of the first order through the calibration parameters, performing three-dimensional reconstruction and detection, and judging the best fitting order of each interval according to the detection values;
step seven, calculating system calibration parameters of each interval according to the optimal fitting order, carrying out three-dimensional detection and reconstruction, and further carrying out sectional polynomial fitting calibration on the object to be detected;
the first step to the third step are pre-preparation processes, and the fourth step to the seventh step are sectional polynomial fitting calibration processes for the object to be measured.
Specifically, height detection is performed at theoretical heights of 10 μm, 20 μm, 30 μm, …, and 500 μm under 1-6 order fitting calibration parameters, and relative errors between actual detection values and theoretical height values are calculated, the relative error change at each order is shown in fig. 6, fig. 6 is a relative error change diagram at each theoretical height position at 1-6 orders, fig. 7 is a fitting order change diagram corresponding to the minimum relative error at each theoretical height, and it can be known from the diagrams that when the heights are less than 90 μm, the minimum relative error is obtained under 4-order fitting calibration; when the height is between 100 and 190 mu m, the minimum relative error is obtained under 3-order fitting calibration; when the height is between 190 and 480 mu m, the minimum relative error is obtained under the 1-order fitting calibration; the minimum relative error was obtained under the 4-order fit calibration when the height was between 480 μm and 500 μm. The smaller the relative error, the higher the three-dimensional reconstruction accuracy.
In order to obtain higher reconstruction accuracy in the range of 500 μm, polynomial fitting calibration can be performed in the range of measurement by adopting different fitting orders in different height ranges according to the information of fig. 7, so as to achieve the purpose of minimizing the relative error of each position in measurement; according to FIG. 7, when the height is smaller than that when the height is smaller than 90 μm, a polynomial of order 4 is adopted for fitting calibration; when the height is between 100 and 190 mu m, fitting and calibrating by using a 3-order polynomial; when the height is 190-480 mu m, fitting and calibrating by using a 1 st-order polynomial; when the height is between 480 and 500 mu m, fitting and calibrating by using a polynomial of 4 th order. In actual detection, the system parameters under 2-order fitting calibration obtained in the second step can be firstly subjected to three-dimensional reconstruction and detection, then the best fitting order of each interval is judged according to the detection values, and finally the system calibration parameters of each interval are used for three-dimensional reconstruction and detection.
According to the interval, the system is subjected to sectional polynomial fitting calibration, then height reconstruction and detection are carried out at the positions with theoretical heights of 20 microns, 40 microns, 60 microns, … microns and 500 microns under the sectional polynomial fitting calibration, the relative error between the reconstructed height and the theoretical height is shown in figure 8, a dotted line is a relative error change diagram of the reconstructed height and the theoretical height under the sectional polynomial fitting calibration, and solid lines are relative error change diagrams of the reconstructed height and the theoretical height under the 1-6 th-order polynomial fitting calibration respectively, so that the three-dimensional reconstruction precision under the sectional polynomial fitting calibration is higher than that under the 1-6 th-order polynomial fitting calibration.
FIG. 9 shows the average relative errors of the reconstructed heights and the theoretical heights at 20 μm, 40 μm, 60 μm, …, and 500 μm under the sectional fitting and the fitting of 1-6 orders, and as can be seen from FIG. 9, the average relative errors at 5 and 6 orders
Average relative error in order 2 and 3
Average relative error in order 1 and 4
Average relative error under segmented fit
Therefore, the accuracy under the sectional polynomial fitting calibration is higher than that under the 1-6 th-order polynomial fitting calibration.
While embodiments of the invention have been disclosed above, it is not intended to be limited to the uses set forth in the specification and examples. It can be applied to all kinds of fields suitable for the present invention. Additional modifications will readily occur to those skilled in the art. The invention is therefore not to be limited to the specific details and embodiments shown herein without departing from the general concept defined by the appended claims and their equivalents.
Claims (6)
1. A calibration method of a phase-height mapping system based on sectional polynomial fitting is characterized by comprising the following steps:
1) obtaining a plurality of height values and phase differences of corresponding positions of an object to be measured, and respectively performing multi-order polynomial fitting on each group of height values and phase differences to sequentially obtain fitting parameters of each order; performing offset and amplification processing on the fitting parameters of each order, and storing the fitting parameters as image data, wherein the fitting of the multi-order polynomial is that of 1-6 orders;
2) performing three-dimensional reconstruction and detection on the standard correction plate by using the fitting parameters of each order in the step 1) to obtain fitting calibration parameters of each order, and simultaneously calculating the fitting measurement height value of each bulge in the standard correction plate under each order and the absolute error between the fitting measurement height value and a standard value, wherein the fitting order with the minimum absolute error is taken as the first order;
3) extracting a group of phase difference values of equal difference intervals in a range to be calibrated, performing polynomial fitting calibration of the first order to obtain calibration parameters of each interval and performing precision verification on the calibration parameters, wherein the interval with the minimum average relative error is a first fitting interval;
4) respectively carrying out height detection under each order of fitting calibration parameters in the step 2) at theoretical heights of the object to be detected with equal difference, and calculating relative errors between actual detection values and theoretical height values;
5) dividing the height of the object to be detected into a plurality of intervals according to the relative error obtained in the step 4), wherein the minimum relative error occurs when each interval is matched by a polynomial of a specific order, and the intervals are 0-90 μm, 100-190 μm, 190-480 μm and 480-500 μm within the range of 500 μm;
6) fitting each interval divided in the step 5) by the first-order polynomial through the calibration parameters in the step 3), performing three-dimensional reconstruction and detection, and judging the best fitting order of each interval according to the detection values; the best fit order for each interval is: fitting by using a 4-order polynomial at 0-90 mu m, fitting by using a 3-order polynomial at 100-190 mu m, fitting by using a 1-order polynomial at 190-480 mu m, and fitting by using a 4-order polynomial at 480-500 mu m;
7) calculating system calibration parameters of each interval according to the optimal fitting order in the step 6), carrying out three-dimensional detection and reconstruction, and further carrying out sectional polynomial fitting calibration on the object to be tested.
2. Calibration method according to claim 1, wherein the formula of the polynomial fit in step 1) is:
wherein the content of the first and second substances,nin order to fit the order of the order,
is composed of
The value of the height of fit of the position,
is composed of
The phase difference of the positions is determined,
fitting coefficients for the polynomial.
3. The calibration method as claimed in claim 1, wherein in step 2), the center of the calibration plate has 16 protrusions, including rectangular protrusions and circular protrusions; and the fitting calibration parameters of each order are respectively subjected to 30 times of height reconstruction and detection repeatability tests.
4. The calibration method as claimed in claim 1, wherein the plurality of height values in step 1) range from-250 to +250 μm; the first order in step 2) is order 2.
5. The calibration method according to claim 4, wherein the range to be calibrated in step 3) is-250 to +250 μm, and the set of equal difference intervals is 10 μm, 20 μm, 30 μm, 40 μm, 50 μm, 60 μm, 70 μm, 80 μm, 90 μm, 100 μm; the first fitting interval is 90 μm and the minimum average relative error is, the system calibration time t =33.884 s.
6. The calibration method as claimed in claim 5, wherein the theoretical height in step 4) is 10 μm, 20 μm, 30 μm, …, 500 μm.
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