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In this Section, numerical examples are carried out to investigate the fitting performance of the PSO-AMLS-based method. Functions similar to the calibration data of the infrared radiometer are used in this section. In the first example, the independent variables are uniform distribution. In the second example, the independent variables are non-uniform distribution. The data were processed in this paper with code written in MATLAB.
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The formula used in the first example is as follows:
A set of points (xi, yi) are chosen, where i=0-20. The independent variables are uniform distribution from 0 to 20 with interval h=1, while the interval of dependent variables is gradually increased and determined by equation (13). The data used in the first example is shown in Table 1.
x y x y x y 0 1 7 3.6458 14 4.7417 1 2 8 3.8284 15 4.873 2 2.4142 9 4 16 5 3 2.7321 10 4.1623 17 5.1231 4 3 11 4.3166 18 5.2426 5 3.2361 12 4.4641 19 5.3589 6 3.4495 13 4.6056 20 5.4721 Table 1. The data used in the first example
The Gaussian function is chosen to be the weighting function. Considering the fitting accuracy and oscillation problem, the polynomial order is 6 for the LS method. The fitting performance is characterized by the fitting error δ between real value yi and fitting value yif
The PSO is used to optimize the MLS parameters dmi, β, and m, where m represents the type of the basic function. The basic function is set to be p1(x) = [1], p2(x) = [1, x] and p3(x) = [1, x, x2] when m=1, m=2 and m=3, respectively. Search space is organized in three dimensions, one for each parameter. Because the independent variables are uniform distribution, there is no AP performed in this example. The fitting errors δ of the LS and PSO-AMLS method when the independent variables are uniform distribution are shown in Figure 2.
Figure 2. The fitting errors δ of the LS and PSO-AMLS method when the independent variables are uniform distribution: (a) nnodes=11; (b) nnodes=21
The parameter “nnodes” is the number of nodes used in the PSO-AMLS method. For the cases when the independent variables of fitting data are uniform distribution, the PSO-AMLS method can acquire better results than the LS method. The PSO-AMLS method can follow the changes of the original function even with low order basis function, while, for the global approximation scheme like the LS, the oscillation phenomenon occurs, which increases the approximation error. When nnodes is 11, an outlier exists where δPSO-AMLS is larger than δLS. The position and amplitude of the outlier are x=1 and δPSO-AMLS=0.19. According to the amplitude δ, the results when nnodes is 21 are much better than the results when nnodes is 11, fewer nnodes means larger dmi is needed. When nnodes is too little, the fitting accuracy may be affected; when nnodes is too much, the regional feature may not be obvious but can be enhanced by the adjustment of parameters in the AMLS.
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In this subsection, a set of points (xi, yi) are used as the calibration points, where i=1-15. The dependent variables are uniformly distributed with interval h=1, while the independent variables are non-uniformly distributed and the intervals are gradually increased. The data used in the second example is shown in Table 2.
y x y x y x 1 1 6 11.5 11 46.5 2 1.5 7 16.5 12 56.5 3 2.5 8 22.5 13 67.5 4 4.5 9 29.5 14 79.5 5 7.5 10 37.5 15 92.5 Table 2. The data used in the second example
The parameters setting in the PSO and the AMLS are the same as in Subsection 3.1. Because the independent variables are non-uniform distribution, logarithmic transformation is performed before the MLS in this example. For the observed fitting results of the LS method are too bad when the independent variables are non-uniformly distribution, the Adaptive Least Squares (ALS) method is used in this subsection to compare with the PSO-AMLS method, where logarithmic transformation is performed before the LS. The fitting errors δ of the ALS and the PSO-AMLS method when the independent variables are non-uniform distribution are shown in Figure 3, in which the x-coordinate is the serial number of points.
Figure 3. The fitting errors δ of the ALS and the PSO-AMLS method when the independent variables are non-uniform distribution
It is shown in Figure 3 that, for the cases when the independent variables of fitting data are non-uniform distribution, the PSO-AMLS method can acquire much better results than the LS method. The oscillation phenomenon occurs both for the ALS and the PSO-AMLS method, but the oscillation amplitude of the PSO-AMLS method is much smaller than that of the ALS method. Combined with the results obtained in Subsection 3.1, it is concluded that the fitting performance of the PSO-AMLS method is superior compared to the LS and ALS method.
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In this Section, based on the infrared radiometer and the data fitting method described above, the calibration of the infrared radiometer has been carried out. The comparison of different data fitting methods is performed to validate the effectiveness of the PSO-AMLS method in the real calibration.
The response voltages V of the infrared radiometer were measured in the calibration, where a blackbody-collimator was used as the standard source. Figure 4 is a diagram of the infrared radiometer. The basic components of the infrared radiometer are the optical lens, chopper, detector, preamplifier, lock-in amplifier, and A/D.
The main specifications of the infrared radiometer are shown in Table 3. The basic components of the blackbody-collimator are the blackbody, collimator, and aperture. The main specifications of the blackbody-collimator are shown in Table 4. The blackbody temperatures T are uniform distribution from 100 ℃ to 700 ℃ with interval h=50 ℃. The corresponding calibration data of T and V are shown in Table 5.
Component Parameter Optical lens Diameter: 70 mm Field of view: ±0.6° Spectral channels MW: 3.7-4.8 µm Detector Type: Insb A/D Precision: 16 Frequency: 100 kHz Preamplifier 106× Table 3. Main specifications of the infrared radiometer
Component Parameter Blackbody Model: HFY-200C Temperature: 5-1000 ℃ Aperture Diameter: 0.1-13.5 mm Collimator Model: HGD-1 Focal length: 650 mm Table 4. Main specifications of the blackbody-collimator
V /V T /℃ V /V T /℃ 0.02 100 1.354 450 0.05 150 1.828 500 0.117 200 2.383 550 0.227 250 2.994 600 0.394 300 3.75 650 0.632 350 4.555 700 0.949 400 Table 5. The calibration data of T and V in the calibration of the infrared radiometer
It is shown from Table 5 that the dependent variables are uniform distribution with equal interval, while the independent variables are non-uniform distribution and the intervals increase gradually. The reasons for this phenomenon are as follows: (1) the response voltage of the infrared radiometer is affected significantly by the standard radiation source; (2) the response of the infrared radiometer is mainly determined by the detection ability of the detector.
For the data characteristic mentioned above and the number limitation of calibration data, conventional data fitting methods commonly used in the calibration of infrared radiometer cannot achieve enough fitting accuracy. The fitting errors δ using the LS method and ALS method are shown in Figure 5. In Figure 5(a), polynomial approximate based on the LS method is carried out with polynomial orders from 5 to 6. In Figure 5(b), logarithmic transformation is carried out before the polynomial approximate with the polynomial orders from 5 to 6.
It can be obviously seen that the ALS method can acquire much better results than the LS method, which means the AP is effective to improve the fitting accuracy and stability when the fitting points are non-uniform distribution. But for the LS and ALS methods, the oscillation phenomenon occurs and the approximation errors change dramatically, especially when the number of calibration points is small and the fitting data are non-uniform distribution.
To improve the fitting accuracy, the PSO-AMLS-based method is applied in the calibration of the infrared radiometer. The fitting errors of the PSO-AMLS method are compared to the ALS method to verify the performance of the PSO-AMLS method. For the PSO-AMLS with LT, only logarithmic transformation is contained in the AP, but for the PSO-AMLS with LT and DT, both logarithmic transformation and data translation are contained in the AP. The fitting errors of the PSO-AMLS and ALS method are shown in Figure 6.
In Figure 6, the R2 of the PSO-AMLS method is 1.8437 compared with 3.6962 of the ALS method, which means better fitting results are achieved by the PSO-AMLS method. The superior performance of the PSO-AMLS method is more obvious in the first half of calibration points, while in the second half, an outlier exists in point 10 where δPSO-AMLS is slightly larger than δALS, and δPSO-AMLS and δALS are almost the same for point 12 and 13. It is supposed that the reason of this phenomenon is the number limitation of calibration points in the second half of the calibration data. Fitting errors δ will be increased when fewer calibration points are used. In the shape function matrix of the PSO-AMLS method, the locality is strong at the beginning, but decline with the increase of voltage. At the end of the calibration points, the locality is weakened but the globality is enhanced. In addition, it is shown that the performance of the PSO-AMLS with LT is only slightly better than the PSO-AMLS with LT and DT, which means the effect of logarithmic transformation is more obvious than data translation in the AP. The PSO-AMLS method should provide a better fitting result but suffer from the number limitation of fitting data, especially in the calibration of the infrared radiometer. As a future line of research, to improve the fitting performance when the samples of fitting data are fewer, more AP methods will be used.
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In this section, the infrared radiometer calibrated using different data fitting methods is applied to the measurement of an infrared imaging simulator in the medium waveband. The measurement parameter is the equivalent blackbody temperature. The measured targets include the target channel and the interference channels. The source of target channel and interference channels are high-pressure xenon lamps, and the output radiation is controlled by the adjustment of the working current. The collimator and apertures used in the experiment are the same as in the calibration of the infrared radiometer. The equivalent blackbody temperature Tt measured by the thermal imaging camera ImageIR-5300 is used as the reference value. Tm measured by the infrared radiometer using two fitting methods is compared to the reference value Tt to obtain the deviation δ*:
The deviation δ* obtained for different fitting methods are shown in Figure 7, where TC represents Target Channel, IC 1, 2, and 3 represents Interference Channel 1, 2, and 3.
It is shown from Figure 7 that for different measured targets, the measured Tm for the PSO-AMLS method is closer to the reference value compared to the ALS method. Besides, compared with the measurement results in section 3.1, more error sources are introduced in this measurement, including the repeatability and reproducibility, which results in the unexpected changes of the measured voltage. However, the most important result is that the PSO-AMLS method is more superior compared to other conventional fitting methods, which is effective for the improvement of measurement accuracy of the infrared radiometer.
Hybrid PSO-AMLS-based method for data fitting in the calibration of the infrared radiometer
doi: 10.3788/IRLA20200471
- Received Date: 2020-12-07
- Rev Recd Date: 2021-03-24
- Publish Date: 2021-08-25
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Key words:
- infrared /
- radiometry /
- calibration /
- data fitting
Abstract: A hybrid PSO-AMLS-based method for data fitting in the calibration of the infrared radiometer was described. The proposed method was based on Particle Swarm Optimization (PSO) in combination with Adaptive Moving Least Squares (AMLS). The optimization technique involved parameters setting in the AMLS fitting, which significantly influenced the fitting accuracy. However, its use in the calibration of the infrared radiometer has not been yet widely explored. Bearing this in mind, the PSO-AMLS-based method, which was based on the local approximation scheme, was successfully used here to get the relationship between the radiation of the standard source and the output voltage of the infrared radiometer. The main advantages of this method were the flexible adjustment mechanism in data processing and the ability in reducing the adverse effect resulting from the non-uniform distribution of fitting data. Numerical examples and experiments are performed to verify the superior performance of the PSO-AMLS-based method compared to the conventional data fitting method.