Wavelet analysis method for noise processing of non-diffractive beam triangulation system
1. Non-diffracting Bessel beam 　　 Laser triangulation is a commonly used method for non-contact precision measurement of geometric quantities such as length, displacement, and surface roughness. However, the traditional laser triangulation method is limited by the depth of focus of the incident beam and requires precise focusing of the incident beam. That is, the measurement system needs to be equipped with a precise focusing mechanism to ensure that the measured object is always in the measurement system during measurement. Within the deep range. This not only complicates the measurement system structure, but also has an impact on the measurement accuracy and the measurement speed. If the non-diffracting Bessel beam is used as the incident beam, the problem of focal depth limitation in the conventional laser triangulation method can be better solved. The exact solution [1,2] is the exact solution of the scalar field propagating at the speed of light c into the free space z ≥ 0 E(x,y,z≥0,t)=exp[i(k11z-ωt)] In the formula, A ( E(x,y,z,t)=exp[i(k11z-ωt)]J0(k⊥r) (3) The ideal non-diffracting beam intensity is Since the beam intensity I is independent of the beam propagation distance z, the ideal non-diffracted beam has a small central spot (up to wavelength order), and its diameter and intensity do not vary with the propagation distance. 2. Diffraction-free beam triangulation system and its measurement error 　　 The working principle of the non-diffractive beam triangulation system implemented by the Axicon method is shown in Fig. 2. The beam emitted by the laser becomes a collimated beam after passing through the lens, and the beam fills the entire aperture of the Axicon prism. After the beam passes through the Axicon prism, it forms a non-diffracted beam, and then is reflected by the object O to the linear array CCD. The linear array CCD will The signal is converted into an electrical signal and the measurement data is obtained after signal processing such as A/D conversion. Since in the non-diffraction range, the longitudinal measurement range of the object O is always within the focal depth Z of the beam, a precise focusing system is not required. Fig. 2 The working principle of the non-diffractive beam triangulation system In a non-diffractive beam triangulation system, the ideal Bessel beam travel distance is long (focus depth is long), the center spot is small, the system can achieve high measurement accuracy in theory, but the noise measurement in the actual measurement will measure the system. Accuracy has a serious impact. 3. Wavelet denoising principle of measurement error 　　 Signal and noise are contradictory. To accurately analyze the signal, noise must be eliminated. Signal-to-noise separation is an effective method to eliminate noise interference and eliminate measurement errors. Therefore, the signal-to-noise separation method with good search effect and wide adaptation has important value in theory and application. Among them, ψmn(x)=2 In the formula, S is the number of decomposition layers, and {hi} and {gi} are already designed wavelet filters. 4. Wavelet Processing of Measurement Error in Non-diffractive Beam Triangulation System 　　 The ideal Bessel beam signal is a low-frequency signal with a smooth curve and large amplitude variation. The variation of the amplitude of the noise is small, and it is in an irregular random state. The variation frequency is also high, that is, the noise error is high frequency. signal. Since the frequency characteristics of the signal and the noise are different, the wavelet analysis method can be used to separate the signal from the noise and eliminate the noise error. Figure 4 Ideal beam signal + random noise Figure 5 Obeys uniformly distributed random noise Figure 6 Wavelet separation noise Fig. 7 Denoised Bessel beam signal Fig. 8 CCD measured Bessel beam Figure 9 shows the measured Bessel beam after wavelet processing. As can be seen from the figure, after the wavelet processing, the beam peak position is prominent and centered. According to the results of multiple measurements, the maximum deviation of the peak value of the measured signal from three pixels to one pixel reduces the accuracy of the corresponding position measurement. Figure 9 Measured Bessel beams after wavelet processing Lawn Mower,Grass Trimmer Machine,Farm Tools Grass Trimmer Machine,Multi Functional Lawn Mower Jiangsu Minnuo Group Co.,Ltd , https://www.jsminnuomachinery.com
The non-diffracting beam is a special set of solutions of the free space scalar wave equation, usually a zero-order or higher-order Bessel function distribution.
In 1987, J.Durnin gave the free space scalar wave equation 
(1) 
A ( 
)exp[ik⊥(xcos +ysin)] 
(2) ) is the angular coordinates Any complex function; kH2+k⊥2=( 
) 2, k11, k⊥ are wave vectors that are parallel and perpendicular to the wave propagation direction, respectively. When k11 is a real number, this solution represents a non-diffractive field. When A ( ) and nothing, especially when A ( )=1, this solution represents a completely axisymmetric diffraction-free field, 
J0 is the first zero-order Bessel function. It is a beam with a non-diffraction optical field distribution, ie, a non-diffracted beam. Because it has a Bessel function distribution, it is also called a non-diffracting Bessel beam, as shown in Figure 1. 
I(x,y,z0)= 
|E(r,t)|=I(x,y,z=0) (4)
The methods for implementing non-diffracting beams include the circular seam method, resonant cavity method, Axicon method, and spherical aberration method [3, 4]. The Axicon method has the advantages of simple structure and high energy efficiency. The processing method of measurement signal and measurement error in a non-diffractive beam triangulation system realized by the Axicon method is discussed below. 
In the actual measurement, the main factors influencing the accuracy of the non-diffractive laser beam triangulation system are the manufacturing error of the Axicon prism, the drift of the laser optical axis, the speckle noise, and the photoelectric noise of the CCD device. Speckle noise and CCD device photoelectric noise are the main noise sources. Speckle noise is the result of the formation of particles in the airspace between scattered light waves on the surface of each surface element when the rough surface is illuminated by coherent light. It is a scattered and interfering image composed of bright spots and dark spots. This noise is determined by hardware methods. Almost impossible to eliminate. The photoelectric noise of the CCD device mainly includes Johnson noise, noise of the charge packet output circuit, charge transfer loss noise, dark current noise, and photosensitive unevenness of each pixel of the CCD, and also belongs to a noise that cannot be completely eliminated by a hardware method. These noises influence the beam profile measured by the CCD, affect the position of the maximum peak of the beam (ie affect the position of the center spot), and thus seriously affect the position measurement accuracy of the measured object, so measures must be taken to remove it.
The filtering method is a more commonly used signal-to-noise separation method. Since signals and noise usually have different frequency characteristics, the use of a filter that satisfies a certain frequency requirement to filter measurement data can achieve the purpose of signal-to-noise separation. When using traditional filtering methods, the design of the filter is a big difficulty. According to the specific SNR data, it is necessary to design a specific filter that can match it. However, because the design of the filter is a difficult professional subject, it often limits the application of filtering methods by engineering technicians.
The development of wavelet analysis method provides a novel filtering method for the signal-to-noise separation technology. It can be applied to the filtering of various SNR data, and does not need to be specially designed to provide a great convenience for its practical application.
In engineering applications, the wavelet series is a very useful mathematical tool. The function f(x) can be expanded into wavelet series. 
(5) 
ψ(2mx-n),m,n∈Za. If m is different, the frequencies of the signals corresponding to ψm,n(x) are also different. It can be seen that the wavelet series can decompose the function f(x) into a combination of different frequency signal components. By eliminating certain frequency components (noise) in the wavelet series, the goal of signal-to-noise separation can be achieved.
However, it is inconvenient to directly use equation (5) for signal-to-noise separation. A simple and effective method is to use Mallat fast decomposition or wavelet packet algorithm. Mallat rapid decomposition method in the wavelet analysis is similar to the role of fast Fourier transform in Fourier analysis.
Let f(n) (n=1, 2, ..., N) be the analyzed data, let C0n=f(n), and decompose f(n) using the following formula: 
(6)
Mallat rapid decomposition first decomposes data f(n) into C1n and d1n, where C1n is the low-frequency signal and d1n is the high-frequency signal. Assuming that the frequency ω corresponding to f(n) satisfies |ω|<Ω, then the frequency corresponding to C1n satisfies 
The frequency corresponding to d1n is satisfied 
. The second layer of decomposition is to decompose C1n again. Repeatedly, f(n) is decomposed into components with different frequencies: d1n, d2n, ..., dsn, Csn. According to the frequency characteristics of the signal and the noise, signals and noise can be easily distinguished from these different frequency components, thereby completing the signal-to-noise separation.
Figure 3 shows a simulated ideal Bessel beam signal, which is superimposed with the random noise that follows the uniform distribution as shown in Figure 5, ie, an analog signal with noise interference is obtained, as shown in Figure 4. The Daubechies orthogonal wavelet filter is used to perform the wavelet decomposition on the signal data of FIG. 4 according to the equation (6), and the signal and the noise can be separated ideally, as shown in FIGS. 6 and 7 . Through multiple simulation experiments with ideal beam noise, the results show that, after wavelet processing, the peak maximum deviation of the noise-added beam is reduced from 6 μm to 1 μm, and the noise reduction effect is significant.
As shown in FIG. 8 , the Bessel beam measured by the CCD contains a noise error, and multiple peaks and peak shifts occur at the center spot, which seriously affect the accuracy of determining the position of the peak. 
