A white light interference three-dimensional reconstruction method based on under-sampling and Hilbert transform

Abstract

This invention claims protection for a three-dimensional reconstruction method for white light interferometry based on undersampling and Hilbert transform, comprising the following steps: selecting the sampling step size for undersampling according to the wavelength and bandwidth of the light source; acquiring the undersampled white light interferometry signal using a white light interferometer and performing Gaussian filtering; simulating the ideal white light interferometry signal according to the wavelength and bandwidth of the light source; performing Hilbert transform on the processed white light interferometry signal to obtain the envelope curve of the undersampled white light interferometry signal (i.e., the measured signal) and the envelope curve of the ideal white light interferometry signal (i.e., the ideal signal); normalizing the measured signal and the ideal signal to obtain the measured sequence and the ideal sequence; and performing sliding correlation calculation on the measured sequence and the ideal sequence to obtain the correlation coefficient r. s Find the correlation coefficient r s The maximum point is the maximum correlation point. Using the maximum correlation point as the starting point of the ideal signal, the ideal signal is replaced in the measured signal, and the envelope peak position of the measured signal is recalculated to solve for the height.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of image processing, and relates to a white light interference three-dimensional reconstruction algorithm based on undersampling and Hilbert transform. BACKGROUND

[0002] White light interferometry is a special interferometry technique that uses white light with a certain spectral width instead of monochromatic light as an interference light source for measurement. It mainly realizes nanoscale scanning of a measured surface through a high-precision piezoelectric displacement device. During the scanning process, the optical path difference between the measurement light and the reference light changes. An image acquisition device is used to collect white light interference light intensity information of each point on the measured surface at different scanning positions (or optical path difference positions) during the scanning process. Since the maximum value of the white light interference light intensity always appears at the position of zero optical path difference, the algorithm is used to calculate the zero optical path difference position from the white light interference signal, and the relative height information of the measured surface can be obtained according to the relative position of the zero optical path difference of each point on the measured surface.

[0003] White light interferometry is widely used in high-precision detection fields. It usually uses vertical scanning interferometry and phase shifting interferometry to measure the sample topography. The phase shifting interferometry has a precision of nanometers, but the range is limited, and the phase corresponding to the height change needs to be limited within the 2π interval. The wrapping phase unwrapping algorithm can expand the range of the phase shifting interferometry, but it is only suitable for smooth surfaces. When the height fluctuation exceeds the limited range of the focal depth or the source coherence length, the interference fringes are blurred or the contrast is lost, and the calculated results will have large errors or even be wrong. The vertical white light interference scanning method overcomes the phase ambiguity within the 2π interval, can measure the shape of a rough surface object, and has high calculation precision and high applicability. However, the vertical white light interference scanning method needs to comply with the Nyquist criterion, that is, at least two samples must be collected within each cycle of the signal, which leads to the need for a large amount of measurement data for measurement, thereby reducing the measurement speed.

[0004] There are more than ten algorithms for white light interferometry. A measurement algorithm based on Fourier transform is proposed in the literature [M, Hart, D, et al. Fast surface profiling by spectral analysis of white-light interferograms with fourier transform spectroscopy. [J]. Applied Optics, 1998.]. The algorithm is based on sampling theory, and the collected white light interference signal is subjected to Fourier transform, the frequency of the sinusoidal modulation term is filtered out, and the coordinates of the coherent peak points of the Gaussian envelope curve of the white light interference data are obtained after inverse transform, so as to obtain the three-dimensional height of the object. A measurement algorithm based on Hilbert transform is proposed in the literature [Larkin K G. Efficient nonlinear algorithm for envelope detection in white light interferometry [J]. Optical Society of America, 1996 (4).]. The algorithm adds a complex signal composed of the Hilbert transform of the virtual signal to the original signal to obtain a signal with all positive frequencies. The process of extracting the signal envelope is completed by Hilbert transform, and the zero optical path difference position point can be obtained by calculating the envelope peak value, so as to obtain the object height. A correction method for non-uniform sampling of white light interference signal based on Hilbert transform is proposed in the literature [Yang Zhongming, Xin Lei, Liu Zhaojun. A correction method for non-uniform sampling of white light interference signal based on Hilbert transform: CN113446930A[P]. 2020.]. The method is based on the Nyquist sampling theorem, and can compensate for the error caused by non-uniform sampling and improve the measurement accuracy.

[0005] The above traditional algorithms use discrete signal processing technology. In order to obtain higher measurement accuracy, narrow sampling interval is required, which follows the Nyquist sampling theorem, thereby increasing the amount of sampling data and the cost of calculation, reducing the scanning speed, increasing the measurement time, and when the Nyquist sampling theorem is violated and undersampling is performed, the collected white light interference signal will have frequency spectrum aliasing, signal distortion, and distorted three-dimensional object morphology. SUMMARY

[0006] The present application aims to solve the problems of the prior art. A white light interference three-dimensional reconstruction method based on undersampling and Hilbert transform is proposed. The technical scheme of the present application is as follows:

[0007] A white light interference three-dimensional reconstruction method based on undersampling and Hilbert transform, comprising the following steps:

[0008] 101. Selecting the sampling step of under-sampling according to the wavelength and bandwidth of the light source, collecting the under-sampled white light interference signal using the white light interferometer, and performing Gaussian filtering on the collected under-sampled white light interference signal; and simulating the ideal white light interference signal according to the sampling wavelength and bandwidth;

[0009] 102. Performing Hilbert transform on the white light interference signal processed in step 101 to obtain the envelope curve of the under-sampled white light interference signal, i.e. the measurement signal, and the envelope curve of the ideal white light interference signal, i.e. the ideal signal;

[0010] 103. Performing correlation calculation on the envelope curve of the measurement signal and the envelope curve of the ideal signal, and performing normalization processing on the measurement signal and the ideal signal to obtain the measurement sequence and the ideal sequence; selecting the reference window of the ideal sequence at intervals of the sampling step;

[0011] 104. Moving the ideal sequence to the right by d for the s-th time, where d < Δ, and Δ is the sampling interval, and performing sliding correlation calculation on the measurement sequence and the ideal sequence to obtain the correlation coefficient r s ;

[0012] 105. Finding the maximum value of the correlation coefficient r s , i.e. the maximum correlation point, taking the maximum correlation point as the starting point of the ideal signal, replacing the ideal signal into the measurement signal, and re-calculating the envelope peak value of the measurement signal to solve the height.

[0013] Further, the function definition formula of the ideal white light interference signal in step 101 is as follows:

[0014]

[0015] wherein, wherein l c is expressed as:

[0016]

[0017] wherein I0 is the background intensity, z is the vertical scanning position along the optical axis, h(x, y) represents the height of the object position, z0 is the length of the reference arm, λc is the central wavelength of the light source, φ(x, y) is the initial phase of a position on the object, λ b is the bandwidth of the light source, and lc describes the coherence length of the light source.

[0018] Further, the Hilbert transform is performed on the processed white light interference signal in step 102, and the formula is as follows:

[0019]

[0020] which can also be expressed as:

[0021] I'(z) = H[I(z)] (4)

[0022] Envelope signal of white light interference signal The envelope signal can be extracted by the following formula:

[0023]

[0024] The sampling step restriction condition of white light interference is studied. The undersampling method is based on the idea that it is unnecessary to reconstruct the cosine oscillation when the high frequency component of the envelope function is very small. The envelope line is feasible to reconstruct because the change of the envelope line is usually much slower than the oscillation shown in Fig. 2. The envelope function is calculated by undersampling the ideal white light interference signal intensity function. The standard deviation of the difference between the envelope function calculated at different sampling intervals and the standard envelope function is calculated as follows: Figure 3 Figure 3 The sampling interval restriction condition of white light interference signal is obtained as follows:

[0025]

[0026] where n is an integer multiple of half the wavelength, λ c is the central wavelength of the light source, λ b is the bandwidth, and Δ is the sampling interval. When the sampling step is within the range of formula (6), it meets the sampling envelope algorithm described in this paper.

[0027] Further, the Gaussian filtering in the step 101 is a linear smoothing filtering. The Gaussian filtering is a process of weighted average of the entire signal. The value of each signal point is obtained by weighted average of its own and other signal points in the neighborhood.

[0028] The template of Gaussian filtering is calculated by Gaussian formula. The one-dimensional Gaussian distribution function is:

[0029] (8)

[0030] where x is the horizontal coordinate of different positions of the signal point, and σ is the standard deviation, which can be calculated by formula (9):

[0031] σ = 0.3 × ((Ksize-1) × 0.5-1) + 0.8 (9)

[0032] The Gaussian filtering window width is 5, and the standard deviation σ is calculated as 1.1.

[0033] ​Further, a suitable method to reduce the amount of sampling data is to undersample the correlation graph. Although the correlation graph is undersampled, the envelope curve can still be reconstructed. The maximum position of the envelope curve defines the height of the corresponding surface point of the object. The correlation method is a suitable method to estimate the maximum position of the envelope curve. For a usual LED light source, the ideal form of the envelope curve can be approximated by a Gaussian function. The advantage is that the ideal form of the envelope can be calculated with an arbitrary fine step. Then, the cross-correlation of the measured envelope curve and the ideal envelope curve can also be calculated with an arbitrary fine step (which can be significantly shorter than the sampling step Delta). The step 104 shifts the ideal sequence to the right by d, d < Delta, Delta is the sampling interval, and the sliding correlation calculation of the measured sequence and the ideal sequence is performed to obtain the correlation coefficient r s , specifically comprising:

[0034] The correlation calculation formula used in step 104 is as follows:

[0035]

[0036] Wherein, And are the average values of the measured signal sequence and the ideal sequence, respectively, I ri is the value of each point in the reference sequence in the ideal envelope signal, I rm is the value of each point in the matching sequence in the measured envelope signal, i is the sequence number, i = 1, 2, 3, … n, and s is the step number of the ideal sequence correlation calculation corresponding to the starting position of the measured sequence.

[0037] Further, the step 105 finds the maximum value of the correlation coefficient r s , that is, the position of the maximum correlation point, and takes the maximum correlation point as the starting point of the ideal signal. The ideal signal is replaced into the measured signal to obtain the corrected measured signal. The envelope peak value of the measured signal is recalculated according to formula (5), the position of the maximum value point of the envelope peak value is found, that is, the position of the zero optical path difference point, that is, the relative position of the object, so as to find the height and reconstruct the object topography.

[0038] The advantages and beneficial effects of the present application are as follows:

[0039] 1. The present application studies the sampling interval limitation condition, and performs undersampling under the condition of violating the Nyquist sampling theorem, increases the sampling step, and improves the speed.

[0040] 2. The use of Gaussian filtering before Hilbert transform of the interference signal weakens the influence of noise, reduces the occurrence of abnormal points in the Hilbert transform of the white light interference signal due to noise, and improves the accuracy of the algorithm.

[0041] 3. In the condition of under-sampling, the sliding correlation calculation of the white light interference measurement signal and the ideal signal is carried out, the ideal signal is used to splice the measurement signal, the signal distortion and the spectrum aliasing problem caused by violating the Nyquist theorem are compensated, and the accuracy of the algorithm is further improved. BRIEF DESCRIPTION OF DRAWINGS

[0042] Figure 1 is a flow chart of the white light interference signal algorithm of under-sampling and Hilbert transform provided by the present application;

[0043] Figure 2 is a correlation graph of white light interference for different sampling intervals;

[0044] Figure 3 is a standard deviation of the difference between the envelope and the ideal envelope function for different sampling intervals;

[0045] Figure 4 is a sliding correlation calculation graph of the measurement signal and the ideal signal;

[0046] Figure 5 is a three-dimensional reconstruction graph of the traditional algorithm and the method provided by the present application;

[0047] Figure 6 is a section graph of the three-dimensional reconstruction of the traditional algorithm and the method provided by the present application. DETAILED DESCRIPTION

[0048] The technical solutions in the embodiments of the present application will be described in detail below with reference to the drawings of the embodiments of the present application. The described embodiments are only a part of the embodiments of the present application.

[0049] The technical solution of the present application to solve the above technical problems is:

[0050] The white light interference three-dimensional reconstruction method based on under-sampling and Hilbert transform of the present embodiment has a flow chart as shown in Figure 1 The specific steps are as follows:

[0051] Step 1: According to the center wavelength and bandwidth of the light source, select a suitable under-sampling sampling step, collect the interference pictures taken by the CCD, and extract the sample all position light intensity transformation curve of the light intensity with the movement of the measurement arm position as the measurement signal in units of pixels; at the same time, generate an ideal white light interference signal according to the center wavelength and bandwidth of the light source;

[0052] In this example, the center wavelength of the light source is 550 nm, and the bandwidth is 100 nm;

[0053] The function formula of the ideal white light interference signal is:

[0054]

[0055] Step two, Gaussian filtering is performed on the measured interference signal;

[0056] Gaussian filtering is a linear smoothing filter. Gaussian filtering is a process of weighted average of the entire signal. The value of each signal point is obtained by weighted average of its own and other signal points in the neighborhood.

[0057] The template of Gaussian filtering is calculated by Gaussian formula. One-dimensional Gaussian distribution function:

[0058] (8)

[0059] Where x is the horizontal coordinate of different positions of signal points, and σ is the standard deviation, which can be calculated by formula (9):

[0060] σ = 0.3 × ((Ksize-1) × 0.5-1) + 0.8 (9)

[0061] In this example, the Gaussian filtering window width is 5, and the standard deviation σ = 1.1 can be calculated.

[0062] Step three, Hilbert transform is performed on the measured signal processed by Gaussian filtering and the ideal signal obtained in step one:

[0063] The Hilbert transform formula used in step three is as follows:

[0064]

[0065] It can also be expressed as:

[0066] I'(z) = H[I(z)] (4)

[0067] Further, the envelope signal of white light interference signal It can be extracted by the following formula:

[0068]

[0069] Step four, correlation method is a suitable method for estimating the maximum value position of envelope. The measured envelope and the ideal form of the envelope are correlated, Figure 4 The calculation method of correlation is shown. For a general LED, the ideal form of the envelope can be approximated by a Gaussian function, which has the advantage that the ideal form of the envelope can be calculated with any fine step. Then, the cross-correlation between the measured envelope and the ideal form can also be calculated with any fine step (which can be significantly shorter than the sampling step Δ), and the ideal envelope and the measured envelope are calculated by sliding correlation according to formula 7).

[0070] Step five, the position coordinate of the maximum correlation coefficient after correlation calculation is extracted, the ideal interference signal corresponding point curve is spliced to the measurement signal to obtain the correction signal, the peak position of the envelope curve is recalculated according to formula (5), and the object height is obtained.

[0071] The three-dimensional physical topography reconstructed by the above method reduces data and improves scanning speed while maintaining the accuracy of the algorithm, as shown in Figure 5 and Figure 6 Figure 5 a is the ideal sampling sampling step length of 100 nm, and the three-dimensional topography of the object recovered by the barycenter method, Figure 5 b, c, and d are sampling step lengths of 750 nm, 1516 nm, and 1915 nm, respectively, and the three-dimensional topography recovered by the method described above. Figure 6 is the processed relative height map at y = 377 pixels. As can be seen from the figure, using the algorithm proposed in this paper, the surface topography of the sample can be quickly and stably observed when the sampling interval is increased to 1500 nm, and the three-dimensional map obtained can clearly observe the shape of the wafer. As the sampling interval increases, the resolution of the reconstructed three-dimensional topography will decrease, but the running rate will be greatly improved.

[0072] The above examples should be understood as only for illustrating the present application and not for limiting the protection scope of the present application. After reading the content of the present application, the skilled person can make various changes or modifications to the present application, and these equivalent changes and modifications also fall within the scope defined by the claims of the present application.​

Claims

1. A white-light interferometric 3D reconstruction method based on undersampling and Hilbert transform, characterized in that, Includes the following steps:

101. Select the undersampling step size according to the wavelength and bandwidth of the light source, use a white light interferometer to collect the undersampling white light interference signal, and perform Gaussian filtering on the collected undersampling white light interference signal; simulate the ideal white light interference signal according to the wavelength and bandwidth of the light source.

102. Perform Hilbert transform on the white light interference signal processed in step 101 to obtain the undersampled white light interference signal envelope curve, i.e., the measured signal, and the ideal white light interference signal envelope curve, i.e., the ideal signal.

103. The measured envelope curve and the ideal envelope curve are correlated and calculated. The measured signal and the ideal signal are normalized to obtain the measurement sequence and the ideal sequence. Select a reference window for the ideal sequence at sampling step intervals; 104. The ideal sequence is shifted to the right by d for the s-th time, where d < Δ, and Δ is the sampling interval. The sliding correlation between the measured sequence and the ideal sequence is calculated to obtain the correlation coefficient r. s ; 105. Find the correlation coefficient r. s The maximum value is the maximum correlation point. Using the maximum correlation point as the starting point of the ideal signal, the ideal signal is replaced in the measured signal, and the envelope peak position of the measured signal is recalculated to solve for the height. The functional definition formula for the ideal white light interference signal in step 101 is as follows: Among them, l c Represented as: Where I0 is the background intensity, z is the vertical scan position along the optical axis, h(x,y) represents the height of the object, z0 is the length of the reference arm, λc is the center wavelength of the light source, φ(x,y) is the initial phase of a position on the object, and λ b It is the bandwidth of the light source, and lc describes the coherence length of the light source; In step 102, Hilbert transforms are performed on the processed white light interference signals, as shown in the following formulas: It can also be expressed as: I′(z)=H[I(z)] (4) Envelope signal of white light interference signal It can be extracted using the following formula: By undersampling the intensity function of the ideal white light interference signal, the envelope function is calculated, and the standard deviation of the difference between the calculated envelope function and the standard envelope function at different sampling intervals is obtained. The sampling interval constraint for the white light interference signal is then derived as follows: and Where n is an integer multiple of half the wavelength, λ c λ is the center wavelength of the light source. b Where Δ is the bandwidth and Δ is the sampling interval, the sampling envelope algorithm is satisfied when the sampling step size is within the interval of equation (6); In step 101, Gaussian filtering is a linear smoothing filter. Gaussian filtering is a process of weighted averaging of the entire signal. The value of each signal point is obtained by weighted averaging of its own value and the values ​​of other signal points in its neighborhood. The template for Gaussian filtering is calculated using the Gaussian formula. The formula for the one-dimensional Gaussian distribution function is: （8） Where x is the abscissa of the signal point at different locations, and σ is the standard deviation, which can be calculated by formula (9): σ=0.3×((Ksize-1)×0.5-1)+0.8 (9) With a Gaussian filter window width of 5, the standard deviation σ is calculated to be 1.

1. The maximum position of the envelope curve defines the longitudinal coordinates, i.e., the height, of the corresponding surface point of the object; the correlation method is a suitable method for estimating the position of the maximum value of the envelope curve; for a typical LED light source, the ideal form of the envelope curve can be approximated by a Gaussian function; then, the cross-correlation between the measured envelope curve and the ideal envelope curve can also be calculated using arbitrarily fine step sizes; the 104 ideal sequence is shifted to the right by d for the s-th time, d < Δ, where Δ is the sampling interval, and the sliding correlation calculation is performed between the measured sequence and the ideal sequence to obtain the correlation coefficient r. s Specifically, it includes: The relevant calculation formulas used in step 104 are as follows: in, and These are the average value of the measured signal sequence and the average value of the ideal sequence, respectively. ri It is the value of each point in the reference sequence of the ideal envelope signal, I rm is the value of each point in the matched sequence in the measurement envelope signal, i is the sequence number, i = 1, 2, 3, ... n, and s is the number of steps in the ideal sequence correlation calculation corresponding to the starting position of the measurement sequence.

2. The white light interferometric 3D reconstruction method based on undersampling and Hilbert transform according to claim 1, characterized in that, Step 105 involves finding the correlation coefficient r. s The location of the maximum value point is the maximum correlation point. The maximum correlation point is used as the starting point of the ideal signal. The ideal signal is replaced in the measurement signal to obtain the corrected measurement signal. The envelope peak value of the measurement signal is recalculated according to formula (5). The location of the maximum value point of the envelope peak value is the location of the zero optical path difference point, which is the relative position of the object. Thus, the height is found and the object shape is reconstructed.

Images