CN111289106A - Spectral noise reduction method based on digital filtering - Google Patents
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Abstract
The invention discloses a spectral noise reduction method based on digital filtering, which comprises the following steps of 1, selecting a spectral signal S to be subjected to noise reduction; step 2, designing a frequency-amplitude response function PF of the spectral signal S to be denoised; step 3, transforming the frequency-amplitude response function PF to obtain a corresponding filtering convolution transfer function H; step 4, calculating the convolution Y of the spectral signal S to be denoised and the convolution transfer function H; step 5, intercepting data from the beginning of Y to the length equal to S, namely the result after noise reduction of S; and 6, selecting parameters Wr and Wp with optimal filtering effect to obtain a corresponding optimal filtering convolution transfer function H, and then obtaining an optimal noise reduction result. The data points obtained by the method are more compact and uniform, the end point is not truncated, and the length of the data sequence can be selected according to the precision requirement; and the method has no polynomial order limitation, no lattice phenomenon under the high-order condition and more precise result.
Description
Technical Field
The invention relates to the field of spectrum instruments, in particular to a spectrum noise reduction method based on digital filtering.
Background
Data smoothing and noise reduction are important means for improving instrument performance and improving precision in modern instrument design and manufacture. Smoothing noise reduction, which is essentially low-pass filtering that filters out high frequency components in the signal, is the key to the choice and design of the filter. Compared with other applied filters, the purpose of the filter adopted by the spectrometer is to reduce noise and filter white noise out of a real signal, on one hand, the intensity of the filtered noise is as high as possible, and on the other hand, the loss of the real signal is as low as possible. A reasonable and efficient filter is of great importance to the performance of an instrument, the performance and the cost of the instrument are directly related in a spectrum instrument, the hardware quality is controlled, a digital filter which is targeted to a spectrum signal can be simply, reasonably and flexibly provided on the software level, and the digital filter has important significance for improving the cost performance of equipment, so that instrument users and manufacturers all provide strong requirements for designing and selecting the digital filter.
Most spectrometers include filtering and noise reduction functions in hardware, software, and hardware integration. In the design of an FIR filter commonly used in the field of signal processing, because a plurality of parameters need to be set, the difficulty is large in actual spectrum application, and the IIR filter has the limitation of difficult design. At present, the commonly accepted application effect is polynomial fitting-least square optimization filtering represented by an S-G filter, and compared with simple Gaussian filtering and moving window filtering, the method has better signal fidelity performance. Polynomial fitting-optimization is actually a simplified FIR filter, and has inherent defects, firstly, rigid parameter setting, including two parameters, the order of the polynomial and the fitting window width are difficult to accurately adapt to specific spectral characteristics; secondly, the problems of polynomial fitting precision and overfitting are solved, the adopted polynomial with lower order has better stability, but higher precision is difficult to obtain, and the higher precision needs higher polynomial order, which can cause the problems of stability damage and overfitting; thirdly, the method defined by S-G reaches the limit of polynomial optimization, and the performance is difficult to further improve.
Compared with other types of signals, such as audio signals and video signals, the spectral signal processing has higher certainty, relatively fixed frequency range and low requirements on phase and time lag, and is more beneficial to adopting simplified FIR thought processing, which is also the reason that S-G filtering is widely accepted. The S-G filtering adopts polynomial fitting signals, a convolution function is obtained through least square optimization, and due to the Runge phenomenon, polynomial fitting is higher in order selection and larger in uncertainty, so that the improvement of the precision of the polynomial method is limited.
The spectrum peak appearance accords with Voigt distribution, the frequency domain amplitude response after Fourier change also basically accords with Voigt distribution or approximately accords with Gaussian distribution, and white noise has the characteristic of uniform distribution on the frequency domain. The invention proposes that the frequency-amplitude distribution can be directly designed according to the difference between the two. And performing Fourier inverse transformation on the designed frequency-amplitude distribution to obtain a function which can be used as a convolution transfer function of FIR filtering, and calculating the convolution of the noise-reduced signal and the convolution transfer function to obtain the noise-reduced signal.
Disclosure of Invention
In order to solve the above problems, the present invention provides a spectral noise reduction method based on digital filtering.
The invention discloses a spectral noise reduction method based on digital filtering, which is realized by the following steps:
the invention provides a spectral noise reduction method based on digital filtering, which comprises the following steps:
step 1, selecting a spectral signal S to be denoised;
step 3, transforming the frequency-amplitude response function PF to obtain a corresponding filtering convolution transfer function H;
and 5, intercepting data from the beginning of the Y to the length equal to that of the S, namely the noise-reduced result of the S.
Further, the frequency-amplitude response function PF comprises two part parameters: wr and Wp;
wherein Wr represents the band stop width of the spectral signal S to be noise reduced; wp denotes the band pass width of the spectral signal S to be noise reduced.
Further, the filtering convolution transfer function H in step 3 is obtained by subjecting the response function PF to inverse Fourier transform to obtain PFNTaking PFNThe left half of the real part, HL, is divided by its sum, sum (HL), to obtain the filter convolution transfer function H.
Further, the method further comprises a step 6 of selecting parameters Wr and Wp with optimal filtering effects through the evaluation indexes, obtaining a corresponding optimal filtering convolution transfer function H according to the optimal parameters, and obtaining an optimal noise reduction result according to the optimal filtering convolution transfer function H.
Further, the frequency-amplitude response function PF is designed according to the following method:
a. generating a half-peak width by Wr, normalizing the peak height and taking the peak width as a band elimination part;
b. generating Wp sequence points equal to 1 as a band pass part;
c. and splicing the band-stop part and the band-pass part into a frequency-amplitude response function PF.
Further, the splicing means that the band-pass part and the band-stop part are spliced end to form a frequency-amplitude response function PF, that is: in the former stage of Wr portion, Wp 1s are added to form an array PF.
Further, the evaluation index is a residual mean square error-kurtosis ratio (VFR) value, and Wr and Wp corresponding to the maximum value of the residual mean square error-kurtosis ratio (VFR) value are the optimal Wr and Wp.
Further, the residual value mean square error-kurtosis ratio (VFR) value is obtained by the following steps:
step i, selecting a filtering algorithm and carrying out filtering on the original measurement signal xbPerforming smooth noise reduction to obtain an output signal xa；
Step ii, finding xbAnd xaThe residual value x of (d);
step iii, calculating the mean square error-kurtosis ratio VFR of the residual value x;
and v, adjusting the filtering parameters and calculating the VFRs corresponding to different parameters.
Further, the mean square error-kurtosis ratio VFR is calculated according to the following formula:
where σ is the mean square error and g is the kurtosis ratio.
Further, the mean square error is calculated using the following formula:
the kurtosis ratio is calculated using the following formula:
in the above formula: the signal sequence contains n data singles, and the residual sequence x is the signal x before filteringbAnd the filtered signal xaRespectively calculating the variance sigma according to the variance and kurtosis formulas2And g.
Compared with the prior art, the invention has the following advantages:
1) the method can directly design frequency-amplitude distribution, the designed frequency-amplitude distribution is subjected to Fourier inverse transformation to obtain a function which can be used as a convolution transfer function of FIR filtering, and the convolution of a noise-reduced signal and the convolution transfer function is calculated to obtain a noise-reduced signal;
2) the data points obtained by the method are more compact and uniform, the end point is not truncated, and the length of the data sequence can be selected according to the precision requirement;
3) the method has no polynomial order limitation, no dragon lattice phenomenon under the high-order condition and more precise result.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
FIG. 1 is a Raman spectrum simulated using Voigt peaks;
FIG. 2 is a graph of the S-G filter convolution function of comparative example 1;
FIG. 3 is a schematic diagram of the frequency-amplitude response function design of example 1 of the present invention;
FIG. 4 is a schematic diagram of the convolution generating transfer function H according to embodiment 1 of the present invention;
FIG. 5 is a schematic diagram of convolution Y in embodiment 1 of the present invention;
FIG. 6 shows the filtering result of embodiment 1 of the present invention;
FIG. 7 is a graph of a convolution function in embodiment 1 of the present invention;
fig. 8 is a comparison graph of ibuprofen raw true signal, comparative example 1S-G filtering and example 2 filtering results based on a sequence of data points, 8(a) being the overall case, 8(b) being the case of sharp peaks, 8(c) being the case of high frequency filtering of flat portions. (ii) a
Figure 9 is a graph comparing the results of ibuprofen original true signal based on raman shift, comparative example 1S-G filtering and example 2 filtering.
Detailed Description
The following description of the embodiments of the present invention is provided for illustrative purposes, and other advantages and effects of the present invention will become apparent to those skilled in the art from the present disclosure.
The invention provides a spectral noise reduction method based on digital filtering, which comprises the following steps:
step 1, selecting a spectral signal S to be denoised;
step 3, transforming the frequency-amplitude response function PF to obtain a corresponding filtering convolution transfer function H;
and 5, intercepting data from the beginning of the Y to the length equal to that of the S, namely the noise-reduced result of the S.
Further, the frequency-amplitude response function PF comprises two part parameters: wr and Wp;
wherein Wr represents the band stop width of the spectral signal S to be noise reduced; wp denotes the band pass width of the spectral signal S to be noise reduced.
Further, the filtering convolution transfer function H in step 3 is obtained by subjecting the response function PF to inverse Fourier transform to obtain PFNTaking PFNThe left half of the real part HL, willHL is divided by its sum (HL) to obtain the filter convolution transfer function H.
Further, the method further comprises a step 6 of selecting parameters Wr and Wp with optimal filtering effects through the evaluation indexes, obtaining a corresponding optimal filtering convolution transfer function H according to the optimal parameters, and obtaining an optimal noise reduction result according to the optimal filtering convolution transfer function H.
Further, the frequency-amplitude response function PF is designed according to the following method:
a. generating a half-Gaussian peak by Wr, normalizing the peak height and taking the peak height as a band stop part;
b. generating Wp sequence points equal to 1 as a band pass part;
c. and splicing the band-stop part and the band-pass part into a frequency-amplitude response function PF.
Further, the evaluation index is a residual mean square error-kurtosis ratio (VFR) value, and Wr and Wp corresponding to the maximum value of the residual mean square error-kurtosis ratio (VFR) value are the optimal Wr and Wp.
Further, the residual value mean square error-kurtosis ratio (VFR) value is obtained by the following steps:
step i, selecting a filtering algorithm and carrying out filtering on the original measurement signal xbPerforming smooth noise reduction to obtain an output signal xa；
Step ii, finding xbAnd xaThe residual value x of (d);
step iii, calculating the mean square error-kurtosis ratio VFR of the residual value x;
and v, adjusting the filtering parameters and calculating the VFRs corresponding to different parameters.
Further, the mean square error-kurtosis ratio VFR is calculated according to the following formula:
where σ is the mean square error and g is the kurtosis ratio.
Further, the mean square error is calculated using the following formula:
the kurtosis ratio is calculated using the following formula:
in the above formula: the signal sequence contains n data singles, and the residual sequence x is the signal x before filteringbAnd the filtered signal xaRespectively calculating the variance sigma according to the variance and kurtosis formulas2And g.
Comparative example 1:
the raman spectrum was simulated with the Voigt peak and then white noise at 2% variance intensity was added to generate a simulated spectral signal with noise. Referring to FIG. 1, FIG. 1a, FIG. 1b, and FIG. 1c respectively show a noise-free signal S0An added noise signal N and an analog spectrum S after the addition of noise.
And (3) adopting S-G filtering to obtain a residual error between the noise-containing signal and a filtering result, and then calculating a correlation coefficient between the residual error and the added noise, wherein the better the correlation is, the better the noise filtering effect is. When the parameters are chosen to be order 6 and window width 59, the optimal correlation coefficient is 0.9512, and the S-G filter convolution function is as shown in FIG. 2.
Comparative example 2
The raman spectrometer was used to collect the true signal of ibuprofen (532nm excitation, 2048 pixel array spectrometer, integration time 5S), and S-G filtering had the best effect with parameters of 9 order and window width 17.
Example 1:
to illustrate the effect of the filter of the present embodiment, the present embodiment uses the same simulation data as in comparative example 1, i.e., a raman spectrum is simulated using a Voigt peak, and then white noise of 2% variance intensity is added to generate a simulated spectrum signal containing noise. Referring to FIG. 1, wherein a, b, and c are noise-free signals S0The added noise signal N, the simulated spectrum S after the addition of noise.
The frequency-amplitude response function contains two parts of parameters: a bandstop width Wr and a bandpass width Wp. (defined by a Gaussian peak hypothesis, Wr is the width of a Gaussian peak, if other peak types such as Voigt peak are adopted, Wr is a parameter related to three peak types, and the Gaussian function can meet most requirements due to simple and accurate comprehensive consideration);
a. generating a half-Gaussian peak by Wr, normalizing the peak height and taking the peak height as a band stop part;
b. generating Wp sequence points equal to 1 as a band pass part;
c. and connecting the band-pass part and the band-stop part end to form a frequency-amplitude response function PF by splicing, namely: in the former stage of Wr portion, Wp 1s are added to form an array PF, as shown in FIG. 3.
Defining Wp and Wr as 100, performing inverse Fourier transform on the response function according to the length of an input signal sequence of 5000, and obtaining PF of 5000 sequence data pointsNThe real part is shown in FIG. 4 (a). The left half HL of FIG. 4(a) is cut, as in FIG. 4 (b). Dividing HL by PFNThe sum (hl) as convolution transfer function H, as shown in fig. 4(c), contains data points that are compact and uniform, and has good continuity, facilitating precise numerical operations.
The convolution Y of S and H is calculated as shown in fig. 5.
The first 5000 data of Y are truncated and output as the result of filtering, as shown in fig. 6.
The optimal filter effect parameters of this embodiment, Wp is 130 and Wr is 89, and the noise-added residual correlation coefficient is 0.9526, as shown in fig. 7, fig. 7 is a convolution function graph of the filter of this embodiment, and it can be seen that the data points of this embodiment are dense and uniform, and the end point is not truncated, and the data sequence length can be selected according to the requirement of accuracy. And the polynomial order limit is avoided, and the dragon lattice phenomenon appearing under the high-order condition does not exist, so that the result is more precise.
Example 2
In order to illustrate the effect of the filter in this embodiment, the same signal as that in comparative example 2, that is, the true signal (532nm excitation, 2048 pixel array spectrometer, integration time 5S) of ibuprofen is collected by using a raman spectrometer as the spectral signal S to be denoised in this embodiment.
The other steps were the same as in example 1.
The filter of the present embodiment has the best filtering effect when Wp is 550 and Wr is 650, as evaluated by the residual mean square error-kurtosis ratio VFR value.
The evaluation indexes used in the embodiment 1 and the embodiment 2 are residual mean square error-kurtosis ratio (VFR) values, the Wr and the Wp corresponding to the VFR values are found out according to the maximum value of the VFR values, the Wr and the Wp are optimal, and the noise reduction result obtained by using the optimal Wr and Wp parameters is the optimal noise reduction result;
wherein, the residual mean square error-kurtosis ratio (VFR) values are obtained by the following steps:
step i, selecting the filtering algorithm of the invention, and carrying out the filtering algorithm on the original measurement signal xbPerforming smooth noise reduction to obtain an output signal xa；
Step ii, finding xbAnd xaThe residual value x of (d);
step iii, calculating the mean square error-kurtosis ratio VFR of the residual value x;
v, adjusting the filtering parameters Wr and Wp, calculating VFRs corresponding to different parameters, and finding out Wr and Wp corresponding to the maximum value of the VFR;
the mean square error-kurtosis ratio VFR is calculated according to the following formula:
wherein sigma is mean square error, g is kurtosis ratio;
wherein, the mean square error is calculated by adopting the following formula:
the kurtosis ratio is calculated using the following formula:
in the above formula: the signal sequence contains n data single values, and the residual error sequence x is the signal before filteringxbAnd the filtered signal xaRespectively calculating the variance sigma according to the variance and kurtosis formulas2And g.
Filtering noise reduction result
1. Comparative example 1 and example 1
As shown in fig. 8, fig. 8 is a comparison of the results of the original real signal, S-G filtering and embodiment 1 filtering of the present invention. Fig. 8(a) shows the entire case, (b) shows the case of a sharp peak, and (c) shows the case of high-frequency filtering of a flat portion. It can be seen that the overall S-G and the filtering mode of embodiment 1 of the present invention have good effects, and in the local amplification effect, embodiment 1 of the present invention has less distortion on the sharp high-frequency peak, and the high-frequency component in the flat portion is filtered more thoroughly, and the curve is smoother.
The filtering effect of embodiment 1 of the invention is superior to the S-G filtering of the prior art.
2. Comparative example 2 and example 2:
figure 9 is a comparison of the results of the original real signal of ibuprofen, S-G filtering and the filter of the present invention. (a) The whole effect is achieved; (b) is the detail of the peak top, the distortion of the filter of this embodiment is smaller than S-G; (c) in the case of the peak bottom, it can be seen that the filtering method of embodiment 2 filters the high frequency signal to a slightly better degree than S-G, and the result is smoother.
Claims (10)
1. A spectral noise reduction method based on digital filtering is characterized by comprising the following steps:
step 1, selecting a spectral signal S to be denoised;
step 2, designing a frequency-amplitude response function PF of the spectral signal S to be denoised;
step 3, transforming the frequency-amplitude response function PF to obtain a corresponding filtering convolution transfer function H;
step 4, calculating the convolution Y of the spectral signal S to be denoised and the filtering convolution transfer function H;
and 5, intercepting data from the beginning of the Y to the length equal to that of the S, namely the noise-reduced result of the S.
2. A method for spectral noise reduction based on digital filtering according to claim 1, wherein the frequency-amplitude response function PF comprises two parameters: wr and Wp;
wherein Wr represents the band stop width of the spectral signal S to be noise reduced; wp denotes the band pass width of the spectral signal S to be noise reduced.
3. A method for spectral noise reduction based on digital filtering according to claim 2, wherein in step 3 said filter convolution transfer function H is obtained by inverse fourier transforming said response function PF to obtain PFNTaking PFNThe left half of the real part, HL, is divided by its sum, sum (HL), to obtain the filter convolution transfer function H.
4. The spectral noise reduction method based on digital filtering according to claim 3, further comprising step 6, selecting parameters Wr and Wp with optimal filtering effect through evaluation indexes, obtaining a corresponding optimal filtering convolution transfer function H according to the optimal parameters, and obtaining an optimal noise reduction result according to the optimal filtering convolution transfer function H.
5. A method for spectral noise reduction based on digital filtering according to any of claims 1 to 4, characterized in that the frequency-amplitude response function PF is designed according to the following method:
a. generating a half-peak width by Wr, normalizing the peak height and taking the peak width as a band elimination part;
b. generating Wp sequence points equal to 1 as a band pass part;
c. and splicing the band-stop part and the band-pass part into a frequency-amplitude response function PF.
6. The spectral noise reduction method based on digital filtering according to claim 5, wherein the splicing is performed by splicing the band-pass part and the band-stop part end to form a frequency-amplitude response function PF, that is: in the former stage of Wr portion, Wp 1s are added to form an array PF.
7. The spectral noise reduction method based on digital filtering according to claim 4, wherein the evaluation index is a residual mean square error-kurtosis ratio (VFR) value, and Wr and Wp corresponding to a maximum value of the residual mean square error-kurtosis ratio (VFR) value are optimal Wr and Wp.
8. The method of claim 6, wherein the residual mean square error-kurtosis ratio (VFR) value is obtained by:
step i, selecting a filtering algorithm and carrying out filtering on the original measurement signal xbPerforming smooth noise reduction to obtain an output signal xa；
Step ii, finding xbAnd xaThe residual value x of (d);
step iii, calculating the mean square error-kurtosis ratio VFR of the residual value x;
and v, adjusting the filtering parameters and calculating the VFRs corresponding to different parameters.
9. A method of spectral noise reduction based on digital filtering according to claim 8, characterized in that the mean squared error-kurtosis ratio VFR is calculated according to the following formula:
where σ is the mean square error and g is the kurtosis ratio.
10. A method for spectral noise reduction based on digital filtering according to claim 9, wherein the mean square error is calculated using the following formula:
the kurtosis ratio is calculated using the following formula:
in the above formula: the signal sequence contains n data singles, and the residual sequence x is the signal x before filteringbAnd the filtered signal xaRespectively calculating the variance sigma according to the variance and kurtosis formulas2And g.
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