Deep Learning-Based Spectrum Reconstruction Method for Raman Spectroscopy

Abstract

Raman spectroscopy, measured by a Raman spectrometer, is usually disturbed by the instrument response function and noise, which leads to certain measurement error and further affects the accuracy of substance identification. In this paper, we propose a spectral reconstruction method which combines the existing maximum a posteriori (MAP) method and deep learning (DL) to recover the degraded Raman spectrum. The proposed method first employs the MAP method to reconstruct the measured Raman spectra, so as to obtain preliminary estimated Raman spectra. Then, a convolutional neural network (CNN) is trained by using the preliminary estimated Raman spectra and the real Raman spectra to learn the mapping from the preliminary estimated Raman spectra to the real Raman spectra, so as to achieve a better spectral reconstruction effect than merely using the MAP method or a CNN. To prove the effectiveness of the proposed spectral reconstruction method, we employed the proposed method and some traditional spectral reconstruction methods to reconstruct the simulated and measured Raman spectra, respectively. The experimental results show that compared with traditional methods, the estimated Raman spectra reconstructed by the proposed method are closer to the real Raman spectra.

1. Introduction

A spectrometer is a basic optical detection instrument that can obtain the spectral information of the measured object. At present, various spectrometers have been widely used in color measurement, gas composition analysis, agricultural medicine, food safety and other fields [1,2,3,4,5,6], and have become among the most important optical detection instruments. Among them, the Raman spectrometer [7,8,9], which is composed of a probe and a spectrometer, is a rapidly developed instrument in recent years. The Raman spectrometer can qualitatively analyze and identify various molecular structures and material types with few samples. In the past decade, the Raman spectrometer has been widely used in biology and analytical chemistry research because of its non-destructive, fast and real-time detection characteristics, and has achieved many significant research results. However, as an indispensable component of the Raman spectrometer, the non-ideality of the spectrometer usually makes it unable to present the real spectrum well. Specifically, due to the influence of the instrument response function, also known as the bandwidth function, the measured Raman spectra commonly contain spectral distortion, which leads to certain measurement error and further affects the accuracy of substance identification combining with noise. In order to overcome the shortcoming, an efficient method is to utilize spectral reconstruction methods to process the measured Raman spectra.

Spectral reconstruction, also known as spectral deconvolution [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], is among the most important issues in the field of spectral instrument signal processing. The purpose of spectral reconstruction is to obtain the estimated spectra closer to the real spectra from the degraded spectra with prior knowledge such as the instrument response function. Researchers have developed a variety of spectral deconvolution methods, with two main types—spatial domain-based spectral deconvolution [13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,29,30,31,32,33,34] and frequency domain-based spectral deconvolution [10,11,12]. Among them, spatial domain-based spectral deconvolution can directly perform more intuitive and understandable operations on the measured spectra in the spatial domain. With the increase in spectral sampling points, early spatial domain-based spectral deconvolution methods usually cause ill-posed problems [21]. Therefore, regularization is necessarily introduced to enforce stability as well as incorporate prior knowledge about the solution. For instance, the authors of [22] upgraded the original Richard–Lucy (RL) method [17], and the improved spectral deconvolution method adopts a stopping condition curve, which is used to control the method to stop iteration before overfitting. Jin et al. [23] proposed a spectral reconstruction method which combines the Levenberg–Marquardt (LM) algorithm [24,25] with a well-known LED spectral model [26] to recover the degraded LED spectra. The specific process of this method is to use the LED spectral model to fit the measured spectra to eliminate high-frequency measurement errors such as noise, and then the LM algorithm is used to deconvolute the preprocessed spectra. The authors of [27] upgrade this method by directly introducing the regularization term, which enables the upgraded method to reconstruct most spectra, rather than be limited to LED spectral reconstruction. A spectral reconstruction method based on the maximum a posteriori (MAP) estimation framework was proposed in [28]. This method uses the Huber–Markov function as the regularization term, which can suppress the noise and retain the high-frequency spectrum, simultaneously. In addition to these methods, there are many spectral reconstruction methods with regularization [29,30,31,32,33], which can effectively solve most of the spectral reconstruction problems.

Nevertheless, Raman spectra often contain many narrowband spectra. Due to the larger spectral distortion, traditional spectral deconvolution methods are often difficult to use to recover the narrowband spectra very accurately. Therefore, compared with general spectral reconstruction, Raman spectral reconstruction is more challenging. At present, many researchers have proposed methods for Raman spectral reconstruction [34,35,36,37]. Angelini et al. [34] proposed a method based on peak identification through FMFs to identify Raman spectra automatically. A denoising method [35] for Raman spectra with a low SNR based on feature extraction was proposed by Zhao et al. Based on the Hilbert Vibration Decomposition (HVD) method, Raman spectra were decomposed into two components. The peaks were located in the first component and compensated by those in the second component. Finally, Gaussian signals are used to reconstruct Raman peaks from strong noise and the baseline. Barton et al. [36] demonstrated how Savitzky–Golay (S–G) smoothing may be enhanced with maximum likelihood estimation in order to prevent significant deviation from the true Raman signal yet retain the robust smoothing properties of the S-G filter. As an emerging technology, deep learning (DL) can employ the collected datasets to train a deep neural network, so as to directly build the restoration model from input data to output data without complex mathematical operations. DL is a branch of machine learning (ML). DL is an algorithm based on an artificial neural network to learn the characteristics of data. Unlike ML, which requires artificial feature extraction, DL has no complex artificial feature extraction process, and the feature extraction process can be completed automatically through a deep neural network (DNN). DL often requires a large number of training datasets to achieve better results. Common deep learning frameworks include TensorFlow, caffe2, Keras, PyTorch, and DyNet. At present, DL has been widely applied to the field of signal processing and classification [38,39,40,41,42,43,44,45], which provides a new possibility for designing spectral deconvolution methods that can effectively reconstruct Raman spectrum. In this paper, we propose a novel spectral reconstruction method which combines the MAP method [28] and deep learning (DL) to recover the degraded Raman spectrum. The proposed method first employs the MAP method to reconstruct the measured Raman spectra, so as to obtain preliminary estimated Raman spectra. Then, a convolutional neural network (CNN) is trained by using the preliminary estimated Raman spectra and the real Raman spectra to learn the mapping from the preliminary estimated Raman spectra to the real Raman spectra. The main advantage of this method over the traditional methods is that it establishes the mapping from the preprocessed spectra to the real spectra, so as to achieve a better spectral reconstruction effect than merely using the traditional methods or a CNN.

The structure of this paper is as follows: Section 2 provides an overview of the background of spectral deconvolution, including the measured spectral model and the spectral deconvolution model. Next, the proposed method is discussed in Section 3. In Section 4, we compare the effect of the proposed method and some traditional spectral reconstruction methods to reconstruct the simulated and measured Raman spectra to prove its effectiveness. The conclusion is presented in Section 5.

2. Basic Concepts

2.1. The Measured Spectral Model

In spectral instruments, the degraded spectrum can generally be expressed as the convolution of the original spectrum and the instrument response function, as well as the noise superimposed on the spectrum, which is mainly produced during the measurement process. Therefore, the process can usually be modeled as

$$M(\lambda )=R(\lambda )\otimes IF(\lambda )+N(\lambda )$$

(1)

where $R(\lambda )$ is the real spectrum, $IF(\lambda )$ is the instrument response function, $N(\lambda )$ is measurement noise, and $M(\lambda )$ is the measured spectrum. Since the pixels of CCD used for sampling are discrete, the mathematical model of the measured spectrum can be written as

$$M(\lambda )={\displaystyle \sum _{i=1}^j\left(R({\lambda }_i)\cdot IF(\lambda -{\lambda }_i)\right)}+N(\lambda )$$

(2)

where j represents the number of sampling points in the measured spectrum. The instrument response function of the spectrometer is the effect of the combination of the response functions of its multiple optical elements. Researchers used to characterize the instrument response function of the spectrometer through a variety of functions, among which the Gaussian function is the closest and most widely used compared to the real instrument response function. For simplification, in this paper, the Gaussian function is selected to estimate the response function of the spectrometer instrument, as shown below:

$$IF(\lambda )=\frac{1}{\sqrt{2\pi }\sigma }\exp \left(-\frac{{(\lambda -\mu )}^2}{2{\sigma }^2}\right)$$

(3)

where $\mu$ and $\sigma$ indicate the wavelength corresponding to the peak point and the standard deviation of the function. In practical applications, Equation (3) is usually used to fit the measured spectrum of the Hg–Ar lamp or laser light source, so as to obtain the instrument response function of the spectrometer. Nevertheless, the instrument response functions corresponding to different wavelengths of the spectrometer are different, that is, the two parameters $\mu$ and $\sigma$ of each instrument response function are different. In view of this, the mathematical model for the measured spectrum can be updated as the form of matrix.

$$M=IFM\cdot R+N$$

(4)

where $IFM$ is the instrument response function matrix including the instrument response functions of all wavelengths of the spectrometer, and its expression is

$$IFM=\left(\begin{array}{cccc}I{F}_1& I{F}_2& \dots & I{F}_j\end{array}\right)={\left(\begin{array}{cccc}I{F}_{11}& I{F}_{12}& \dots & I{F}_{1j}\\ I{F}_{21}& I{F}_{22}& \dots & I{F}_{2j}\\ ⋮& ⋮& \ddots & ⋮\\ I{F}_{j1}& I{F}_{j2}& \dots & I{F}_{jj}\end{array}\right)}_{j\times j}$$

(5)

where j is the number of sampling point of the spectrometer. Thus, $M$, $R$ and $N$ in Equation (4) can be redefined as $M={(M_1,M_2,\cdots ,M_j)}^T$, $R={(R_1,R_2,\cdots ,R_j)}^T=$ and $N={(N_1,N_2,\cdots ,N_j)}^T$.

Researches show that the noise [46] of the spectrometer can be classified into four main parts: readout noise, dark noise, fixed pattern noise, and photoelectron noise. Although the reasons for these kinds of noise are different, they are all Gaussian white noise. Considering that these types of noise are independent of each other, the overall noise can be regarded as the superposition of each kind of noise, which is also Gaussian white noise.

2.2. Spectral Reconstruction Model

The purpose of spectral reconstruction is to obtain the estimated spectra closer to the real spectra from the degraded spectra with prior knowledge such as the instrument response function. Generally, spectral reconstruction can be converted into the minimization of the spectral reconstruction model. The spectral reconstruction model, also known as the objective function, can be obtained by the least-square principle, and can be written as

$$E(R)=\frac{1}{2}{‖F(\widehat{R}, \lambda )-M(\lambda )‖}^2$$

(6)

where $\widehat{R}$ represents the estimated spectra of real spectra, and $F(\widehat{R}, \lambda )=IFM\cdot R$.

Nevertheless, spectral deconvolution is ill-posed and extremely sensitive to the noise in the measured spectrum. Even very small noise may cause uncertainty in the solution of the estimated spectrum, resulting in a large amount of artificial noise generated in the estimated spectrum. To solve this, regularization is necessarily introduced to enforce stability as well as incorporate prior knowledge about the solution.

Therefore, a new spectral reconstruction model with a regularization term can be rewritten, and its expression is Equation (7).

$$E(R)=\frac{1}{2}{‖F(\widehat{R}, \lambda )-M(\lambda )‖}^2+\alpha Rt(R)$$

(7)

where $Rt(\widehat{R})$ is a regularization term and α indicates the coefficient of regularization term. Researchers have proposed many regularization methods, such as Tikhonov regularization, Gaussian–Markov regularization, wavelet transforms regularization, and total variation regularization. Since so many types of regularization terms have different constraint functions, the appropriate regularization term is usually selected according to the specific reconstruction requirements. In addition, a non-negative constraint can be added to the estimated spectrum.

3. Raman Spectrum Reconstruction Method

3.1. Traditional Methods

As mentioned above, the estimation of the real spectrum can be obtained by selecting the appropriate regularization term to construct the objective function and solving it. The objective function can be solved by many traditional methods, among which, a MAP-based deconvolution method has great advantages in dealing with the broadening effect of narrowband spectrum [28]. The method uses the Huber–Markov regularization term, which can suppress the noise and restore the high-frequency spectrum (narrowband spectrum) by reasonably selecting µ, simultaneously.

$$Rt(R)=\left\{\begin{array}{l}{R}^{\prime }{}^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left|R^{\prime }\right|\le \mu \hfill \\ 2\mu \left|R^{\prime }\right|-{\mu }^2\hspace{1em}\left|R^{\prime }\right|>\mu \hfill \end{array}\right.$$

(8)

The algorithm uses Euler–Lagrange equation and the Neumann boundary condition to minimize objective function with respect to R, and the expression is shown in Equation (9):

$$\frac{\delta E}{\delta R}={\displaystyle \sum \left(\frac{\partial F}{\partial R}-\frac{d}{d\lambda }\left(\frac{\partial F}{\partial R^{\prime }}\right)\right)}$$

(9)

The expression of $\partial F/\partial R$ and $d\left(\partial F/\partial R^{\prime }\right)/d\lambda$ in Equation (9) can be easily obtained by calculation, as shown in Equations (10) and (11), respectively.

$$\frac{\partial F}{\partial R}=IFM\cdot \left(IFM\cdot R-M\right)$$

(10)

$$\frac{d}{d\lambda }\left(\frac{\partial F}{\partial R^{\prime }}\right)=\left\{\begin{array}{l}2\alpha R^{\prime }{}^{\prime }\hspace{1em}\hspace{1em} \left|R^{\prime }\right|\le \mu \hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left|R^{\prime }\right|>\mu \hfill \end{array}\right.$$

(11)

Therefore, the estimated spectrum can be obtained by using a successive iteration, as shown in Equation (12).

$${\widehat{R}}^{j+1}={\widehat{R}}^j+t^j\left(-\frac{\delta E}{\delta R^j}\right)$$

(12)

where t^j is the time step parameter used for controlling the convergence rate.

3.2. Proposed Method

Due to the employ of traditional deconvolution methods, the reconstructed Raman spectrum has been much improved compared with the measured Raman spectrum. However, the reconstruction effect of these methods on the measured Raman spectrum is still unsatisfactory, the peak value of reconstructed narrowband spectra is often lower than the real spectra. To overcome these limitations of traditional methods, a novel spectral reconstruction method which combines the MAP method and a CNN framework to enhance the spectral reconstruction effect of degraded Raman spectrum is proposed. The proposed method first employs the MAP method to reconstruct the measured Raman spectra, so as to obtain preliminary estimated Raman spectra. Then, a CNN is trained by using the preliminary estimated Raman spectra dataset and the real Raman spectra dataset to learn the mapping from the preliminary estimated Raman spectra to the real Raman spectra. The main advantage of this method over the traditional methods is that it establishes the mapping from the preprocessed spectra to the real spectra, so as to achieve a better spectral reconstruction effect than merely using the traditional methods or a CNN. Figure 1 illustrates the schematic of the proposed method. There are three parts of this system: CNN architecture, training and predicting.

The CNN architecture consists of eight learnable layers, including five convolution layers and three fully connected layers. Convolution layers are employed for feature extraction in non-linear mapping between the preliminary estimated Raman spectra and real Raman spectra. The fully connected layers are employed for synthesizing the features extracted by the convolution layers, so as to reconstruct the real Raman spectrum. Moreover, Figure 1a shows the detailed information of learnable kernels. After each convolution layer, the ReLU and max pooling which is used for down-sample are employed. Then, the output data of convolution layers are flattened and used as the input data of fully connected layers. ReLU and dropout operations are performed on the first two fully connected layers, while the third fully connected layer is followed by a linear activation function, which will directly return the data of the third fully connected layer.

In the CNN training stage, first of all, it is necessary to establish a real Raman spectra dataset, which can be downloaded from the website or constructed using multiple Lorentz functions. Then, the measured Raman spectra dataset can be simulated from the real Raman spectrum dataset combined with the instrument response function and noise. Next, the measured Raman spectra dataset is preprocessed by the traditional method such as the MAP method, and the preliminary estimated Raman spectra is used as the input dataset of CNN. Details of the construction of these datasets will be described in next section. Finally, Raman spectra in the output dataset are compared with the corresponding spectra in the real Raman spectra dataset to adjust the structural parameters of a CNN to minimize the loss function, so as to obtain the trained CNN. There are many kinds of loss functions, in this paper, the mean square error is selected:

$$Loss=\frac{1}{n}{\displaystyle \sum _{i=1}^n{‖R_i-{\widehat{R}}_i‖}_2^2}$$

(13)

where $R_i$ is the i-th term of the real spectrum R, ${\widehat{R}}_i$ is the i-th term of the reconstruction spectrum $\widehat{R}$, and ${‖\cdot ‖}_2^2$ represents the square of L2 norm.

In the CNN prediction stage, the measured Raman spectrum needs to be deconvoluted by the MAP method first, and then the preliminary estimated Raman spectrum is used as the input of the CNN to obtain final estimated Raman spectrum. It should be noted that in addition to the MAP method, other traditional methods can also be employed to deconvolute the measured Raman spectrum, but the traditional method selected in the CNN predicting process should be consistent with the method in the training stage. Compared with traditional methods, the main advantage of the proposed method is that it builds the mapping from the preliminary estimated Raman spectra to the real Raman spectra, rather than just deconvolution of the measured Raman spectra, so that the preliminary reconstruction results are more similar to the real Raman spectra. Compared with directly using a CNN to establish mapping relationship, the reconstruction method in this paper first employs the MAP method to extract more spectral features, which improves the quality of CNN input data, thus enhancing the ability of the CNN to obtain better estimates.

4. Simulations and Experiments

To prove the effectiveness of the proposed spectral reconstruction method, we employed the proposed method and some classical traditional spectral reconstruction methods to reconstruct the simulated and measured Raman spectra, respectively.

4.1. CNN Training Stage

Two synthetic Raman spectra datasets were established, both of which contain approximately 2000 Raman spectra. The real spectra of the first synthetic dataset were established by the combination of multiple Lorentz functions. We randomly generate multiple Lorentz functions within the spectral measurement range, and their peaks, central wavenumbers and full width at half maximum (FWHM) were also randomly determined within a certain range. More details of the first synthetic Raman spectrum dataset are shown in

Table 1. Details of the first synthetic Raman spectrum dataset.

Parameter	Details
Spectral function type	Lorentz function
Numbers of peaks	9~15 randomly generated
FWHM	20~200 randomly generated
Peak intensity	2000~30,000 randomly generated
Peak position	Randomly generated within the measurement range

.

The real spectra of the second synthetic dataset were obtained by denoising and smoothing Raman spectra in the KnowItAll Raman spectral dataset [47]. The real spectra as the CNN output datasets. Then, the measured Raman spectra dataset was generated by employing the real spectra combined with the measured spectrum model (Equation (2)). Finally, the traditional method (MAP method) was employed to deconvolute the measured spectra, and the preliminary estimated Raman spectra are used as the input datasets of a CNN.

The number of training, validation, and test spectra were randomly assigned in the ratio of 5:1:1 for both of the real Raman spectra datasets and CNN input datasets, respectively. The validation spectra were used to estimate the number of epochs and adjust the hyperparameters. The Adam optimizer with the batch size of 32 implemented in TensorFlow 2.0 (Google, Mountain View, CA, USA) was employed, and all calculations were performed on NVIDIA GeForce RTX 3070Ti (NVDIA, Santa Clara, CA, USA) graphics processing unit (GPU). The whole training stage took about 30 min. After a CNN is trained, the time for reconstructing the input spectrum is very short, but considering that it takes approximately 10 s for the MAP method to reconstruct the Raman spectrum, the process of reconstructing the spectrum with the proposed method lasts approximately 10 s.

4.2. Simulations

According to the instrumental characteristics of a self-developed Raman spectrometer, a measured Raman spectrum was simulated by combining a real Raman spectrum which was developed based on multiple Lorentz functions and the instrument response functions. For simplicity, the instrument response functions corresponding to all wavelengths are set to the same value, and the spectral range of the Raman spectrum is between 200 and 4000 cm⁻¹. The simulated measurement was obtained by convolving the real Raman spectrum with the instrument response function and adding noise. The real Raman spectrum and the measured Raman spectrum are shown in Figure 2a,b, respectively.

As shown in Figure 2, owing to the effect of the instrument response function, the narrow band parts of the measured Raman spectrum are degraded, causing the decline of spectral resolution, and even the degradation of three spectral peaks into one peak and five spectral peaks into three peaks. This leads to a certain measurement error, and further affects the accuracy of substance identification in combination with noise.

Then, the measured Raman spectrum was reconstructed by the LM method [27], the MAP method [28] and proposed method. In addition, to show the superiority of the proposed spectral reconstruction method over using a CNN directly, a CNN is also trained with measured Raman spectra dataset and real Raman spectra dataset to estimate the real Raman spectrum. The estimated spectra of the four methods are shown in Figure 3a–d.

Figure 3a–d show that these methods have a certain correction effect on the measured Raman spectrum. Compared with the measured spectrum, the estimated spectra of all the four methods are more similar with the real spectrum. The effect of the two traditional methods is similar, a CNN method is slightly weaker, while the proposed method is undoubtedly the best of the four methods. The main reason for this phenomenon is due to the proposed method is a deconvolution model based on the MAP method, which is equivalent to reconstructing the estimated spectrum of the MAP method towards the real spectrum again. To further analyze the reconstruction effect of the four methods on the Raman spectrum, the real Raman spectrum is subtracted from their reconstructed spectra, respectively, and the reconstruction errors are obtained, as shown in Figure 4.

It can be seen from the figure that the reconstruction error of the proposed method is significantly lower than that of the other three methods, especially near the peak points of the Raman spectrum. Moreover, root mean square errors (RMSEs) and normalized mean square errors (NMSEs) of the reconstruction results of these methods are calculated to quantify the reconstruction effect of each method. These two parameters can be calculated by Equations (14) and (15).

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(R_i-{\widehat{R}}_i)}^2}}$$

(14)

$$\mathrm{NMSE} =\frac{{\displaystyle \sum _{i=1}^n{\left(R_i-{\widehat{R}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{R}_i{}^2}}$$

(15)

where $R_i$ is the i-th term of the real spectrum R, and ${\widehat{R}}_i$ is the i-th term of the reconstruction spectrum $\widehat{R}$.

Table 2. RMSEs and NMSEs of the reconstruction results of four methods.

Method	Parameter
	RMSE	NMSE
LM	2.82 × 102	3.95 × 102
MAP	3.02 × 102	4.38 × 102
CNN	3.17 × 102	4.41 × 102
Proposed	1.43 × 102	2.03 × 102

shows the calculation results of these two parameters. The results show that the two parameters calculated from the reconstruction results obtained by the proposed method are much smaller than those obtained by other methods, which is consistent with the results in Figure 3.

4.3. Influence of Noise

The measured Raman spectrum is always accompanied by noise. Although noise has been added to the simulated spectrum in Section 4.2, the influence of the intensity of noise on the proposed reconstruction method is not considered. After generating the measured Raman spectrum without noise, add the noise of levels 10, 50, 100 and 200 in turn to obtain four measured Raman spectra with the same degree of spectral degradation but different noise levels. Next, the four methods were employed to obtain estimated spectra. The four simulated Raman spectrum are shown in Figure 5, and the estimated spectra of the measured spectrum (noise level = 200) obtained by the four methods are shown in Figure 6, the reconstruction errors of the four methods are shown in Figure 7.

In Figure 6, the proposed method still shows an excellent reconstruction effect when the noise level is 200, as the reconstructed spectrum is essentially consistent with the real spectrum. Compared with the low noise situation, the estimated spectra obtained by the traditional methods have a greater degree of under correction, which is because these methods increase the regularization parameters to repress the noise. In addition, the estimated spectrum of the proposed method is smooth in the low-frequency regions, while other methods still have some residual noise. A similar conclusion can be drawn from Figure 7. In addition, comparing Figure 4 with Figure 7, we can find that the reconstruction error of the proposed method does not increase significantly when reconstructing the Raman spectrum with more noise, while the reconstruction error of the other three methods increases significantly. We also calculated the two parameters of the estimated spectra corresponding to the measured spectra, and the results are shown in

Table 3. RMSEs and NMSEs of the reconstruction results of measured Raman spectra with different noises by four methods.

Noise Level	Method	Parameter
		RMSE	NMSE
10	LM	2.32 × 102	3.27 × 102
	MAP	2.62 × 102	3.70 × 102
	CNN	2.75 × 102	3.84 × 102
	Proposed	1.45 × 102	2.04 × 102
50	LM	3.42 × 102	4.78 × 102
	MAP	3.62 × 102	5.14 × 102
	CNN	3.90 × 102	5.49 × 102
	Proposed	1.68 × 102	2.35 × 102
100	LM	4.17 × 102	5.84 × 102
	MAP	4.08 × 102	5.71 × 102
	CNN	4.66 × 102	6.57 × 102
	Proposed	1.76 × 102	2.48 × 102
200	LM	5.94 × 102	8.32 × 102
	MAP	4.97 × 102	7.00 × 102
	CNN	4.56 × 102	6.29 × 102
	Proposed	2.02 × 102	2.85 × 102

. As can be seen from the table that with the increase in noise, both of the parameters of the estimated spectra obtained by the four methods increase. Even so, compared with other methods, the proposed method has more satisfactory results, which can simultaneously repress noise and reconstruct the narrowband spectrum effectively.

4.4. Experiments

Moreover, we carried out experiments to explore the effectiveness of this method on the real measured spectra, which were measured from several drug samples through a self-developed Raman spectrometer with a 785 nm laser source. All these measured Raman spectra were baseline corrected.

Figure 8a displays the measured spectrum of caffeine (C₈H₁₀N₄O₂) from 2200 to 200 cm⁻¹. A linear spectrum corresponding to approximately 1400 cm⁻¹ position of the Raman spectrometer was obtained by a tunable laser, and the instrument response function of the Raman spectrometer is obtained by Gaussian function fitting. Next, three methods were used to reconstruct the measured Raman spectrum of caffeine, respectively. The reconstruction results show that, the resolution of the estimated spectra obtained by the three methods has been improved, especially the overlap peak can be split into multiple peaks. However, the estimated spectra obtained by the two traditional methods can be observed obvious residual noise, while the reconstructed spectrum of proposed method is smooth in the flat region. In addition, Raman spectra of other three drug samples include ketamine (C₁₃H₁₆ClNO), methamphetamine (C₁₀H₁₅N) and ibuprofen (C₁₃H₁₈O₂) were collected, and estimated spectra were obtained by three methods, and shown in Figure 9, Figure 10 and Figure 11, respectively. The estimated spectra of each sample obtained by three methods are similar to Figure 8. Therefore, from the perspective of spectrum reconstruction, the performance of the proposed method is the best of three tested methods.

For these measured Raman spectra, the real spectra are unknown. In order to quantify the effects of various methods, the spectra in the sample dataset are selected as the reference spectra. The reference Raman spectra are subtracted from their reconstructed spectra, respectively, and the reconstruction errors are obtained, as shown in Figure 12.

From the reconstruction errors of Raman spectra by each method, it can be seen that although the reconstruction effect of the three methods is superior, the estimated spectra obtained by the proposed method are the closest to the reference spectra in the sample dataset, and the reconstruction errors are significantly smaller than the other two methods. Similarly, taking the reference spectra as the standard spectra, we calculated the RMSEs and NMSEs of the reconstruction results of these methods, and the results are shown in

Table 4. RMSEs and NMSEs of the reconstruction results of different measured Raman spectra by four methods.

Sample	Method	Parameter
		RMSE	NMSE
Caffeine	LM	1.11 × 102	1.58 × 102
	MAP	1.02 × 102	1.44 × 102
	Proposed	7.24 × 101	1.02 × 102
Ketamine	LM	1.00 × 102	1.41 × 102
	MAP	9.26 × 101	1.31 × 102
	Proposed	7.02 × 101	9.83 × 101
Methamphetamine	LM	9.47 × 101	1.33 × 102
	MAP	8.76 × 101	1.24 × 102
	Proposed	7.04 × 101	9.79 × 101
Ibuprofen	LM	1.05 × 102	1.48 × 102
	MAP	1.00 × 102	1.42 × 102
	Proposed	6.99 × 101	9.86 × 101

. Moreover, the parameter correlation coefficient (CC) is also employed to quantify the reconstruction performance of these methods, which is defined as

$$\mathrm{CC}=\frac{{\displaystyle \sum _{i=1}^n\left({\widehat{R}}_i-\overline{\widehat{R}}\right)\left({\widehat{S}}_i-\overline{\widehat{S}}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left({\widehat{R}}_i-\overline{\widehat{R}}\right)}^2\cdot {\displaystyle \sum _{i=1}^n{\left({\widehat{S}}_i-\overline{\widehat{S}}\right)}^2}}}}$$

(16)

where ${\widehat{R}}_i$ is the i-th term of the reconstruction spectrum $\widehat{R}$, ${\widehat{S}}_i$ represents the i-th term of the corresponding spectral data in the sample dataset, $\overline{\widehat{R}}$ is the average of $\widehat{R}$, and $\overline{\widehat{S}}$ is the average of $\widehat{S}$. The calculation results of the parameter CC are shown in

Table 5. The results of the CC of each method.

Analytes	Measured	Method
		LM	MAP	Proposed
caffeine	0.8143	0.9814	0.9866	0.9979
ketamine	0.8024	0.9883	0.9917	0.9986
methamphetamine	0.8091	0.9910	0.9932	0.9983
ibuprofen	0.7962	0.9846	0.9884	0.9992

.

Table 4 shows that the RMSE and the NMSE calculated from the reconstruction results obtained by the proposed method are much smaller than those obtained by other methods, which is consistent with the results in Figure 12. The results in Table 5 show that, the parameter CC of the three methods is improved. Even so, compared with other methods, the proposed method has more satisfactory results, the value of the CC is closest to 1. All the experimental results prove the superiority of the proposed method.

5. Conclusions

The Raman spectrometer is a rapidly developed instrument in recent years. The Raman spectrometer can qualitatively analyze and identify various molecular structures and material types with few samples. However, as a component of the Raman spectrometer, the non-ideality of the spectrometer usually makes it unable to present the real spectrum well. Due to the influence of the instrument response function, the measured Raman spectra commonly contain spectral distortion, which leads to certain measurement error and further affects the accuracy of substance identification combining with noise. In this paper, we propose a novel spectral reconstruction method which combines the MAP method and the DL to recover the degraded Raman spectrum. First, the MAP method is employed to reconstruct the measured Raman spectra, so as to obtain preliminary estimated Raman spectra. Then, a CNN is trained by using the preliminary estimated Raman spectra and the real Raman spectra to learn the mapping from the preliminary estimated Raman spectra to the real Raman spectra. The main advantage of this method over the traditional methods is that it establishes the mapping from the preprocessed spectra to the real spectra, so as to achieve a better spectral reconstruction effect than merely using the traditional methods or a CNN. To prove the effectiveness of the proposed spectral reconstruction method, we employed the proposed method and some traditional spectral reconstruction methods to reconstruct the simulated and measured Raman spectra, respectively. The RMSE, the NMSE and the CC were used to quantify the reconstruction effect. The experimental results show that compared with traditional methods, the estimated Raman spectra reconstructed by the proposed method are closer to the real Raman spectra.