© The Author(s), published by EDP Sciences, 2025

1 Introduction

Infrared radiation signals have interferences from various factors during atmospheric transmission; this aspect restricts the processing of space remote sensing image information and the accuracy of detection and tracking of the weak infrared targets. Therefore, identifying targets with weak radiation signals against a strong background environment is difficult. The infrared radiation generated during the combustion of propellants is used for the identification of rocket-type aircraft in current space-based infrared early warning systems [1–5]. The methods for target identification based on the infrared radiation characteristics of the tail flame include the processing of the tail flame radiation spectral line data via the line-by-line integration method [6], the use of the fuzzy algorithm to analyse the infrared radiation spectrum signal [7] of the tail flame and the establishment of a k distribution model to perform numerical calculations on the tail flame radiation signal [8]. These methods are based on the strong infrared radiation characteristics of the tail flame for target detection. However, with the development of technology, the stealth requirement of low emissions has been proposed for flying targets. The addition of a flame suppressant to the propellant can effectively reduce the infrared radiation intensity of the tail flame [9]; this increases the difficulty of identification and causes the detection system to have a higher false alarm rate. Therefore, research on new detection methods and technologies is needed to improve the identification ability.

The solid particulate matter in the tail flame is excited at high temperatures during combustion, and a spectrum is generated that can reflect its intrinsic attributes. The spectral information of the different components vary. The use of spectral characteristic information in the field of target detection can increase the accuracy of identification [10, 11]. Studies have shown that potassium salt is a basic ingredient in flame suppressant formulations [12], and during combustion, two distinctive characteristic spectral lines for potassium are observed. Therefore, developing a method for rocket target identification via spectral signal detection of potassium salt combustion instead of infrared radiation characteristic detection is feasible [13–15]. Spatial heterodyne spectroscopy (SHS) is a new type of ultrahigh-resolution remote sensing spectral detection technology. Compared with traditional Fourier transform spectrometers, SHS has greater luminous flux, resolution and a unique advantage in the fine detection of weak spectral signals [16–18]; its application in the detection of potassium salt combustion characteristic spectral lines has practical importance. In actual detection, the detection of potassium combustion strongly interferes with the background signal of the sky. Under long-distance conditions, the intensity difference between the two is 3~4 orders of magnitude or greater, and the potassium combustion signal is often hidden in the atmospheric background and difficult to distinguish. The extraction and identification of the weak signals of this type of aliasing spectrum is the focus of this research. In this study, the sky was used as the background, and a potassium lamp light source was used to simulate the potassium combustion signal. The extraction of the potassium characteristic signal from the strong background spectrum was experimentally investigated, and the principal component analysis (PCA) method was used on the measurement data [19–21]. The principal components of the mixed signals were separated by reducing the dimensionality of the spectral data and the information redundancy between the data. The potassium signal was restored and extracted via the Principal Component Regression (PCR) method. Combined with the Non-Local Means (NLM) denoising algorithm, the qualitative identification and quantitative analysis of potassium lamp signals were ultimately achieved.

2 Methods

2.1 SHS principle

The SHS uses a new type of modulated spatial interference spectrometer that has the characteristics of small volume, high throughput, and a large field of view; moreover, it has a clear advantage in the detection of the characteristic peaks of spectral lines with weak signals, and its structure is illustrated in Figure 1.

Fig. 1 Basic principles of SHS.

Under ideal circumstances, the interferogram collected by the SHS can be expressed by formula (1):$$I\left(x\right)={\int }_{-\infty }^{\infty }B\left(\sigma \right)\{1+\cos [8\pi \left(\sigma -{\sigma }_0\right)x\tan \theta \left]\right\}\mathrm{d}\sigma ,$$(1)where x represents the position of the detector pixel, B(σ) represents the spectrum of the incident light, σ represents the wavenumber of the incident light, σ ₀ represents the Littrow wavenumber, and θ represents the angle between the grating normal and the optical axis.

In actual detection, the expression of the original interferogram collected by the SHS is shown in formula (2):$$I\left(x\right)=A\left(x\right)+{\int }_{-\infty }^{\infty }B\left(k\right)\exp \left(\mathrm{ikx}\right)\mathrm{d}k,$$(2)where A(x) is the background noise and B(k) is the characteristic spectrum of potassium.

In this study, SHS is used to detect the weak potassium signal in the tail flame. However, formula (2) shows that the original spectral information of the tail flame radiation is affected by the background signal, and a large amount of background noise affects the accuracy of the feature signal identification. Therefore, the potassium signal needs to be further extracted.

2.2 PCA-NLM algorithm

In applications, the measurement spectrum is usually composed of a mixture and superposition of the spectra of multiple different components. In studies, the spectra of one or a few components are often the focus, and the spectra of other components are considered interference. An important part of this research is to effectively separate the spectrum of the target component of interest from the interference. PCA is a method of dimensionality reduction and can separate complex datasets using orthogonal transformation. Its application to the analysis and processing of the spectral data can reduce the correlation among the original variables of the spectrum and decompose the information from the different components in the original spectrum into different principal components. The information contained in each principal component can also be quantitatively analysed based on the eigenvectors of each principal component.

The raw spectral data can be expressed as an n-dimensional random variable X. The dataset X is analysed via PCA. After orthogonal transformation, the original linearly correlated variables in the spectrum are converted to new linearly independent variables to achieve separation of the datasets. n new variables from the matrix Y are expressed as follows:$$Y=E·X,$$(3)where E is the coefficient matrix. New variables Y ₁, Y ₂, … Y _n are represented as the 1st to nth principal components, respectively, each Y _i are independent of each other, and each principal component contains information from a different component in the original spectrum; thus, this process effectively decomposes the spectral information.$$Y_i={\vec{e}}_{i1}{x}_1+{\vec{e}}_{i2}{x}_2+\dots {\vec{e}}_{\mathrm{in}}{x}_n,$$(4)where ${\vec{e}}_{i1}, {\vec{e}}_{i2}, \dots {\vec{e}}_{\mathrm{in}}$ are the eigenvectors of the i-th principal component and represent the proportion of each original variable on the component. The projection size of the variables contained in the original data on the different principal components can be obtained from the value of the eigenvector. The eigenvalue of each principal component can be obtained via formula (5), and the eigenvalue represents the amount of information contained in each principal component.$${\lambda }_i=\mathrm{var}\left({\vec{e}}_i^{T}x\right).$$(5)

To more visually express the percentage of the information contained in each principal component in the total spectral information, the contribution rate μ _i of the i-th principal component can be calculated via formula (6):$${\mu }_i=\frac{{\lambda }_i}{\sum _1^n{\lambda }_i}.$$(6)

The Non-Local Means Filtering approach relies on the intrinsic redundancy of signals, employing a patch-matching strategy to assess the correlation among various components of the signal and, consequently, determining the weighting coefficients within the filter. This methodology does not necessitate reliance on external information; rather, it achieves noise reduction solely through the integration of internally similar features within the signal, while simultaneously preserving the integrity of edge features to a certain extent.

Given a discrete two-dimensional noisy signal v = {v(i)|i ∈ I}, where I represents the entire signal domain, the NLM estimate NL[v](i) for any point i within v can be derived by computing a weighted average of all points in the two-dimensional signal. This is mathematically formulated as [22–24]:$$\mathrm{NL}\left[v\right]\left(i\right)=\sum _{i\in I}^{}w\left(i,j\right)v\left(j\right),$$(7)wherein, the computation of the weights {w(i, j)} _j depends on the similarity between points i and j, and satisfies conditions 0 ≤ w(i, j) _j  ≤ 1 and ∑ _j w(i, j) = 1.

The similarity between two points i and j is measured based on the similarity between vectors v(N _i ) and v(N _j ) within their respective fixed-size square neighborhoods centered at point p. This similarity is determined by calculating the weighted Euclidean distance $\left|\right|v\left(N_i\right)-v\left(N_j\right)|{\mathrm{}|}_{2,a}^2$ between the two neighborhood vectors, where the weights in the weighted Euclidean distance are given by a Gaussian kernel with a standard deviation of a(a > 0).$${\Vert v\left(N_i\right)-v\left(N_j\right)\Vert }_{2,a}^2=\sum _{k\in Q}^{}{G}_a\left(k\right){\left|v\left(i+k\right)-v\left(j+k\right)\right|}^2.$$(8)

Here, a region Q centered at the coordinates (0, 0) is defined, with its size consistent with the previously mentioned square neighborhood. Additionally, a two-dimensional Gaussian kernel G _a with a standard deviation of a is introduced. The calculation of weights is performed according to the following formula [22–24]:$$w\left(i,j\right)=\frac{1}{Z\left(i\right)}{e}^{-\frac{{\Vert v\left(N_i\right)-v\left(N_j\right)\Vert }_{2,a}^2}{h^2}},$$(9)wherein, $Z\left(i\right)=\sum _j^{}{e}^{-\frac{{\Vert v\left(N_i\right)-v\left(N_j\right)\Vert }_{2,a}^2}{h^2}}$, h are filter coefficients.

This paper proposes an algorithm that integrates PCA with NLM filtering. The primary steps involve firstly separating the target signal from background noise using the PCA method. Subsequently, the extracted feature signal, which may still be affected by noise, undergoes further optimization of the parameter configuration of the Non-Local Means filtering method to achieve optimal denoising effects, thereby obtaining a clear and effective feature signal.

To validate the effectiveness of the PCA-NLM algorithm, simulation experiments were conducted in this paper. Firstly, potassium lamp signals were simulated, with characteristic peaks located at 766.49 nm and 769.89 nm, as shown in Figure 2a. Subsequently, to simulate the impact of environmental disturbances, noise was added to this ideal potassium lamp signal, generating a noisy potassium lamp signal as depicted in Figure 2b. The corresponding interferogram obtained through the simulated spatial heterodyne spectrometer is presented in Figure 2c. By applying Fourier transform, potassium lamp spectral information was extracted from the noisy signal, but at this point, the spectral information was significantly affected by noise, as shown in Figure 2d. Subsequently, Principal Component Analysis was employed to process the spectral information, with the extracted results displayed in Figure 2e. While the PCA effectively attenuates the influence of noise to some degree, residual noise that is challenging to remove still persists in the spectra. Therefore, the NLM denoising algorithm was further applied. Ultimately, the spectral information shown in Figure 2f was obtained. To quantitatively evaluate the denoising performance of the proposed method, we calculated the peak signal-to-noise ratio (PSNR) of the spectra processed by the PCA-NLM method and by PCA alone. The results were 22.20 dB and 17.35 dB, respectively. This comparison further validates the effectiveness of the proposed PCA-NLM algorithm. Additionally, the full width at half maximum (FWHM) of the peaks in Figures 2e and 2f were calculated to be 0.176 nm and 0.0586 nm, respectively. These results indicate that the PCA-NLM algorithm enhances the quality of the signal. The experimental results demonstrate that the PCA-NLM algorithm can significantly suppress noise and successfully extract the spectral characteristic peaks located at 766.49 nm and 769.89 nm.

Fig. 2 Simulation analysis of PCA-NLM algorithm for ideal potassium lamp (a) Spectral characteristic peaks of potassium lamp; (b) Spectral characteristic peaks of potassium lamp with noise interference; (c) Interferogram; (d) Noisy spectrogram; (e) Spectrogram processed by PCA method; (f) Spectrogram processed by PCA-NLM Method.

3 Experimental system and measurement data

In the experiments, a potassium lamp was used to simulate the potassium combustion signal in the tail flame of an aircraft. In the scene of placing the potassium lamp against the sky background, the hybrid data from the potassium lamp and the sky background were collected via remote measurements. The experimental setup is shown in Figures 3a and 3b. The data was collected using SHS (HEP-756-S, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences). The actual spectrometer is shown in Figure 3c. The main parameters of the instrument are listed in Table 1. The potassium lamp used was a JH-B cathode lamp from Beijing Shuguangming Electronic Lighting Instrument Co., Ltd.; the actual lamp is shown in Figure 3d.

Fig. 3 Experimental setup and equipment. (a) Structure of the actual measurement optical path; (b) Schematic diagram of the experimental system; (c) HEP-756-S SHS; (d) Potassium lamps and activators.

In the experiments, different intensities of the tail flame radiation were simulated by changing the current of the potassium lamp. Due to limitations, six groups of interference data with different current intensities were used in this study. Table 2 lists the experimental parameters.

4 Data processing and analysis

Subsequently, under darkroom conditions in the laboratory, the experimental setup was configured as shown in Figure 3. Interference pattern data from a potassium lamp light source was collected, with the results presented in Figure 4a. After Fourier transform processing, the spectrum of potassium was successfully obtained, as shown in Figure 4b. However, the spectral signal obtained at this point was significantly affected by environmental disturbance noise. To further enhance the accuracy of potassium spectrum extraction, PCA method was employed for in-depth processing of the potassium spectrum, with the results displayed in Figure 4c. It clearly demonstrates that PCA can effectively distinguish between the potassium spectrum and environmental disturbances, but a trace amount of background noise remains difficult to completely eliminate. Next, based on the NLM denoising algorithm described in equations (7)–(9), further fine processing was conducted on the spectral data in Figure 4c, ultimately yielding the potassium lamp spectrum shown in Figure 4d. Experimental results indicate that the NLM denoising algorithm can further reduce the trace noise remaining after PCA. Meanwhile, Figure 4c clearly shows significant characteristic peaks at wavelengths of 766.49 nm and 769.89 nm. In subsequent analysis of mixed signals from the sky background, these two characteristic peaks will serve as key indicators for assessing the success of signal extraction. This experiment fully confirms the effectiveness of the PCA-NLM algorithm adopted in this paper for potassium lamp spectrum extraction.

Fig. 4 Interferogram and average spectrogram of potassium lamp in dark laboratory environment. (a) Interferogram of potassium lamp; (b) Noisy spectrogram; (c) Spectrogram obtained by PCA method; (d) Spectrogram processed by PCA-NLM Method.

Six groups of mixed data were collected in the experiment. Figure 5a shows an interferogram of the potassium lamp under the conditions of a current of 6 mA and a sunny atmosphere as the background. The bright line with a zero optical path difference in the middle of the interferogram was slightly inclined in the vertical direction. This occurred because the grating in the interferometer was shifted from its ideal position, and the component in the y direction was slightly tilted [25]. In addition, the interferogram also had dark spots and uneven intensity distributions; thus, performing PCA on the interferogram dimension led to erroneous results. Here, the analysis was carried out mainly in the spectral dimension. The size of a single interferogram was 2048 × 2048 pixels, and each row represented a set of one-dimensional interferometry data. One-dimensional spectral data was obtained from the single-row data of the interferogram after apodization, first-order difference removal of the baseline, and one-dimensional Fourier transform. To reduce the influence of the high-frequency random noise in the single-line spectrum, an averaging process was performed on the spectral data of 2048 lines. Figure 5b shows the average spectrum after the above preprocessing.

Fig. 5 Measured interferogram and average spectrogram under an atmospheric background environment. (a) Interferogram of the actual measurement; (b) Average spectrum of the measured interferogram after pretreatment.

To provide a closer view, Figure 6 shows a partially magnified portion of Figure 5b. Under a potassium lamp current of 6 mA, a small peak at 766.49 nm was present; however, this peak was similar to the other small peaks at the trough positions of the adjacent periods, could be an intrinsic signal of the atmospheric background and did not likely reflect the potassium signal. A smooth downward curve was observed at 769.89 nm, with a small valley near the peak; thus, this characteristic peak of potassium could not be observed either. These results indicated that the potassium signals from the lamp were hidden in the atmospheric background signal. To identify the potassium target, the potassium signals need to be extracted from the atmospheric background.

Fig. 6 Average spectral diagram corresponding to the interferogram with a measuring distance of 442 cm and potassium lamp at a current magnitude of 6 mA. (a) Average spectral diagram; (b) Magnified view of the local region.

In this study, the PCA was performed on six sets of interferometric data obtained under clear-sky conditions to extract the potassium lamp signal from the mixed spectra. The potassium lamp spectrum measured in the laboratory was used solely as a reference for comparison with the final results and did not directly participate in the data processing. The extraction steps of potassium signals via the PCA are as follows: the interferogram collected by the SHS was obtained by preprocessing the interferogram to obtain the average spectrum; then, the above steps were repeated for the six pieces of hybrid interferogram collected in the experiment. The average spectral data from the six groups were used and reconstructed as a new matrix T = (t ₁, t ₂, … t _i ), i = 1, 2, …6. After the reconstructed spectral data matrix was processed via the PCA algorithm, six principal components Y ₁, Y ₂, … Y _i , the projection value of each principal component S _ij , and the eigenvalue λ _i were obtained. The PCA decomposes the mixed spectra into principal components with varying contribution rates through a dimensionality reduction approach. Each principal component corresponds to distinct signal components, as illustrated in Figure 7. Figure 7a shows the atmospheric background signal, and Figures 7b and 7c show the potassium signal from the lamp, and the remaining three principal components were random noise. Therefore, by performing PCA on the six groups of mixed spectra that were hidden by the strong atmospheric background, separation of the atmospheric background signal and the potassium signal were successfully achieved.

Fig. 7 Spectral diagram of each principal component. (a) First principal component spectrum; (b) Second principal component spectrum; (c) Third principal component spectrum; (d) Fourth principal component spectrum; (e) Fifth principal component spectrum; and (f) Sixth principal component spectrum.

The amount of information contained in the spectrum of the different principal components can be reflected in the contribution rate. According to formulas (3) and (4), the contribution rate of each principal component was obtained and are listed in Table 3.

Table 3 shows that the contribution rate of the first principal component was 99.77%, and this component was the atmospheric background signal and corresponded to the strong background characteristics of the measurement mixed signal. The contribution rates of the second and third principal components were 0.17% and 0.047%, respectively. They both reflected the characteristics of the potassium signals, but their contribution rates differed by an order of magnitude; these results indicated that the spectrum of the potassium signal was mainly contained in the second principal component. Additionally, the potassium signal in the third principal component was caused mainly by the difference in the intensity from the two characteristic peaks of the potassium signals under different current intensities. The fourth to sixth principal components corresponded to random noise, and their contribution rates were even lower; thus, they were not considered in the potassium spectral analysis.

To analyse the intensity relationship of the extracted signals, the noise signals and the potassium signals in the six groups of hybrid data needed to be separated and restored using the specific principal components and their corresponding projection values for linear superposition. Figure 8a shows the noise spectrum of the potassium lamp recovered by the first, fourth, fifth, and sixth principal components at 6 mA. Figure 8b displays the potassium lamp spectrum recovered through PCA method under a current condition of 6 mA. The analysis results indicate that the PCA method can effectively isolate the potassium lamp signal from complex background signals, but the extracted spectral components still contain a certain amount of residual noise. Based on this, Figure 8c presents the potassium lamp spectrum further recovered by the PCA combined with NLM denoising algorithm. The results demonstrate that the PCA-NLM method can significantly further reduce the noise components in the potassium lamp spectrum, enhancing the purity of the spectrum and improving the signal-to-noise ratio (SNR).

Fig. 8 (a) Recovered noisy spectrogram; (b) Spectrogram obtained by PCA method; (c) Spectrogram processed by PCA-NLM method; (d) Relationship between current magnitude and signal intensity.

To illustrate the effectiveness of the PCA-NLM algorithm for signal extraction, the SNR of the potassium spectrum was defined as follows:$$\mathrm{SNR}=\frac{S}{N},$$(10)where S is the potassium signal intensity and N is the atmospheric background intensity. Based on the restored potassium spectral intensity at 6 mA shown in Figure 8c and compared with the data in Figure 6, the SNR of the potassium signal in the original measurement results was 0.1310; these results indicated that a weak potassium signal was successfully extracted at this low SNR. The SNR of the restored potassium spectrum in Figure 8c was 16.9019; these results further illustrated the effectiveness of the PCA-NLM method.

Although the second principal component and the third principal component both contained potassium signals, since the contribution rates of the two were different by an order of magnitude, the projection value of the second principal component S _2j (j = 1, 2, 3 … 6) could be used to characterize the intensities of the potassium signals in the six groups of mixed data. Linear regression was performed on S _2j and the potassium lamp current, and the results are shown in Figure 8c. The intensity of the restored potassium lamp signal was linearly related to the corresponding current; here, the correlation coefficient was 0.9823, and the regression model was as follows: y = −0.39547 + 0.181115x. This trend was consistent with the theoretical result that a larger current correlated to a greater signal intensity; these results could provide a basis for subsequent distance estimation using a constant signal intensity.

5 Conclusions

The addition of flame suppressant substances to the jet fuel of a spacecraft can reduce the intensity of the infrared radiation; thus, the detection of the tail flame in the infrared band is difficult. Therefore, another feasible method involves the identification of the spacecraft tail flame using the characteristic spectrum generated by the excitation of the potassium atoms in the flame suppressant. Due to highly complex radiation noise in the atmospheric background, the weak potassium signal is easily hidden in the strong background spectrum and cannot be directly identified. In this study, an extraction algorithm for the potassium signals in the spatial heterodyne spectrum was proposed using PCA. The results showed that the algorithm could effectively extract the weak signals against a strong background, and simultaneously, the signal intensity had a good linear relationship with the light source current intensity. However, at present, the algorithm cannot completely remove the residual noise signal around the potassium characteristic peak; this could result in the loss of some effective information. Furthermore, the further application of the NLM denoising algorithm effectively addresses the limitations of the PCA method. The experimental results fully confirm the remarkable effectiveness of the PCA-NLM algorithm employed in this paper in terms of accurately extracting target spectral components and efficiently suppressing noise.

Funding

National Key Research and Development Project (2022YFB3901803), supported by the National Natural Science Foundation of China (41961050, 41975033), Key Laboratory of General Optical Calibration and Characterization Technology, Chinese Academy of Sciences, Graduate Education Innovation Program of Guilin University of Electronic Science and Technology (2024YCXS220).

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Author contribution statement

XinQiang Wang: Methodology, Investigation, Writing-Original Draft Preparation, Writing – review & editing. SiQian Yang: Conceptualization, Writing – review & editing. Wei Xiong: Validation, Writing – review & editing. FangYuan Wang: Conceptualization, Validation, Writing – review & editing. Song Ye: Writing – review & editing.

References

All Tables

All Figures

	Fig. 1 Basic principles of SHS.
In the text

	Fig. 2 Simulation analysis of PCA-NLM algorithm for ideal potassium lamp (a) Spectral characteristic peaks of potassium lamp; (b) Spectral characteristic peaks of potassium lamp with noise interference; (c) Interferogram; (d) Noisy spectrogram; (e) Spectrogram processed by PCA method; (f) Spectrogram processed by PCA-NLM Method.
In the text

	Fig. 3 Experimental setup and equipment. (a) Structure of the actual measurement optical path; (b) Schematic diagram of the experimental system; (c) HEP-756-S SHS; (d) Potassium lamps and activators.
In the text

	Fig. 4 Interferogram and average spectrogram of potassium lamp in dark laboratory environment. (a) Interferogram of potassium lamp; (b) Noisy spectrogram; (c) Spectrogram obtained by PCA method; (d) Spectrogram processed by PCA-NLM Method.
In the text

	Fig. 5 Measured interferogram and average spectrogram under an atmospheric background environment. (a) Interferogram of the actual measurement; (b) Average spectrum of the measured interferogram after pretreatment.
In the text

	Fig. 6 Average spectral diagram corresponding to the interferogram with a measuring distance of 442 cm and potassium lamp at a current magnitude of 6 mA. (a) Average spectral diagram; (b) Magnified view of the local region.
In the text

	Fig. 7 Spectral diagram of each principal component. (a) First principal component spectrum; (b) Second principal component spectrum; (c) Third principal component spectrum; (d) Fourth principal component spectrum; (e) Fifth principal component spectrum; and (f) Sixth principal component spectrum.
In the text

	Fig. 8 (a) Recovered noisy spectrogram; (b) Spectrogram obtained by PCA method; (c) Spectrogram processed by PCA-NLM method; (d) Relationship between current magnitude and signal intensity.
In the text